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A technique for calibrating carbon resistor thermometers below 2 K
The usual method o f calibrating carbon resistor thermometers against the vapour pressure of He3 and He4 is often inconvenient due to the nature of the experimental apparatus. Culbert and Sungaila have proposed a heat capacity technique in which the resistancetemperature relation is deduced by comparing the calculated internal energy-temperature function o f a known sample to the measured internal-resistance function for the same sample. An improvement on their method utilizing holmium metal as a thermometric device is presented. The advantages, disadvantages, and reliability of this technique are discussed and a new carbon resistor thermometry equation of the form R = A + B/7" + C/T 2 is proposed.
A technique for calibrating carbon resistor thermometers below 2 K D. A. Goer, E. F. Starr, G. R. Little, and R. A. Erickson
Resistor thermometers are widely employed in the temperature range of 0.2 K to 4 K. They are usually calibrated by comparing their resistance values against the vapour pressure of either He4 or He 3 liquid in the temperature region above 0.5 K and a three parameter fit of temperature versus resistance is obtained which is then extrapolated for use in all temperature regions. However, the instability of carbon resistors due either to treatment or passage of time requires the experimenter to obtain calibration points each time he uses his apparatus if he wishes to have reliability, and even then, the extrapolation to temperatures outside the actual region of calibration is questionable. The accuracy of the thermometry will therefore be improved if calibration is done as part of the primary experiment in which the thermometer is to be used and if the calibration is made over the entire temperature range of interest. In a cryostat constructed for use in neutron diffraction experiments, vapour pressure determinations below about 1.6 K were not easily obtained and thus a new calibration technique was desirable. Culbert and Sungaila 1 have suggested a method of calibrating resistance thermometers using the known specific heat of a pure metal. In their technique, the heat capacity of a sample of pure (99.999%) platinum is measured above 0.75 K and is used to calculate the internal energy-temperature function for the sample. The internal energy-resistance relation is then obtained by feeding known amounts of electrical energy into the sample and measuring the resultant change in resistance. The temperature-resistance relation is then deduced by comparing the two functions. This technique has advantages over pure vapour pressure techniques in that the calibration is done for all temperatures accessible to the cryostat and hence no calibration extrapolation is necessary. The use of a pure metal standard allows the extrapolation of the heat capacity to be made with reliability and also allows the usage of pub-
The authors are with the Department of Physics, The Ohio State University, Columbus, Ohio, USA. EFS is at present with the Systems Research Laboratories Inc, Dayton, Ohio, USA. Received 16 July 1973.
CRYOGENICS. J A N U A R Y 1974
fished specific heat parameters. However, the specific heat of most pure metals decreases as T and hence the internal energy decreases as T 2. Furthermore, the specific heat of metals is generally small even near 1 K. Therefore, the use of a normal metal as a standard requires that there be a minimum amount of other materials (superconducting solders and construction material alloys, for example) in the calibration cell. It would then be virtually impossible to employ this technique as part of a primary experiment and even if performed in a special cell the decreasing heat capacity would cause the calibration reliability to diverge at low temperatures where accuracy is often most desired. Clearly these difficulties could be overcome if a substance of large specific heat were used as the standard. For work at temperatures accessible to He 3 cryostats, the rare earth metal holmium is an excellent choice as a calibration standard. Due to alignment of nuclear spins, the specific heat of holmium exhibits a well known and abnormally large Schottky-type anomaly near 0.4 K, the maximum value of 6.98 J mole "1 K"1 occuring at 0.3 K. The heat capacity of a relatively small amount of holmium would easily mask that due to other materials in the test cell, thereby allowing the calibration to be made as part of the primary experiment. Furthermore, since the specific heat of holmium increases with decreasing temperature, the accuracy of calibration is enhanced at low temperatures. If enough holmium is added to the sample region so that all the other component materials contribute but a small fraction to the total internal energy of the system, the errors in the thermodynamic functions of the remaining components (for example, alloys) are reduced. This is especially true if the apparatus contains stainless steels since the specific heat of stainless steel is not well known. Furthermore, the presence of a substance with such a high specific heat adds a great deal to the temperature stability of the system. The He 3 cryostat used consists of four stages. A two-stage nitrogen-hydrogen liquefier (Air Products Cryo-Tip Model AC2-112) maintains 80 K and 20 K flanges from which heat shields are attached for thermal isolation. Stage III is a liquid He 4 bath which can be pumped to about 1.6 K. In the final stage, stage IV, liquid He 3 is cryo-
15
Stainless steel tube
Primer coil 304 S flanges ~
S
~
(]]I1=3 - - I n d i u m seal
__I
[
Cobalt cup
Copperring
Holmium R;c m ~s~l u[ g ~ ~ [ ~
R6
= ~
,om[ E[
F
Diffraction region
____J
Silverring " Indiumseal
~------------- Aluminium heat shield
Fig.1 Schematicdiagramof stageIV of the cryostat.
pumped by a charcoal pump to temperatures of 0 . 3 0.4 K. A detailed description of this cryostat will be published elsewhere. Stage IV of the cryostat is schematically shown in Fig. 1. A cobalt cup is vacuum sealed to a stainless steel flange by an indium gasket. Two rods of pure (99.9%) holmium metal (0.0352 moles) are sandwiched between copper and silver rings and the silver ring is indium soldered to the bottom of the cobalt cup. The holmium rods are thermally connected to the silver ring by indium discs and are held tightly to the indium coated copper ring by brass screws under tension. Carbon resistor thermometers R5, R6, and R9 are encased in tightly fitting copper or silver mounts and are positioned as shown in Fig. 1. To reduce rf heating 0.001/2F capacitors are attached across each resistor. The manganin lead wires are thermally anchored to the silver ring. R9 is a Speer's 220 ~ (1/2 W grade 1 002) carbon resistor and R 5 and R6 are Speer's 300 ~2 (1/4 W grade 1 002) carbon resistors. A 'primer' system rapidly cools the thermally isolated stage IV and also permits vapour pressure measurements to be made in the temperature region above 1.6 K. In stage IV the primer consists of a 5 in long coil of 0.062 in x 0.042 in copper tubing which is indium soldered to the stainless steel flange. The input side of the primer is thermally connected to a He4 bath (not shown) and the output side of the primer can be connected either to a roughing pump or a Wallace-Tiernan pressure gauge. After the He4 bath is pumped below the X-point, He 4 gas is condensed in the primer coils, thus establishing a superfluid He II link between the bath and stage IV. This cools stage IV to the temperature of the He 4 bath, about 1.6 K. The temperature of the He 4 bath can be controlled and static vapour pressure readings can be taken of stage IV by using
16
the primer, thereby obtaining calibration points for the three thermometers above 1.6 K. The primer is then evacuated using a roughing pump and stage IV is taken to its lowest temperatures by cryopumping He 3 liquid which drips down into stage IV from a condensing coil located in the He4 bath. The cryopumping is carried out by a charcoal pump which is in thermal contact with the He4 bath. Temperatures of 0.3-0.4 K are obtained. After the He 3 liquid has been pumped away (typically about 1/80 mole) the low temperature calibration commences. Using R6 as a heater, measured amounts of energy are fed into stage IV. Typical power inputs, of about 100/~W, caused R6 to warm to about 2.3 K. After a heating interval the resistors thermally relax in a few minutes to equilibrium drift rates that are associated with background heating or cooling as shown in Fig.2. The three thermometers were read using a Hewlett-Packard Model 3440A DVM and the readings were printed every sixty seconds on a Beckman Model 1453-4 printer. The heating process is repeated until a temperature within the vapour pressure calibration region is obtained. The vapour pressure calibration is used to assign a temperature TO to this final equilibrium resistance R0. TO is chosen to lie close to the hpoint since the vapour pressure calibration work is most accurate in this region.
A U(R) function,
defined as the amount of energy input necessary to take stage IV from some equilibrium resistance R to the final equilibrium resistance R0, is obtained by summing the Joule heating energy inputs, Uj, from R to R 0 and adding to this the integrated background heat leak, UBG, as shown in Table 1. Since the principal heat leak into stage IV is due to metallic conduction, the background heat leak is adequately described by the empirical equation: (~BG = C( T2 -
r2)
(1)
where c is the background heating coefficient, ("2.3/aW K'2), measured in the vapour pressure calibration region, TB is the temperature of the He 4 bath, and T is the temperature of stage IV determinec] from extrapolating the vapour pressure calibration. QBG is of the order of 5-7/aW at the lowest temperature.
i
_ .O.--O~ O"]
195(2
720
,4
19OC C~
d I
18OC
I
I
I I !
175C 17OC O
700
/o-"o- o
185C
2
4 6 Time r rain
680
O 2
4 6
568
8
IO
12
Time ~ rain
Fig.2 Typical thermal relaxation rate curves for the heater, R6, in two different temperature regions. At point A the heating w a s started and at point B the heating was stopped
CRYOGENICS. JANUARY 1974
Table 1. Results from a typical calibration run
R 5, ,.Q
R 6, ,.Q
R 9, ~
Uj ,
UBG,
U(R) = Uj + UBG,
mJ
mJ
mJ
U N (R) = U ( T o) - U(R), mJ
T, K
l/T, K1
696.4 748.3 805.4 880 961 1 051 1 138 1 221 1 305 1 390 1 488 1 597 1 704 1 841 1 971 2 073
671.8 720.8 776 848 927 1 013 1 097 1 176 1 256 1 336 1 424 1 523 1 625 1 742 1 866 1 973
493.7 531 573.4 628 686 753 815 876 934 995 1 065 1 140 1 210 1 302 1 387 1 452
0 23.3 43.1 64.4 93.2 100.8 115.6 127.8 138.6 148.1 157.6 167.2 175.3 183.4 190.2 194.3
0 -1.5 -1.9 -1.6 -0.9 0.1 1.3 2.5 3.7 5.2 6.9 8.5 10.1 12.0 14.0 15.9
0 21.8 41.2 62.8 82.3 100.9 116.9 130.3 142.3 153.3 164.5 175.7 185.4 195.4 204.2 210.2
354.6 332.8 313.4 291.8 272.3 253.7 237.7 224.3 212.3 201.3 190.1 178.9 169.2 159.2 150.4 144.4
2.187 1.833 1.554 1.291 1.096 0.941 0.827 0.743 0.675 0.617 0.562 0.512 0.469 0.427 0.392 0.368
0.457 0.546 0.644 0.775 0.913 1.063 1.209 1.346 1.483 1.621 1.779 1.955 2.132 2.342 2.553 2.717
T is t h e t e m p e r a t u r e at w h i c h
UN(R)
= U ( T ) ; To = 2 . 1 8 7 K a n d
U(To) = UN(Ro)
In order to calibrate R as a function of T it is necessary to know the internal energy of the sample region as a function o f temperature. U(T) is obtained by summing the contributions to the internal energy of the component materials o f stage IV. Cobalt and holmium have, in addition to their electronic and lattice contributions to the internal energy, an energy term arising from the interaction o f the nuclear spins with the electromagnetic field o f the electrons, The Hamiltonian o f the nuclear spins can be represented by 2
= 3 5 4 . 6 mJ
1 1 U(T) - U0 = - A 1T 2 + - A 2 T4 2 4
/2'K T
7/2 7
+taR
" Ei/k exp(-Ei/k T)
i=+7/2
(4)
-7/2
exp(-Ei/kT)
-a'Iz+P
12--I(I+1)
k
+P'(I2-12)
(2)
3
where the P and P ' terms are due to the electric quadrupole interaction and the a' term is due to the magnetic dipole interaction. I is the nuclear spin and I z is its azimuthal component. For holmium the contribution to the energy levels due to the P ' term of the quadrupole interaction is two orders o f magnitude less than the P term 2 and for cobalt the quadrupole interaction may be neglected entirely. 3 Thus the resultant energy levels are o f the form
-a'i+P k
i 2-
I(I+I)
(3)
3
with P = 0 for cobalt where i = I z. The values of a ' and P have been reported from either nmr measurement of the effective field at the nucleus or from direct specific heat measurements. For holmium a' = 0.320 K and P = 0.007 K. 4 In cobalt a ' = 1.09 x 10 -2 K 4 and for T >> a ' the internal energy may be expanded in inverse powers of T, with the leading term of - K / T and K = 1/3 Ra'2I(I + 1) = 5.02 mJ mole "l K. 5 Thus the internal energy o f stage 1V can be detennined from an equation o f the form (for T > 0 . 1 K)
CRYOGENICS . J A N U A R Y 1974
where A 1 and A 2 result from the electronic and lattice terms o f the component materials,/x' and/2 are respectively the number o f moles o f cobalt and holmium present, R is the molar gas constant, k is Boltzman's constant, and U0 is an arbitrary constant. Corrections were made for the superconducting transitions in both indium and aluminium, but they have no noticeable effect on U(T). U(T) and U(T) - Ulto are shown in Fig.3 and the values of the parameters used are listed in Table 2. The calibration procedure is depicted in Fig.4. The calculated U(T) is equated to a renormalized U:v (R) = U(To) U(R) where TO is the temperature associated with the final equilibrium resistance value, RO. This determines a temperature T for any equilibrium resistance value R. The results obtained from carrying out this procedure for a typical calibration are shown in Table 1. In the temperature region above 1.6 K the R (T) points from vapour pressure measurements are fitted to an equation of the form b R = a + -T
(s)
17
runs was only 0.5 K and the sample region was not warmed above nitrogen temperatures.
600
500 --3 E
4OO
l
As can be seen from Fig.5, our suggested thermometry equation fits the data as well as or better than the other suggested equations and does much better than the log R equation in this temperature region. The form of the suggested equation makes it easy to determine what resistance value one should observe for a particular temperature though the inverse process is a bit more inconvenient as it involves solving a quadratic equation in 1/T. Due to the small quadratic term in 1/T 2, C must be determined to several significant figures if the inverse process is to be used with confidence. However, this is not a serious problem as a graphical determination of temperature from resistance values is particularly easy since R versus 1/T is a straight line over much of the temperature range. (R 5 is well represented by a straight line in 1/T down to about 0.8 K, as shown in Fig.6.) One interesting feature of our suggested equation is that significant variations of the parameters can result in almost no change in the fit. For example, if the parameters for R9 are changed to A = 297.72 ~ , B = 427.73 ~. K, C = +0.347 ~ K 2 (obtained by weighting the experimental points with T), the differences in calculated temperatures using the two different sets of parameters are all within 1%. This suggests that there is a rather broad minimum in the standard deviation as a function of the parameters A, B, C.
J
•.• 3OO 200 I00 O
Fig.3
I
I
I
0.5
0.6
1.5
I I 2.0 2.5 TtK
I 3.0
1 3.5
4.0
Internal energy of stage IV as a function of temperature.
U(T) is the total internal energy of stage IV and U(T) - U H . is the
contribution to the internal energy of all the components offstage IV except for holmium. U 0 is taken to be 393.3 mJ
to ~0.2% in A TIT. This equation is used to correlate TO with R 0 and to calculate an approximate temperature of stage IV for heat leak purposes for use in (1). The error resulting from this approximation is unimportant since the approximation, (5), breaks down only at low temperatures where the heat leak is determined primarily by the bath temperature. If measured values of R are plotted against lIT as determined from U(T) the resulting graph suggests that an appropriate equation to try as a fit would be B
R = A +-
C +-T T2
Some doubt has existed concerning the reproducibility of the temperature dependence of a given resistor over periods of time. Our data, as mentioned above, indicates that a calibration is good over a 24 hour period providing the thermometers are not warmed above 80 K. However, our experience is that significant changes do occur over longer periods of time depending on how the resistor is treated. A history of R6 over a seven month interval is shown in Table 4. Typically, the shifts in the calibration are a few percent. Changes occurred often after soldering lead wires, but in some cases they seemed to occur spontaneously. These changes occurred for all temperature ranges including temperatures which were directly measured by vapour pressure and hence do not reflect on the reliability of our technique for calibrating carbon resistors. This question of reproducibility makes it desirable to be able to calibrate carbon resistors each time an experiment is performed.
(6)
A least square fit of the above equation is good to within 0.5% in AT/T for the 1/4 W resistors and to within 1% for the 1/2 W resistor. The higher deviations of the 1/2 W resistor are thought to be due to a poor thermal contact of the thermometer with its case. Other suggested thermometry equations 10,11,12 are also fitted for the sake of comparison (see Fig.5). The constants found for our suggested equation are given in Table 3. The equations obtained from the fitted data points were applied to two other runs taken during the same 24 hour period. The results agreed to within 1% for both the 1/4 W and 1/2 W resistors, but the minimum temperature achieved on these
Table 2. Parameters used in calculating U(T)
Material
Mass, g
T, mJ mole "1 K "2
0, K
Cu Ag AI In Co Ho 304 stainless steel
16.038 24.780 3.127 1.5 15.5 5.797 21.66
0.695 0.650 1.35 1.69 4.72 10.0 *
344.5 226 427.7 111 460 95.1 *
Reference 6 6 7 8 5 4
Contribution to A1, mJ K "2
Contribution to A 2 x 102, mJ K "4
0.175 0.149 0.157 0.022 1.241 0.352 10.4
1.20 3.87 0.29 1.86 0.53 7.94 0.80
For 15.5 gm of cobalt, k = 1.320 mJ K; in the nuclear c o n t r i b u t i o n of h o l m i u m to U ( T ) , a " = 0.320 K and P = 0.007 K * D u Chatenier, et al 9 su~lgests t h a t a valid specific heat equation below 10 K f o r stainless steel is Css = 0 . 4 6 T + 3.7 x 10-4T 3 mJ g-! K "l. A T coefficient o f 0.48 mJ g", K" 2 was experimentally determined to be more correct for the 304 stainless steel used in our apparatus
18
CRYOGENICS . JANUARY 1974
Table 3. Values of the constants A , B, C, as d e t e r m i n e d f r o m a least-square fit of the equation: R = A + B / T + C / T 2
R,n 7OO
I OOO
--T
500
1500
I
2000
[
~t
iI
I
I
% .....
_~_
R5 R6 R9
3OC g
2oc IOO
O
/5 /;'
+
,
,
i
I
[
I
I
I.O
1.5
2.0
i
[
I
I
2.5
3.0
3.5
Fig.4
Graphical method for determining temperature from The energy axis is common to both curves. For a particular resistance, R 1, one finds U N(R 1) and moves horizontally to a point [T1, U(T1)] on the U(T) curve such that U(T l) = U N (RI). This temperature T l is then associated with Rb Two examples of this procedure are given for thermometer R s. R s = 696 ~2 corresponds to a temperature of 2.19 K and Rs = 1650 gZ corresponds to a temperature of 0.49 K
UN(R)=U(T)-U(R)andU(T).
• R . a + a i rc + c l r J
~s.2%f
o liT .A+BR~+CR o re . (AR,a) I (R-C)
ku /
C,~K
432.65 414.44 289.04
568.81 559.10 443.76
13.2 2 4.38 -5.33
2
R U JNt
~ K
2.0
B,~K
_ _ + __'2"y~__ - _ ~ ;
O.5
3.0
A,~
• C/Fc - l o g R + ( A / I o 9 R ) - B
However, it must be pointed out that all errors in determining the heat leak contribution to U(R) are cumulative so that care should be taken in determining the heat leak coefficient, c. In the present experiment, an error o f 10% in the background heating coefficient (c) will cause an error o f about 2% in determining the value of the minimum temperature reached. Since the contribution of the heat leak to the total U(R) depends on the total elapsed time of the calibration process, its importance will be reduced if larger power inputs are made to shorten the time o f calibration. The presence of more holmium will also reduce this difficulty considerably. In a new five slug (0.0837 mole) holmium system an error o f 10% in determining c results in less than a 1% error in T minimum. Also the increased holmium increases the accuracy of the U(T) determination since the remaining components of stage IV are much less important. Finally it should be noted that other values o f a ' and P for holmium have been suggested. Krusius et al 13 find a ' = 0.319 K and P = 0.004 K while Van Kempen et al 2
1.0 O
&&
O •
q
•
2000
-I.O
-2.0 i
0.5
L
I
1
I.O
1.5
2.O
1500
r,K
Fig.5 Differences between the measured temperature, Tm. Tc is calculated using a least squares fit to the data for resistor R5 for each of the listed expressions 1(3OO
Using the internal energy o f the sample region as a thermometric device has an advantage over the standard vapour pressure technique. It is difficult to obtain vapour pressure measurements with accuracy below 1 K for He 4 and below 0.6 K for He 3, whereas in this technique calibration is easily obtained for any region in which the internal energy is well known, and hence extrapolation to a noncalibrated region is not required. While this technique requires some computation to determine the temperature, no thennomolecular corrections are needed and the actual measurements are very simple to take, Furthermore, due to the large heal capacity o f holmium, the temperature stability o f the sample region is increased.
CRYOGENICS
. JANUARY
1974
5OO
0.4
I I.O
I 2.O I
7,
_
I 3.0
K-I
Fig.6 R s versus 1 IT, illustrating the linear behaviour of the resistor over much of the temperature range in lIT. The standard deviation for a two-parameter least squares fit is 6 1 ~ which compares to 1.6 D, for the three parameter fit
19
Table 4. A history of R 6
Date
A,~
B, fZ K
C,£Z K 2
T = 0.5 K R, E2 AT, K
T = 1.0K R,£ AT, K
T = 1.5K R, f2 AT, K
T = 2.0 K R,~2 AT, K
14/7/70 24/7/70 5/8/70 14/8/70 2/9/70 1/11/70 14/2/71
428.4 427.8 438.9 430.4 412.7 414.4 427.9
514.2 510.3 490.1 511.3 543.5 559.1 535.9
13.6 16.9 31.0 24.0 18.7 4.4 15.0
1 511 1 516 1 543 1 549 1 575 1 550 1 560
956 955 960 966 875 978 979
777 776 779 782 783 789 792
689 687 692 692 689 695 700
+0.018 +0.015 +0.003 +0.001 -0.011 -0.004
+0.040 +0.042 +0.033 +0.021 +0.005 -0.002
+0.050 +0.054 +0.041 +0.029 +0.025 -0.011
+0.044 +0.059 +0.022 +0.022 +0.044 -0.034
A, B, and C are the parameters obtained b y a least-square f i t of the data f o r that day. R 6 is determined at f o u r temperatures from these parameters.
A T = T c -- 7", where T c is the calculated temperature based on the w o r k done on November 1970
25
Lounasmaa's but not with Krusius'. The discrepancy between Krusius' and Lounasmaa's values for a' and P can be seen in Fig.7 in which R versus A T = TL - TK is plotted for both sets of parameters. In a neutron diffraction experiment we will be performing shortly utilizing the hyperfine interaction in CoO it may be possible to determine which set of holmium parameters Is correct.
20
~o
is
k" I
~."
I0
References 5
o
/
,i ,soo
,ooo
, ooo
R.fi Fig.7 Comparison of Lounasmaa's (a " = 0.320 K and P = 0.007 K) and Krusius' (a " = 0.319 K andP = 0.004 K) data for holmium. T K -- T L is the difference in the temperature deduced from the U(T)'s derived from their data. T L is indicated for various resistance values of R 5
find a ' = 0.320 K and P = 0.008 K. We have adopted Lounasmaa's values o f a ' and P in our work for two reasons. First his values o f a ' and P were determined from a specific heat measurement made between 0.38 K and 4.2 K and thus they are entirely in the region of our calibration work. Secondly, Van Kempen's data are in agreement with
20
10 11 12 13
Culbert, H. V., Sungaila, Z. Cryogenics 8 (1968) 386 Van Kempen, H., Miedma, A. R., Huiskamp, W. J. Physica 30 (1964) Freeman, A. J., Frankel, R. B. (ed) Hyperfine Interactions, (Academic Press Inc, New York, 1967) Lounasmaa, O. V. PhysRev 128 (1962) 1136 Cheng, C. H., Wei, C. T., Beck, P. A. PhysRev 120 (1960) 426 Furukawa, G. T., Saba, W. G., Reilly, M. L. 'Critical analysis of heat capacity data of the literature and evaluation of the thermodynamic properties of Cu, Ag and Au from 0 to 300 K', NSRDS Report NBS18 (1968) Dixon, M., Hoare, F. E., Holden, T. M., Moody, D. E. Proc Roy Soc (London) A 285 (1965) 561 O'Neal, H. R., Phillips, N. E. PhysRev 137 (1964) 748 DuChantenier, F. J., Boerstoel, B. M., DeNobel, J. Physica 31 (1965) 1061 Hetzler, M. C., Walton, D. Rev Scilnst 39 (1968) 1656 Clement, J. R., Quinnell, E. H. Rev Sci Inst 23 (1952) 213 Lounasmaa, O. V. PhilMag 3 (1958) 652 Krusius, M., Anderson, A. C., HolmstrOm, B. Phys Rev 177 (1969) 910
C R Y O G E N I C S . J A N U A R Y 1974
A technique for calibrating carbon resistor thermometers below 2 K D. A. Goer, E. F. Starr, G. R. Little, and R. A. Erickson
Resistor thermometers are widely employed in the temperature range of 0.2 K to 4 K. They are usually calibrated by comparing their resistance values against the vapour pressure of either He4 or He 3 liquid in the temperature region above 0.5 K and a three parameter fit of temperature versus resistance is obtained which is then extrapolated for use in all temperature regions. However, the instability of carbon resistors due either to treatment or passage of time requires the experimenter to obtain calibration points each time he uses his apparatus if he wishes to have reliability, and even then, the extrapolation to temperatures outside the actual region of calibration is questionable. The accuracy of the thermometry will therefore be improved if calibration is done as part of the primary experiment in which the thermometer is to be used and if the calibration is made over the entire temperature range of interest. In a cryostat constructed for use in neutron diffraction experiments, vapour pressure determinations below about 1.6 K were not easily obtained and thus a new calibration technique was desirable. Culbert and Sungaila 1 have suggested a method of calibrating resistance thermometers using the known specific heat of a pure metal. In their technique, the heat capacity of a sample of pure (99.999%) platinum is measured above 0.75 K and is used to calculate the internal energy-temperature function for the sample. The internal energy-resistance relation is then obtained by feeding known amounts of electrical energy into the sample and measuring the resultant change in resistance. The temperature-resistance relation is then deduced by comparing the two functions. This technique has advantages over pure vapour pressure techniques in that the calibration is done for all temperatures accessible to the cryostat and hence no calibration extrapolation is necessary. The use of a pure metal standard allows the extrapolation of the heat capacity to be made with reliability and also allows the usage of pub-
The authors are with the Department of Physics, The Ohio State University, Columbus, Ohio, USA. EFS is at present with the Systems Research Laboratories Inc, Dayton, Ohio, USA. Received 16 July 1973.
CRYOGENICS. J A N U A R Y 1974
fished specific heat parameters. However, the specific heat of most pure metals decreases as T and hence the internal energy decreases as T 2. Furthermore, the specific heat of metals is generally small even near 1 K. Therefore, the use of a normal metal as a standard requires that there be a minimum amount of other materials (superconducting solders and construction material alloys, for example) in the calibration cell. It would then be virtually impossible to employ this technique as part of a primary experiment and even if performed in a special cell the decreasing heat capacity would cause the calibration reliability to diverge at low temperatures where accuracy is often most desired. Clearly these difficulties could be overcome if a substance of large specific heat were used as the standard. For work at temperatures accessible to He 3 cryostats, the rare earth metal holmium is an excellent choice as a calibration standard. Due to alignment of nuclear spins, the specific heat of holmium exhibits a well known and abnormally large Schottky-type anomaly near 0.4 K, the maximum value of 6.98 J mole "1 K"1 occuring at 0.3 K. The heat capacity of a relatively small amount of holmium would easily mask that due to other materials in the test cell, thereby allowing the calibration to be made as part of the primary experiment. Furthermore, since the specific heat of holmium increases with decreasing temperature, the accuracy of calibration is enhanced at low temperatures. If enough holmium is added to the sample region so that all the other component materials contribute but a small fraction to the total internal energy of the system, the errors in the thermodynamic functions of the remaining components (for example, alloys) are reduced. This is especially true if the apparatus contains stainless steels since the specific heat of stainless steel is not well known. Furthermore, the presence of a substance with such a high specific heat adds a great deal to the temperature stability of the system. The He 3 cryostat used consists of four stages. A two-stage nitrogen-hydrogen liquefier (Air Products Cryo-Tip Model AC2-112) maintains 80 K and 20 K flanges from which heat shields are attached for thermal isolation. Stage III is a liquid He 4 bath which can be pumped to about 1.6 K. In the final stage, stage IV, liquid He 3 is cryo-
15
Stainless steel tube
Primer coil 304 S flanges ~
S
~
(]]I1=3 - - I n d i u m seal
__I
[
Cobalt cup
Copperring
Holmium R;c m ~s~l u[ g ~ ~ [ ~
R6
= ~
,om[ E[
F
Diffraction region
____J
Silverring " Indiumseal
~------------- Aluminium heat shield
Fig.1 Schematicdiagramof stageIV of the cryostat.
pumped by a charcoal pump to temperatures of 0 . 3 0.4 K. A detailed description of this cryostat will be published elsewhere. Stage IV of the cryostat is schematically shown in Fig. 1. A cobalt cup is vacuum sealed to a stainless steel flange by an indium gasket. Two rods of pure (99.9%) holmium metal (0.0352 moles) are sandwiched between copper and silver rings and the silver ring is indium soldered to the bottom of the cobalt cup. The holmium rods are thermally connected to the silver ring by indium discs and are held tightly to the indium coated copper ring by brass screws under tension. Carbon resistor thermometers R5, R6, and R9 are encased in tightly fitting copper or silver mounts and are positioned as shown in Fig. 1. To reduce rf heating 0.001/2F capacitors are attached across each resistor. The manganin lead wires are thermally anchored to the silver ring. R9 is a Speer's 220 ~ (1/2 W grade 1 002) carbon resistor and R 5 and R6 are Speer's 300 ~2 (1/4 W grade 1 002) carbon resistors. A 'primer' system rapidly cools the thermally isolated stage IV and also permits vapour pressure measurements to be made in the temperature region above 1.6 K. In stage IV the primer consists of a 5 in long coil of 0.062 in x 0.042 in copper tubing which is indium soldered to the stainless steel flange. The input side of the primer is thermally connected to a He4 bath (not shown) and the output side of the primer can be connected either to a roughing pump or a Wallace-Tiernan pressure gauge. After the He4 bath is pumped below the X-point, He 4 gas is condensed in the primer coils, thus establishing a superfluid He II link between the bath and stage IV. This cools stage IV to the temperature of the He 4 bath, about 1.6 K. The temperature of the He 4 bath can be controlled and static vapour pressure readings can be taken of stage IV by using
16
the primer, thereby obtaining calibration points for the three thermometers above 1.6 K. The primer is then evacuated using a roughing pump and stage IV is taken to its lowest temperatures by cryopumping He 3 liquid which drips down into stage IV from a condensing coil located in the He4 bath. The cryopumping is carried out by a charcoal pump which is in thermal contact with the He4 bath. Temperatures of 0.3-0.4 K are obtained. After the He 3 liquid has been pumped away (typically about 1/80 mole) the low temperature calibration commences. Using R6 as a heater, measured amounts of energy are fed into stage IV. Typical power inputs, of about 100/~W, caused R6 to warm to about 2.3 K. After a heating interval the resistors thermally relax in a few minutes to equilibrium drift rates that are associated with background heating or cooling as shown in Fig.2. The three thermometers were read using a Hewlett-Packard Model 3440A DVM and the readings were printed every sixty seconds on a Beckman Model 1453-4 printer. The heating process is repeated until a temperature within the vapour pressure calibration region is obtained. The vapour pressure calibration is used to assign a temperature TO to this final equilibrium resistance R0. TO is chosen to lie close to the hpoint since the vapour pressure calibration work is most accurate in this region.
A U(R) function,
defined as the amount of energy input necessary to take stage IV from some equilibrium resistance R to the final equilibrium resistance R0, is obtained by summing the Joule heating energy inputs, Uj, from R to R 0 and adding to this the integrated background heat leak, UBG, as shown in Table 1. Since the principal heat leak into stage IV is due to metallic conduction, the background heat leak is adequately described by the empirical equation: (~BG = C( T2 -
r2)
(1)
where c is the background heating coefficient, ("2.3/aW K'2), measured in the vapour pressure calibration region, TB is the temperature of the He 4 bath, and T is the temperature of stage IV determinec] from extrapolating the vapour pressure calibration. QBG is of the order of 5-7/aW at the lowest temperature.
i
_ .O.--O~ O"]
195(2
720
,4
19OC C~
d I
18OC
I
I
I I !
175C 17OC O
700
/o-"o- o
185C
2
4 6 Time r rain
680
O 2
4 6
568
8
IO
12
Time ~ rain
Fig.2 Typical thermal relaxation rate curves for the heater, R6, in two different temperature regions. At point A the heating w a s started and at point B the heating was stopped
CRYOGENICS. JANUARY 1974
Table 1. Results from a typical calibration run
R 5, ,.Q
R 6, ,.Q
R 9, ~
Uj ,
UBG,
U(R) = Uj + UBG,
mJ
mJ
mJ
U N (R) = U ( T o) - U(R), mJ
T, K
l/T, K1
696.4 748.3 805.4 880 961 1 051 1 138 1 221 1 305 1 390 1 488 1 597 1 704 1 841 1 971 2 073
671.8 720.8 776 848 927 1 013 1 097 1 176 1 256 1 336 1 424 1 523 1 625 1 742 1 866 1 973
493.7 531 573.4 628 686 753 815 876 934 995 1 065 1 140 1 210 1 302 1 387 1 452
0 23.3 43.1 64.4 93.2 100.8 115.6 127.8 138.6 148.1 157.6 167.2 175.3 183.4 190.2 194.3
0 -1.5 -1.9 -1.6 -0.9 0.1 1.3 2.5 3.7 5.2 6.9 8.5 10.1 12.0 14.0 15.9
0 21.8 41.2 62.8 82.3 100.9 116.9 130.3 142.3 153.3 164.5 175.7 185.4 195.4 204.2 210.2
354.6 332.8 313.4 291.8 272.3 253.7 237.7 224.3 212.3 201.3 190.1 178.9 169.2 159.2 150.4 144.4
2.187 1.833 1.554 1.291 1.096 0.941 0.827 0.743 0.675 0.617 0.562 0.512 0.469 0.427 0.392 0.368
0.457 0.546 0.644 0.775 0.913 1.063 1.209 1.346 1.483 1.621 1.779 1.955 2.132 2.342 2.553 2.717
T is t h e t e m p e r a t u r e at w h i c h
UN(R)
= U ( T ) ; To = 2 . 1 8 7 K a n d
U(To) = UN(Ro)
In order to calibrate R as a function of T it is necessary to know the internal energy of the sample region as a function o f temperature. U(T) is obtained by summing the contributions to the internal energy of the component materials o f stage IV. Cobalt and holmium have, in addition to their electronic and lattice contributions to the internal energy, an energy term arising from the interaction o f the nuclear spins with the electromagnetic field o f the electrons, The Hamiltonian o f the nuclear spins can be represented by 2
= 3 5 4 . 6 mJ
1 1 U(T) - U0 = - A 1T 2 + - A 2 T4 2 4
/2'K T
7/2 7
+taR
" Ei/k exp(-Ei/k T)
i=+7/2
(4)
-7/2
exp(-Ei/kT)
-a'Iz+P
12--I(I+1)
k
+P'(I2-12)
(2)
3
where the P and P ' terms are due to the electric quadrupole interaction and the a' term is due to the magnetic dipole interaction. I is the nuclear spin and I z is its azimuthal component. For holmium the contribution to the energy levels due to the P ' term of the quadrupole interaction is two orders o f magnitude less than the P term 2 and for cobalt the quadrupole interaction may be neglected entirely. 3 Thus the resultant energy levels are o f the form
-a'i+P k
i 2-
I(I+I)
(3)
3
with P = 0 for cobalt where i = I z. The values of a ' and P have been reported from either nmr measurement of the effective field at the nucleus or from direct specific heat measurements. For holmium a' = 0.320 K and P = 0.007 K. 4 In cobalt a ' = 1.09 x 10 -2 K 4 and for T >> a ' the internal energy may be expanded in inverse powers of T, with the leading term of - K / T and K = 1/3 Ra'2I(I + 1) = 5.02 mJ mole "l K. 5 Thus the internal energy o f stage 1V can be detennined from an equation o f the form (for T > 0 . 1 K)
CRYOGENICS . J A N U A R Y 1974
where A 1 and A 2 result from the electronic and lattice terms o f the component materials,/x' and/2 are respectively the number o f moles o f cobalt and holmium present, R is the molar gas constant, k is Boltzman's constant, and U0 is an arbitrary constant. Corrections were made for the superconducting transitions in both indium and aluminium, but they have no noticeable effect on U(T). U(T) and U(T) - Ulto are shown in Fig.3 and the values of the parameters used are listed in Table 2. The calibration procedure is depicted in Fig.4. The calculated U(T) is equated to a renormalized U:v (R) = U(To) U(R) where TO is the temperature associated with the final equilibrium resistance value, RO. This determines a temperature T for any equilibrium resistance value R. The results obtained from carrying out this procedure for a typical calibration are shown in Table 1. In the temperature region above 1.6 K the R (T) points from vapour pressure measurements are fitted to an equation of the form b R = a + -T
(s)
17
runs was only 0.5 K and the sample region was not warmed above nitrogen temperatures.
600
500 --3 E
4OO
l
As can be seen from Fig.5, our suggested thermometry equation fits the data as well as or better than the other suggested equations and does much better than the log R equation in this temperature region. The form of the suggested equation makes it easy to determine what resistance value one should observe for a particular temperature though the inverse process is a bit more inconvenient as it involves solving a quadratic equation in 1/T. Due to the small quadratic term in 1/T 2, C must be determined to several significant figures if the inverse process is to be used with confidence. However, this is not a serious problem as a graphical determination of temperature from resistance values is particularly easy since R versus 1/T is a straight line over much of the temperature range. (R 5 is well represented by a straight line in 1/T down to about 0.8 K, as shown in Fig.6.) One interesting feature of our suggested equation is that significant variations of the parameters can result in almost no change in the fit. For example, if the parameters for R9 are changed to A = 297.72 ~ , B = 427.73 ~. K, C = +0.347 ~ K 2 (obtained by weighting the experimental points with T), the differences in calculated temperatures using the two different sets of parameters are all within 1%. This suggests that there is a rather broad minimum in the standard deviation as a function of the parameters A, B, C.
J
•.• 3OO 200 I00 O
Fig.3
I
I
I
0.5
0.6
1.5
I I 2.0 2.5 TtK
I 3.0
1 3.5
4.0
Internal energy of stage IV as a function of temperature.
U(T) is the total internal energy of stage IV and U(T) - U H . is the
contribution to the internal energy of all the components offstage IV except for holmium. U 0 is taken to be 393.3 mJ
to ~0.2% in A TIT. This equation is used to correlate TO with R 0 and to calculate an approximate temperature of stage IV for heat leak purposes for use in (1). The error resulting from this approximation is unimportant since the approximation, (5), breaks down only at low temperatures where the heat leak is determined primarily by the bath temperature. If measured values of R are plotted against lIT as determined from U(T) the resulting graph suggests that an appropriate equation to try as a fit would be B
R = A +-
C +-T T2
Some doubt has existed concerning the reproducibility of the temperature dependence of a given resistor over periods of time. Our data, as mentioned above, indicates that a calibration is good over a 24 hour period providing the thermometers are not warmed above 80 K. However, our experience is that significant changes do occur over longer periods of time depending on how the resistor is treated. A history of R6 over a seven month interval is shown in Table 4. Typically, the shifts in the calibration are a few percent. Changes occurred often after soldering lead wires, but in some cases they seemed to occur spontaneously. These changes occurred for all temperature ranges including temperatures which were directly measured by vapour pressure and hence do not reflect on the reliability of our technique for calibrating carbon resistors. This question of reproducibility makes it desirable to be able to calibrate carbon resistors each time an experiment is performed.
(6)
A least square fit of the above equation is good to within 0.5% in AT/T for the 1/4 W resistors and to within 1% for the 1/2 W resistor. The higher deviations of the 1/2 W resistor are thought to be due to a poor thermal contact of the thermometer with its case. Other suggested thermometry equations 10,11,12 are also fitted for the sake of comparison (see Fig.5). The constants found for our suggested equation are given in Table 3. The equations obtained from the fitted data points were applied to two other runs taken during the same 24 hour period. The results agreed to within 1% for both the 1/4 W and 1/2 W resistors, but the minimum temperature achieved on these
Table 2. Parameters used in calculating U(T)
Material
Mass, g
T, mJ mole "1 K "2
0, K
Cu Ag AI In Co Ho 304 stainless steel
16.038 24.780 3.127 1.5 15.5 5.797 21.66
0.695 0.650 1.35 1.69 4.72 10.0 *
344.5 226 427.7 111 460 95.1 *
Reference 6 6 7 8 5 4
Contribution to A1, mJ K "2
Contribution to A 2 x 102, mJ K "4
0.175 0.149 0.157 0.022 1.241 0.352 10.4
1.20 3.87 0.29 1.86 0.53 7.94 0.80
For 15.5 gm of cobalt, k = 1.320 mJ K; in the nuclear c o n t r i b u t i o n of h o l m i u m to U ( T ) , a " = 0.320 K and P = 0.007 K * D u Chatenier, et al 9 su~lgests t h a t a valid specific heat equation below 10 K f o r stainless steel is Css = 0 . 4 6 T + 3.7 x 10-4T 3 mJ g-! K "l. A T coefficient o f 0.48 mJ g", K" 2 was experimentally determined to be more correct for the 304 stainless steel used in our apparatus
18
CRYOGENICS . JANUARY 1974
Table 3. Values of the constants A , B, C, as d e t e r m i n e d f r o m a least-square fit of the equation: R = A + B / T + C / T 2
R,n 7OO
I OOO
--T
500
1500
I
2000
[
~t
iI
I
I
% .....
_~_
R5 R6 R9
3OC g
2oc IOO
O
/5 /;'
+
,
,
i
I
[
I
I
I.O
1.5
2.0
i
[
I
I
2.5
3.0
3.5
Fig.4
Graphical method for determining temperature from The energy axis is common to both curves. For a particular resistance, R 1, one finds U N(R 1) and moves horizontally to a point [T1, U(T1)] on the U(T) curve such that U(T l) = U N (RI). This temperature T l is then associated with Rb Two examples of this procedure are given for thermometer R s. R s = 696 ~2 corresponds to a temperature of 2.19 K and Rs = 1650 gZ corresponds to a temperature of 0.49 K
UN(R)=U(T)-U(R)andU(T).
• R . a + a i rc + c l r J
~s.2%f
o liT .A+BR~+CR o re . (AR,a) I (R-C)
ku /
C,~K
432.65 414.44 289.04
568.81 559.10 443.76
13.2 2 4.38 -5.33
2
R U JNt
~ K
2.0
B,~K
_ _ + __'2"y~__ - _ ~ ;
O.5
3.0
A,~
• C/Fc - l o g R + ( A / I o 9 R ) - B
However, it must be pointed out that all errors in determining the heat leak contribution to U(R) are cumulative so that care should be taken in determining the heat leak coefficient, c. In the present experiment, an error o f 10% in the background heating coefficient (c) will cause an error o f about 2% in determining the value of the minimum temperature reached. Since the contribution of the heat leak to the total U(R) depends on the total elapsed time of the calibration process, its importance will be reduced if larger power inputs are made to shorten the time o f calibration. The presence of more holmium will also reduce this difficulty considerably. In a new five slug (0.0837 mole) holmium system an error o f 10% in determining c results in less than a 1% error in T minimum. Also the increased holmium increases the accuracy of the U(T) determination since the remaining components of stage IV are much less important. Finally it should be noted that other values o f a ' and P for holmium have been suggested. Krusius et al 13 find a ' = 0.319 K and P = 0.004 K while Van Kempen et al 2
1.0 O
&&
O •
q
•
2000
-I.O
-2.0 i
0.5
L
I
1
I.O
1.5
2.O
1500
r,K
Fig.5 Differences between the measured temperature, Tm. Tc is calculated using a least squares fit to the data for resistor R5 for each of the listed expressions 1(3OO
Using the internal energy o f the sample region as a thermometric device has an advantage over the standard vapour pressure technique. It is difficult to obtain vapour pressure measurements with accuracy below 1 K for He 4 and below 0.6 K for He 3, whereas in this technique calibration is easily obtained for any region in which the internal energy is well known, and hence extrapolation to a noncalibrated region is not required. While this technique requires some computation to determine the temperature, no thennomolecular corrections are needed and the actual measurements are very simple to take, Furthermore, due to the large heal capacity o f holmium, the temperature stability o f the sample region is increased.
CRYOGENICS
. JANUARY
1974
5OO
0.4
I I.O
I 2.O I
7,
_
I 3.0
K-I
Fig.6 R s versus 1 IT, illustrating the linear behaviour of the resistor over much of the temperature range in lIT. The standard deviation for a two-parameter least squares fit is 6 1 ~ which compares to 1.6 D, for the three parameter fit
19
Table 4. A history of R 6
Date
A,~
B, fZ K
C,£Z K 2
T = 0.5 K R, E2 AT, K
T = 1.0K R,£ AT, K
T = 1.5K R, f2 AT, K
T = 2.0 K R,~2 AT, K
14/7/70 24/7/70 5/8/70 14/8/70 2/9/70 1/11/70 14/2/71
428.4 427.8 438.9 430.4 412.7 414.4 427.9
514.2 510.3 490.1 511.3 543.5 559.1 535.9
13.6 16.9 31.0 24.0 18.7 4.4 15.0
1 511 1 516 1 543 1 549 1 575 1 550 1 560
956 955 960 966 875 978 979
777 776 779 782 783 789 792
689 687 692 692 689 695 700
+0.018 +0.015 +0.003 +0.001 -0.011 -0.004
+0.040 +0.042 +0.033 +0.021 +0.005 -0.002
+0.050 +0.054 +0.041 +0.029 +0.025 -0.011
+0.044 +0.059 +0.022 +0.022 +0.044 -0.034
A, B, and C are the parameters obtained b y a least-square f i t of the data f o r that day. R 6 is determined at f o u r temperatures from these parameters.
A T = T c -- 7", where T c is the calculated temperature based on the w o r k done on November 1970
25
Lounasmaa's but not with Krusius'. The discrepancy between Krusius' and Lounasmaa's values for a' and P can be seen in Fig.7 in which R versus A T = TL - TK is plotted for both sets of parameters. In a neutron diffraction experiment we will be performing shortly utilizing the hyperfine interaction in CoO it may be possible to determine which set of holmium parameters Is correct.
20
~o
is
k" I
~."
I0
References 5
o
/
,i ,soo
,ooo
, ooo
R.fi Fig.7 Comparison of Lounasmaa's (a " = 0.320 K and P = 0.007 K) and Krusius' (a " = 0.319 K andP = 0.004 K) data for holmium. T K -- T L is the difference in the temperature deduced from the U(T)'s derived from their data. T L is indicated for various resistance values of R 5
find a ' = 0.320 K and P = 0.008 K. We have adopted Lounasmaa's values o f a ' and P in our work for two reasons. First his values o f a ' and P were determined from a specific heat measurement made between 0.38 K and 4.2 K and thus they are entirely in the region of our calibration work. Secondly, Van Kempen's data are in agreement with
20
10 11 12 13
Culbert, H. V., Sungaila, Z. Cryogenics 8 (1968) 386 Van Kempen, H., Miedma, A. R., Huiskamp, W. J. Physica 30 (1964) Freeman, A. J., Frankel, R. B. (ed) Hyperfine Interactions, (Academic Press Inc, New York, 1967) Lounasmaa, O. V. PhysRev 128 (1962) 1136 Cheng, C. H., Wei, C. T., Beck, P. A. PhysRev 120 (1960) 426 Furukawa, G. T., Saba, W. G., Reilly, M. L. 'Critical analysis of heat capacity data of the literature and evaluation of the thermodynamic properties of Cu, Ag and Au from 0 to 300 K', NSRDS Report NBS18 (1968) Dixon, M., Hoare, F. E., Holden, T. M., Moody, D. E. Proc Roy Soc (London) A 285 (1965) 561 O'Neal, H. R., Phillips, N. E. PhysRev 137 (1964) 748 DuChantenier, F. J., Boerstoel, B. M., DeNobel, J. Physica 31 (1965) 1061 Hetzler, M. C., Walton, D. Rev Scilnst 39 (1968) 1656 Clement, J. R., Quinnell, E. H. Rev Sci Inst 23 (1952) 213 Lounasmaa, O. V. PhilMag 3 (1958) 652 Krusius, M., Anderson, A. C., HolmstrOm, B. Phys Rev 177 (1969) 910
C R Y O G E N I C S . J A N U A R Y 1974