Estimation of thin-film platinum thermometer precision in the range 13.8–273.16 K and common mathematical description above 70 K

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1996 Elsevier

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Estimation of thin-film platinum thermometer precision in the range 13.~ 273.16 K and common mathematical description above 70 K D.A. Dimitrov,

J.K. Georgiev,

A.L. Zahariev and I. Bivas

Bulgarian Chaussee

Academy of Sciences-Institute blvd., 1784 Sofia, Bulgaria

Received

14 November

1995; revised

of Solid State Physics, 72 Tzarigradsko

16 January

1996

The temperature-resistance dependence of 10 thin-film platinum sensors was experimentally studied between 5 K and 300 K. A method was proposed for evaluating data errors in both coordinates. The orders of magnitude of the errors for the investigated thermometers were f(5-8) mK for temperature and f(2-3) mR for resistance. Common mathematical descriptions of 77R) and R( 77 functions above 70 K and a two-point method for individual calibration are proposed. 0 1996 Elsevier Science Limited Keywords:

resistance;

temperature;

standard

The precise determination of temperature is an important part of different measurements and technological processes. For example, the main experimental error in measuring thermal properties of solids comes from uncertainties in the thermometer and its temperature stability; hence the importance of improving the accuracy of the thermometers used. One of the most precise instruments for temperature measurements (especially below 273.16 K) is the standard platinum resistance thermometer (SPRT), with a measurement accuracy of f1.5 mK’. However, SPRTs are large and fragile and the high price restricts their widespread use. Thus, an industrial version of the platinum resistance thermometer (IPRT) is admissible in many cases as they are robust, much smaller and cheaper. Different methods can be applied to determine their uncertainties more accurately. One of them is to improve their mathematical description. Thus, it is important to know the limit of precision that can be achieved with an IPRT and to find out how some reliable points in its R(T) function can be used with a higher weight in its description. A review of investigations into IPRT precision below 0°C shows that they are not sufficient and that results are to some extent contradictory. Thulin’ reported stability tests on IPRTs in the range 50- 100°C. He found that some ageing procedures are necessary for improving reproducibility. Knobler et al.’ have cycled four samples between 77 and 293 K without observing any systematic resistance deviations. They found that a two-point calibration method provided an accuracy of +80 mK. Magnum and Evans4 investigated the stability upon thermal cycling and handling of 60 IPRTs and found that most of them exhibited calibration

deviation;

thin-film

platinum

sensor

drifts, as well as instability caused by hysteresis. Besley and Kemp’ reported measurements made on the stability of seven IPRTs between 77 K and 373 K. They described a two-point calibration method which gave an accuracy of f35 mK. These studies reveal complicated problems in determining the behaviour of IPRTs and show the necessity for a complete investigation of sensors fabricated in different places and at different times.

Experimental description

results and mathematical

The studied sensors consist of a thin-film platinum layer of thickness 1500-2000 A deposited on a polycrystalline A&O, substrate, produced by Pribor Ltd, Koprivshticza, Bulgaria under a technology described in reference 5. Our measurements have been carried out on a computer-driven experimental set-up. The block scheme is shown in reference 6 and the cryogenic aspects are fully described in reference 7. A computer-driven thermal regulator supplies an automatic approach in a narrow interval of +50 mK to any preset temperature. The thermal process control automation increases the stability of the chosen temperature maintenance to fl mK/30 min. An Rh-Fe resistance thermometer (produced by VNIIFTRI, Russia) gives temperature data with a reproducibility better than f3 mK in the 0.5-400 K range. Hence the maximum measurement error of the experimental set-up is less than &4 mK. Ten thin-film platinum sensors (TFPSs) of the abovementioned type were investigated after having previously undergone the ageing procedure described in reference 6.

Cryogenics

1996 Volume

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599

Estimation

of thin-film

platinum

thermometer:

D.A. Dimitrov

type as the standard R(T) polynomial to ITS-90:

They were numbered from 03 1 to 042 (without 037 and 039). Their R(T) dependencies have been obtained using the method described in reference 7. The sensors were calibrated 12 times in the temperature interval 5-320 K. Their resistances were measured at 63 temperature levels in the increasing direction only (from 5 K up to 320 K). Some results from the last calibration cycle are presented in Table 1. The quantities AR in the last column are the maximum relative differences in the resistance values of the investigated sensors at the temperatures presented in the first column, calculated for the ith temperature with the equation ARi = lOO(Ry

- Rmin)lRTaX%

of pure Pt according

M W = R/R, = exp c BJln( T/T,,)I1.5 + 1I’-’ 1 ,=I I

(4)

Here Ai and Bi are polynomial coefficients, W is the resistance ratio R/R0 and M is the number of polynomial terms (M=7 forequation (2), M=5 for Equation (3), M=6 for Equation (4)), To = 273.16 K and R,, is the resistance at T = To). Equations (2) and (3) describe the experimental data with a standard deviation of about kO.5 K and Equation (4) describes them with a standard deviation of about f0.2 0. The maximum deviation was about 1.2 K or 0.5 CI at 75-80 K for sensor 033 (see Figure I). The obtained values of the coefficients Ai for Equations (2) and (3) and coefficients Bj for Equation (4) are presented in Table 2. Statistical research on a large number of TFPSs must be carried out to define them more precisely. The differences AW, (j is the sensor number) between the experimental W; data and the thermometric polynomial W(T) describing the ensemble of thermometers (Equation (4)) are shown in Figure I to illustrate the goodness of fit. Different symbols are used to present the data points of every sensor to eluci-

(1)

They show that the resistances of the investigated thermometers at 270 K differ with 0.6 a, i.e. 0.6% of the maximum resistance Ry at this temperature (the resistance of sensor 035 in this case), while at 5 K the difference becomes 0.8 0, i.e. about 60% of RF. It is evident that any sensor possesses an individual R(T) function. On the other hand, our analysis of the experimental data above 65 K (see Table I) showed that it was possible to find a general mathematical T(R) description of this type of sensor in the range 70-273.16 K with a maximum error of +1.5-2 K. For this reason the experimental data between 65 K and 320 K from the last three calibration cycles for all investigated TFPSs were treated by using a least-squares method to obtain a common mathematical description of the T(R) and R(T) functions. It was found that a seventerm conventional polynomial M-1 T = c A,W’ i=o

et al.

6

I

I

4-

I

!

I

.

031.

o

032,

.

033,

.

034,

.

035.

v

036,

A

036.

"

040.

0

041,

n

042.

*

_

2

m 0

(2)

$Q

0

-2

or a five-term polynomial of the same type as the standard T(R) polynomial of pure Pt according to ITS-90: -6

T = 5 A;[( W’” - 0.65)/0.35]‘-’ i=l

One-cycle experimental

data

I,

I.

I

100

150

200

I

I

250

300

Temperature, K

Figure 1 Dependencies on temperature of the differences AW, between the experimental W, data and the Wj calculated from Equation (4) in the range 65-320 K for all thermometers

can be used to describe the T(R) function. The R(T) function can be described by a six-term polynomial of the same Table 1

I, 50

(3)

(T,/?) for all investigated thermometers R, (R)

T(K) +3 mK

031

032

033

034

035

036

038

040

041

042

AR (%I

5.008 9.998 15.004 20.007 30.010 40.007 50.004 60.016 65.022 70.026 79.989 100.018 150.018 200.008 240.018 270.017

1.1614 1.1943 1.3251 1.6297 2.9841 5.4414 8.7779 12.6512 14.6930 16.7930 21.0313 29.5880 50.4470 70.6586 86.5209 98.2678

1.3765 1.4108 1.5478 1.8655 3.2600 5.7577 9.1231 13.0143 15.0619 17.1661 21.4081 29.9667 50.8136 70.9996 86.8367 98.5628

0.6456 0.6788 0.8048 1.0989 2.4391 4.9082 8.2700 12.1720 14.2278 16.3411 20.6053 29.2092 50.1719 70.4822 86.4281 98.2334

1.3857 1.4194 1.5549 1.8699 3.2543 5.7370 9.0860 12.9618 15.0022 17.0995 21.3274 29.8628 50.6637 70.8078 86.6105 98.3103

1.1753 1.2098 1.3461 1.6619 3.0555 5.5621 8.9415 12.8485 14.9032 17.0148 21.2722 29.8572 50.7709 71.0188 86.9080 98.6720

0.8928 0.9277 1.0628 1.3765 2.7762 5.3014 8.6971 12.6116 14.6680 16.7789 21.0312 29.5956 50.4305 70.5914 86.4116 98.1200

1.4696 1.5038 1.6409 1.9590 3.3509 5.8382 9.1878 13.0630 15.1023 17.1986 21.4253 29.9551 50.7437 70.8747 86.6742 98.3718

1.1581 1.1909 1.3212 1.6248 2.9759 5.4303 8.7620 12.6324 14.6735 16.7726 21.0085 29.5619 50.4231 70.6341 86.5015 98.2541

0.8656 0.9000 1.0327 1.3400 2.7187 5.2261 8.6147 12.5324 14.5932 16.7097 20.9735 29.5734 50.5172 70.7913 86.7028 98.4845

0.8708 0.9058 1.0385 1.3471 2.7310 5.2443 8.6347 12.5521 14.6119 16.7273 20.9884 29.5781 50.4909 70.7317 86.6166 98.3770

60.77 59.58 55.64 48.35 30.26 17.47 10.70 7.13 6.00 5.12 3.92 2.51 1.13 0.55 0.57 0.56

600

Cryogenics

1996 Volume

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Estimation Table2

Coefficient

values for Equations Coefficient

(2) and (3)

Equation (2)

Equation (3)

Equation (4)

1 2 3 4 5 6 7

29.365 254.103 -129.636 348.801 -422.162 252.662 -59.977

47.578 61.784 66.029 47.887 49.885

2.0316 2.8834 -1.4124 -0.6379 -1.411 E-2 -6.305E-2

the differences between their R(T) functions. Regarding the common description of an ensemble of TFPSs, the standard deviation cannot be used to characterize the precision of a single sensor (see for example the data for sensor 033 in Figure I). In this case the maximum deviations are more important characteristics. Hence, if R, is known, it can be assumed that Equations (2) or (3) and Equation (4) can be applied for a single sensor with an accuracy better than f1.5 K and f0.6 R, respectively. When R, is unknown, a value of 100 R can be used instead and the accuracy of Equations (2) or (3) and Equation (4) will decrease to +2.0 K and f0.8 a, respectively. Figure I shows that the A Wj( T) are almost linear above 70 K. This linearity allows a simple two-point calibration method to be applied for the TFPSs in the interval 70273.16 K. If the resistance values at 273.16 K and 77.4 K or some other temperature TX in the range are known for a sensor, the following additive corrections to Equations (2) and (3): date

AZJ(W)=A7;(T,)(l-W)l[l-W(T,)] and to Equation

(5)

(4):

AW;( T) = AW,( T,)( To - T)l( To - TX)

(6)

can be applied. Here ATj( W) is the difference between T,(W) and T(W), calculated from Equations (2) or (3), AWj( T) is the difference between Wj( T) and W(T), calculated from Equation (4), and W(T,) is the value of W at temperature T,, calculated from Equation (4). The sign of the corrections depends on the sign of AW,( TX) and A7;.( TX).

0

50

100

150

200

250

platinum

thermometer:

D.A. Dimitrov

et al.

The proposed corrections to Equations (2), (3) and (4) increase their accuracy to 0.1 K and 0.04 Q, respectively. Figure 2 illustrates that the same accuracy can be attained if known T(W) and W(T) functions of any TFPS are used instead of Equations (2)-(4). It is important to compare W,(T) for the investigated sensors with W(T) for pure platinum calculated according to ITS-908. The dependence A W,,( T), defined as

values

Number of coefficients

of thin-film

300

is shown in Figure 3. There is again a maximum in A Woj( T) in the region of 50 K. The pattern of the curves between 14 K and 50 K elucidate a higher sensitivity of the TFPSs in comparison with the SPRT. The application of these sensors to measurements with higher accuracies (including the region below 70 K) is possible if their individual T(R) and R(T) functions are known.

Estimation

of sensor precision

A very important property of the TFPSs is their precision. It is also necessary to determine an adequate individual mathematical description of any sensor. For this reason, the data of each thermometer from the above-mentioned 12 calibrations between 5 K and 320 K were subjected to statistical investigations in order to determine the upper limit of the deviations in resistance and temperature for any preset point. As a comparative method of calibration was used, our experimental results include the uncertainties in the experimental set-up as well as uncertainties in the investigated sensors. Every experimental point is defined by two coordinates (R,, T;), each of them being determined with some experimental errors. It is difficult to estimate these errors correctly from independent experiments. For this reason a statistical approach was applied’.” to quantify the upper limits G”“(T,) and c(Ti) of the resistance and temperature standard deviations a, and V~ for each of the investigated temperature levels. Groups of 12 experimental points (less in some cases), which settled in a narrow interval of 100 mK, were available for each level. This permitted a straight line fit to be obtained for the points in each group.

_osk , 0

1

50

Temperature, K

Figure2 Dependencies on temperature of the differences AW, between the experimental W, data and the W, of sensor 033 in the range 5-320 K for all thermometers

,

I

100

I

150

,

4

I

I

I

200

250

300

Temperature. K

Figure 3 Dependencies on temperature of the differences AW, between the WI 7-l calculated from the ITS-90 polynomial of the pure platinum in the range 14-320 K for all thermometers

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1996 Volume

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Estimation

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platinum

For each of these ensembles was defined as

thermometer:

(RJ,)

the function

D.A. Dimitrov

et al.

2 (a,b)

I

I

I

I

I

I

a

.mnx $(a,&

(Rj,-a;~,-b,)‘l[(a,)2+a~($T>2]

(8)

j,=l

where Jy was the number of points in the group. The function 2 (~,b) was first minimized with respect to variables ai and bj in order to calculate their values. After replacing these values of ai and b, in Equation (8), d becomes a known function of the expression [ (4)’ + a’( c$)*]‘” and of the number of degrees of freedom i,,, - 2. Let & be a value of 2 for which the probability of 2 > & is W,,,,,. A quantity c corresponding to W,,, exists with the prop-

1

b

erty W{[(0$)2 + u2( CJ#] “2 I C$=} = w,,, Evidently, C(T,)

one upper limit c

0 o

(9)

of a, is

= e

(10)

obtained for the case when c$ = 0. Analogously, limit v(T,) of oJ> is

one upper 0

v

E”

= a,.&

50

100

150

200

250

300

(11) Temperature, K

obtained in the case when C$= 0. Later on we used the value 0.9 for W,,,,, and the calculated numerical values c( 7;) and @Y( 7’;) corresponding to this W,,,,,. The experimental data set obtained was subjected to the foregoing procedure, in order to determine the probability distribution limits F( Ti) and a;-‘““(T(i). The p(T;) and c( Ti) results for all investigated thermometers are shown in Figures 4a and 4b, respectively. The maximum value of p is about 8 mK (about 4 mlR for G”“(T,)). It can be shown that 85% of the points settle below 6 mK (2.5 mlR for c(T,)). Therefore, for each of the investigated sensors the corresponding T(R) function can be obtained with a mean accuracy of the order of 55 mK. Some of the results shown in Figure 4 have high d’( Ti) and c( Ti) values because they are obtained for ensembles with quite a small number of experimental values in the vicinity of the corresponding temperature point. Hence, these were removed and the rest are shown in Figure 5. In this figure the dependence of @‘““(T,) and w(7;) and some peculiarities in temperature intervals lower than 14 K and between 60 K and 100 K are seen. These properties can be visualized if we fit the data with a polynomial. Figures 5u and 5b show slight differences in fl”( T) and e”(T) dependencies with temperature. While c(T) can be described with a straight line for temperatures higher than 100 K, e”“(T) shows second-power dependence in the same interval. Evidently, the fit has only a qualitative meaning. Figure 5u illustrates the rapid increase of c$““(T~) values in the region below 14 K. This is a consequence of the fast deceleration of the R variation with T (the parameter a, from Equation (8) is the reason for this increase). This deceleration results from the impurities included in the platinum layer. The maximum of d;““(T) is placed in the same interval (60-100 K) where the maximum of the first derivative dRldT appears.

602

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1996 Volume

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Figure4 Deviations flax (a) and fl”

(b) for all investigated

thermometers

a

I

0

50

,

1,

I

100

150

I,

I

200

250

I

,

I

300

Temperature, K Figure5 Peculiarities on temperature

of flax

(a) and Gax (b) dependencies

Estimation Our results clarify that the mean value of e(T,) at temperatures between 13.8 K and 75 K is about 1.3 mK, while in the interval 100-300 K it varies from 4 mK to 8 mK. The I dependence exhibits the same picture in the same intervals: 0.4 mR and 2-3 ma, respectively. These results may be used to define adequate mathematical descriptions of the T(R) and R(T) functions, in particular in the case when a weighted mathematical approach is applied. Polynomials of the same type as the standard polynomials of pure Pt according to ITS-90 can be used for this.

of thin-film 3

D.A. Dimitrov

et al.

References 1 Besley, L.M. and Kemp, R.C. Cryogenics (1983) 23 26 2 3 4

6

Conclusions The results from this paper show that the investigated TFPSs can be applied as temperature sensors:

8

1

9

2

thermometer:

in the range 13.8-273.16 K with an accuracy of f(58) mK after an appropriate individual calibration.

5

in the range 70-273.16 K with an accuracy of +1.52 K without any calibration; in the range 70-273.16 K with an accuracy better than 0.1 K after two-point calibration;

platinum

10

Thulin, A. J. Phys. ( 197 1) E4 764 Knohler, C.M., Honeywell, W.I. and Pings, CJ. Rev Sci Instrum (1963) 34 1437 Mangum, B.W. and Evans, G.A. Temperature, its Measwement and Control in Science and Industry, American Institute of Physics, New York (1982) 5 795 Dimitrov, D.A., Terzijska, B-M., Guevezov, V. and Kovachev, V.T. Cryogenics ( 1990) Xl 3486 Zahariev, A.L., Dimitrov, D.A. and Geargiev, J.K. Cryogenics (in press) Dimitrov, D.A., Zahariev, A.L., Georgiev, J.K., Kolev, G.A., Petrinski, J.N. and Ivanov, Tz. Cryo,qenics ( 1994) 34 487 Supplementary Information for the International Temperature Scale of 1990, Bureau International des Poids et Mesures, Pavilion de Breteuil, F-923 10, Sevres Lavrenchik, V.N., Postanovka Fizicheskogo Experimenta I Statisticheskaya Obrabotka Ego Rezultatov, Energoatomizdat, Moskva, 1986 Press, W.H., Teukolsky, S.A., Veterling, W.T. and Flannery, B.P. Numerical Recipes University Press, Cambridge (1992) 650-683

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