THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING TEMPERATURE MEASUREMENTS IN CRYOGENICS John A. Clark January, 1969 IP-830

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TABLE OF CONTENTS Page LIST OF TABLES...............ej 0.*.e.e**eV~ ** iv LIST OF FIGURES..V....................... I INTRODUCTION.....s................e..................... 1 II THE CONCEPT OF TEMPERATURE.............................. 4 III THE ABSOLUTE (THERMODYNAMIC) TEMPERATURE SCALE......... 10 IV THE ABSOLUTE (GAS) TEMPERATURE SCALE.............. 14 V THE INTERNATIONAL TEMPERATURE SCALE (ITS)............... 20 VI TEMPERATURE SCALES BELOW 90~K................e0........ 30 VII THERMOMETERS FOR CRYOGENIC TEMPERATURES.....oo....... 45 VIII THERMOCOUPLES......................0................ 45 IX RESISTANCE THERMOMETRY.......................... 5)4 X MAGNETIC THERMOMETRY........................... o 97 REFERENCES..................................................... 114 iii

LIST OF TABLES Table Page I Defining Fixed Points for the International Temperature Scales *.............* *...........**..... 23 II Official Interpolation Procedures for the 1927 and 1948 International Temperatures Scales......................... 24 III Differences Between ITS (1927) and IPTS (1948).... 25 IV Relation Between the International Practical Scale and the Thermodynamic Scale............................. 26 V Fixed Points Below 90~K.............................. 33 VI Vapor Pressure of He4 (1958 Scale) in Microns (10-3) Mercury at O0C and Standard Gravity (980.665 cm/sec2)..... 36 VII He3 Vapor Pressure on the 1962 He3 Scale at O0C and Standard Gravity, 980.665 cm/sec2 42 VIII Comparison of Cryogenic Temperature Measuring Devices.... 45 IX Thermoelectric Potential Differences in Microvoltsfor Several Thermocouple Combination.......................... 49 X Inhomogeneity of Thermoelectric Voltages Obtained from Dip Tests............. e * 50 XI Thermoelectric Potential Differences in Microcolts for Gold-Cobalt and Constantan vs. Copper Thermocouples....... 51 XII Report of Calibration of Platinum Resistance Thermometer, L & N No. 1653433.................oooooooooo.....00 00.... 61 XIII August 1965 Table for Platinum Resistance Thermometer 1653433.....63 XIV Resistance-Temperature Characterisitcs of Germanium Thermometers Fabricated from Arsenic-Doped Crystal VIII-899 -N.................... 92 XV Temperatures Attained by Adiabatic Demagnetization of Various Paramagnetic Salts...................... 98 XVI Properties of Paramagnetic Salts.....................99 iv

LIST OF FIGURES Figure Page 1 Arbitrary Scales of Temperature.................. 6 2 Comparison of Low Temperature Scales (10 to 900K), Including PSU, NPL, PRMI, CCT-64, and NBS(2-20) 1965, with Respect to NBS(55) Scale............................ 11 3 Constant Volume Gas Thermometer Used for Calibraction of a Platinum Resistance Thermometer........................ 17 4 Thermometer Bulb, Shields, and Cryostat.................. 18 5 Gas Imperfection Corrections for Constant Volume Helium Gas Thermometer......................................... 19 6 Temperature Differences Between Thermodynamic and International Temperature Scales............................. 28 7 Comparison of Measurements of TTS - IPTS...............e 29 8 Comparison of Temperature Scales........................ 31 9 Oxygen Vapor-priessure Thermometer for Calibrating Working Thermometers............................ 34 10 Phase Diagram for Helium.................... 39 11 Deviation of Earlier Helium Vapor Pressure Scales from the 1958 He4 Scale.....,............................ 40 12 Vapor-pressures of He3 and He4...... 41 13 Estimated Reproducibilities of Various Actual and Postulated Temperature Scales................ 44 14 Temperature Ranges Normally Associated With Various Low Temperature Thermometers.......................... 47 15 Thermoelectric Power as a Function of Temperature for Various Thermocouple Combinations....................... 53 16 Resistance Ratio of Platinum as a Function of Temperature 57 17 Resistance-temperature Relationship for Various Resistance Type Temperature Sensors-high Range................ 58 18 Capsule-type, Strain-free Resistance Thermometer........ 59 19 Mueller Bridge with a Four-lead Platinum Resistance Thermometer....................................... 60

LIST OF FIGURES (CONT'D) Figure Page 20 Resistance-temperature Relationship for Various Resistance-type Temperature Sensors-low Range............ 65 21 Resistance-temperature Curve for Two Allen-Bradley Carbon Resistors.................. * 67 22 Schematic Diagram of the L&N Type K-3 Universal Potentiometer Circuit............................................ 68 23 Typical Calibration Curves for Carbon Resistance Thermometer..................... 69 24 Temperature-resistivity Relationship of Insulators, Semi-conductors and Good Conductors......... 72 25 Logarithm of the Conductivity of Various Specimens of Silicon as a Function of Inverse Absolute Temperature.... 74 26 Logarithm of the Conductivity of Various Specimens of Cuprous Oxide as a Function of Inverse Absolute Temperature,.............................................. 74 27 Thermistor Response to Room Temperature Variations....... 76 28 Logarithm of the Specific Resistance of Two Thermistor Materials as a Function of Inverse Absolute Temperature.. 78 29 Effect of Aging in 105C Oven on Thermistor Characteristics; Materials 1 and 2.................................. 82 30 Calibration Curve for A Germanium Thermometer............86 31 Equilibrium Resistance As A Function of the Number of Accumulated Cycles T = 4.20K............................ 88 32 Equilibrium Resistance As A Function of the Number of Accumulated Cycles for Resistor D........................ 89 33 A Plot of the Resistance-temperature Calibration Data for Resistors 1, 2 and 3...................... e..90 34 Sensitivity dR/dT Typical Standard Cryo Resistor....... 91 35 Encapsulated Germanium Thermometer —Model II, with Cover Removed................................. 0 0 0 93 36 Resistance-temperature Characteristics of Germanium Thermal Sensing Elements.............................95 vi

LIST OF FIGURES (CONT'D) Figure Page 37 Variation of dR/dT with Resistance at 4.2~K............. 96 38 Temperature-entropy Diagram for a Paramagnetic Salt Under the Influence of Different Magnetic Fie lds.............. 101 39 Typical Leiden Demagnetization Cryostat, One Fifth of Real Size...................................... 103 40 Typical Paramagnetic Salt Samples.................g...105 41 Plot of Susceptibility per gm vs Reciprocal Temperature for Powdered CuSO4 K2SO4'6H20, Showing the Curie Law Temperature Dependence...............,,,,.,...... 106 42 Plot of Magnetic Moment vs H/T for Spherical Samples of Potassium Chrominum Alum, Ferric Ammonium Alum, and Gadolinium Sulfate Octahydrate........................... 107 43 Electrical Circuit for Magnetic Thermometer............... 108 44 DeKlerk's Result in the Adiabatic Demagnetization of Chromium Potassium Alum.......................... 110 45 Deviation of Curie Temperature T* from Thermodynamic Temperature for Several Paramagnetic Salts.............. 112 46 Entropy of Potassium Chrome Alum as a Function of Absolute Temperature and of T*........C.0.....00.0....... 113 vii

TEMPERATURE MEASUREMENTS IN CRYOGENICS * By John A. Clark Professor and Chairman Department of Mechanical Engineering The University of Michigan Ann Arbor, Michigan, U.S.A. I. INTRODUCTION The word cryogen is derived from the two Greek words kryos- and -en, meaning literally "the production of icey-cold". More simply, a cryogen is a refrigerant. Cryoenic is the adjective form of the noun, and signifies physical phenomena below -150~C (1230K). This is the approximate temperature at which physical properties of many substances begin to show significant variation with temperature (1). Of natural importance to research, engineering design and operation at the low temperatures is the measurement of the temperature itself. The purpose of this paper is to discuss this question in a reasonably broad context within the framework of the 1968 state-of-the-art. The selection of a suitable temperature sensing element depends on a number of important considerations. Perhaps most fundamental of all is the accuracy required in the measurement. Entirely different techniques will be employed, for example, if *Prepared for Measurement Techniques in Heat Transfer, AGARD Publication, Edited by Professor E. R. G. Eckert, 1969. -1

-2an accuracy of 0.001 K is necessary or if 1.00K is sufficient. Other significant factors include the influence of transient effects, sensitivity, type of readout, nature of signal, availability or desirability of recording and control, durability, stability and ruggedness of the sensing element, and of course, replacement, interchangeability and cost. As of this writing (1968) no internationally accepted standard for temperature measurement exists below the defining fixed point temperature for oxygen (90.18 K, -182.97~C)*. This situation is expected soon to be resolved, however, with the adoption of a uniform scale sometime during 1968 or 1969 which will extend the range of the present international scale to 13.80~K (4). Below this temperature no international scale will exist for some time although convenient and practical methods for measurement of temperatures to 0.2~K and lower have been developed during the past decade. These will be included in this discussion. This paper will cover the following topics: the concept of temperature, the absolute thermodynamic temperature scale, the gas thermometer, the International Practical Temperature Scale, temperature scales below 900K, thermocouples, resistance thermometry, and magnetic thermometry or adiabatic demagnetization. The basic principles as well as practical considerations of measurement will be presented. It will probably be of value at the outset if the principal sources of reference for cryogenic temperature measurement are listed. Most of these will be cited from time to time in the body of this paper and hence are also included among the references. However, in view of their value it is important to have them conveniently listed, as follows: *In 1960, the 11th General Conference on Weights and Measures dropped the notation "Fundamental" and "Primary" fixed points and adopted instead the terms "defining fixed points" and "secondary reference points" for the various two-phase reference states for temperature calibration (2, 3).

1. Temperature, Its Measurement and Control in Science and Industry, Reinhold Publishing Co., American Institute of Physics. (i) Vol. I, 1941, 1343 pages. (ii) Vol. II, Edited by Hugh C. Wolfe, 1955, 451 pages. (iii) Vol. III, Charles M. Hertzfeld, Editor-in-Chief, 1962. a. Part I, "Basic Concepts, Standards and Methods", F. G. Brickwedde, Editor, 838 pages, 1962. b. Part II, "Applied Methods and Instruments", A. I. Dahl, Editor, 1087 pages, 1962. c. Part III, "Biology and Medicine", J. D. Hardy, Editor. 2. R. B. Scott, Cryogenic Engineerin, Van Nostrand Co., 1959. 3. R. W. Vance and W. M. Duke, Editors, Applied Cryogenic Engineering, John Wiley & Sons, 1962. 4. R. W4. Vance, Editor,Cryogenic Technoloy, John Wiley & Sons, 1963. 5. Encyclopedia of Physics, Edited by S. Flrigge, Vol. XV, Low Temperature Physics II. "Adiabatic Demagnetization", by Dirk De Klerk. 6. Journal of Research, NBS, Section A (Physics and Chemistry); Section C, (Engineering and Instrumentation). 7. Advances in Cryogenic Engineering, K. D. Timmerhaus, Editor, Vol. 1-13, 1955-68 (to date). 8. Cryogenics, Vol. 1-8, 1960-68 (to date). 9. Metrologia, Vol. 1-4, 1964-68 (to date), Published under the auspices of the International Committee of Weights and Measures. 10. The International Temperature Scale (ITS) a. 1927 ITS G. F. Burgess, "International Temperature Scale", J. Research NBS, Vol. 1, p. 635-37, 1925. See also, Temperature, Vol. 1, 1941, p. 21-23. b. — 1948 ITS (i) J. A. Hall, Temperature, Vol. 2, 1955, p. 115-141. (ii) H. F. Stimson, "The International Temperature Scale of 1948", J. Research NBS, Vol. 42, p. 209, 1949. c. 1960 Text Revisions of the 1948 ITS H. F. Stimson, "International Practical Temperature Scale of 1948. Text Revision of 1960", J. Research, NBS, Vol. 65A, No. 3, 1961. See also, H. F. Stimson, "The Text Revision of the International Temperature Scale of 1948", Temperature, Vol. 3, Part I, 1962, p. 59-67.

II THE CONCEPT OF TEMPERATURE The concept of temperature is old and doubtless stems from the desire to attach definite numerical quantities to a feeling of hotness or coldness. Galileo (1600) was one of the earliest to experiment with the design of an instrument to which a scale was attached for the purpose of indicating a numerical temperature. These early instruments were called "Thermoscopes" and were said to have measured "degrees of heat". Today, 370 years later, we find the art of temperature measurement highly developed but under continued study. The basic standard instrument presently employed is the Gas Thermometer in some respects similar to the first of Galileo, but registering in degrees of absolute temperature, not in degrees of heat. This distinction was made only after the discovery of latent heats by Joseph Black and James Watt (1764) and the enunciation of the Second Law of Thermodynamics (about 1850). A sensation of warmth or cold is of little value to the physical world when measurement and reproducibility of temperature is required. While it is of small value in measurement, the subjective sense might be used to indicate an equivalent "hotness" or "coldness" of two separate bodies. It is common experience, for example, that two blocks of iron, one taken from an ice bath, and the other from a furnace, will approach the same feeling of warmth if they are brought into thermal contact with one another. Of course, great doubt might reasonably be raised concerning the validity of the conclusion of equal "hotness" if the indications were taken by touching the blocks with the skin of the hands. However, other schemes could be used which are less subjective and would produce the same result. One might not use the hands but use, for instance, a small rod of silver and place it in intimate thermal contact with each iron block. The increase or decrease in the length of the silver rod could then serve as an indicating device since by experience it is known that this dimension will

change as the rod is heated or cooled; also, when the heating or cooling ceases the changes in length also cease. Hence, after some period of time following the bringing of the two blocks together very careful observations of the length (say with a powerful microscope) would show no subsequent change in length of the rod when it was placed successively in contact with each block. Furthermore, if the block and the silver rod were mutually in thermal contact with each other and with nothing else, the length of the rod would be the same when it was attached to each block. This would define the measureable state known as the Equality of Temperature. The silver rod might also be called a temperature meter or, more simply, a thermometer. The process just described has led to a generalization, or law, called the Zeroth Law of Thermodynamics. This law, which is the logical basis of all temperature measurement, may be stated as follows: Two bodies (the iron blocks) at the state of equality of temperature with a third body (the silver rod) are in a state of equality of temperature with each other. Returning to the silver rod, which has been called a thermometer, one might be led to attempt to assign a sort of numerical scale to its length so as to convert its elongation or contraction into some definite, reproducible scale of temperature. It is apparent that this could be done with no particular mechanical difficulty, although some amplification of the changes in the length might be necessary for convenience in use. The selection of the type and magnitude of the units on this scale is wholly arbitrary. It is known that when the rod is placed into a bath of ice and liquid water or in a bath of saturated steam at constant pressure and allowed to reach a state of equality of temperature with the bath in each instance, it does not change in length, but has a greater length in the steam than in the ice. Because the length

-6_ 100 A - ARBITRARY 80 - B LINEAR C- PARABOLIC B 60 A I 40 0 0 (A2/AtOOO) STD MATERIAL 1.0 Figure 1. Arbitrary Scales of Temperature.

-7is greatest in the steam bath, it would seem reasonable to assign it the greatest level of temperature, although this is arbitrary and the reverse has been done, as in the Celsius Scale (1740). The temperature of the ice bath, for convenience, then could be established as zero, while that of the steam bath could be taken as 100. This would define a difference of 100 degrees of temperature between these two fixed points. The final decision to be made in the construction of a temperature scale is the selection of an interpolation device to be employed to obtain the level of temperature and the magnitude of the unit degree of temperature in this interval from observations of the length of the rod. Freedom of choice is to be had in this selection also. That is, any curve connecting the 0 and 100 degree points in Figure 1 may be chosen as the interpolation device.* Some curves obviously are more convenient than others. One would find a multivalued curve quite inconvenient to use, for example, Three curves or interpolating devices are shown in Figure 1. The abcissa is the ratio Anl, the difference between the length of the rod at intermediate levels of temperature and at the ice bath temperature to Alo0, the difference between its length in the steam bath and the ice bath. Clearly, there are an infinite number of different scales one could select. One could take curve B which is a straight line joining the fixed points of temperature, and have a linear scale which, if the silver rod were used as the thermometer, could be called the linear silver scale of temperature. Denoting this as the O scale, the relationship between length changes and e is the simple one: A- l100 Al100' *The ordinate in Figure 1 is divided arbitrarily into 100 equal divisions between 0 and 100. A scale of temperature is established both by the curves and the scale of the ordinate, as is evident, but freedom of choice is preserved if one fixes the scale of the ordinate and allows the curves to have an arbitrary shape.

-8A question which might be raised now is this: Is this 4 scale fundamental or if only the material in the rod is changed, will a different scale result? It turns out that this scale, indeed, is not fundamental and different temperatures would be obtained with, say a linear copper scale. This important consequence of physics may be demonstrated as follows. Consider that a linear silver scale has been adopted for reference with the silver rod being selected as the primary standard thermometer. All temperatures will be referred to this scale which we have called 0. We may take any property of matter which is measureable and which changes with heating and cooling as the indicating quantity. Changes in such properties may be called =p = PPo where p is the magnitude of the property p at the level of temperature of the ice bath. In a rod of any other material the temperature coefficient of p based on the linear silver scale of temperature is defined as: a = X (1) ae ~ Hence, AP = f ade, (2) Also, p100 100 0(3) 80, Up _ Ori ado (4) nploo 100go 10P ioo~

Each of the integrals in equations (2) and (3) may be expressed as follows: f a dG = 100 f a d t - 100, 0 100 where ( is the average value of ca in a range of temperature. Equation (4) is then written: Ap p 100 (5) P00oo 100 100 It is a matter of experience that any property which changes with heating or cooling (such as length) does not do so at a constant rate of change in terms of the temperature of the scale of temperature employed to measure it. The exception, of course, is the property of the material used to define the linear scale of a selected standard temperature scale. But this is trivial. A linear scale based on AP of any other material would be written: znsp _ t (6) IPlo0 100' where t is the temperature on the linear scale pertaining to this other material. Combining equations (5) and (6) we have: _a e t 0oo (100) =lo or LOO t al (7)

Equation (7) is the expression relating the linear scales of temperature (4 and t) of two different materials. We find therefore, except in that rather improbable instance of o6OO 6;09 being universally unity for all materials, a fundamental difference must be expected between scales of temperature defined in the manner outlined here. It is only for the case of a constant value of the temperature coefficient of change of a temperature dependent property that would produce a value of ~/00/6C of unity and exact agreement between all linear scales of temperature. This circumstance cannot reasonably be expected in nature. We conclude from these arguments that thermometers constructed after the fashion described, while useful as operational tools to measure and reproduce levels of "temperature", would each produce different values of temperature when used to measure the state of a given system. This is roughly the present state of thermometry for temperatures below 900K. In this range there presently exists no accepted standard thermometer nor scale although many working groups have defined their own "wire" scales (5). However, these scales all differ from each other. This is demonstrated by Hust (6) in Figure 2, where temperature scales from several different laboratories in the U.S., Canada and Europe are compared with a scale created by the National Bureau of Standards (NBS-55) in the range 10~K to 90 K. This scale is formed by lowering all temperatures on a previous scale (NBS-39), references 7 and 8, by 0.010K. III THE ABSOLUTE (THERMODYNAMIC) TEMPERATURE SCALE In 1848 Kelvin extended the reasoning of Carnot and demonstrated the existence of a scale of temperature which, unlike those shown in Figure 1, would be completely independent of a thermometric substance. Hence, such a scale is called an "absolute" scale and being deduced from the laws of thermodynamics only, it is known as the

.06 I I.IOOA} ~, (O=T PRMI)(o =TPSU) (t= TN PL) } O1RLOVAE TAL OPEN MEASURED AT NPL. (9).05 _ CLOSED-MEASURED AT PRMI. (O( I Vy- FURUKAWA AND REILLY (TSMOOTH) (10) + - PLUMB (T2-20(1965)) (11).04 | * -P~ - - X- MOESSEN (TPSU) (12) -- - LOVEJOY (TCCT-64) (6) ~* -- - BOROVICK- ROMANOV (TPRMI) (i3).03 I __"_-_} BARBER ( ---.TPsU)( —-.TNPL)(14-t$) I..002 *- RODER (TSMOOTH) (?7).02 "' " I -.04 -. -.o0 i 0,O~~~. A6 Uniersity; NP = National Physical Laboratory (England); PRM USSR; CCT

-12Absolute Thermodynamic Temperature Scale. Actually, there are an infinite number of such scales of temperature possible, the final one selected being a matter of convenience. The scale we employ today is not the first scale Kelvin proposed but rather his second proposed scale. His first scale had -400 and -Q0 as the upper and lower bounds of temperature, which is inconvenient, but his second scale remedied this having bounds at 0 and +4W. From the Second Law of Thermodynamics, Kelvin was able to show that the ratio of the heat quantities from a cyclicly operating reversible heat engine could be written: Q1 f(tl) f(t0) (8) where f(t) denotes an unknown but arbitrary function of temperature alone. The form have an infinite number of possible forms. When a specific form is chosen a scale of temperature is then defined which is independent in its definition of any thermometric substance. The reversible engine becomes the thermometer but the nature of such a device, as well as the ratio of its heat quantities, is quite independent of the fluid - the thermometric substance - employed to operature the engine. Kelvin selected as his second scale the simple function f(t) = T, where T is called the Absolute Thermodynamic Temperature and is given in degrees Kelvin. Hence equation (8) becomes Q1 T % T (9) wQO T0 where T1 and T, are the Kelvin temperatures of the heat source and the heat sink, respectively.

-153The complete definition of the Kelvin scale, including the establishment of the size of the degree, is dependent on a single arbitrary constant. Originally this was accomplished by defining the difference between the steam and ice points as 100 degrees Kelvin, exactly. The absolute temperature of the ice point was then determined experimentally using a gas thermometer. Since 1954, however, the arbitrary -onsxtant selected for defining the scale has been the temperature of the triple point of water, taken to be 273.1.6~K exactly* (2, 3). The size of the degree in the interval between the ice and steam points now must be determined experimentally. This interval will certainly not be exactly 100~K but will probably be within 0.0010K or less of this value. The dependence of the Kelvin scale on a single arbitrary constant may be shown by a simple argument. For fixed thermal states at, say, T1 (steam point) and To (ice point) the ratio of the heat quantities in equation (9) is established by the constraints of nature. Thus, Qa'. T - r 1 -- constant. (10) Qo T0 If we define AT 01 as T1 - 1T, then the temperature of the ice point TO may be expressed in terms of AT01 as, TO T01 Hence, since r is fixed by the thermal states (To, T1 whose numerical temperatures may be yet unknown), TO is determined by the value of AT0o or ATo0 is determined *In 1854, one hundred years earlier, Kelvin had stated that the triple point of water "must be adopted ultimately" as a defining fixed point (3). The triple point has been adopted to replace the ice point owing to a greater reliability in establishing its temperature experimentally and the fact that the triple point temperature is not pressure sensitive as are two-phase states.

by the value of TO. Thus, one constant only, either TO or AT01, is sufficient to define the scale. Prior to 1954 AT01 was chosen as 100~K and T determined by the gas thermometer. Since 1954 the reverse procedure has been adopted and TO is defined as the triple point of water at 273.160K*. Now, the thermodynamic temperatures of all defining fixed points and secondary reference points must be determined by the gas thermometer. IV THE ABSOLUTE (GAS) TEMPERATURE SCALE The-practical use of the Kelvin scale would require the operation of reversible engines. This, of course, is quite impossible and it is necessary to find an approximation to the Absolute Thermodynamic Scale. This is found from the properties of an ideal gas** for if such a fluid is employed in a reversible engine it may be shown that: Q O(12) where 4 is a "temperature" on a new scale called the Absolute (Gas) Temperature Scale and would be determined from pressure and volume measurements on the gas, as: R X * (13) Further, if the difference 4 - 4 or 0 are defined to be identically the same as T1 - TO or To, respectively, then it follows from equations (9) and (10) that: 41 = T1 1 t1 eo: TO (14) or, 4 = T *The trigle goint of water is defined by the International Practical Temperature Scale as +0.01o ( C being degrees Celsius) which gives the ice point an absolute temperature of 273.15 K (3). **An ideal gas is defined as a substance having an internal energy (u) which is a function only of temperature and an equation of state written as pv =.

-15In words, the Absolute (Gas) Temperature Scale is identical with the Absolute (Thermodynamic) Temperature Scale if an ideal gas is employed in a reversible engine or any other device permitting the measurement of p and v. Such a device (which now replaces the reversible engine) is known as a gas thermometer and is the primary standard thermometer used to determine and establish the absolute thermodynamic temperature scale. Beattie (18) and Barber (19) describe the gas thermometer and the techniques of its use. Gas thermometers are either of constant pressure or constant volume. Although both types give essentially identical results independent of the type of gas used, the constant volume type appears to be the more widely used. The M.I.T. thermometer used by Beattie (18,20,21) is of the constant volume type and uses nitrogen gas in a very pure state as the thermometric substance. Gas thermometry also, has used H2, He,.Ne, A and Air as thermometric substances. It was found by equation (14) that the requirement of a reversible engine as a thermometer was unnecessary to realize the Kelvin scale, if one could substitute a gas thermometer using an ideal gas. However, owing to the departure of the properties of a real gas from conditions of ideality, it becomes necessary to emperically correct the readings taken from a gas thermometer in order to determine the absolute temperature. Such temperatures are not Kelvin temperatures, actually, but owing to the necessary corrections must be looked upon as approximations. They are called Absolute (Gas) Temperatures and represent the closest approximations to the absolute scale of Kelvin. The nature of these corrections is well established and is outlined by Keyes (22). However, it will be appropriate to indicate the nature of the result for a constant-volume gas thermometer. From thermodynamic theory and the properties of real gases it may be shown that for V = V CT _ 1 T T J 1d( (15)

-16The integral is the gas scale correction and is evaluated from measurements of the properties of real gases. Having once obtained the absolute temperature T of the triple point of water, for example, by definition at 273.160K, then equation (15) can be used to obtain the absolute temperature T on the gas scale of any other thermodynamic state from gas thermometer measurements of p and p* and the real properties of the gas. A simple constant-volume gas thermometer for cryogenic application is illustrated by Barber (19) and shown in Figure 3. This gas thermometer is used for the calibration of platinum resistance thermometers in the liquid hydrogen temperature range. A typical gas bulb with platinum thermometer receptacles and auxiliary apparatus is shown in detail in Figure 4 taken from Moessen, et al (12). Figure 5 shows temperature scale corrections for the non-ideality of the helium gas scale and the absolute temperature scale. This indicates that gas thermometer temperatures from various laboratories may differ from each other in the cryogenic temperature range by as much as 0.03 K. These differences are attributed to the various methods used to account for gas imperfectability and do not include the influence of dead space volume, gravity and other deviations from ideal measuring conditions. A negative absolute temperature may seem to be a suitable subject for a discussion of cryogenic temperatures. The concept of a negative absolute temperature is presented by Ramsey (24) and Hertzfeld (25). Such a condition may be described whenever -the population of an energy state of higher energy is greater than the population of a lower energy state (25) under conditions which permit a statistical mechanical interpretation of thermodynamics. Hertzfeld (25) lists these conditions as: "(1) The system must be fairly well isolated from its surroundings, but must come to internal equilibrium rapidly; (2) the states of the system must be quantized; and (3) the system must have a highest state in the same sense that a system has a

-17PHOSPHOR-BRONZE DIAPHRAGM 2HI X1 II ~ —-- CAPACITY PROBE VACUUM PUMP a LIQUID ei r ThermoMERCURY NITROGEN-n' 1 1 ~-1 ~~MANOMETER LIQUID HYDROGEN A B STAINLESS STEEL CAPILLARY PLATINUM BALLAST HELIUM VACUUM RESISTANCE VOLUE THERMOMETER GAS THERMOMETER BU LB Figure 3. Constant Volume Gas Thermometer Us.9d or Calibration of a Platinum Resistance Thermometer. 19

-18CRYOSTAT TUBE __- - l 1 | | DEWAR FOR HYDROGEN FILLING TUBE CAPILLARY - tLEAD0//" BLOCK' - -- ark F XCRYOSTAT CAN -181J ]; alD —-COPPER RING -___ / ______ - CABLE OF LEAD WIRES SHIELD RESISTANCE THERMOMETERS VACUUM SPACE 1 DIFFERENCE COUPLE _________-_____ JUNCTIONS ABSOLUTE LIQUID OR SOLID-// THERMOCOUPLE \\ A- HYDROGEN- — // JUNCTIONS Figure 4. Thermometer Bulb, (12) Shields, and Cryostat.

-19120 RE F. o Keyes (22) 100 l Van Dijk(23) A Henning + Keesom and Onnes 80 x Keesom and Tuyn 60 40 -20 -300 -200 -100 0 100 200 300 400 TEMPERATURE,~C Figure 5. Gas Imperfection Corrections for Constant Volume Helium Gas Thermometer.(6)

lowest state." Examples of such systems are the nuclear magnetic moments of the constituents of certain crystals. No violation of the principles of thermodynamics is envisioned in this phenomenon. Interestingly, states at negative absolute temperatures are "hotter" than those at infinite temperature. This results from the fact that at T equal to OC all states are equally populated. However, if higher energy states have larger populations than lower energy states the system must be "hotter" than those states having a temperature of infinity. This naturally removes the discussion from the cryogenic range! V THE INTERNATIONAL TEMPERATURE SCALE (ITS) Owing to its size and complexity the gas thermometer is impractical to use for laboratory or industrial measurement of temperature. Most gas thermometers occupy a room-sized space and require elaborate and time consuming preparations for their use. For these reasons a simple, reproducible and convenient secondary temperature standard is required even for precise laboratory measurements. The gas thermometer remains, of course, the primary standard and is employed to determine the gas scale absolute temperature of the various defining fixed points and to calibrate precision laboratory secondary standard thermometers. The definition of the secondary standard thermometers is established by international agreement by the General Conference on Weights and Measures, a body consisting of scientific representatives from 36 nations which meets every six years. This parent body is assisted by a smaller executive group known as the International Committee on Weights and Measures which consists of 18 elected members from the various nations and normally meets every two years. It is this group that supervises the International Bureau of Weights and Measures at Sevres, near Paris, France. This

-21committee also oversees the publication of Metrologia*, a journal devoted to original papers on research "directed towards the significant improvement of fundamental measurements in any field of physics". An Advisory Committee on Thermometry (Comite Consultatif de Thermometrie), the CCT, actively assists the International Committee on Weights and Measures in all matters relating to temperature measurement. International Temperature Scales (ITS) have been adopted in 1927 (26) and in 1948 (27,28). In 1960 a textual revision of the 1948 scale was made (2,3). This revision did not significantly affect the numerical values on the scale. The principal changes were: a. Replace the ice-point with the triple point of water (0.01~C, exactly). b. Use zinc point (pressure insensitive) instead of the sulphur point (pressure sensitive). c. Change the name to the International Practical Temperature Scale (IPTS). The ITS (1927 and 1948) is a Celsius ( C, 1948) scale and no mention was made in the text of an International Practical Kelvin Temperature Scale (IPKTS). However, in practice this conversion was made on the basis of the accepted value of T0, the Kelvin scale temperature of the ice-point. The 1960 revision of the ITS (1948) specifically mentions the IPKTS and relates it to the IPTS by T K(IPKTS) - t C(IPTS) + TO, (16) where TO is now defined to be 273.150~K. This was done so that the IPKTS will have the same value (273.16 K) for the triple point of water as the Kelvin Absolute Tem*A publication of Springer-Verlag.

-22perature Scale. The IPTS and the Celsius scale both are defined to have a value of 0.01~C as the triple point of water. Stimson (2) observes that for precision no greater than 0.0010C, the zero on the 1948 Celsius scale (ITS 1948) may be realized with an ice bath as described in the 1948 ITS. The International Practical Temperature Scale (IPTS) specifies four things: (a) the gas scale temperatures of reproducible defining fixed points and the secondary reference points at which instruments are calibrated, (b) the types of instruments to be used in realizing the scale, (c) the equations to be used for interpolating or extrapolating from the fixed points, and (d) the experimental procedures recommended for both measurement and calibration. A summary of the IPTS and a comparison of the 1927, 1948 and the 1960 revision of the 1948 scale are given in Tables I, II and III, taken from Hust (6). The complete range of temperatures from -182.970C to 10630C is included for completeness. Twenty-two secondary reference points from -78.50C to 33800C including their vapor temperature-pressure relations are given by Stimson (3). The relationship between the International Practical Temperature Scale (1948) and the thermodynamic scale is shown in Table IV (2). Because of the polynomial form of the interpolating equations used to describe the International Practical Temperature Scale (IPTS) between the defining fixed points, an inherent difference exists between the IPTS and the thermodynamic temperature scale (TTS). This difference is determined by comparing gas thermometer readings with those from standard thermometers as prescribed by the IPTS. Hust (6) reports the data of several investigators who examined the differences between the TTS and the IPTS in the range -1900C to OOC. These results are shown in Figure 6. As may be noted, fairly large discrepancies exist between these various results probably because of differences in the gas thermometers measurements as suggested

TABLE I Defining Fixed Points for the International Temperature Scales. Rust (6) Description (all at 1 atm) except 0C (Int 1927) 0C (Int 1948) 0C (Int 1948) triple point of water ITS ITS 1960 text rev. IPTS Oxygen point: Equilibrium between liquid and gaseous oxygen -182.97 -182.97 -182.97 Ice point: Equilibrium between ice and air saturated liquid water 0.000 0.000 Triple point of water: Equilibrium between ice liquid water and gaseous water 0.010 Steam point: Equilibrium between liquid and gaseous water 100.00 100.000 100.000 Sulfur point: Equilibrium between liquid and gaseous sulfur 444.60 444.60 (444.60) Zinc point: Equilibrium between solid and liquid zinc 419.505* Silver point: Equilibrium between solid and liquid silver 960.5 960.8 960.8 Gold point: Equilibrium between solid and liquid gold 1063.0 1063.0 1063.0 *Recommended to replace the sulfur point

TABLE II Official Interpolation Procedures for the 1927 and 1948 International Temperature Scales. Hust (6) (t in 0C- T in 0K; T is temperature of ice point, T0 273.150K) 1927 Scale 1948 Scale -190 to QO -182.97 to 00C platinum resistance thermometer platinum resistance thermometer Rt =R0 [1 + At + Bt2 + C(t - 100) t3 Rt R[+At+Bt +C(t 100) t3 calibrate at 022 ice, steam, S-points calibrate at 02, ice, steam, S-points 0 to 6600C 0 to 630.50C 2 2) Rt R (I + At + Bt Rt R0 (l-At=Bt R /R >JR 1.390; R R / 2.645 R JR > 1.3920 100 0 444.60 1000 calibrate at ice, steam, S-points calibrate at ice, steam, S-points 660 to 10630C 630.5 to 1063 C Pt-Pt 10% Rh thermocouple Pt-Pt 10% Rh thermocouple e = a + bt + ct2 e = a + bt + ct2 calibrate at Sb, Ag, Au-points calibrate at Sb, Ag, Au-points above 10630C above 1063 0C monochromatic optical pyrometer monochromatic optical pyrometer J ~~~ ~~~~~~~~~~~exp t C2 1 1 exp (tAu + Tt log I o og u 1336 t+T0 Au C2 exp - (t + T0) with c =1.432 cm deg c = 1.438 cm deg 2 2cdg2

TABLE III Differences Between ITS (1927) and IPTS (1948). Hust (6) Platinum Thermometer Range The temperature differences in this range are negligible. The changes were made primarily to make the scale more reproducible and definite. Note that the range on the 1948 scale is restricted to -182.97 to 630.50C instead of -190 to 6600C. Standard Thermocouple Range Radiation Law Range It is difficult to determine exact differences of t, ~C (Int) 1948 ~ C (Int) 1948 - ~C (Int) 1927 the ITS and IPTS in this range because of the variability of A; the wavelength of the radiation on 630.5 0.00* the 1927 scale is restricted only to the visible 650.08* spectrum and is not restricted at all on the 1948 700.24 scale. The following table contains the differences 750.45 calculated at A =0.4738xlO- 4cm and 2=0.65x104cm 800.42 1 2 850.43 according to Corruccini (29). 900.40 950.32 t nt 1000.20 1050.05 0C (Int) 1948 0C (Int) 1948 - C (Int) 1927 1063.00 A * These values are uncertain since platinum thermometers are defined only up to 630.5 C 1063 0 0 on 1948 scale (see Corruccini (29). 1500 - 2 - 2 2000 - 6 - 6 2500 -12 -12 3000 -19 -20 3500 -28 -30 4000 -38 -43

TABLE IV Relation Between the International Practical Scale and the Thermodynamic Scale* International Practical Scales Celsius Absolute Names Internationa 1 Interna tiona 1 Practical - Practical Kelvin Temperature Temperature Symbols tint Tint tint + TO Designations ~C (Int. 1948) OK (Int. 1948) degrees Celsius degrees Kelvin internationa 1 internationa 1 practical 1948 practical 1948 Thermodynamic Scales Celsius Absolute Names The rmodynamic Thermodynamic Celsius _ Kelvin temperature temperature Symbols t = T — 0 T Designations 0C (therm.) OK degrees Celsius degrees Kelvin thermodynamic (TO = 273.15~) NOTE- For the international practical temperature, the subscript "int" after t may be omitted if there is no possibility of confusion. * Adopted 11th General Conference on Weights and Measures, October 1960.

-27by the data in Figure 5. In order to examine and define these differences systematically, Preston-Thomas and Kirby (35) redetermined part of the TTS, in terms of platinum resistance thermometer readings, by means of a constant volume helium gas thermometer of reasonably high accuracy. These authors expect to extend similar measurement to -219~C, the triple point of oxygen. Their measurements in the range -1830C to 1000C are given in Figure 7. As may be seen by the data in Figures 6 and 7, the IPTS and the TTS differ by a maximum of about 0.04~C in the cryogenic range.

.05 0 - HEUSE AND OTTO (30),04 I - KEESOM AND DAMMERS(31) A -BARBER AND HORSFORD(32) V - RODER(33) A X - BOROVICK - ROMANOV, ET AL. (13).02 -FURUKAWA AND REILLY(10) A 0.02 - -X —-CT-66 TERRIEN AND A eX -01 ___ -.0 -1 -.0160 -14 -12 -100 -0 - -.02 O, Figure 6. Temperature Differem s Between Thermodynamic and International Temperature Scales.

0.05 0.04 0.03 o 0.02 0 U) 0.01 cn Cf 0 a-.O I- -0.02 -0.03 -0.04 -0.05 -200 -160 -120 -80 -40 0 40 80 120 Temperature 0C Figure 7. Comparison of Measurements of TTS - TPTS.(35)

VI TEMPERATIURE SCALES BELOW 90~K Below the oxygen point (-182.97 C, 90.18 K) no International Temperature Scale presently exists. During the past few years, however, a great deal of study has been devoted to this problem by the CCT and a number of proposals for extending the IPTS to 13.8~K, the triple point of equilibrium hydrogen, have been made. It now seems probable that the IPTS in force since 1948 will be abandoned and replaced with a new scale (4,5). This new scale, which may take effect as early as late 1968 or perhaps during 1969, will conform to the best experimental values of the thermodynamic temperatures now available. If these events transpire as expected, it will mark a period of 20 years between the new scale and the 1948 scale which itself replaced the 1927 scale after about a similar 20 year tenure. This will, of course, leave the important range below 13.80K undefined by an international standard. Since this includes the entire region of He4 and He3 it can only be hoped that similar efforts by the CCT will bring about a standard scale at these very low temperatures. Because of the absence of an international standard below 90 K several "national" or "laboratory" scales have been developed. Each of these is different from the others and from the thermodynamic scale. They are based on the resistance characteristics of platinum calibrated against a gas thermometer. Several of these scales were compared in Figure 2. Scott (36) compares several other scales as indicated in Figure 8. The NBS (1939) scale has been superseded by the NBS (1955) scale formed by lowering all temperatures on the NBS (1939) scale by 0.010C. These scales have been the basis for all NBS calibrations in the interval 120 to 900K since 1939. The agreement of the NBS (1939) scale with the thermodynamic scale was + 0.02~C in the range 120 to 90~K.' It is interesting to note that the scale identified as "Calif (1927)" in Figure 8 is formulated on the basis of a copper-constantan thermocouple which was stable over a period of 3 years having an estimated accuracy of 0.05 K in the range 12~ to 90~K (37).

+ 00 CD 50 N E. 0 Calif (1927) PTR (936) ~ 50 -c00 0 10 20 30 40 50 60 70 80 90 100 TOK Figure 8. Comparison of Temperature Scales. (3) Calif —University of California(37) NBS — National Bureau of Standards(7,8) PSU — Pennsylvania State University(12) PTR — Physikalische Technische Reichsanstalt(36)

-32In 1964 the CCT established a provisional temperature scale to be considered as a replacement for the IPTS below 273.15 K. This scale, in the form of a resistancetemperature table for platinum thermometers, is referred to as CCT-64 and extends from 100K to 273.150K. The derivation for the range 900K to 273.150K is given by Barber and Hayes (38). Certain modifications currently are being considered in this scale prior to its recommendation as an international scale (5). However, any modification in CCT-64 will doubtless be small and the low temperature part is assumed (6) to be the best approximation to the thermodynamic scale in this region. Calibration of thermometers below 90 K may be accomplished using a number of multi-phase equilibrium states, called fixed points. Timmerhaus (39) lists several of these states which are given here in Table V. These are not presently "secondary reference points" as prescribed by an International Temperature Scale, but represent the best literature values. An equilibrium cell or vapor pressure thermometer is employed to determine these states and would be similar to the oxygen vapor-pressure thermometer described by Timmerhaus (40) shown in Figure 9. For precise calibration of thermometers in the range 0.20 K to 5.2 K the vapor pressure scales of He4 and its light isotope He3 are available. For temperatures between 10K and 5.20K the International Committee on Weights and Measures in October 1958 recommended for international use a scale based on equilibrium between He4 liquid and its vapor now known as the "1958 He4 scale of temperatures". This scale 4 is described by Brickwedde, et al (41) where the vapor pressure of He is tabulated for intervals of 0.001 K from 0.500K to 5.220K. Clement (42) concludes that the He4 1958 scale is accurate within 0.001 to 0.0020K with a roughness less than 0.00010K. Values of the vapor pressure of He in microns (10 mm Hg) for intervals of 0.010K are given in Table VI, taken from reference 41.

-33TABLE V Fixed Points Below 900K (39) Temp., Point ~K Lambda point of helium 2.173 Boiling point of helium (1 atm) 4.21.5 Triple point of equilibrium hydrogen 13.81 Triple point of normal hydrogen 13.95 Boiling point of equilibrium hydrogen (1 atm) 20.27 Boiling point of normal hydrogen (1 atm) 20.39 Triple point of neon 24.57 Boiling point of neon (1 atm) 27.17 Triple point of oxygen 54.36 Triple point of nitrogen 63.14 Boiling point of nitrogen (1 atm) 77.35

-34thermometer hrmoouplee shield Oxygeno bubbler liquid oxygen Dewar Heavy copper cylinder Pure iiquid oxygen Mercury o o manometer Figure 9. Oxygen Vapor-pressure Thermometer for Calibrating Working Thermometers. (40)

A comparison of the 1958 He4 scale with previous scales is shown in Figure 11, taken from Hust (6). The identification of the various scales is given by Brickwedde, et al (41). The acoustical thermometer of Plumb and Cataland (11) provides an interesting comparison with the results of the vapor pressure scale. The phase diagram for He4 is shown in Figure 10. Below the X-point at 2.172 K liquid helium experiences a transition to its superfluid state. The tendency of superfluid helium to flow makes vapor pressure measurements difficult below the A point. Furthermore, below 10K the vapor pressure of He4 is less than 120 microns which adds additional problems in measurement. To overcome both of these drawbacks the vapor pressure of the light isotope He3 was determined and developed into the He3 scale. A comparison of the vapor pressures of He3 and He4 is given in Figure 12, taken from Arp and Kropschot (43). For comparison it will be noted that the vapor pressure of He3 at 1 K is 8,842 microns and that of He4 at the same temperature is only 120 microns. The International Committee on Weights and Measures in 1962 recommended the use of the He3 vapor pressure temperature data for international use. This scale, which is known as the "1962 He3 scale of temperatures", is tabulated in intervals of 0.0010K from 0.200K to 3.3240K by Sherman, et al (44). Vapor pressure data in intervals of 0.01 K for He3 are given in Table VII,taken from the summary table of Sydoriak, et al (45). Preston-Thomas and Bedford (5) have examined the reproducibilities of various actual and postulated temperature scales in the range 1 K to 1063 0K. Their results -2 -3 are given in Figure 13 indicating a general reproducibility of 10 to 10 over the full range.

TABLE V! Brickwedde, et. al. (41) Vapor pressure of He4 (1958 scale) in microns (10-3 ram) mercury at O~C and standard gravity (980.668 cm/sec2)..... ~.~~:~.~ —; —~-~........ -... _. ~l J,,1,,,:~,.L L',:" ~' L _ I,,,,, j,,,.''.......~.. _ ~.,,~, IlL,,l It,..._ ~' T~K Microns T~K Microns T~K Microns T~K Microns T~K Microns T~K Microns T~K Microns 0,50 0.016342 1.30 1208,51 2.00 23767.4 2.70 112175 3.40 314697 4.10 680740 4.80 1263212.51.022745 1,31 1284.81 2,01 24470.9 2,71 114145 3,41 318659 4.11 687399 4,81 1273414.52.031287 1.32 1364,83 2.02 25188.1 2.72 116139 3.42 322654 4,12 694103 4.82 1283673.53.042561 1,33 1448,73 2.03 25919.2 2.73 118156 3.43 326684 4.13 700851 4.83 1293991,54.057292 1.34 1536.61 2.04 26664.2 2.74 120198 3,44 330747 4,14 707643 4.84 1304367,55.076356 1.35 1628.62 2,05 27423.3 2.75 122263 3,45 334845 4.15 714479 4.85 1314802.56.10081 1.36 1724.91 2,06 28196.3 2.76 124353 3.46 338976 4.16 721360 4.86 1325297.57.13190 1.37 1825.58 2.07 28983.2 2.77 126465 3.47 343141 4.17 728285 4.87 1335850,.58.17112 1.38 1930.79 2.08 29784.2 2.78 128603 3.48 347341 4.18 735255 4.88 1346462 c~ I,59.22021 1,39 2040.67 2.09 30599.1 2.79 130765 3,49 351575 4,19 742269 4.89 1357136 0.60 0.28121 1,40 2155,35 2o10 31428,1 2.80 132952 3.50 355844 4.20 749328 4.90 1367870 ~ 61.35649 1.41 2274.99 2,11 32271,1 2,81 135164 3.51 360147 4,21 756431 4.91 1378662 ~ 62.44877 1.42 2399.73 2.12 33128.0 2.82 137401 3.52 364485 4.22 763579 4.92 1389516.63.56118 1.43 2529.72 2.13 33998.6 2.83 139663 3.53 368860 4.23 770772 4.93 1400429 ~ 64.69729 1.44 2665.09 2.14 34882.8 2.84 141949 3.54 373269 4.24 778010 4.94 1411404 ~ 65.86116 1.45 2805.99 2.15 35780.3 2.85 144260 3.55 377714 4.25 785294 4.95 1422438.66 1.0574 1.46 2952.60 2.16 36690.9 2.86 146597 3.56 382194 4.26 792623 4.96 1433533.67 1.2911 1.47 3105.04 2.17 37614.3 2.87 148961 3.57 386710 4.27 799999 4.97 1444690,68 1,5682 1.48 3263,48 2.18 38550,2 2.88 151349 3,58 391262 4.28 807422 4.98 1455911 ~ 69 1.8949 1.49 3428,07 2.19 39500.3 2.89 153763 3.59 395849 4.29 814893 4.99 1467191,70 2.2787 1.50 3598,97 2.20 40465.6 2.90 156204 3.60 400471 4.30 822411 5.00 1478535,71 2.7272 1.51 3776,32 2.21 41446.6 2.91 158671 3.61 405130 4.31 829978 5.01 1489940 ~ 72 3.2494 1,52 3960.32 2.22 42443,5 2,92 161164 3,62 409825 4.32 837592 5.02 1501409.73 3.8549 1.53 4151,07 2.23 43456~5 2.93 163684 3.63 414556 4,33 845255 5.03 1512940

TABLE V{ (cont.) -- ~ - -::: ~,,~__. _, ~ ~,,,,[,,:...:.. -........................,,,,,, m, J ~ t L J,,,,,,,,L, I I,,,,t,_,, ~,,,,, T~K Microns T~K Microns T~K Microns T~K Microns T~K Microns T~K Microns T~K Microns.74 4.5543 1.54 4348.79 2.24 44485.7 2.94 166230 3.64 419324 4.34 852966 5.04 1524535.75 5.3591 1.55 4553.58 2.25 45531.3 2.95 168802 3.65 424128 4.35 860725 5.05 1536192.76 6.2820 1.56 4765.68 2.26 46593.5 2.96 171402 3.66 428968 4.36 868533 5.06 1547912.77 7.3365 1.57 4985.18 2.27 47672.5 2.97 174028 3.67 438846 4.37 876390 5.07 1559698.78 8.5376 1.58 5212.26 2.28 48768.6 2.98 176682 3.68 438760 4.38 884296 5.08 1571546.79 9.9013 1.59 5447.11 2.29 49881.8 2.99 179364 3.69 443713 4.39 892252 5.09 1583458.80 11.445 1.60 5689.88 2.30 51012.3 3.00 182073 3.70 448702 4.40 900258 5.10 1595437 ~ 81 13.187 1.61 5940.76 2.31 52160.2 3.01 184810 3.71 453729 4.41 908313 5.11 1607481 ~ 82 15.147 1.62 6199.90 2.32 53325.8 3.02 187574 3.72 458794 4.42 916418 5.12 1619589.83 17.348 1.63 6467.42 2.33 54509.2 3.03 190366 3.73 463897 4.43 924573 5.13 1631761.84 19. 811 1.64 67 43.57 2. 34 55710.5 3. 04 193187 3.7 4 469038 4.44 932778 5 ~ 14 1644000, k.N ~ 85 22.561 1.65 7028.47 2.35 56930.0 3.05 196037 3.75 474218 4.45 941033 5.15 1656305 I.86 25.624 1.66 7322.31 2.36 58167.8 3.06 198914 3.76 479435 4.46 949338 5.16 1668673.87 29.027 1.67 7625.21 2.37 59423.8 3.07 201820 3.77 484691 4.47 957693 5.17 1681108.88 32.800 1.68 7937.40 2.38 60698.8 3.08 204755 3.78 489985 4.48 966099 5.18 1693612 ~ 89 3.6.974 1.69 8259.02 2.39 61992.0 3.09 207719 3.79 495317 4.49 974556 5.19 1706180.90 41.581 1.70 8590.22 2.40 63304.3 3.10 210711 3.80 500688 4.50 983066 5.20 1718817 ~ 91 46.656 1.71 8931.18 2.41 64635.2 3.11 213732 3.81 506098 4.51 991628 5.21 1731521.92 52.234 1.72 9282.06 2.42 65985.4 3.12 216783 3.82 511547 4.52 1000239 5.22 1744290.93 58.355 1.73 9643.02 2.43 67354.8 3.13 219864 3.83 517036 4.53 1008905.94 65.059 1.74 10014.3 2.44 68743.5 3.14 222975 3.84 522564 4.54 1017621.95 72.386 1.75 10395.9 2.45 70152.0 3.15 226115 3.85 528132 4.55 1026390.96 80.382 1.76 10788.2 2.46 71580.2 3.16 229285 3.86 533739 4.56 1035213.97 89.093 1.77 11191.2 2.47 73028.1 3.17 232484 3.87 539387 4.57 1044087 ~ 98 98.567 1.78 11605.1 2.48 74496.0 3.18 235714 3.88 545075 4.58 1053014.99 108.853 1.79 12030.1 2.49 75984.2 3.19 238974 3.89 550805 4.59 1061995 1.00 120.000 1.80 12466.1 2.50 77493.1 3.20 242266 3.90 556574 4.60 1071029

TABLE VI (cont.) ToK Microns TOK Microns T0K Microns T~K Microns T~K Microns T~K Microns T~K Microns 1.01 132.070 1.81 12913.7 2.51 79022.2 3.21 245587 3.91 562383 4.61 1080114 1.02 145.116 1.82 13372.8 2.52 80572.2 3.22 248939 3.92 568234 4.62 1089254 1.03 159.198 1.83 13843.6 2.53 82142.9 3.23 252322 3.93 574126 4.63 1098449 1.04 174.375 1.84 14326.1 2.54 83734.6 3.24 255736 3.94 580059 4.64 1107699 1.05 190.711 1.85 14820.7 2.55 85347.2 3.25 259182 3.95 586034 4.65 1117002 1.06 208.274 1.86 15327.3 2.56 86981.2 3.26 262658 3.96 592051 4.66 1126359 1.07 227.132 1.87 15846.3 2.57 88636.7 3.27 266166 3.97 598110 4.67 1135772 1.08 247.350 1.88 16377.7 2.58 90313.8 3.28 269706 3.98 604210 4.68 1145239 1.09 269.006 1.89 16921.7 2.59 92012.6 3.29 273278 3.99 610352 4.69 1154761 1.10 292.169 1.90 17478.2 2.60 93733.4 3.30 276880 4.00 616537 4.70 1164339 1.11 316.923 1.91 18047.7 2.61 95476.0 3.31 280516 4.01 622764 4.71 1173972 1.12 343.341 1.92 18630.1 2.62 97240.8 3.32 284183 4.02 629033 4.72 1183662 1.13 371.512 1.93 19225.5 2.63 99028.2 3.33 287883 4.03 635345 4.73 1193407 1.14 401.514 1.94 19834.1 2.64 100838 3.34 291615 4.04 641700 4.74 1203209 1.15 433.437 1.95 20455.9 2.65 102669 3.35 295380 4.05 648099 4.75 1213066 1.16 467.365 1.96 21091.1 2.66 104525 3.36 299178 4.06 654541 4.76 1222981 1.17 503.396 1.97 21739.7 2.67 106403 3.37 303008 4.07 661026 4.77 1232955 1.18 541.617 1.98 22402.0 2.68 108304 3.38 306871 4.08 667554 4.78 1242983 1.19 582.129 1.99 23077.9 2.69 110228 3.39 310768 4.09 674125 4.79 1253069 1.20 625.025 1.21 670.411 1.22 718.386 1.23 769.057 1.24 822.527 1.25 878.916 1.26 938.330 1.27 1000.87 1.28 1066.67 1.29 1135.85

-39CRITICAL POINT 1000 X-LINE -a LIQUID HELIUM I LIQUID HELIUM II _: I I /-SATURATION E 100 - CURVE E D r 37.8 mmHg —-- -OPOINT ( I I Uf) W 2.i73 OK X!0 -. HELIUM GAS 0 1 2 3 4 5 TEMPERATURE (o K) Figure 10. Phase Diagram for Heliumx.

120 12~~~~~~ - - -~~~~~ I I i I I II __x~ 0~~O —Plumb and Cataland (11) X t00 100 -- >, _ - Brickwedde(4t) X 0 - TL55 *- T55E 80 0- T48 ~~X X B- TBS a) )( z~ T37 60 A-T32 0) XX A- T$2 _=~~~~ -)(~~~~~ +- T29 X~~~~~ I'~~~~~~~~~- 20 A ~~~~~~~~~~X _( _A lX-l I -- I ~ ~., AB~ " b~~~- j AA C A X -20 F- ER -40 - 0 1 2 3 4 5 6 TEMPERATURE, ~K Figure 11. Deviation of Earlier Helium Vapor Pressure Scales from the 1958 He4 Scale.(6)

1000 H04 He Li/ 0;0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Temperature, OK Figure 12. Vapor-pressures of He3 and He4. (43)

TABLE VII* He3 vapor pressures on the 1962 He3 scale at 0 C and standard gravity, 980.665 cm/sec2 -3 0 The units of pressure are microns (10 mm) of mercury below 1 K and millimeters of mercury at higher temperatures T 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.20 0.012 0.024 0.046 0.084 0.144 0.239 0.382 0.592 0.891 1.308 0.30 1.8'77 2.636 3.633 4.921 6.561 8.619 11.173 14.304 18.105 22.673 0.40 28.11 34.54 42.08 50.86 61.01 72.68 86.02 101.17 118.31 137.61 0.50 159.2 183.3 210.1 239.8 272.5 308.5 347.9 391.1 438.0 489.1 0.60 544.4 604.3 668.9 738.4 813.0' 893.0 978.7 1070.1 1167.6 1271.4 0.70 1381 1498 1622 1753 1892 2038 2192 2355 2525 2704 0.80 2892 3089 3295 3511 3736 3971 4216 4472 4739 5016 0.90 5304 5603 5914 6237 6572 6918 7277 7649 8034 8431 1.00 8.842 9.267 9.704 10.156 10.622 11.102 11.597 12.106 12.631 13.170 1.10 13.725 14.295 14.881 15.484 16.102 16.737 17.388 18.056 18.741 19.443 1.20 20.163 20.900 21.655 22.428 23.220 24.029 24.857 25.704 26.571 27.456 1.30 28.360 29.285 30.229 31.193 32.177 33.181 34.206 35.252 36.319 37.407 1.40 38.516 39.646 40.799 41.973 43.169 44.388 45.629 46.893 48.179 49.489 1.50 50.822 52.178 53.558 54.961 56.389 57.840 59.316 60.817 62.342 63.892 1.60 65.467 67.068 68.694 70.345 72.022 73.726 75.455 77.211 78.993 80.802 1.70 82.638 84.501 86.391 88.309 90.254 92.228 94.229 96.258 98.315 100.402 1.80 102.516 104.660 106.833 109.035 111.266 113.527 115.818 118.138 120.489 122.870 1.90 125.282 127.724 130.197 132.701 135.236 137.803 140.401 143.031 145.692 148.386 2.00 151.112 153.870 156.661 159.485 162.342 165.232 168.155 171.112 174.102 177.126 2.10 180.184 183.276 186.403 189.564 192.760 195.990 199.256 202.557 205.894 209.266 2.20 212.673 216.117 219.597 223.113 226.665 230.255 233.881 237.544 241.244 244.982 2.30 248.757 252.570 256.420 260.309 264.236 268.202 272.206 276.249 280.331 284.452 2.40 288.613 292.813 297.053 301.333 305.653 310.013 314.414 318.855 323.337 327.861 2.50 332.425 337.031 341.679 346.368 351.100 355.874 360.690 365.549 370.450 375.395 2.60 380.383 385.414 390.489 395.608 400.771 405.978 411.230 416.526 421.868 427.254 2.70 432.686 438.164 443.687 449.256 454.872 460.534 466.242 471.998 477.801 483.651 2.80 489.549 495.495 501.488 507.531 513.622 519.762 525.951 532.189 538.477 544.815 2.90 551.203 557.642 564.131 570.672 577.264 583.907 590.602 597.349 604.149 611.002

TABLE VIIrr (cont.) T 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 3.00 617.907 624.866 631.879 638.945 646.066 653.241 660.472 667.757 675.098 682.496 3.10 689.949 697.459 705.026 712.650 720.332 728.072 735.871 743.728 751.644 759.620 3.20 767.656 775.753 783.910 792.128 800.408 808.750 817.155 825.622 834.153 842.747 3.30 851.406 860.130 868.918 877.773 *From Sydoriak, et al. (45)

-440- D 0 0. A K 0 100 314 420 630 1063 Temperature Figure 13. Estimated Reproduci i ities of Various Actual and Postulated Temperature Scales. 5) A. Vapour Pressure Measurement. 1958 He- Scale; B. Estimated Reproducibility of a Ge Resistance Thermometer "Wire Scale"; C. Estimated Reproducibility of Any of the Pt Resistance Thermometer "National" Scales; D. Reproducibility of the Barber van Dijk Scale(46) on the (Optimistic) Assumption of the Fixed Points being Rea.lisable to 0.2 Millikelvin; E. IPTS in the 90 6 to 273 K Range Assuming the Indeterminacy Shown by Barber 32); F. IPTS in the 90 K to 273 K Range Assuming the Indeterminacy Shown by Lovejoy(47)(F1) or Alternatively the Use of an Additional Fixed Point (C02 Point) in a Modification of the IPTS (F2); G. IPTS in the Range 00C to 630~C; H. Reproducibility of a Modification of the IPTS in the Manner Suggested by McLaren(4 ); J. Estimated Reproducibility of a Pt Resistance Thermometer Scale in the 630'C to 10630C Range; K. Reproducibility of the IPTS (Using Pt 10 Rh/Pt Thermocouples) in the 630oC to 1063'C Range.

VII. Thermometers for Cryogenic Temperatures The remaining sections of this paper will consider the principal devices which are employed to measure cryogenic temperatures. These will include thermoelectric, electrical resistance and magnetic thermometers. A survey of available low temperature thermometers is illustrated in Figure 14 for the temperature range 0.05~K to 3000~1. This is an extension of a similar chart presented by Timmerhaus (40). A comparison of the performance of several of these devices has been prepared by Corruccini (63) in a survey of temperature measurements at cryogenic temperatures. His results are given in Table VIII. TABLE VIII Comparison of Cryogenic Temperature Measuripnj Devices Type Range,~K Best Reproducibility,~K Best Accuracy, ~K -3 -4 -2 -4 1. Platinum Resistance 10-900 10 to 10 10 to 10 to -3 -2 -3 2. Carbon 1-30 10 to 10 10 to 10 -t -2 -3 3. Germanium 1-100 103 10 10 to 10 o 4. Gold-Cobalt 4-300 101 to 10-2 0.10 vs Copper Thermocouple VIII. Thermocouples The familiar thermo-electric circuit - the thermocouple - in which an EMF is produced by subjecting the junctions of dissimiliar metallic combinations to different temperatures is commonly used in the cryogenic temperature range. In circumstances where measurement accuracy is from 0.50 to 1.00F a thermocouple may even be the preferred temperature sensing element. There are several reasons for this. A thermocouple is easily made, is small and can be mounted relatively

simply in remote and fairly inaccessible locations, requires only standard laboratory or industrial measuring instruments, can be made rugged and relatively insensitive to environmental disturbances, and is inexpensive. Other desirable characteristics that can be obtained using thermocouples are: a large net thermal EMF, a monotonic or linear EMF-temperature characteristic, a stable EMIFtemperature characteristic, resistance to chemical corrosion, including the effects of both oxidizing and reducing atmospheres, uniformity of the wire material in large batches and high thermal response. The influence of the environment often must be reduced by the use of protection tubes. The common thermo-electric elements in use today at temperatures from 40K to 3000K are the gold-cobalt vs copper and the constantan vs cooper junctions. These will be described below. Measurements made at the liquid nitrogen, liquid oxygen and liquid hydrogen temperatures probably have used the constantan vs copper thermocouple with greater frequency than any other single combination. As was mentioned earlier, the "Calif (1927) scale in Figure 8 was formulated on the basis of a copper-constantan thermocouple (37). This particular thermocouple was found to be stable for a period of 3 years in the temperature range 12~K to 900K with an accuracy of 0.05~K. Although thermocouples are commonly used at temperatures below 3000K the International Practical Temperature Scale is not specified in terms of thermo-electric systems in the cryogenic range of temperatures. The principal reasons for this are accuracy and reproducibility as compared with the platinum resistance thermometers. The general principles of thermo-electric thermometry and the various thermo-electric circuits and instrumentation are, of course, important to design for the installation of thermocouples. In view of space limitations here and the generally wide availability of this kind of information, it will not be

4 — _ I1T77777772=777777 777777 zzzzzzz CARBON RESISTANCE THERMOMETER 02 VAPOR PRESSURE F7777777777777777 777 7 77 777"~ He3 VAPOR PRESSURE N2 VAPOR PRESSURE He4 VAPOR PRESSURE H2 VAPOR PRESSURE GERMANIUM RESISTANCE THERMOMETER PT RESISTANCE THERMOMETER - MAGNETIC THERMOMETER THERMOCOUPLE GAS THERMOMETER O.050K O.100K 10K 100K 1000K 3d800 Liquids —. X-He He H2 N2 02 Rcom temperature Temperature scale, OK Figure 14. Temperature Ranges Normally Associat~ed With Various Low Temperature Thermometers.

-48included in this discussion. Finch (49) has given a thorough presentation of the principles of thermoelectricity. Some improved reference tables for iron-constantan, chromel-alumel, copper-constantan and chromel-constantan thermocouples are presented by Benedict and Ashby (50). Caldwell (51) discusses the properties of various materials that could be used as thermocouple elements at temperatures above O0C. The use of thermocouples in engineering measurements and their circuits are given by Weber (52), Baker, et al (53) and Dike (54), among others. The behavior of a thermocouple element is usually characterized by its thermoelectric potential or EMF, E, and its thermoelectric power, dE/dT. Its EMIF, E, is always related to an arbitrarily selected reference temperature. At cryogenic temperatures the common thermoelectric combinations are goldcobalt (Au + 2.11 atomic'percent Co) vs copper, copper vs constantan (60% Cu and 40% Ni), gold-cobalt vs normal silver (Ag + 0.37 atomic percent Au), Iron vs constantan and chromel-P (90% Ni and 10% Cr) vs Alumel (95% Ni and 5% (Al, Si, Min)). The most frequently used thermocouples, however, are the goldcobalt vs copper and the copper vs constantan combinations. These thermocouples have been used to temperatures as low as 0.20K. Their best accuracy in the temperature range 4 to 3000K is 0.100K for gold-cobalt vs copper and 0.500K for copper vs constantan. The thermoelectric potential differences for these 5 thermocouples is given in Table IX, taken from Powell, et al (55).

TABLE IX Thermoelectric Potential Differences in Microvolts for Several Thermocouple Combinations (55) Thermocouple Combination Gold- Normal Constantan Cobalt Silver Iron Chromel P Temp. vs VS. vs vs vs. vs. ~K Copper Copper Copper Constantan Alumel 4-20 57.8 171.4 0.2 59 41 20-76 646.9 1562.5 37.9 805 616 76-273 5545.6 8123.2 133.7 8252 6182 The principal advantage of the use of the gold-cobalt vs. copper combination is evident from these data as it has a significantly higher thermal EMF. However, owing to inhomogeneities in its chemical constituency this combination produces irregular EMF's that are uncompensated for in its calibration. This, of course, gives rise to measurement errors. In fact, this lack of homogeneity in composition is the single greatest defect in the Au-Co vs. Cu thermocouple. When the chemical metallurgy of the Au-Co wires can produce a product having a constant, controllable and stable composition this thermocouple will come into much wider use at very low temperatures. The effects of inhomogeneity are usually greatest when the measuring and reference junctions are at widely different temperatures. In such cases the thermocouple lead wires are subjected to steep temperature gradients and at the points of greatest temperature change in the wire chemical inhomogeneity will produce an EMF. Thus, the Au-Co vs. Cu combination is best used where temperature differences to be measured are small. A convenient and practical application is the use of the Au-Co vs. Cu combination as a differential thermocouple in an installation where the temperature differences are small or zero,

as in constant temperature baths, cryostats and equilibrium cells for temperature calibration. The Fe vs. Con and Ch vs. Al thermocouples are infrequently used at low temperatures principally because of voltage uncertainties resulting from inhomogeneities in the wires. An estimate of the inhomogeneity of thermoelectrical voltages obtained by placing one section of a wire sample in a cryogen while the two ends of the wire are attached to a potentiometer is reported by Powell, et al (56) and shown in Table X. TABLE X Inhomogeneity of Thermoelectric Voltages Obtained from Dip Tests (50) Bath temperatures 4-3000K 76-3000K Samples | Voltage (uV) Voltage (uV) Maximum Average Maximum Average (1) Cu 4*5 2.5 2.0 0.8 (2) Cu 1.8 0.7 1.0 C.3 (3) Constantan 0.5 0.2 0.5 0.2 (4) Au-Co 5.0 3.0 4.0 2.5 (5) Au-Co 5.5 3.5 4.0 2.5 (6) Ag-Au 2.2 1.2 1.2 0.8 Samples were: (1) Instrument grade copper, 32 A.W.G.; (2) Thermocouple grade copper, 36 A.W.G.; (3) Thermocouple grade constantan, 36 A.w.g.; (4) Gold-cobalt, Bar 9, 36 A.w.g. (1960); (5) Gold-cobalt, Bar 5, 36 A.w.g. (1958); (6)'Normal' silver, 36 A.w.g. As is evident from this sampling, the gold-cobalt wire is subject to the greatest voltage uncertainty. Thermocouple grade copper and constantan exhibit the least voltage uncertainty owing to inhomogeneity.

Powell, et al (55, 56) have studied the thermoelectric characteristics of several thermocouple combinations in the temperature range 40K to 3000K. A summary of their results for the Au-Co vs. Cu and Cu vs. Con thermocouples is given in Table XI for a 0~K reference temperature. An extensive tabulation of the thermoelectric potentials and thermoelectric power in 10K intervals for these combinations from 0~K to 3000K is found in reference 56. These results represent the best average or smoothed data for a family of thermocouple combinations. Thus, they mav be used as the standard reference data for each thermocouple. Such data are of great value in thermocouple calibration, as is discussed later. TABLE (I Thermoelectric Potential Differences in Microvolts for Gold-Cobalt and Constantan vs. Copper Thermocouples (56) Temp., Con- Temp., ConOK Au-Co stantan ~K Au-Co stantan 0 0.00 0.00 90 2246.8 946.7 2 2.09 0.66 95 2433.3 1038.5 4 8.22 2.62 100 2622.6 1133.7 6 18.20 5.83 110 3008.5 1333.7 8 31.83 10.26 120 3402.8 1546.4 10 48.93 15.88 130 3804.1 1771.7 12 69.30 22.64 140 4211.2 2009.5 14 92.75 30.50 150 4623.2 2260.0 16 119.1 39.43 160 5039.1 2522.7 18 148.1 49.40 170 5458.4 2797.1 20 179.6 60.40 180 5880.4 3083.1 25 269.1 92.31 190 6304.6 3380.3 30 372.5 130.3 200 6730.6 3688.6 35 483.0 173.9 210 7158.0 4007.7 40 614.2 222.9 220 7586.4 4337.4 45 749.9 276.8 230 8015.7 4677.5 50 893.9 335.6 240 8445.5 5027.8 55 1045.2 398.8 250 8875.6 5388.0 60 1202.9 466.2 260 9305.9 5757.9 65 1366.2 537.5 270 9736.2 6137.3 70 1534.5 612.7 280 10166.3 6526.0 75 1707.1 691.2 290 10596.1 6923.7 80 1883.5 773.0 300 11025.5 7330.2 85 2063.4 858.1

The thermoelectric power, dE/dT, in microvolts per degree Kelvin for the five thermocouple combinations in Table IX is given in Figure 15 as a function of temperature. At temperatures from 40K to about 2000K, the Au-Co vs. Cu thermocouple is clearly the superior combination from the standpoint of thermoelectric power. Below about 400K the Fe vs. Con, Ch vs. Al and Cu vs. Con thermocouples have about the same thermoelectric power. Normal silver vs. copper produces an almost insignificant thermoelectric power at low temperatures. The thermoelectric power, dE/dT, in Figure 15 may be used to give an indication of the sensitivity of temperature measurement when it is related to the measurement sensitivity of the potentiometer AE* in microvolts, used to measure the thermal EMF of the thermocouple. That is, the uncertainty in temperature indication, aT*, may be written AE* AT* =., E/ (17) (dE/dT) Thus, thermocouples having large thermoelectric power will enable smaller measurement uncertainty for a given measuring instrument. The calibration of a thermocouple is conveniently done by establishing its deviation characteristic as compared with "Standard" thermocouple EMF. Standard thermocouple potentials at cryogenic temperatures are given in Table XI in summary form for Au-Co vs. Cu and Cu vs. Con and may be found in much greater detail in reference 56. The thermocouple'Deviation" is defined as AE E (18) DEV STD OBS where ESTD is the standard EMF corresponding to the temperature of the thermoSTD couple for which E is its "observed' EMF. A deviation plot is usually CBS constructed by obtaining several corresponding values of EDEV and EBS over a range of temperatures. Generally, when these data are plotted as EDEV vs. EOBS a smooth, frequently almost linear, curve may be used to join the data points. This is a result of the fact that while each individual thermocouple

-53Constantan 50 ~~~~~~~~~~~~~~~~~iron: 40 0 0 04 IC~~~- ~ ~ Temperature, iK Figure 15. Thermoelectric Power as a ction of Temperatue for Various Thermocouple Combinations. t Au3)

wire combination will differ slightly from others of similar composition, the EMF-temperature characteristics of a "family" of similar wires are essentially parallel. Thus, their'deviations" will be almost linear and exactly zero at a common reference temperature. The use of a deviation plot provides a very convenient method for making accurate temperature measurements with thermocouples. Owing to the essential linearity of a deviation curve interpolation between a minimum number of calibration points may be done with confidence. The determination of an unknown temperature in measurement is made by computing ESTD from Equation (18) using EDEV taken from the deviation plot corresponding to the EOBS for the thermocouple. The unknown temperature is found from the "standard" table of ESTD vs. temperature. STD The "standard" table is usually formulated in great detail and represents the best, smoothed data for a family of thermcouple combinations. A potentiometer is probably the most satisfactory instrument for precision temperature measurement using thermocouples. These instruments are described in detail elsewhere (52, 53, 57). Instruments presently available by the Leeds and Northrup Co., the K-5 (facility) and K-3 have sensitivities 0.02-0.1 (AV and 0.5 eV, respectively. The Wenner potentiometer has a sensitivity of 0.1 pV. A potentiometer manufactured by the Minneapolis Honeywell Co. also has a sensitivity of 0.1 MV. Similar potentiometers are produced by other companies. Comparing these sensitivities with the thermoelectric power of the Au-Co vs. Cu thermocouple, temperature sensitivities will range from 0.0020K to 0.01~K at a level of temperature of 100K.:X. RESISTANCE THERMOMIETRY The variation of electrical resistance with temperature provides a very convenient, accurate and practical method for temperature measurement. This method is enhanced when the material from which the thermometer is made has a stable and easily reproducible composition. Otherwise, the method becomes impractical

owing to inherent instabilities in the resistance-temperature characteristic and consequent uncertainties in the temperature. The basic measurement required is that of electrical resistance and this can be done with great precision using available resistance bridges or potentiometers. Hence, with a stable material and present instrumentation a resistance thermometer can be used to measure temperature to a high degree of accuracy (52, 53, 58-60). For precision measurements the platinum resistance thermometer is the most widely used temperature measuring device in the range 1iK to 3000K. As mentioned in Section V, the International Practical Temperature Scale (1948) is defined in terms of the resistance characteristics of platinum from -183~C to 630.5~C. Undoubtedly, when the new International temperature scale is introduced (4, 5, 6) some time in the period 1968-69, as expected, it also will employ the resistance characteristic of platinum as the standard below 3000K, as well as above that temperature. The reason for this, of course, is the unusually high degree of purity that can be achieved in the production of platinum, the reproducibility of the purity from batch to batch, its monotonic resistance-temperature curve in the strain-free, annealed state, and its inertness to chemical contamination. Its cost is high which may be a factor in its use. Other materials which also are used include copper, nickel, carbon, germanium and certain semi-conductors known as thermstors. These will be discussed later. The resistance-temperature characteristic of platinum is shown in Figure 16. Above 500K this relationship is essentially linear. The 1948 IPTS requires that the resistance ratio in Figure 16 be equal to or greater than 1.3920 at 373.150K (100~C) to insure purity in the platinum wire. The resistance-temperature characteristics of platinum, a platinum film on a non-conducting substrate, nickel, tungsten and copper are shown in Figure 17. Precision platinum-resistance thermometers are made of a fine coil of highly purified, strain-free platinum wire would around a non-conducting frame. A

typical method of construction is shown in Figure 18. The ice-point resistance of these thermometers is commonly set at approximately 25.5 absolute ohms. The platinum thermometer is usually manufactured as a capsule (Figure 18) or as a cane. In each case four lead wires are provided for resistance measurement. Precision resistance is best measured using a Mueller Bridge with a four lead wire thermometer as shown in Figure 19. The accuracy of this bridge circuit is 10 ohms (39) which would correspond to approximately 0.003~K at 120K and 0.000090~K at 1000K. Except at very low temperatures accuracies from 0.001~K to O.00010K can be obtained using a platinum thermometer and Mueller Bridge. As indicated in Figure 19, the four lead wire circuit provides a means for reversing lead wire connections during *a measurement. This technique permits the complete cancellation of lead wire resistance so that the net measured resistance is that of the platinum resistance thermometer wire itself. Potentiometric methods for resistance measurement are summarized by Dauphinee (61). Calibration of a platinum thermometer'can he made using a gas thermometer, another standard thermometer or using the defining fixed-points and a polynomial equation between resistance and temperature, such as the Callendar or CallendarVanDusen equations (28) which are equivalent to those given in Table II. In the United States calibration is frequently done by the National Bureau of Standards, Institute for Basic Standards. Typical calibration data for a platinum thermometer is given in Table XII. The constants o, g and r were found from the Callendar-VanDusen formula: Rt + R (19) t t + 0_ )~ ) + 5( t3

-571.0 0.8 0 0.6. 0.4 0.2 0 100 200 300 Temperature, OK Figure 16. Resistance Ratio of Platinum as a Function of Temperature.

-581.20 1.00 Platinum film 0.80 / Copper a Platinum Nickel o Tungsten 0 0.60 0:C~ 1 0.40 0.20 -200CC IOOOC OOC 500C OOK OOO1K 2000K 3000K 3500K Figure 17. Resistance-temperature Relationship for Various Resistance Type Temperature Sensors-high Range. Sources of Data: Platinum, Mean of NBS Calibration; Tungsten; Copper; and Nickel. (Courtesy of Rosemount Engineering Company.)

Fu 8. Notched mica cross Leads 0.1 mm platinum wire Figure 18. Capsule-type, Strain-free Resistance Thermometer.(40

GALVANOMETER ADJUSTMENT TO MAKE RATIO EXACTLY MEASURING DIALS ~/\ ~ COMMUTATOR NORMAL )A 1 C REVERSED D C B A THERMOMETER Figure 19. Mueller Bridge with a Four-lead Platinum Resistance Thermometer.

-61TABLE XII Report of Calibration of Platinum Resistance Thermometere N No. 1653433 Submitted by The University of Michigan (62) Constant Value ta 0.003925780 6 1.49168 0.11116 (t below 0~C) 0 (t above 0~C) R 25.5510 abs. ohms Additional qualifying information was provided by the NBS for this calibration as follows: "The value of 8 was estimated using the assumption, based on experience with similar thermometers, that the product Q,8 is a constant. The uncertainty in the estimated value of 6 is equivalent to an uncertainty at the sulfur point of less than ~ 0.01 deg C. The other values given are determined from measurements at the triple point of water, the steam point, and the oxygen point. The uncertainty of the measurements at these points, expressed in temperature, is less than + 0.0003, + 0.0015 and + 0.005 deg C respectively. About one-half of each of these uncertainties is an allowance for systematic errors, including the differences among national laboratories, the remaining part representing the effect of random errors in the measurement process. The effects of these uncertainties on other measured temperatures are discussed in Intercomparison of Platinum Resistance Thermometers between -190 and 4450C, J. Research NBS 28, 217 (1942), During calibration the value of R changed by the equivalent of 5 x 10 deg C. These results indicate that this thermometer is

-62Satisfactory for use as a defining standard in accordance with the text of the International Practical Temperature Scale." Resistance-temperature data on the thermometer described in Table XII are listed in Table XIII for a small range of temperatures above 9P0K. These data are computed from the following equation and represent a few of the numerical results abstracted fromn the original calibration. = + -2t [1+(1- )10 + t21- t )10-6] (20) Ro 100 100 The first column is the temperature in ~K (IPTS 1948), the second column is the thermometer resistance in absolute ohms and the third column gives the inverse (reciprocal) of the difference between each two successive values in the second column. These reciprocal first differences are included to facilitate interpolation. The error introduced by using linear interpolation will be less than 0.00010C. The third column may also be expressed as dT/dR, ~K/ohm, as the tabular difference in the first column is 1.00K. The thermometer described in Table XII was also calibrated and the results tabulated in 0.10K intervals from 11~K to 920K by the NBS using the NBS-1955 temperature scale. This temperature scale was referred to L% Section II and Figure 2 and defines the temperature in terms of the electrical resistance of platinum in the range 100K to 90~K. An important class of low temperature thermometers are those whose electrical resistance increases with decrease in temperature, rather than the opposite, as is the case with platinum. Below 200K these thermometers become most practical. This class of thermometers includes carbon, germanium, and the semi-conductors (thermistors) and are the most sensitive resistance elements to temperature changes at low temperatures available. The electrical resistance characteristics of these materials is shown in Figure 20 in comparison with platinum, tungsten and indium.

TABLE XIII August 1965 Table for Platinum Resistance _Thermometer 1653433 TEMP. RESISTANCE INVERSE..TEMP. R.ESISTANCE INVFR... DEG.K ABS CHMS DIFF. DEG.K ABS OHMS DIFF. 139'11.56169 9,300 90 6.20974 1 40 11.66916 9.305.91 6.32084 9.001'14. 11.77656 9.311 92 6.43186 9.008 142 11.88390 9. 316 93 6.54279 9. 015 143.11.99118 9~322 94 6.65364 9.'022 144. 12.09843 9.327 95 6.76440 9.029 145 12.20556 9.332 96 6.87507 9.035 146 12.31265.9.337 97. 6,98567.9.042 147 12.41969 9.343 98 7.'096t7 9.049 148 12.52667 9.3.48 99 T.20660 9.056 149 12.63358 9.353 100 7.3'1695 9.063 150 12.74044 9.358 o101 7427/21' 9...9 151 12.84724 9.363 102 7.53739 9.076 152 12.95399 9.368 103 7.64749 9.0'83 153 13.06067 9.373 104 7.75751 9,089 154 13. 16730 9.379 105 7.86745 9.096 155 13.27387 9.384 106 7.97732 9.102 156 13. 38038 9.389 107. 8.08710 9. 109 157 13.48684 9.393 108. 8. 19681 9.115 i58 13.59324 9.398 109 8.30644 9.122 159 13.69959 9.403 110 6.41599 9.128 160 13.80588 9.408 11i1 8.52546 9. 135 161 13.91212 9.413 112 8.63486 9.141 162 14.01830 9.418 113 8.74419 9.147 163 14.12443 9.423 114 8.85343 9.153 164 14.23050 9.427 115 8.'96261 9. 160 165 14.33653 9.432 116 9.-07171 9.'166 166 14.44249 9.437 117 9..18074 9. 172 167 14.54641 9.441 118 9.28969 9.178 168 14.65428 9.446 119 9.39857 9.184 169 14.76009 9.451 120 9.50738 9.t190 170 14.86585 9.455 121 9.61612 9.196 171 14.97156 9.460 122. 9.72479 9.202 172 15.07722. 9.464 123 9.83338 9.208 173. 1518283 9.469.124 9.94191 9.214 174 15.28839 9.473 125 10.05036 9.220- 175 15.39390 9.478 126 10. 15875 9.226 176.15.49936 9.482 1-27 10.26707.9.232 177 15.60477 9.487 128 10.37532 9.238 178 15.71013 9.491 129 10.48350 9.244 179 15.81545 9.495 130 10.59162 9.249 180 15.92071 9.500 131 10.6996 9.255 181 16.02593 9.504. 132 10.80765 9.261 182 16.13110 9.508 133.10.91556 9.267 183 16.23623. 9.513 134 11.02341 9.272 184 16.34130 9.517' 135 11. 13120..9.278 185 16.44633 9.521 136.11.23692 9.283 186 16.55132 9.525 137 11.34657 9;289 187 16.65625 9.529 138 I 1.45417 9. 294

The most common resistance element which also is readily available and inexpensive, is the conventional carbon radio resistor. In addition to its high thermal sensitivity at low temperatures, the carbon resistor can be made small, is rather insensitive to magnetic fields and has a small heat capacity for rapid thermal response. It is slightly pressure sensitive, having temperature changes of 0.310K at 200K and 0.02~K at 40K for an increase in pressure of 1000 psi (64), and is subject to thermal instabilities or ageing. This lack of reproducibility is particularly significant after the resistor has been exposed to thermal cycling. Carbon in the form of thin graphite coatings has been used as a thermometer (65). This type of thermometer is especially useful where high response is required, as in low temperature (Q.1~K) adiabatic demagnetization experiments. Lindenfeld (66) reports on the use of carbon and germanium thermometers between 0.300K and 20~K. One problem in the use of carbon radio resistors below 10K is the difficulty in measuring their high resistance. Maximum power dissipated in these resistors is about 10 watts for temperatures 10K and higher. Using a Wheatstone Bridge temperature changes of 10 to 10-6 K can be detected. The use of the carbon resistor in measurement is greatly aided if a reasonably simple and accurate formula can be written relating resistance to temperature. Clement and Quinnell (67) found that Allen-Bradley Company cylindrical carbon radio resistors has a resistance temperature relationship below 200K which could be expressed to within + 1/2% by a semi-empirical expression of the form K B log10R + log R A T (21) 10 The constants K, A and B are determined by a calibration of the resistor at a minimum of three known temperatures. Typical resistance-temperature curves for two Allen-Bradley carbon resistors are shown in Figure 21. Schulte (68) calibrated an Allen-Bradley 0.1 W, 270 ohm carbon resistor between 4~K and 296~K and found his

-65 — 5.0'I~ ~!~ ar' gar Commercially available JI~ I;;germanium thermometers. I I1-'~ i _ I, f1, I, i 54oat200K K ~I I.s38. at 20faoK K/ I ~ ~i Low tamp I-~~~ I /*-therml'tor 5'1, 2.0 Carbon~ ~ ~ * * O27OK 2600C -2500C -2400C 00K $00IOK 200K 300K 350K Figure 20. Reitneteprtr Relationship for Various ResiStance..ty-pe temperature Sensorslow Range. Sources of Data: Platinum, Mean of NBS Calibrations, Carbon Resistor; GermaniUm Thermometer; Thermistor; Tungsten; I~~~4 I and Indium. (Courtesy of Rosemout Engineering Company. ct~~~~~:%t I l\ \\I ~~~~~~~~~~~~i \ I - ~00 -26000 -25000 -24000 O~K ~.0OK ~.0OK 3OK 35OK Figure 20. Resistance-temperature Relationship for Va~rious Resistanice-type temperature Sensorslow Ranlge. Sources of Data. Platinum~ Mean of NBS Calibrations, Carbon Resistor; Germanium Thermzometer; Thermistor; Tungsten; and Indium. (Courtesy of Rosemount Engineering Company. )

results to correlate within 7 percent of Equation (21). For a range of temperatures from 20K to 200K Mikhailov and Kaganovskii (69) also found that Equation (21) gave satisfactory results for carbon thermometers. In this case the constants in Equation (21) were determined from calibrations at 20K, 4.20K and 20.40K. This permitted temperatures to be calculated with an accuracy of a few hundredths of a degree in the range 20K to 4.2~K. After 100 heating and cooling cycles between 3000K and 770K, uncertainty in the temperature measurements in the same 2.20K interval did not exceed 0.010K. Measurement of the resistance of a carbon thermometer may be made with a resistance bridge, as in Figure 19, or with a potentiometer using an accurately calibrated monitoring resistor of known resistance. A schematic diagram of this latter method is shown in Figure 22 as used by Greene (70). He calibrated a carbon resistor having a nominal resistance of 82 ohms with a measuring current of 10 -a. The results of this calibration are given in Figure 23 which illustrates the influence of thermal cycling, the reproducibility of the calibration before and after a calibration run and heat conduction along the thermometer lead wires. The ordinate in Figure 23 is the voltage drop across the resistor for a 10 FI-a current. During any one calibration the accuracy amounts to + 0.020K and is within the precision of the measurements. Although thermal cycling did produce a shift in the calibration curve, its slope remains constant. Thermal conduction along the lead wires raised the calibration curve by approximately 0.100K in this instance. The use of carbon resistors for field measurement where laboratory precision is not demanded has been studied by Herr, et al (71). Allen-Bradley Co. 0.1 watt, 100 ohm (+ 5% at 300~K) resistors were found to be reproducible within + 1% of the absolute temperature in the range 19.5~K to 55.5~K (359R to 100~R). The measurement of resistance ratio rather than absolute resistance was found to be a more satisfactory method owing to drift in the resistance values of the carbon resistor.

106 104 03 40 C 0 1.0 2,0 3.0 4.0 5.0 Temperature, ~ K Figure 21. Resistance-temperature Curve for Two AllenBradley Carbon Resistors. (39)

MONITORING r —-.... RESISTOR RL - CURRENT L2 I CARBON LIMITING AUX. POT 2 ESISTANEOE RESISTO BINDING POSTSTHEMOME a~ N TYPE K- ~3 ~RCRTI UNIVERSAL CRT - +V POTENTIOMETER I I ~ EMIF | |L3 BINDING POSTS IRITIgure -I Figure 22. Schematic Diagram of the L&N Type K-3 Universal Potentiometer Circuit.(7~)

-6980 60 O 5/10/66 Therm. immersed 40 in liquid A 6/7/66 Therm. immersed in liquid-Cycle 2 20 * 5/25/66 Therm. immersed in liquid- Cycle i X 5/25/66 Therm. in vapor 3000 above liquid A 5/10/66 Therm. immersed 80 - in liquidReproducibility Check 60 C) t 40 > 20 0 c: 2900 80IL_ 60 - w 40 20 2800 - 80 60 40 I I I I I I I - I I I I 14.0 14.5 15.0 15.5 TEMPERATURE - OK Figure 23. Typical Calibration Curves for Carbon Re istance Thermometer (Nominal Resistance 82 OHMS).(70)

-70The word''thermistor" is a trade name for a class of semi-conducting solids having a large negative temperature coefficient of electrical resistance. It is a name derived from the word combination thermal-sensitive-resistor. In a physical description these substances are classed as electronic semi-conductors whose characteristics have been given much theoretical and experimental examination since World War II. Semi-conductors may be classed with those substances having electronic conductivities in the range 10 to 10 (ohm-cm), or resistivities falling between 103 and 10 ohm-cm (72). This can be compared with the pure metals and metallic alloys -4 whose resistivity (73) are generally less than 10 ohm-cm or with the electrical insulators, as mica and quartz, having resistivities above 106 ohm-cm at ordinary temperatures. Figure 24 shows these relationships. The important difference between semi-conductors and metals for thermal sensitive uses is not, however, their orders of magnitude of resistivity but the great differences in the change of resistivity with temperature as compared with the metals. This may be illustrated by a typical thermistor which will increase in resistance from 780 to 17,800 ohms for a temperature change from +30 to -30~C. This is a total change of approximately 17,000 ohms or a percentage change of about 2000%. Compared with standard platinum and copper resistance thermometers the corresponding change is about 6 and 100 ohms, respectively, over the same range of temperatures, both changing about 20%. The thermistor, then, undergoes a percentage change in resistivity of about 100 times that of the metals in this range of temperature. Should a greater interval of temperature be examined, as in Figure 24, the percentage change for the thermistor might be as large as 6 2 x 10. Possibly of greater significance inthe field of thermal measurements is (dR/DT), the rate of change of resistance with temperature, of this thermistor as a function of temperature. At 25~C, for example, dR/dT is 44 ohms/~C and at -30~C it is 1120 ohms/~C, while for a standard 25-ohm platinum resistance thermometer, dR/dT is about 0.10 ohms/~C in this same range of temperature. This means that if

one is able to measure changes in resistance, say, to 0.01 ohm, the temperature change capable of detection with this thermistor is 0.00020C at 25%C and 0.0000090C at -30~C. Some commercially available thermistors have sensitivities 100 to 1000 times greater than this. The ordinary platinum resistance thermometer would detect a temperature change of 0.10C under these same circumstances. It is quite generally true that thermistors have greatest sensivitity at lower temperatures. For absolute temperature measurement other considerations, naturally, are necessary, not least among which is the thermal stability of the thermistor element, a property possessed in the highest degree by an annealed, strain free platinum resistance thermometer. Thermistors are available from the manufacturers in a variety of shapes and sizes: discs, beads, rods, washers, and wafers. The shape selected depends on the use to be made of the element. The sizes range from 0.006 inch to 0.10 inch diameter. For beads, 0.2 to 0.75 inch diameter and 0.040 to 0.500 inch thick for discs, wafers and washers, and for rods from 0.01 to 0.50 inch diameter, 0.25 to 2 inches long. Lead wires of various lengths and diameters consist of platinum, platinum-iridium alloys, or copper which can be butt-soldered, wrapped and soldered or fired in place on the thermistor element. Silver paste contacts are available to which the user can soft-solder lead wired, if desired. Washer type elements have terminals which may be mechanically clamped into place against the faces of the element. Protective coatings are frequently placed over the thermistor to prevent or retard atmospheric attack. These consist of a thin or thick layer of glass or enamel coating. For certain applications the element can be placed in an evacuated or gas-filled bulb. The recommended maximum temperature for continuous service varies but it can be as high as 300~C, some manufacturers recommend a temperature no greater than 150~C, however. To a large extent this will depend upon such things as accuracy

-72ALUMINIUM 1012 10 IC IN SU LATORS \,ORCELAIN 10 E 6 _ _ PLURANIUM OXIDE 10 2 -100 0 100 200 300 400 500 NICKEL Ud SEMI-CONDUCTORS tC0 1)4 GOOD CONDUCTORS PLATINUM COPPER -TE00 0 MPERAT00 200 30 400 500 TEMPERATURE ~C Figure 24. Temperature-resistivity Relationship oS Insulators, Semi-conductors and Good Conductors.( )

-735 required, the atmosphere surrounding the element and the melting point of the solder, if any, used to fasten the lead wires to the element. In any event, the thermistor is used to its greatest advantage, from a thermal-sensitive consideration, at lower temperatures. Most thermal-sensitive semi-conductors (thermistors) are manufactured by sintering various mixtures and combinations of metallic-oxides, the common materials being the oxides of manganese, nickel, cobalt, copper, uranium, iron, zinc, titanium, and magnesium. For the commercial thermistors the oxides of manganese, nickel and cobalt, however, are the most commonly used substances for the mixtures. The result of this type of manufacturing process is a hard, dense ceramic type of material. Other materials (72, 74) which may be classed with the semi-conductors and which possess a large negative temperature coefficient of electrical resistance include chlorides such as NaC1l, some sulphides like Ag2S, CuS, PbS, CaS and some iodides, bromides, and nitrides. Lead sulfide has been used as a detector of infra-red radiation in a radiation pyrometer and is marketed commercially. Its response is high (10,000 cps) and can detect temperatures as low as-300~F. The uses of thermistors in a radiation-type pick-up is reported (77) for measurement of sub-zero temperatures. Some pure materials such as silicon, tellurium, germanium and selenium (74) which are monatomic become semi-conductors in the presence of certain impurities. This effect is shown qualitatively in Figures 25 and 26 for silicon containing an unknown impurity and for cuprous oxide with varying amounts of oxygen in excess of the stoichiometric. Figure 25 taken from Becker, et al (75), shows a 107 increase in the conductivity of pure silicon by the addition of a foreign impurity. A similar large increase in conductivity is seen in the case of cuprous oxide, Figure 26, also taken from reference (75) where the increase is due to an excess

tO 3 0 to-___1F 10.. _ —. 10' ___ Eo~'~l " 10 - O~~~~~~~~~~~~~~~~~~~~~~~~~~ o __' ()~;E _o _ _~ _" _ E _ \' _"_' _ o-_ _ _ _ 0 _ _ _ _ - _ _ Z~~~~~~~~~~~~~~~~~~~~~~~ t0~~~~~~~~~~~~~~~clJ-~ I, I - \1- zO\ _ 0o S i s Si a VS o tu 0o- _ \ _ __ _, __ __o-..., -- %"" n t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00o, o 0 10 \40 ~ 0 o o 0 1 - -.. - INVERSE ABSOLUTE TEMPERATURE,oK- INVERSE ABSOLUTE TEMPERATURE, ~ Figure 25. Logari-thm of the Conductivity of Figure 26. Logarithm of the Conductivity of Various Specimens of Siicon as ao Function of Tnverse Absolute as a Fnction of Invers Temperature.(84) Temperature. (84)

-75of oxygen up to 1%. These effects vary greatly with the type of impurity, its amount, its dispersion within the solid, and the heat treatment of the solid. Generally speaking, a thermistor can be considered for use in any application requiring a thermal sensitive electrical resistance element. The obvious and perhaps most widely employed application is that of temperature measurement. As was pointed out earlier, it is possible to detect very minute changes in temperature with a thermistor owing to the large change in its electrical resistance with temperature. Brown (76) employed a Western Electric 17A thermistor to measure small changes in air temperature. The device was used in a bridge circuit, the output of which was amplified and fed into a recording oscillograph. During the initial measurements it was found that the thermistor was so sensitive that it recorded with fidelity the fluctuations in air temperature resulting from atmosperic turbulence. Figure 27 taken from this work shows a typical oscillograph record. Changes in temperature could be measured to an estimated 0.00070C. Theoretical work of Wilson (78, 80) and others has lead to the following expression for the electronic conductivity of a semi-conductor = Ae -B/T (22) Since the conductivity o is the reciprocal of the resistivity p we may write P = Ce or, 0 = POe T T (237 Also, since the electrical resistance is a geometric extension of the resistivity Equation (23) may be written B( - T) (24) R= Re o

-76100 SEC. o! I ZERO ADJUSTED o l I _ _ _ Fgre 7 hritrRsos oRo eprtr aitos(4 Figure 27. Thermistor Response to Room Temperature Variations. (84)

-77Because of the form of Equations (23) and (24), the logarithm of the resistivity or resistance is frequently plotted against the reciprocal of the absolute temperature, as shown in Figure 28, in order to demonstrate the electrical characteristics of a thermistor and to compare it with others. These data are experimental and are taken from Becker, et al (75). The experimental curves in Figure 28 are almost straight, as required by Equation (23). However, close inspection will disclose a slight curvature which may be shown to increase linearly with increase in level of temperature (75). Hence, the equation is sometimes modified as p = ET D/T (25) where C is a small number compared with D or B and may be positive, negative or zero depending on the material (75). For our present purpose we shall employ Equation (24), since if the interval T - T is not too great this equation will adequately represent the data and it is somewhat easier to handle mathematically. As was mentioned above the relationship of resistance to absolute temperature, given by Equation (24), has the same shape as the curve shown in Figure 28. From such a curve several important characteristics may be obtained relative to the suitability of a thermistor as a temperature sensing element. A curve of R vs. 1/T is also a convenient chart for comparing several different thermistors for use in temperature measurement. By taking logarithms and differentiating Equation (24) the following equations are obtained: dR B dR dT, (26) R 2 or,'f = — (27) R dT 2 and, dR R dT B2 (28)

-7810 to6 10 10~ 102 C) Ev 3 i-o" I-1 10 ~0~/,, /, a: _ 10'0 0 2 3 4 5 x1o INVERSE ABSOLUTE TEMPERATURE - Figure 28. Logarithm of the Specific Resistance of Two Thermistor Materials as a Function of Inverse Absolute Temperature. (84)

-79Equation (28) may be interpreted in relation to a curve similar to Figure 28 or Log R vs. 1/T. It will be noted that the slope of a curve onisuch a chart is written d (Log R) dR/R d(l/Tg) - dT — (slope of Log R - l/T curve) (29) Comparison of Equations (28) and (29) disclosed that the right hand side of Equation (29), the slope of a curve plotted as Log R vs i/T, is equal to the parameter B in Equation (24). Hence B = (slope of Log R vs. 1/T curve). (30) Equation (28) is then rearranged to dR _ (slope) R2 (31) Interpretation of Equation (31) is as follows. For use as a temperature sensing element it is desirable that a thermistor have as large a value of dR/dT as possible in order that it be sensitive and capable of detecting small changes in temperature for any given resistance measuring system. From Equation (31) it follows that at any given temperature, that thermistor which has the greatest slope on a Log R vs. 1/T plot and the greatest resistance will also be the most sensitive as a temperature sensing element. In this way, therefore, a series of thermistors can be very rapidly evaluated as to their thermal sensitivity. Another method for evaluation of thermistors consists of plotting Log R vs. B, where B is determined from experimental thermistor data in the region of To, which 0 may be taken to be 0~C; R0 is then the resistance of the thermistor at 0~C. Because most thermistors have similar characteristics it will be generally true that a thermistor with superior thermal sensitivity at 0C will also have superior sensitivity at other temperatures. In any event the resistance-temperature characteristics of the thermistor can be obtained approximately from Equation (24) or from the

_80manufacturer's published data. Equation (24) is approximate owing to the non-linear nature of Log R vs. l/T, as mentioned above in connection with Equation (25) and Figure 28. The technical literature does not contain a large body of data on the stability or ageing effects of thermistors. What is reported here are heterogeneous results of a number of observers on a few isolated tests. It may be generally concluded, however, that an ageing effect may be expected which usually is of the nature of an increase with time of the electrical resistance. This increase is not linear but logarithmic, resulting in smaller percentage cShanges in resistance with increased time. Preageing may be accomplished by heating or by the passage of higher than service current through the thermistor (79). These have the effect also of accelerating the ageing if the temperature is high enough. The change of electrical resistance is sometimes attributed to a rearrangement in the distribution of the components of the mixture of oxides making up a semiconductor. Heat treatment is believed to play a major role in the dispersion of the components. Hence, ageing and pre-ageing usually involve some kind of heat treatment. MHuller and Stolen (81) tested two Western Electric 14A thermistors at 25~C over a period of six months. They report a decrease in resistance of about 50 ohms out of a total of approximately 100,000 ohims. This corresponds to an ageing effect of about 0.0120C. Figure 29 shows ageing data (75) taken on three quarter-inch diameter discs of material No. 1 and No. 2 (No. 1 is composed of manganese, nickel oxides; No. 2 is composed of oxides of manganese, nickel and cobalt) with silver contacts and soldered leads. These discs were measured soon after production, were aged in an

oven at 105~C and were periodically tested at 240C. The percentage change in resistance over its initial value is plotted versus the logarithm of the time in the ageing overn. It is to be noted that most of the ageing takes place in the first day or week. If these discs were pre-aged for a week or a month and the subsequent change in resistance referred to the resistance after pre-ageing, they would age only about 0.2 per cent in one year. In a thermistor thermometer, this change in resistance would correspond to a temperature change of 0.05~C. Thermistors mounted in an evacuated tube or coated with a thin layer of glass age even less than those shown in the figure. For some applications such high stability is not essential and it is not necessary to give the thermometers special treatment. Thermistors have been used at high temperatures with satisfactory ageing characteristics. Extruded rods of material No. 1 have been tested for stability by treating them for two Mnonths at a temperature of 3000C and -75%C for a total of 700 temperature cycles, each lasting one-half hour. The resistance of typical units changed by less than one per cent. In order to determine the life of 1A thermistor, Pearson (82) placed it in a circuit where an off-and-on current of 10 mA. a.c. was repeated 30 seconds over an extended period of time. Resistance measurements were made on the units periodically in order to determine their stability with time. The general trend was a rise in resistance during the first part of its life, after which the resistance became quite constant. Over a period of 15 months, during which time the thermometer was put through 650,000 heating cycles, the cold resistance did not increase by more than 7%. The resistance of the thermistor when hot was found to be equally stable. The characteristics of both thermistors and thermocouples shift when exposed to high temperatures for lengthy periods of time (83). For thermistors the resistance change varies slogarithmically wth time with higher tmperatures accelerating the change. This suggests that if thermistors are subjected for several

-82LLJ z t 5 L__ MATERIAL _ 0.5 z 0.5. z i D 1 WEEK i MONTH 6 MONTHS 1 YEAR 2YRS S YRS [,J t0t 102 403 TIME IN HOURS AT t050 C Figure 29. Effect of Aging in 105 C Oven on Thermistor Characteristics; Materials 1 and 2. (84

-83days or weeks to temperatures somewhat higher than those to be encountered in actual use, the major portion of the change would have occurred. For thermocouples, the change in voltage output becomes greater as the exposure time to high temperature is increased. Over a three month period in which thermistors and thermocouples were exposed to 200~F for about 15 hours, the thermistor shifted a maximum of 0.2~F, while the thermocouple shifted 0.3~F. However, when new elements were tested and aged for 100 hours at 500~F, the thermistors still shifted only 0.2~F while the thermocouples shifted twice as much or 0.6~F. It was found by M1uller and Stolen (81) that if the exciting potential is left impressed across the thermistor, a steady state is reached. This implied a resistance change of less than 1 ohm on daily measurement. The cold resistance of this thermistor at 0~C was 350,000 ohms. Short range stability of a Western Electric 14A thermistor measured at five minute intervals at 25~C (in ohms) is shown in the table below. The authors (81) used the thermistor to measure small temperature difference in a laboratory experiment. Thermistor A Thermistor B 1. 96,234.0 1. 96,234.7 2. 96,234.6 2. 96,234.6 3. 96,234.2 3. 96,234.9 4. 96,235.0 4. 96,234.4 5. 96,234.8 5. 96,234.4 6. 96,234.8 6. 96,234.8 7. 96,234.7 7. 96,234.2 8. 96,234.8 8. 96,234.7 9. 96,234.6 9. 96,234.0 10. 96,235.0 10. 96,234.6 The conclusion is that no significant change in resistance was detected which could not be attributed to measurement uncertainty. To obtain a stable thermistor the following steps are generally thought to be necessary (75). By these precautions remarkably good stabilities can be attained.

-841. Select only semi-conductors which are pure electronic conductors. 2. Select those which do not change chemically when exposed to the atmosphere at elevated temperatures. 3. Select one which is not sensitive to impurities likely to be encountered in manufacture or in use. 4. Treat it so that the degree of dispersion of the critical impurities is in equilibrium or else that the approach to equilibrium is very slow at operating temperatures. 5. Make a contact which is intimate, sticks tenaciously, has an expansion coefficient compatible with the semi-conductor, and is durable in the atmosphere to which itwill be exposed. 6. In some cases, enclose the thermistor in a thin coating of glass or a material impervious to gases and liquids, the coating having a suitable expansion coefficient. 7. Pre-age the unit for several days or weeks at a temperature somewhat higher than that to which it will be subjected. Clark and Kobayashi (84, 85) have studied the general characteristics of thermistors to be used for temperature measurement. This includes the theory of their conductance properties, the dynamic response and steady-state error of the thermistor temperature-sensing element, their stability and the resistance-temperature characteristics of approximately 300 commercially available thermistors from 8 different manufacturers. Friedberg (79) describes a semi-conducting film of ZnO used as a thermometer at 20K. This device had an electrical resistance of 5(10 ) ohms at liquid helium temperatures and had a sensitivity of approximately 5(10 4) ohms per degree K at 20K. Germanium, with impurities consisting variously of arsenic, gallium or indium, has become one of the most satisfactory materials for thermal resistance elements

in the range 0.20K to 200K. This material possesses a negative temperature coefficient of resistance, a moderate level of resistance, high sensitivity of resistance change to temperature change, high reproducibility and stability to thermal cycling and is readily manufactured and fabricated. The impurities are included in the germanium in controlled quantities to influence both the resistance-temperature characteristics and the sensitivity. A typical resistance-temperature curve for germanium "doped"r with 0.001 At% indium is shown in Figure 30 for the temperature range 10K to 50K. Thiis particular element was found to be highly reproducible over a period of several months. The thermometer was subjected to a number of warming and cooling cycles following which its resistance-temperature characteristic could be reproduced to within + 0.0010K. The measuring current used was 0.01 ma although the author reports an increase of current to 0.1 ma did not appreciably influence the R-T characteristic (79). Edlow and Plumb (86, 87) studied the reproducibility and temperature-resistance characteristics of a number of commercially available germanium thermometers. The germanium had either arsenic or gallium as the impurity. Their purpose was to find out if a germanium thermometer was sufficiently stable to be used as a basic secondary standard thermometer. As a consequence of their study the NBS adopted the germanium resistance thermometer as the basis for the NBS scale from 20K to 200K and used it for basic temperature calibration in this range. The determination of reproducibility was made by cycling the resistance element from 4.20K to 3000K and measuring the resistance change at 4.20K. This change in resistance was then related to the corresponding temperature change. Two typical heating-cooling cycle tests are shown in Figures 31 and 32. In each case the reproducibility is within + 0.001~K. In the case of resistor D, Figure 32, the reproducibility is within 0.0005~K after 86 cycles. Because of this high degree of stability the resistor of Figure 32 became one of the NBS standard thermometers. This result is quite typical

-8612,000 8,000.I 6,000 0 o 6,000 - (n 4,000,) 2,000 - 0 L I I, I I 1 1 2 3 4 5 TEMPERATURE (~K) Figure 30. Calibration Curve for A Germanium Thermometer.(79)

-87of that found by others. Kunzler, et al (88), for example, cycled arsenic "doped" germanium encapsulated in helium-filled thermometers as many as 50 times and found no evidence of calibration change of as much as 0.00010K. Furthermore, they report two such thermometers in use for 3 years on low temperature experimental apparatus with no observable change in calibration. From results such as these it seems safe to conclude that germanium'doped" with a selected impurity is a suitable material for low temperature thermometers below 200K. The resistance-temperature calibration data for a number of encapsulated, hermetically sealed, arsenic "doped' germanium thermometers was determined by Enlow and Plumb (86) in the range 2.10K to 5.00K. The resistance was measured at temperature intervals of 0.10K in a pressure-controlled helium liquid-vapor equilibrium cell. Other measurements were made in a calibration comparator apparatus. The results agreed to within 0.0010K. The basic standard temperature reference was the INBS 1958 4 He scale, Table VI. Some typical data are given in Figure 33. A polynomial function was derived for each thermometer to represent its resistance-temperature calibration in the range 2.10K to 5.00K. The sensitivity of a germanium thermometer, dR/dT, manufactured by Cryo Cal, Inc. (89), is shown in Figure 34 for the temperature range 20K to 280K. At 200K the sensitivity of this thermometer is 3 ohms per 9K which may be compared with a sensitivity of 0.0185 ohms per ~K for a platinum thermometer at the same temperature. The very large increase in sensitivity for germanium at temperatures below 200K is characteristic of this type of resistance thermometer. The use of arsenic-doped germanium prepared from a single germanium crystal is reported by Kunzler, et al (88). The germanium element is cut into the form of a'bridge" of dimensions 0.06 x 0.05 x 0.52 cm with side arms near each end for electrical connections. An encapsulated thermometer design is illustrated in Figure 35. Wlen covered with a platinum case it is filled with helium gas which limits its lowest useful temperature to about 0.25~K. "Bare" bridges have also

* JULY 1963 O STARTED JAN.7, 64(RESOLDERED O STARTED JAN. 12, 64 2628.6 A ENDED FEB. 12, 64 1.0 C,) o 2628.2 < 2627.8 0.5 LU a 2627.4 2627.0 0 - I 0 30 60 90 120 150 180 210 240 270 300 NUMBER OF ACCUMULATED CYCLES Figure 31. Equilibrium Resistance As A Function of the Number of Accumulated Cycles T = 4.20K.(87)

2566.6 I 2566.2 0 I. 2565.8 ~C2 2565.4 — 2565,0 4 0 10 20 30 40 50 60 70 80 90 NUMBER OF ACCUMULATED CYCLES Figure 32. Equilibrium Resistance As A Function of the Number of Accumulated Cycles for Resistor D.(7) T = 4.20K.

-9o - 1 5,000 0 ox 0 0 10,000 9,000 0 8,000 C 7,000 0 6,000 C 5,000 4,000 0?* c0. 3,000 F RESISTOR IDENTIFICATION QA * RESISTOR NO. 1 0 " "2,0X " "3! 2,000 1,500 I I I 1.5 21 3 4 5 6 T(~K) Figure 33. A Plot of the Resistance-tem erature Calibration Data for Resistors 1, 2 and 3. (86D Temperatures were Derived from Liquid Helium-4 Vapor Pressures.

r r~-91 1,000.. 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 TEMPERATURE (OK) Figure 54. Sensitivity dR/dT Typical Standard Cryo Resistor. (Courtesy Cryo. Cal, inc.)(89 Fiue3. Snitvt Rd yia anad oResor ~- ~ ~~~~~(ores ro aIc)8

-92been used in applications such as adiabatic demagnetization experiments where a thermometer with minimum heat capacity is required to give high response. In this case a lag time between the thermometer and sample was 0.1 second.- Higher currents are permitted with the "bare" bridge than with encapsulated models owing to the improved cooling permitted by the exposed germanium element. The resistancetemperature characteristics of the encapsulated model were found to be unaffected (within ~ 0.00010K) by thermal cycling or aging over a period of several years. Cycling of the'"bare" bridge between 4.20K and 2930K several times produced only a few thousandths of a degree change in its calibration. The resistance-temperature characteristics of four typical encapsulated thermometers are shown in Table XIV and Figure 36. TABLE XIV Resistance-Temperature Characteristics of Germanium Thermometers Fabricated from Arsenic-Doped Crystal VIII-899-N (88) Sample no. T~K 9-2R 15-2R 21-2R 27-2R R ohms dR/dT R dR/dT R dR/dT R dR/dT 273 1 1 1 tl 77 2.0 1.9 1.8 1.7 35 4.6 0.16 4.3 0.13 3.9 0.11 3.4 0.07 20 8.0 0.5 7.0 0.3 5.9 0.22 4.9 0.15 15 10.5 0.8 9.0 0.6 7.3 0.4 5.9 0.25 10 18.3 2.8 14.1 1.8 10.7 1.1 8.0 0.67 4.2 101 50 53 15.5 29 -7.7 16.7 2.6 2 789 1000 216 200 77 46 29.5 11.2 1.5 2300 450 120 36.5 The R-T characteristics of a carbon thermometer is shown for comparison. At low temperatures the germanium thermometers have widely different electrical properties because the arsenic impurity concentration is not the same in each sample even though they were cut from the same germanium crystal. This is a good example of the

-93 - Pt-GLASS SEAL RETAINER PINS ENCLOSED IN REN ER Pt COLLAR Ge"BRIDGE PHENOL FER ~~ "BR DGE' SUPPORT /O7C GOLDW025 CM SOLDER COLLAR I Pt WIRE SOLDER COLLAR. 007 CM GOLD WIRE AND Au-Ge BOND.004 CM IMM Pt WIRE Figure 35. Encapsulated German) The with Cover Removed.

extreme sensitivity of the electrical properties of these resistors to impurity concentration. However, in the range 20K to 350K the ratio (dR/R)(dT/T) is of the order of unity for all samples, a result found with germanium bridges cut from other crystals as well. This characteristic is responsible for the tremendous temperature sensitivity of the resistance thermometers. The sensitivity dR/dT at 4.20K of several thermometers, including those in Table XIV, is shown in Figure 37. An increased sensitivity may be achieved roughly according to R3/2, by selecting a thermometer of higher resistance providing the instrumentation is compatible with the selected resistor. The resistance of germanium is influenced by a magnetic field. The variation of the magnetoresistance with temperature was studied by Kunzler, et al (88) using a germanium bridge having a zero field resistance (RI=0) of 200 ohms at 4.2~K The magnetoresistance (AR/RH=0) at 18 kilogauss was found to be 0.16 at 4.2~K and gradually increased with decreasing temperature, reaching a maximum value of 0.28 at 1.9~K. Below 1.9~K the magnetoresistance decreased to a value of R/Rt1=0 of 0.21 at 1.20K. At 4.20K the corresponding change in temperature calibration as a result of the magnetic field would be approximately 0.200K. The magnetoresistance of germanium is also slightly anisoptrpic, being somewhat less than 10% of the total magnetoresistance over 180 angular degrees at 4.20K and in an 18 kilogauss field. Other papers treating the germanium resistance thermometers have been published by Low (90) and Orlova, et al (91). Antcliffe, et al (92) report the use of germanium thermometers below 1~K. Their lowest temperature was 0.400K obtained by a He bath and the range of temperatures investigated was 4.200K to 0.400K. Between 4.20~K and 1.200K the data were fitted to -A -B/T I (32) with representative values of the constants as A = 0.507 - 1.004, -B = 1.245 - 1.399 and C = 71.89 - 112.8 for three resistors calibrated.

-954000 I D ANCE DISTANCE THERMO- p 250~C FROM 2000 METER IN SEED END NO. OHM CM IN MM 9 -2R'0.017 54 1000 15-2R "0.016 59 21 -2R 0.016 64 \600 - - _ l 27- 2R'0.015 69 600 \2A-19 FROM DIFFERENT 400 - — 4B3-59 CRYSTAL 200 _ 0 iO i 60 W 40 Q 20 WT 10 D oO.Q CARBON RESISTOR x 1/10 0.2 A0.40.6 1.0 2 4 6 10 20 40 60 100 TEMPERATURE IN DEGREES KELVIN Figure 36. Resistance-temperature Characteristics of Germanium Thermal Sensing Elements. The Curve Extending to Temperatures Below 10K are for Elements from a Different Crystal than those Extending to the Higher Temperatures. The Characteristics at Low Temperature are Very Sensitive to Arsenic Concentration. However, the Resistivity-temperature Characteristics of an Element are Defined Approximately by its Resistivity at 4.20K.(88)

-961000 600 /2A-48 400 200 (-9 t00 _el A-48 3B-47 0D 100 3A4 LL,''./, Ia~i 60 ___ ~3A-48 A-47 60 92R Ll 40 20 / I/ 20:z: ~15-2R 1 — i -'-21-2R 6..... 6 CONST. /~R dT 7-2R 1 2 4 6 10 20 40 700 200 400 1000 RESISTANCE, R, IN OHMS Figure 37. Variation of with Resistance at 4.20K. The variadT tion of the Sensitivity of Thermometers with Their Resistance at 4.20K is Approximately 1 dRR1/2 (88) R dT

-97X. MAGNETIC THERMOMETRY The practical minimum temperature which may be produced by pumping helium is about 10K for He4 and 0.50K for He3. Below these temperatures the vapor pressure is too low to be maintained for most useful experimental purposes. To produce as well as measure temperatures below 0.5~K the properties of paramagnetic substances, usually paramagnetic alums, are used. The low temperatures are achieved by demagnetizing these salts adiabatically from an initial state of a high magnetic field and a temperature of approximately 1~K. The process of adiabatic demagnetization produces a rapid drop in the temperature of the salt owing to a decrease in its energy by the work of demagnetization. This is analogous to the drop in temperature of a compressed gas as it expands isentropically or adiabatically while doing work on its environment. Garrett(93) reports the limit of cooling by this process (electronic) is about 104 ~K but if the energy of nuclear spin is involved a temperature of 10 -6K is thought possible. A two stage demagnetization of a diluted chromium alum has produced a final temperature of 10-3 ~K with a field of 9000 gauss. To reach this temperature in a single demagnetization from 10K a field of 25,000 gauss would be required. Temperatures of a few hundredths of a degree absolute can be achieved without exceptional difficulties and those of the order of one-thousandths of a degree can be obtained with somewhat greater effort (94, 95). Table XV, prepared by Zemansky (96), summarizes some results of adiabatic demagnetization experiments and identifies the paramagnetic salts used. Certain properties of commonly used paramagnetic salts are given in Table XVI (96).

-98TABLE XV Temperatures Attained by Adiabatic Demagnetization of Various Paramagnetic Salts (96) Initial Initial Final Experimenters Date Paramagnetic salt field, temp. magnetic oersteds ~K temp. T*~K Giauque and 1933 Gadolinium sulfate 8,000 1.5 0.25 MacDougall De Haas, Wiersma, 1933 Cerium fluoride 27,600 1.35 0.13 and Kramers Dysprosium ethyl 19,500 1.35 0.12 sulfate Cerium ethyl sulfate 27,600 1.35 0.085 De Haas and Wiersma 1934 Chromium potassium 24,600 1.16 0.031 alum 1935 Iron ammonium alum 24,075 1.20 0.018 Alum mixture 24,075 1.29 0.0044 Cesium titanium alum 24,075 1.31 0.0055 Kurti and Simon 1935 Gadolinium sulfate 5,400 1.15 0.35 Maganese ammonium sulfate 8,000 1.23 0.09 Iron ammonium alum 14,100 1.23 0.038 Iron ammonium alum 8,300 1.23 0.072 Iron ammonium alum 4,950 1.23 0.114 MacDougall and 1936 Gadolinium nitrobenGiauque zene sulfonate 8,090 0.94 0.098 Kurti, Laine, Rollin, and Simon 1936 Iron ammonium alum 32,000 1.08 0.010 Kurti,Laine, and Simon 1939 Iron ammonium alum 28,800 9.5 0.36 Ashmead 1939 Copper potassium 35,900 1.17 0.005 sulfate DeKlerk (95) 1956 Chromium Potassium - -- 0.0029 alum

-99Table XVI Properties of Paramagnetic Salts (96) Gram- Density, Curl const., ionic cm3 dem Paramagnetic salt ionic.. C cm3 de weight cm gm ion i' (gm) Cerium magnesium nitrate. 765 -- 0.318 2Ce (NO3) 33Hg (NO3) 2241{20 Chromium potassium alum. 499 1.83 1.86 Cr2 (SO4) 3K2S0424H20 Chromium methylammonium alum. 492 1.645 1.87 Cr2 (SO4) 3CH3N3SO424H 20 Copper potassium sulfate. 442 2.22 0.445 CuSO K SO H20 0 4 2 4 2 Iron ammonium alum. 482 1.71 4.35 Fe2 (S04) 3 (NH4) 2 S04 2420 Gadolinium sulfate. 373 3.010 7.85 Gd2 (SO4) 38H20 MIanganese ammonium sulfate. 391 1.83 4.36 M!nS04 (NI14) 2SO46H20 Titanium cesium alum. 589 01 2 0.118 Ti2 (SO4) 3Cs2S04 24IH2 At liquid helium temperatures the orientation of the magnetic ions in the faramagnetic salts are influenced by a magnetic field in a significant way and contribute to both the energy and the entror:y of the salt. Lattice vibrations also have energy and entropy contributions but at the low temperatures (< (1~K) associated with adiabatic demagnetization experiments these effects are small. The partial spacial ordering of the paramagnetic ions in the presence of a magnetic field at constant temperature results in a decrease of the system entropy, as would be expected in an isothermal transition from a less-ordered to a greater ordered state. Thus, the effect of an increase in the magnetic field on a paramagnetic salt is exactly analogous to tile

-100isothermal compression of a fluid or the isothermal extension of an elastic substance. This is illustrated in Figure 38 which shows the temperature-entropy diagram of a paramagnetic salt for magnetic fields of strength H. States of lower entropy at a given temperature correspond to the magnetic fields of greater strength, i.e., H4 > H3, etc. The process of magnetic cooling consists first of cooling a sample of paramagnetic salt to as low a temperature as possible in the absence of any significant magnetic field (H O 0) shown as state A, Figure 38. This is usually accomplished in a helium cryostat pumped to a temperature T1 of approximately 10K. While maintained at this temperature by the helium bath a magnetic field is introduced into the system which causes the entropy to decrease to state B, Figure 38. In state B the paramagnetic salt is removed from the immediate influence of its cooling bath, usually accomplished by pumping away the helium surrounding the salt, and the magnetic field switched off. With the removal of the field the paramagnetic ions are reoriented to a state of greater disorder in a reversible-adiabatic process with a corresponding flow of work to the environment by virtue of the magnetic rearrangement. The consequence of this is an isentropic drop in energy of the salt to a state of zero magnetic field and lower temperature T, shown as state C, Figure 38. A cryostat for doing this is described by DeKlerk and Steenland (97) and shown in Figure 39. The time required to reduce the magnetic field is about 1 second which may be compared with -9 the spin-spin relaxation time of 10 seconds and the spin-lattice relaxation time -3 of 10 seconds (93, 98). The temperature T to which the paramagnetic salt was cooled is computed from "Curie's Law", C M = - H, (33) or M - ~H. (34)

-101Hi 0 Hp LC w. T Ti Temperature Figure 38. Temperature-entropy Diagram for a Paramagnetic Salt Under the Influence of Different Magnetic Fields. (0)

-102with 9( = C, (35) where M is the magnetization or magnetic moment, C is Curie's constant, H1 is the magnetic field strength and % is the magnetic susceptibility. The magnetic susceptibility is related to the permeability, p, and the magnetic flux intensity, B, by B = pH, (36) and 4~X = x - 1. (37) Substances are classified according to the value of X: diamagnetic for X < 0, paramagnetic for X > O, and ferromagnetic X >> 0. Departures from Curie's Law result from the effects of the shape of the paramagnetic salt and are expressed by the Curie-Weiss Law, C X = T-A (38) where A is the Curie-Weiss constant and is equal to zero for a spherical sample. For this reason spherical samples are used, as in Figure 39, if possible or, if not, the results are corrected to that of a spherical sample. Typical spherical and spheroidal sample tubes are illustrated by DeKlerk (94) in Figure 40. The magnetic temperature computed from Equations (35) or (38) is not a true thermodynamic temperature owing to the emperical constants C and A which do not follow from considerations of the second law of thermodynamics. These temperatures will be denoted as T* and will be related to the thermodynamic temperature T later. For either spherical or spheroidal samples the defined magnetic temperature is written (94) as, T* = C (39) sphere Xsphere

-103high vacuum pump heliumump manometer radiation traps ~ ~ liquid hydrogen ---- hquid helium primary coil secondary coil sample thin wailed foot Figure 39. Typical Le den Demagnetization Cryostat, One Fifth of Real Size. (97)

where Xsphere Xspheroid/[l + (47/3 - ) Xspheroid] (40) 1-e2 a= 4n (1e2) T2e loge ( — 2 ~ (41) and 2 1/2 e = ( 1_e2) (42) E is defined as the eccentricity of the spheroid as outlined by Maxwell (99). The magnetic behavior of four paramagnetic salts showing their conformance with Curie's Law at liquid helium temperatures and higher is given in Figures41 and 42, taken from Kittel (100). At low temperatures saturation effects cause departure from Curie's Law at a certain magnetic field strength, as illustrated in Figure 42. The determination of T'*phere, or T - A, is made by measuring the magnetic inductance produced by the paramagnetic salt in an electrical measuring circuit, as shown in Figure 43. The magnetic inductance is proportional to the magnetic susceptibility, X. The magnetic susceptibility of the salt is determined at 10K and 40K 4 using the 1958 lHe scale and then extrapolated to lower temperatures for use during demagnetization experiments. T*here is a good approximation to T for a few tenths of a degree below 10K. At the lowest temperatures it fails to represent T since for all known salts a maximum in)( has been found which produces a minimum in T* sphere' and higher T* sphere. In this range T* and T may differ by an order of magnitude. sphere sphere The final temperature T*he for a chromium potassium alum as a function of magnetic sphere field strength is shown in Figure 44. In this case a minimum T*nhere is approached asympotically for progressively higher fields. The thermodynamics of a paramagnetic salt indicate (94) that the entropy S is a

-105I igure 40. Typical Paramagetic Salt Samples.( Figure 40. Typical Paramagnetic Salt Samples.(94)

80 70 - 60 50 40 30 - 20 10 o 0 I i.. I 0 20 40 60 80 1000 o 4 T Figure 41. Plot of Susceptibility per gm vs Reciprocal Temperature for Powdered CuS04K2S04 6H2, 100) Showing the Curie Law Temperature Dependence.(lOO)

-1077,00'l I 1 I I l oo~_ t ~ ~S=- 7/2(Gd3+) III _ 6.00 C.o 5 4.00 C: *00 0 10 20 30 40 H/T x 10-3 oersted/deg Figure 42. Plot of Magnetic Moment vs H/T for Spherical Samples of,(I) Potassium Chrominum Alum, (II) Ferric Ammonium Alum, and (III) Gadolinium Sulfate Octahydrate. Over 99.5% Magnetic Saturation Is Achieved for 1.3'K anld About 50,tOOO Gauss. 00) Gauss.' 100)

PRECISION VARABLE MUTUAL INDUCTANCE DETECTOR 0 o o PRIMARY COIL o 0 o o o,' o o o COMPENSATING A-C INPUT o 0 0 0 o o o o o o o o 0 o 0 o O o o o 0o o o SPECIMEN o oSECONDARY o O ~dK PARAMAGN ETIC o o o o o 0 Electrical Circuit for Magnetic Thermometer Figure 43. Electrical Circuit for Magnetic Thermometer.(40)

-109function of T and H. Thus, S = f (T, H). (43) This may also be written (94) Tds = C dT dH (44) H )dT()H__ where H = ()H T)H (45) For the isentropic demagnetization, state B to C, Figure 38, then equation (44) indicates that H4 T aM d~ T-T1 = T H (46) 0 H T (46) Since, (aM/aT) < 0 by Equation (33), the temperature T will always be less H than T1 for adiabatic demagnetization processes. The relation between T*phere (to be identified as T* hereafter) and the thermo sphere dynamic temperature T is determined from the thermodynamic definition T d= Od (47) dSRev This may also be written T = (d/dT*) H 0 (48) (dS/dT*) H 0 Now (dQ/dT*) may be found from heating experiments in which dT* = d T (49) For a detailed account of the thermodynamic analysis of paramagnetism the work of DeKlerk (94) or Garrett (93) should be consulted.

-110-.10.09.08 Chromium potassium 0D alum 1.07 - Cr K(SO4)2-12H20.0 X Ca E.03.025.04 C o00.020 20,000 25,900 10,000 15,000 20,000 25,000 Initial magnetic field Hi in oersteds Figure 44. DeKlerk's Results in the Adiabatic Demagnetization of (96) Chromium Potassium Alum. (Initial Temperature = 1.170K)

-111in which, m is the mass of a sample of paramagnetic salt and h is its enthalpy per unit mass. The quantity (dS/dT*)H=0 is determined from a series of demagnetization experiments from T1, Figure 38, and a number of different magnetic fields such as H1, H2, H3 and H4. The entropies corresponding to the isotherm T1 and the various fields is determined from Equation (44) as H a~~~~~~~~~M ~(50) S(HT1) - S(O,T1) = f (ar ) d The corresponding temperatures are then computed from Equation (46). This gives a curve of S vs. T* for a zero field (H - 0). From this the slope (dS/dT*) may be derived and T computed from Equation (48). The relationship between T and T* for several paramagnetic salts is shown in Figures 45 and 46.

-1121.0 0.8 - 0.6 GADOLINIUM SULFATE 7/X 0.4 0.2 E MANGANESE 04T m AMMONIUM SULFATE'E 0.08 0.06 --'0.06~/' IRON AMMONIUM O~0.04.0 ALUM 0 E 0'04 I/ / 0.02 / /. (CR/i3 AL) / / POTASSIUM ALUM 0.01' I _ 0.01 0.02 0.04 0.06 0.1 0.2 0.4 0.6 1.0 Figure 45. Deviation of Curie Temperature T* from Thermodynamic Temperature for Several Paramagnetic Salts.(40)

-113S L)2 2' 0.001 0.01 0.1 1 10 T Figure 46. Entropy of Potassium Chrome Alum as a Function of Absolute Temperature and of T*.(93)

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