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Variable temperature dewar for infrared absorption studies MacPherson, Ronald William 1965

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A V A R I A B L E T E M P E R A T U R E D E W A R F O R I N F R A - R E D A B S O R P T I O N S T U D I E S by R O N A L D W I L L I A M M A C P H E R S O N B . Sc. , University of B r i t i s h C o l u m b i a , 1964 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in the Department of P h y s i c s We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A D e c e m b e r , 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o lumbia, I agr e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s representatives„ I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date January 12, 1966 i i ABSTRACT A variable temperature dewar has been designed and built at the solid state laboratory of the University of British Columbia for infra-red absorption experiments at low temperatures. The dewar features a rotatable sample holder which brings any one of three samples into the light beam for comparison measurements. A removable tail section permits easy access to the samples. The dewar has been successfully operated over the 4°K to 50°K range. It can maintain sample temperatures to within 1°K for periods up to two hours. Approximately ten minutes is required to achieve any desired temperature in the operating range. The construction and calibration of a copper vs. constantan thermocouple is described. The thermocouple was used to measure the sample temperatures to an accuracy of about 1°K. Temperature variations of a few tenths of a degree could be detected. As a demonstration of the dewar's capabilities, an experiment was performed to measure the line width dependence of hole transitions from the ground state of the P3/2 band to the first excited state in the Pj^band in boron doped silicon. The experiment showed that the line broadening starts at about 20°K and increases at higher temperatures. Several methods of improving the performance of the dewar are suggested. i i i T A B L E OF CONTENTS Abstract ' . . ii Table of Contents i i i List of Tables v List of Figures vi Acknowledgement viii Section I: Introduction 1 Section II: Proposed Solution to the Problem 2 Section III: Theory and Construction of the Dewar , 1. Heat Flow 3 2. Design and Operation of the Dewar 9 Section IV: The Measurement of Temperature 1. Errors in Thermometers 13 2. Types of Thermometers 14 3. Construction of the Thermocouples 21 4. Calibration of the Thermocouples 24 Section V: Experimental Results 1. Temperature Measurement 28 2. Running Time 30 3. Absorption Experiment 35 Section VI: Conclusions 1. Performance of the Dewar . . . . 38 2. Suggested Improvements 39 iv Notes 40 Bibliography 47 Appendices: A. Heat Flow in the Stainless Steel Tube between the Sample Holder and the Helium Container on Cooling 50 B. Heat Flow in the Stainless Steel Tube Between the Sample Holder and Helium Reservoir on Applying a Constant Supply of Heat at the Sample Holder 54 C. IBM Computer Programme for the Thermocouple Calibration 57 V LIST OF T A B L E S Table I. Some Approximate Characteristics of the Most Widely Used Classes of Thermometers 15 Table II. Inhomogeneity of Thermocouple Voltages Obtained from Drip Tests 19 Table III. Estimated Calibration Error s in Temperature Determination with A E = 9 microvolts at 4. 2 ° K 27 vi ,LIST OF FIGURES Fig. 1 : Sample Holder and Heater 4 Fig. 2 : Bar in which Heat is Flowing 5 Fig. 3 : First Dewar Model 7 Fig. 4 : Variable Temperature Dewar 8 Fig. 5 : Modified Helium Container 10 Fig. 6 : Thermal Conductivity of Stainless Steel 12 Fig. 7 : Variation of Indicated Temperature of a IN 144 Diode on Thermal Cycling Between Liquid Nitrogen and Room Temperature 17 Fig. 8 : Thermocouple Construction 23 Fig. 9 : Correction Curve for the Copper vs. Constantan Thermocouple 29 Fig. 10 : Heater Power Required for a Specified Operating Temperature 31 Fig. 11 : Sample Temperature over a Period of Several Hours During a Typical Experiment 33 Fig. 12 : Dewar Running Time as a Function of Operating Temperature 34 Fig. 1.3 : Observed Internal Level Absorption of the Boron Doped Silicon 36 Fig. 14 : Temperature Dependence of Line Width at Half Maximum Absorption . . . 37 vii Fig. 15 : FORTRAN Source List 58 Fig. 16 : Sample Computer Input and Output 59 v i i i A C K N O W L E D G E M E N T The w r i t e r i s grateful for the assistance provided by the following persons: Dr. J . W. B i c h a r d , under whose supervision the work was undertaken, for his guidance throughout the experimental work and i n the preparation of this thesis; Dr. J . B. Brown, Dr. P. W. Matthews and Dr. T. Timusk for their many helpful suggestions; Mr. W. M o r r i s o n for his construction of the dewar; and Mr. R. R. Parsons for his assistance i n performing and evaluating the line width experiments. A V A R I A B L E T E M P E R A T U R E DEWAR FOR INFRA-RED ABSORPTION STUDIES Section I: Introduction Many experiments performed in the so l i d state laboratory require the cooling of samples to low temperatures. Often a wide range of temperatures i s needed, for example, f r o m 4°K to room temperature. In many cases the temperatures obtained by pumping on l i q u i d nitrogen (63°K to 77°K) , l i q u i d oxygen (54°K to 90°K), liq u i d neon (c. 27°K) , l i q u i d hydrogen (10°K to 20°K), and l i q u i d helium (1°K to 4.2°K) are sufficient. Dewars for cooling samples to these temperatures are r e l a t i v e l y simple. The specimens are mounted on the side of a container which is f i l l e d with the appropriate coolant. However , if temperatures outside the vapour pressure ranges of the available l i q u i d coolants are required, other methods must be used. These methods usually maintain the sample at a higher temperature than that of the surrounding bath. The temperature i s maintained by introducing a small heat leak to the sample to balance the loss of heat f r o m the sample. This supply of heat i s controlled manually or automatically depending on the p a r t i c u l a r requirements of the experiment. Only liquids oxygen, nitrogen, and helium are available i n the so l i d state laboratory at the U n i v e r s i t y of B r i t i s h Columbia so there i s a large gap of temperatures between 4°K and 55°K which must be bridged. E x p e r i -ments are now needed which require sample temperatures in this range. A special dewar built to meet these temperatures must be capable of maintaining the temperature of the samples to within +_ 1°K for periods up to two hours, n. The dewar must have windows for an i n f r a - r e d beam, 2 access for several electrical leads, and provision for changing samples during the course of an experiment. .Furthermore, the samples must be maintained in a vacuum, since the experiments performed are mainly absorption experiments and the presence of liquid coolant around the specimens may greatly affect the results. A means of determining the sample temperatures to within 4^ 1°K must also be found. Section II: Proposed Solution to the Problem The University of British Columbia solid state laboratory has three dewars which satisfy all the requirements except the ability to provide sample temperatures in the 4°K to 55°K range. These dewars are metal and were designed for mounting two samples at one time. Each sample may be brought into the light beam by rotating the sample mount through a 90° angle. The dewars consist of a jacket containing liquid nitrogen to act as a radiation shield for an inner liquid helium container onto which the samples are mounted. The infra-red light passes through ports of suitable alkali halides or like material, held in a removable tail section to permit the mounting of the samples. The radiation shield jacket and the helium container are suspended by stainless steel tubing inside an outer vacuum jacket which is evacuated to c. 10"^ mm. Hg to eliminate heating of the samples and the cold reservoirs by conduction and convection. ,Covar seals are mounted in the outer jacket to provide vacuum tight electrical connections between the samples and the outside. 3 The variable temperature feature was introduced by modifying the design of these dewars. Instead of mounting the samples directly on the helium container , the sample holder was effectively isolated from the helium by a length of stainless steel tubing. A heater, built into the sample holder, then raised the temperature of the samples to any desired value in the 4°K to 55°K range. This temperature was maintained by supplying just enough heat with the heater to balance that lost by conduction along the stainless steel tube. The heater consists of about thirty turns of two mil diameter enamel covered Advance wire wound on the three quarter inch dameter top of the specimen holder. ^  The total resistance is approximately 400 ohms and varies less than 10% over the 4° K to 300°K range. Each of the two terminals for the winding consists of a single turn of #22 B.S. plastic covered copper wire wound on each side of the heater. The ends are twisted together to make a snug fit and then soldered. The electrical connections were made to these joints. The winding and the terminal wires were given a coating of Eastman 910 adhesive to hold them in place. Figure 1 shows the construction of the holder. The sample holder was also modified to hold three samples at one time. A 60° rotation of the helium container brings each sample in turn into the light beam. Section III: Theory and Construction of the Dewar 1. Heat Flow The transfer of heat,from the sample is primarily through the 4 SECTION A A' •625"x 28 T.P.I, (to fit SS tube) 30 Turns 2 mil D. Advance Wire Heater ~7212 / FT. 'i \v»>j»rA^r„„A\ Indium Metal Pad Terminal Wires SCALE 2x FULL SIZE MATERIAL: COPPER Figure I Sample Holder and Heater. 5 process of conduction in the stainless steel since the effects of radiation and convection are made negligible by means of the vacuum and the radiation shield. To a good approximation, the conduction of heat from the sample to the helium bath along the stainless steel tubing is a one dimensional heat flow problem. The following derivation of the heat flow equation is based on the arguments of Sokolnikoff and Redheffer. 2 Consider a section cut from an insulated,uniform bar by two parallel planes AX units apart as shown in Figure 2. 0 T T + AT Jl Figure 2. Bar in Which Heat is Flowing The temperature at x is T and at x + Ax is T f AT. From experiments and elementary considerations, it is known that the rate of heat flow perpendicular to these planes is approximately given by, Rate of heat flow = KA _AT (1) A x where A is the cross sectional area of the rod and K is the thermal conductivity of the material. In general, K is a function of T. Equation (1) is rigorously true in the limit that AT and Ax approach zero. If ^> is the density of the rod and c is the specific heat capacity, the amount of heat in the volume bounded by the two planes is cpATAx, where T is the average temperature over the length x to x + Ax. The increase i n the a m o u n t of h e a t c o n t a i n e d i n the v o l u m e b o u n d e d b y t h e s e p l a n e s d u r i n g the t i m e f r o m t to t •+ A t i s g i v e n b y c o A T ( x , t + At) A x - c ^ A T ( x , t ) AX a n d t h i s i s e q u a l to the net f l o w of h e a t i n t o the v o l u m e i f no h e a t i s g e n e r a t e d o r l o s t i n s o m e w a y i n t h i s r e g i o n of the r o d . T h a t i s , cpAT ( x , t 4 A t ) A x - c p A T ( x , t ) A x = KA AT ^ A x H e n c e , At - KA A T A t . x + A x A x l x cp T ( x , t + A t ) - cpT(x . t ) ~^ A t * In the l i m i t that A t , A x , a n d AT, a p p r o a c h z e r o , ( K~C)T\ K AT - K AT 1 Ax x f Ax Ax | x A x ( C O T ) X (2) If K, p , .and c a r e c o n s t a n t , e q u a t i o n (2) r e d u c e s , to the d i f f u s i o n e q u a t i o n d t X K e c (3) 2- 3 w h e r e C< i s c a l l e d the t h e r m a l d i f f u s i t i v i t y of the m a t e r i a l . T h e f i r s t m o d e l of the v a r i a b l e t e m p e r a t u r e d e w a r h a d a s t a i n l e s s s t e e l tube t h e r m a l l y c o n n e c t i n g the s a m p l e h o l d e r to the l i q u i d h e l i u m c o n t a i n e r a s s h o w n i n F i g u r e 3. T h e s a m p l e h o l d e r i s e q u i v a l e n t to a f i n i t e heat r e s e r v o i r at the e n d of the s t a i n l e s s s t e e l t u b e . I n i t i a l l y , the w h o l e s y s t e m i s at s o m e t e m p e r a t u r e T j . ^ W h e n the e n d of the tube a t the h e l i u m c o n t a i n e r i s s u d d e n l y r e d u c e d to 4°K b y f i l l i n g the h e l i u m r e s e r v o i r w i t h l i q u i d h e l i u m , the t i m e a n d s p a t i a l d e p e n d e n c e of the t e m p e r a t u r e , T , of the tube i s g i v e n b y E q u a t i o n (4). T = 2(Tj - T Q) oo L sin(^x/£) exp -(o< f 9/£) 2t + T Q (4) The ^ are solutions of tan jg1^ = £/bj9^  , where ^ is the length of the stainless steel tube and b is the ratio of C|^ , the heat capacity of the sample holder, to C, the heat capacity per unit length of the stainless steel tube, Figure 3. First Dewar Model Stainless steel tube Sample holder Note that as t goes to infinity, T approaches T Q (as expected) , and that after a sufficiently long time, T - T Q is approximately proportional to exp -(t/'tr) , where - ( tf/cX^)^ is the "relaxation time" for cooling. Substitution of the physical values for the dewar gives a value of about one hour for "fc. For T to approach to within 1°K of T Q requires a time of the order of 4"fc . A long cooling time of this order was observed in the first experiments and seriously reduced the running time for the dewar since the total cold time is limited to 6 or 8 hours before all the helium boils off. 8 to He return line 0.125 SS. thin wall tubing Transfer Siphon Sprays LHe over S.S. Tube to fill He Container first SEE ® Covar Seat Reference Junctions Vacuum Jacket Radiation Shield Jacket SS Tube Sample Radiation Shield Alkali Halide Window to Vacuum Pump Helium Container Heater ^Infrared Beam Port Sample Holder SCALE ~ % FULL SIZE Figure 4 Variable Temperature Dewar. 9 2. Design and Operation of the Dewar. To avoid the difficulty of a long cooling time, the helium container was modified after an idea of Timusk.^ A long stainless steel tube was axially inserted into the cylindrical one litre reservoir. The top of the tube was left open to the helium container and the lower end, which extends four inches below the bottom of the container , was closed off with a copper plug. This end of the tube was threaded to accept various copper sample holders. A pad of indium, placed as shown in Figure 1 , between the sample holder and the copper plug assured a good thermal contact. In operation, a special transfer siphon, illustrated in Figure 4 , filled the reservoir with liquid helium. When the reservoir was full, the helium overflowed into the stainless steel tube to cool the sample holder rapidly to 4°K. The intermediate temperatures between 4°K and 5 5 ° K were reached by warming with the electric heater wound on the top of the sample holder as shown in Figure 1 . After the liquid helium has boiled away from the inside of'the tube , the sample holder is effectively isolated from the reservoir by the high thermal resistance of the stainless steel tube. Thus this system provides isolation of the sample holder from the liquid helium and at the same time permits a good initial thermal contact between the sample holder and the coolant without the use of any thermal switches. Figure 5 illustrates the modified helium container and the long stainless steel tube. This should be compared with the original idea shown in Figure 3 . The complete drawing of the 10 dewar is given in Figure 4. Figure 5. Modified Helium Container o — I x JL — After the liquid helium has been transferred, the entire system is at an initial temperature of T Q. The temperature of the stainless steel tube and the sample holder remainsthe same after the heater, at x = is turned on until all the coolant has boiled out of the tube. At this instant, denoted by t * 0 , the energy from the heater goes into warming the sample holder. The time and spatial dependence of the temperature of the stainless steel tube is given by Equation (5) , where Q is the power g input to the sample holder, starting at t = 0 . T = T Q Hr (Q x /KA) oo •+2(Q^/KA) ^ sin ( g„x/ I ) s j n Bn exp -(CX&./P, ) Zt (5) ( k + (3, Note that Equation (5). has the same time dependence as the cooling case; that is, the relaxation time "t = ( C. /o<|3 ) * s s a m e . There is some control over the time required to reach an equilibrium temperature He •Long stainless steel tube -Sample holder 0 0 0 11 since Q may be made large at first, then reduced as the temperature approaches the desired value. The temperature pf the sample holder is T( £,t) of Equation (5). The high thermal conductivity of the copper sample holder assures a low temperature gradient over the samples. Unfortunately, while the temperature dependence of p for solids is negligible, the temperature dependence of K and c are not, but depend strongly on T. Approximate temperature dependences of K and c for 2 3 1 ( metals at low temperatures are 1/K = AT + B/T and c = c QT respectively. Figure 6 gives the temperature dependence of the thermal conductivity of the stainless steel used in the dewar. ^  There are approximate methods 12 for solving Equation (2) for variable K and c. These are tedious, usually require the use of computers, and do not necessarily provide much more needed insight into the problem. The parameter f is still approxi-mately given by ( t /otpj^ and it is seen that making c<? = K/ p c large will reduce the time required to reach equilibrium. However , the equilibrium temperature is given by T » T Q + (Q x/KA) 1 3 (6) so that any attempt to reduce t b y making K large requires a large Q to maintain high temperatures and consequently a higher rate of helium consumption. Since the time to reach equilibrium is somewhat controllable without changing K, a sample to bath material'with low thermal conductivity was chosen to minimize the helium consumption. Stainless steel has an average conductivity of 38 mw/cm °K over the 4°K to 6.0°K range compared with about 10,000 mw/cm °K for copper and so satisfies this requirement well. 1 4 12 0 10 20 30 4-0 SO 60 70 80 .90 TEMPERATURE (°K) F i g u r e 6. Ther m a l Conductivity of Stainless Steel. This i s a composite curve drawn f r o m the data of P o w e l l and B l a n p i e d ^ f o r steel types B. -Stainless (7.9% N i , 18.9% C r , 1% T i , 0.7% S i , 0. 1% C, Austenite grains about 0. 1 mm across), E s t i 303 (18% C r , 9 % N i , 0.15%C), E s t i 347 ( 1 8 % C r , 10% N i , 0.5% Nb, 0.08% C), P.Z.J. 304 (18. 68% C r , 8 . 8 4 % N i , 1.12%Mn, 0.43% S i , 0.06% Cu, 0.05% C, 0. 0 3 % N, 0.0 2 % P ) , and P. Z. J. 347 (17.88% C r , 1 0 . 2 8 % N i , 1.24% M n . , 0.85%Nb, 0.57% S i , 0.26% C r , 0.06% C, 0.03% N, 0.02% P). 13 F o r s m a l l variations i n the preset temperature T, the parameters K and c are roughly constant so that Equation (6) may be applied to determine a stability c r i t e r i o n . A controlled temperature change of less than A T at the sample holder requires that AQ < (KA AT/£). Hence for a heater resistance of R ohms and a heater current of I amperes, A l </( KAR AT/£)'or a percentage regulation of A I/I < / A T / T ' . Therefore, to obtain a regulation of +1°K at 4°K requires I be kept constant to within +50% while at 50°K I must be kept constant to within +_ 1.4%. This has been obtained by using a stable current source. Section IV: The Measurement of Temperature 1. E r r o r s i n Thermometers In a l l types of thermometers e r r o r s may re s u l t f r o m two major sources. The f i r s t , e r r o r s due to ca l i b r a t i o n and interpolation, w i l l be discussed for the individual types of thermometers. The second source of e r r o r s re s u l t s f r o m the problem of bringing the thermometer to the same temperature as the object being measured. Heat i s transmitted to thermometers by solid conduction i n the supports or by radiation. The radiation heating may be reduced by using radiation shields. In addition, heat flow down the e l e c t r i c a l leads to the thermometer can be significant. This i s p a r t i c u l a r l y true for the resistance type of thermometer where the high e l e c t r i c a l conductance leads are unavoidably accompanied by high thermal conductance. Heating duertoconductance i n the supports and e l e c t r i c a l leads may be 14 minimized by bringing the supports and leads to approximately the temperature being measured for some distance away f r o m the sensing element. In a l l types of resistance thermometers, joule heating must 15 be kept to a minimum. Matthews,et a l . have studied heat leaks due to e l e c t r i c a l leads and the joule heating of such leads when ca r r y i n g current. 2. Types of Thermometers. The measurement of temperature requires a thermometer of s m a l l s i z e , rapid response, and accuracy and r e p r o d u c i b i l i t y to within +1°K after thermal cycling to room temperature. In addition, the thermometer should be ea s i l y c alibrated, introduce a negligible amount of heat to the sample during the process of measurement, and operate i n the vacuum of the dewar. A good thermal contact between the sample and the thermometer must be re a d i l y established if r e l i a b l e r e s u l t s are to be obtained. Most temperature measurements are made with resistance thermometers , thermocouples, or vapour pressure thermometers although some more recent developments are p r o m i s i n g . ^ Table I i s a summary of the approximate c h a r a c t e r i s t i c s of the more common 17 thermometers. B u l k i n e s s , slow response time, and l i m i t e d ranges make the vapour pressure types unsuitable for this application. Copper resistance and platinum resistance thermometers have the disadvantages of low r e s i s t i v i t y , low sensitivity below 20°K, and 15 variations in the resistance-temperature relationships between individual 1 8 thermometers at low temperatures. The low resistivity requires the use of long thin wires which results in a bulky, delicate resistor coil with the associated problems of supporting and insulating it. Table 1. Some Approximate Characteristics of the Most Widely Used .Classes of Thermometers. ^ Best Best, > Response Relative — — — ( a ) C rr:\ Range Reproducibility Accuracy Time _Size_j ' (°K) (°K) (sec.) Type (OK) Resistance Thermometers Platinum 10--900 io--3_ 10" -4 io- 2-10" 4 0 . 1-10 3 Carbon 1--30 10" -2_ io--3 10 -2-10 -3 0 . 1-10 2 Germanium 1--100 10--3. 10" -4 10 -2-10 -3 0 . 1-10 2 Copper (c) 20--350 10 -3 IO"3 4 t Thermocouple s 1 (d) (e) Gold-Cobalt; copper 4--300 10" -1 _ 10" -2 c . 0.1 1 or les s Constantan; copper 20--600 1.0' -1 _ 10 -2 c . 0.1 1 or less *J Vapour Pressure «) Helium 1--5 io--3_ 10 -5 .002 0 . 1-100 4] Hydrogen 14--33 c. 10 -3 .02 0 .1- 100 4 I Nitrogen Oxygen 63-126 54-155 10-2-10^ 10-2-10"3 02 02 0.1-100 0.1-100 (a) Including non-reproducibiLity, calibration errors, and temperature scale uncertainty. (b) From 1 (smallest) to 4 (largest). (c) Data for copper resistance thermometer was obtained from Dauphinee^ et al. (d) Bare (e) Down to a few mils diameter. (f) 1 cm 3 and up. 16 The most widely used carbon resistor in thermometry is the Allen Bradley resistor. These resistors deviate very little from the relation log R +. K/log R = A + B/T. 1 9 The principal disadvantages are that their calibration is not reproducible on thermal cycling and they are insensitive above c. 30°K. Obtaining a good thermal contact and electrical insulation between the resistor and the sample is also a significant problem. Of the semiconductor materials, germanium has received the most 20 attention. Germanium resistance thermometers are now available 21 commercially. They are rugged and. reproducible , but must be calibrated against a primary standard since no simple mathematical expression exists which fits the data well over extended temperature ranges. The appreciable magnetoresistance of germanium is also a disadvantage if measurements must be made in a magnetic field. 2? Barton has shown that junction diodes and transistors can be adapted for thermometry for temperatures in the liquid helium to room temperature range. An experimental check of such devices showed the reproducibility to be poor. The reproducibility of the temperature indication was determined by immersing a diode or transistor, suspended on thin wire leads, in liquid nitrogen. After thermal equilibrium was established a measurement was taken with the device still in the nitrogen. Between each measurement, the diode or transistor was warmed to room temperature with a stream of warm air , then quickly dropped back into the coolant. Figure 7 shows the results of 21 such cycles. The vertical 17 0 2. 4- 6 © 10 IZ /4 /6 /8 20 NUMBER OF CYCLES F i g u r e 7. V a r i a t i o n of Indicated Temperature of a 1N144 Diode on Thermal Cycling Between L i q u i d Nitrogen and Room Temperatures. The v e r t i c a l bar indicates the e r r o r due to the resolution of the measuring device. 18 bar in the figure indicates the error in the measurements due to the resolution of the measuring instrument. For the particular diode used in obtaining Figure 7, the reproducibility of the device was +2°K. Other diodes and some transistors that were tested showed similar variations. Such a reproducibility on thermal cycling was considered inadequate for 23 use in the dewar. However, Cohen,et al. reported that their specially constructed GaAs mesa diodes were reproducible to 0.01°K on thermal cycling between liquid nitrogen temperatures and room temperatures. Piezoelectric thermometers based on the temperature variation of 24 the resonant frequencies of quartz crystals show promise. These thermometers have large sensitivity, no self heating errors, and are amenable to digital readout and display. The linear response is limited to restricted temperature regions (e.g. from 20°K to 40°K or from 50°K to 70°K etc.). Accordingly, it is necessary to use multiple sensors to cover wide temperature ranges. The best choice of thermometer for this problem appears to be the 25 thermocouple. In general, thermocouples are extremely local, have a very small size and heat capacity, generate a negligible amount of heat during the measuring process, have a.-rapid response time, permit the use of a single thermometer for the entire range of temperatures, and are 26 reproducible to +0.05°K with precautions. The small voltages produced may be somewhat of a disadvantage but they can be measured with available potentiometers such as the Leeds and Northrup Model 7622 used here. 19 A serious disadvantage is that they are susceptible to errors due to parasitic e.m.f.s. caused by inhomogeneities in the materials. Table II shows the experimental values of spurious voltages obtained by R. L. Powell, 27 et al. The voltages were measured across the ends of various samples of wires as the samples were dipped into liquid nitrogen and liquid helium. 28 The procedure recommended by Powell, et al. to minimize the stray voltages due to inhomogeneities is to maintain the reference junction temperature as closely as possible to the sensing junction temperature and to avoid taking the wires through regions where the temperature differs greatly from that of either junction. Table II. Inhomogeneity of Thermocouple Voltages Obtained from Dip Tests Samples * Bath Temperatures - 300°K 76°K - 300°K Voltage Max. (microvolts) Av. Voltage ( Max. microvolts) Av. 1. Cu 4. 5 2.5 2. 0 0.8 2. Cu 1.8 0.7 1.0 0.3 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. .(I960); (5) Gold-cobalt, Bar 5 , 36 A..w.g. (1958); (6) "Normal" silver, 36 A.w.g. 20 Gold -2.1 atomic % cobalt vs. copper and constantan vs. copper thermocouples are perhaps the most common types used at low temperatures. The former has a larger thermoelectric power than the constantan vs. copper type but tends to be less homogenious than the latter. Constantan vs. copper permits good homogeneity and the effects of the high thermal conductivity of copper can be reduced by using fine wires and by thermally anchoring the leads to the reservoirs before they reach the junctions. The high thermal conductivity of copper at low temperatures affects the temperature of the junctions because the junction temperature is determined partly by the heat conducted down the wires and partly by the heat conducted through the contact between the junction and the object whose temperature is being measured. Transition metal impurities have a large effect on the thermoelectric power of copper. A. V. Gold, et al.^9 have shown that variations in the thermoelectric power as high as six microvolts per degree at 10°K may occur among samples of high purity copper. This effect occurs only for temperatures up to about 20°K so that temperature measurements more accurate than +_1°K below 20°K are difficult. The thermocouple advantages of small size, rapid response, wide temperature range for a single unit, reproducibility on thermal cycling, and the requirement of having to determine the temperature to within only -H°K has made the copper vs. constantan thermocouple the choice ofthermometer for the dewar. 21 3. Construction of the Thermocouples. Two thermocouples were constructed. Each one consisted of about two feetOf.#30B. S. thermocouple grade constantan wire and four feet of on #37 B. S. copper w i r e . P r i o r to being used, the wire samples were dip tested i n li q u i d nitrogen for stray e.m.f. s. due to inhomogeneities. The copper wires produced no measurable voltages ( i . e . l e s s than one microvolt) even when strained by kinking and bending and the constantan wires produced approximately two m i c r o v o l t s , but only after severe straining. Unstrained samples, which produced no stray voltages when tested, were used i n the construction of the thermocouples. Figure 8 i l l u s t r a t e s the construction of one of the thermocouples. Timusk suggested sandwiching the temperature sensing junction between two sapphire wafers with a f i l l i n g of indium metal to provide e l e c t r i c a l insulation and, at the same time, a good thermal contact with the sample 31 holder. The junction was f i r s t soldered together with ordinary tin-lead radio solder then'soldered" to the sapphire wafers to make the sandwiches The "tinning" of the sapphire was accomplished by f i r s t cleaning the wafer with absorbent cotton saturated i n toluene, then by immediately rubbing indium metal onto the surface with a low power soldering i r o n just hot enough to melt the indium. The wires leading to the junction were fed through two feet of spaghetti tubing and emerged at the reference junction. The reference junction was prepared by soldering with radio solder as was done with the sensing junction but it was not ..-.enclosed i n a sapphire sandwich. The leads f r o m the reference junction were contained in 22 spaghetti tubing and were terminated at the covar seals leading to the outside of the dewar. The thermocouples were installed i n the dewar by soldering the reference junctions to the outside of the radiation shield jacket near the bottom as shown i n F i g u r e 4. A phosphor bronze plate bolted to the jacket shielded the junctions from room temperature radiation. The leads to the sensing junction were wrapped around the jacket for about nine inches then taped to the bottom and wound once around the sample s h i e l d before passing through a slot i n the neck to reach the stainless steel tube. Two or three turns of the leads around the stainless steel tube before mounting the sensing junctions on the sample holder permitted a 1209 rotation of the helium container for sample changing without damage to the leads. The lead wires f r o m the reference junctions to the outside were wrapped once around the radiation shield jacket then taped to the inside of the vacuum container before being terminated at the covar seals which in turn were connected to the external binding posts numbered 5, 6 and 7, 32 8. Coaxial cable patch cords provided the connection to the Leeds and Northrup potentiometer. The junctions formed at the binding posts, covar, seals and potentiometer terminals produced no stray e.m.f. s. if they were a l l at the same temperature. E.m.f. s. of the order of ten m i c r o v o l t s were observed when li q u i d nitrogen was poured over these connections but these voltages disappeared after the terminals were allowed to reach thermal eq u i l i b r i u m with the room. Mechanical movement of the terminals produced no noticeable effects. 23 RADfO SOLDER- JUNC7VOM COMPL £ TE THERMOCO UpL E SAPPHIRE SA PPHIRE 5ANDW/CH FIGURE 8. THERMOCOUPLE CONSTRUCTION. 24 4. Calibration of the Thermocouples Calibration of the thermocouples was accomplished by first mounting the sensing junctions directly onto the copper plug in the bottom of the stainless steel tube with a phosphor bronze clip. A mixture of vacuum grease and fine grit silver powder was applied to improve the thermal 33 contact between the plug, sapphire sandwiches, and clip. Liquid oxygen was used in the radiation jacket to provide a reference temperature of 90. 18°K at 760.0 mm Hg pressure. Oxygen was chosen over liquid nitrogen for a reference coolant since the latter tends to become contaminated with oxygen from the air and the slightest trace of such contamination greatly affects the boiling point of the nitrogen. Even the liquid oxygen will acquire traces of nitrogen with similar results but since nitrogen boils at a lower temperature than oxygen, it is believed that contamination of the oxygen by nitrogen is less than the contamination of nitrogen by oxygen. The thermocouples were calibrated at six points; five in the 90°K to 73°K range and one point at 4°K. The point at 4°K was made with liquid helium and the others by pumping on liquid oxygen in the helium container. The pressure over the oxygen was held constant to within 35 1+ mm Hg by a Cartesian Diver manostat and the temperature was 36 determined from a vapour pressure table. A modified version of the "standard" temperature-e.m.f. function table of Powell,et al. 3? was used, with a correction curve obtained from the calibration, to determine the temperatures. The correction curves 25 were made by plotting A E , the difference between the observed and the modified standard table e.m.f. , against the modified table e.m.f. at the six c a l i b r a t i o n points. The value of A E shown by such a co r r e c t i o n curve 38 was subtracted f r o m the observed e.m.f. to give the true e.m.f. reading. A table based on an a r b i t r a r y reference point i s easily made by shifting the standard table voltages by an amount equal to the difference in e.m.f. between the tabulated and the actual reference temperatures. Hence , the modified table was made by subtracting the reference voltage (949.9 mi c r o v o l t s for liquid oxygen at 90. 18°K) f rom a l l the values i n the standard table. D a i l y variations i n the atmospheric pressure altered the boiling point of the liquid oxygen and therefore caused the reference voltage to change.'. These small changes were of the order of one or two m i c r o v o l t s . Corrections were made in the modified table to account for such shifts in the reference voltage. If these corrections were not made , e r r o r s larger than 1°K resulted for temperatures below 10°K. S i m i l a r precautions were taken when the temperatures were determined during the temperature measuring experiments. A ca l i b r a t i o n of two thermocouples was attempted. Good thermal contact with the copper plug was achieved with only one of them. F o r the successful thermocouple, deviations f rom the modified table were random and less than two mi c r o v o l t s i n the 73°K to 90°K range. Since the uncertainties i n the deviations were also about two mi c r o v o l t s and the average deviation was nearly zero, the thermocouple followed the modified 26 table to within measureable accuracy. At 4°K the deviations f rom the table were -8 and -10 m i c r o v o l t s on two separate ca l i b r a t i o n experiments and 4 to 9 mic r o v o l t s on several later temperature measuring experiments when the thermocouple was mounted on the sample holder. The large deviations at helium temperatures may be attributed to two things: the thermal contact between the junction and the copper plug was poor during \ the ca l i b r a t i o n experiments at 4°K but good at the higher temperatures, and second, the thermocouple m a t e r i a l s ( p a r t i c u l a r l y the copper) had some anomaly at the lower temperatures. The anomaly may be due to im p u r i t i e s in the m a t e r i a l s or to strains produced by a tighter mounting of the junction to the sample holder than to the copper plug. It was assumed that the points obtained during the ca l i b r a t i o n experiments were i n v a l i d . The points obtained from the temperature experiments were accepted as being a better estimate of the temperature-voltage function. Since the anomalies usually occur below 20°K, it was assumed that no c o r r e c t i o n to the table was necessary down to 20°K and that a linear anomaly exists f r o m 20°K down to 4°K. This c o r r e c t i o n curve i s given in F i g u r e 9. If this assumption i s not approximately v a l i d ( i . e . if the dashed curve of F i g u r e 9 i s more correct) , e r r o r s of up to 1. 5°K may result for temperatures near 20°K. The e r r o r s decrease rapidly at the higher temperatures. An estimate of the c a l i b r a t i o n e r r o r s i n the determination of temperature is given in Table III. F o r this table it was assumed that A E was nine mi c r o v o l t s at 4°K. In this case i f , for example, the temperature were determined to be 20°K, the actual temperature could be i n the range f r o m 18. 5°K to 20.2°K. 27 Table III Estimated Calibration Errors* in Temperature Determination With A E - 9 microvolts at 4. 2°K. Temperature (°K) Positive Error (°K) Negative Error(°K) 4 1.0 0. 0 6 0.6 0.9 8 0.4 0.7 10 0.3 0.9 15. 0.2 1.3 20 0.2 1.5 25 0 . 1 1.4 30 0. 1 1.0 40 0. 1 0. 7 50 0.08 0.5 60 0.07 0. 2 *These estimates include errors caused by the +1 microvolt uncertainty in reading the potentiometer. The calculation to convert the observed thermocouple voltages into temperatures is somewhat tedious since the reference voltage and the helium point voltage vary slightly from one experiment to the next. Details of a computer programme , which does this calculation, are given in Appendix C. The programme also provides a calibration table for the individual thermocouple for each experiment that is performed. This calibration table is calculated from Equation (6) which uses the correction curve of Figure 9. 28 ,E(T) = E Q ( T r e f ) - 'Eo(T) JE 0 ( 4 0 K ) - Eo(Tref) + E(4°K)][E o(20OK) - E 0(T)] if T 4 2 0 ° K and E o(20° K) - E 0(4°K) = Eo( Tref) " Eo(T) if T > 20°K (6) The observed thermocouple voltage at temperature T is E(T). The standard table voltage is E Q(T) , and the reference junction temperature is T r e£. The calibration takes into account variations in the reference voltage and the helium point. For the temperature measuring experiments , the calibrated thermo-couple was secured to the sample holder with a metal plate bolted to one of the sample mounts. Vacuum grease with a suspension of silver powder was used to assure a good thermal contact. Section V: Experimental Results 1. Temperature Measurement. The calibration results have been discussed in the preceding section. During the temperature measuring experiments with the same sample holder, the termocouple voltage was reproducible to within two microvolts at 4°K. A shift of three microvolts was observed when a new sample holder was installed. These variations were taken into account by re-adjusting the 4°K calibration point on the correction curve to correspond to the value for the particular experiment. Although the absolute temperature was determinable to only about 1. 5°K in the worst places, fluctuations in temperature smaller than approximately 0. 1°K could be 29 HO -V o v. K o HI o o +5" 0>'r 0 O X X 9-© & Ye -5 (79°A-) _ J B (Z0°K) (66°r0 (SD°f<) (33°J<) (4°K) J 1 I l I t i L 0 ZOO 400 600 800 /OOO MODIFIED TABLE l/OLTAGE (yU I/) F i g u r e 9. C o r r e c t i o n Curve for the Copper vs. Constantan Thermocouple. The solid line i s the co r r e c t i o n applied to the modified table to obtain the voltage-temperature re l a t i o n for the thermocouple. The paucity of points i n the 300 to 950 mi c r o v o l t region makes the dashed line also a p o s s i b i l i t y for the curve. Reasons for choosing the solid curve are given i n the text. The c i r c l e d points were obtained on the f i r s t c a l i b r a t i o n experiment, the crossed points on the second/ The squared point represents two points obtained with the new sample holder. The triangles (There are two points at A E = 6 microvolts) were obtained with the old holder. In the 0-300 mi c r o v o l t region the mean co r r e c t i o n , A E , i s 0.077 mic r o v o l t s with a standard deviation of 2.4^ volts. The numbers i n parentheses are approximate temperatures corresponding to the table voltages. The v e r t i c a l bar represents the uncertainty i n measuring ,AE. 30 detected above 25°K, and variations of 0. 1°K to 0.9°K could be observed in the 25°K to 4°K range. 2. Running Time. If the only means of heat loss from the sample is that due to conduction along the stainless steel tube, then the power required to maintain an equilibrium temperature T is found by integrating. Equation (1) over the length of the tube. i.e. Q = — \ K(T)dT (7) £ J T 0 « The required power is Q, the temperature of the helium reservoir is T Q, A is the cross sectional area of the stainless steel tube, is its length, and K(T) is the thermal conductivity of the steel. A numerical integration of the thermal conductivity from Figure 6 was used to calculate the required heater power for a given temperature. A plot of this power , which has been corrected for the effect of convection of the helium in the tube , against the operating temperature is given in Figure 10 along with the experimental values of Q for two heaters on different sample holders. The large discrepancy between the experimental and theoretical points indicates that the sample holder was losing energy by some other means than by conduction along the steel. However, the evaporation rate of the liquid helium shows that the energy is eventually absorbed by the helium. The slope of the experimental curve is about 13 mw/°K. Thus the heater power must be held to within + 13 mw to maintain the temperature to 31 Figure 10. Heater Power Required for a Specified Operating Temperature. The solid curve is based on an integration of the thermal conductivity given in Figure 6 (See text, p. 30). The circled and crossed points are experimental values for the old and new heaters respectively. The vertical bar represents the range of error in the heater power. The dashed line through the experimental points.isnearly linear with a slope of 13 mw/°K. 32 within + 1 K. This was done by measuring the input power with Avometers which could show variations in current and voltage of about 0.5% of the full scale readings. Figure 11 shows the temperature of the sample holder over a period of several hours during a typical experiment. The numbers with the arrows indicate the heater power in milliwatts and the time at which the value shown was applied. The spike at fifteen minutes into the running time occurred when the helium boiled out of the stainless steel tube. The heat of vapourization of liquid helium at 4°K and one atmosphere pressure is 3060 joules/liquid litre. After the helium has left the steel tube, up to 0.74 litres of liquid remain in the helium container. Approxi-mately 630 mw-hour of energy is required to boil off this amount of helium. The helium evaporates at a rate of about 0.13 litre/hour when no heat is supplied by the heater. This evaporation is due to radiation and heat conduction down the electrical leads and steel supporting tubes and is equivalent to a heat input of 110 miEiwatts. If this heat leak is assumed to be constant at all temperatures, then the total heat input into the helium reservoir at an operating temperature T is Rate of heat input = Q(T) + 110 mw. Hence the running time before all the helium evaporates is Running time = 630/(Q(T) + 110) hours, (8) where Q(T) is the power required to maintain the temperature T. This calculated running time is plotted in Figure 12 along with the experimental values. Note that the experimental running times are about ten minutes 5c <fc I UJ -J 60 so 4 5 \-40 3 0 | -2 5 2 0 Z5- -/0 -H*. 60A/£ I 0000 O O o O O O © ^ t 0 r t o / 2 3 4-RUNNING TIME (HOURS) Figure 11. Sample Temperature Over a Period of Several Hours During a Typical Experiment. The numbers with the arrows indicate the heater power in milliwatts and the time at which the values shown were applied. This spike at fifteen minutes into the run occurred when the helium boiled out of the stainless steel tube. 34 Fi g u r e 12. Dewar Running Time as a Function of Operating Temperature. The s o l i d curve i s the running time calculated f r o m the known amount of helium left i n the container and the power input at each temperature. The crossed points are the experimental values. The c i r c l e d points are the experimental extended run times. The dashed curve represents the total expected running time. 35 shorter than the calculated times. This i s because the experimental values do not include the ten minutes required to reach equilibrium at the desired temperature. Fortunately the running time could be extended by decreasing the power to the heater after a l l the l i q u i d helium had evaporated. The temperature was maintained in this way unt i l the helium container and the gas inside i t absorbed enough heat to rai s e their temperature above the operating temperature. In p r a c t i c e , this amounted to an extra thir t y to forty minutes. Experimental values of this extended running time are also shown i n F i g u r e 12. The dashed curve i s an estimate of the total running time that may be expected when the heater i s gradually powered down after the helium boils off. 3. Absorption Experiment. 39 I n f r a - r e d Experiments performed by M r . R. R. Parsons i n the soli d state laboratory at the U n i v e r s i t y of B r i t i s h Columbia required the maintaining of samples at temperatures i n the 4°K to 50°K range. P a r t of the investigation involved the study of the temperature dependence of the i n f r a - r e d absorption line widths i n p-type s i l i c o n . The line studied i s due to hole transitions f r o m the ground state of the P3 / 2band to the 40 f i r s t excited state of the P j / ^ band i n boron-doped s i l i c o n . Two samples of various impurity concentrations and one i n t r i n s i c sample were mounted on the sample holder at one time. The i n f r a - r e d absorption spectra of the e x t r i n s i c samples were compared at various temperatures with the spectrum of the i n t r i n s i c sample to determine the 36 tZO 81.1 82.2 823 82A 82.S 8U 8Z7 ©2-8 823 8&0 83.1 INCIDENT PHOTON ENERGY (MlLLIELECTRON VOLTS) Figure 13. Observed Internal Level Absorption of Boron Doped Silicon. The numbers on the curves refer to run and sample numbers. The impurity concentration was 2. 7x10^ impurities /cm^. 37 F i g u r e 14. Observed Temperature Dependence of Line Width at Half Maximum Absorption. 38 absorption due to the transition. The resulting absorption curves are shown i n F i g u r e 13. In F i g u r e 14, the observed line width at half maximum i s plotted against the sample temperature. The onset of the broadening appears to occur at about 20°K, a temperature which has been unavailable with the older dewars. Section VI. Conclusions. 1. Performance of the Dewar. The variable temperature dewar has been successfully used for i n f r a - r e d absorption experiments i n the 4°K to 50°K region. The temperature dependence of the width of an absorption line due to i n t e r n a l l e v e l transitions i n boron-doped s i l i c o n was studied to demonstrate the capabilities of the dewar. The observed temperature dependence i s given i n F i g u r e 14. No attempt i s made here to explain the mechanism 41 of this transition. The results are quoted m e r e l y to show the usefulness of the dewar f o r i n f r a - r e d absorption experiments i n the 4°K to 50°K range. The desired temperature could be obtained within ten minutes of the start of a run and could be maintained f o r periods of up to one and one-half to two hours. The heater power req u i r e d to maintain the operating temperature was much l a r g e r than the power calculated f r o m heat flow considerations. No explanation f o r this discrepancy could be found. The running time, however, agrees very w e l l with the values calculated f r o m the known i n i t i a l amount of helium i n the dewar and the heater power required f o r each temperature. 39 The temperatures of the samples were measured with a copper vs. constantan thermocouple. The greatest uncertainties in the temperature determination were about +_ 1. 5°K and occurred in the region around 20°K and below. The conversion of the thermocouple voltages to the correspon-ding temperatures was mads b/ihe University's IBM 7040 computer. The liquid helium required for each run was from three to four litres. This amount is comparable with the coolant requirements of older, fixed temperature dewars in use in the laboratory. 2. Suggested Improvements. The running time may be increased by a factor of about two by reducing the wall thickness of the stainless steel tube supporting the sample holder. This reduces the power required to maintain a given temperature and hence reduces the helium consumption. An electronic thermostat for the heater would free the operator from 4Z manually controlling the temperature and give better stability. Ferrie describes a simple thermostat that may be adapted for use with the dewar if a suitable thermistor can be found. The determination of the temperatures could be made more accurate by obtaining a few calibration points in the liquid hydrogen region (10°K to 20°K). The use of finer wires of thermocouple grade material in all partsof the thermocouples should also improve the accuracy of temperature determination. 40 NOTES 1. The Advance wire i s available f r o m the D r i v e r H a r r i s Company, H a r r i s o n , New Jersey. 2. Mathematics of Ph y s i c s and Modern Engineering, New York, M c G r a w - H i l l Book Co. , Inc. , 1958, p. 455. z 3. The constant o< was called the thermal d i f f u s i t i v i t y of the m a t e r i a l by K e l v i n and its thermometric conductivity by C l a r k Maxwell. See H, S. Car slaw and J . C . Jaeger , Conduction of Heat in Solids, London, Oxford Un i v e r s i t y P r e s s , [l959] , 2nd edition, p. 9. 4. The temperature T j i s usually the precooling temperature of about 80°K. 5. Equation (4) i s derived i n Appendix A . 6. "Optical Absorption and Luminescence of the (X-Centre i n K B r ," J . Phys. Chem. Solids, v o l . 26 (May 1965), p. 850. 7. A rough calculation indicates that the helium gas inside the tube contributes a heat l o s s , due to convection and conduction, of about 20% to 100% that due to the steel. The greatest contribution occurs at the lower temperatures. F o r the method of calculation, see Mark W. Zemansky, Heat and Thermodynamics, New York, M c G r a w - H i l l Book Co. , Inc. , fourth edition, 1957, p. 91. 8. Equation (5) i s derived i n Appendix B. 9. Compare Equations (4) and (5). A l s o note the s i m i l a r i t y of the x-dependent terms. 41 10. Zemansky, Heat and Thermodynamics, pp. 374 and 267 respectively. 11. R. L. Powell and W. A. Blanpied, "Thermal Conductivity of Metals and Alloys at Low Temperatures" , National Bureau of Standards  Circular 556, (September 1, 1954) , pp. 43-44. The steels listed in Figure 6 have a similar composition and nearly the same thermal conductivity in the regions in which the data of Powell and Blanpied overlap. Since the Atlas Steel type 304 (Cr 18.00 - 20.00%, Ni 8.00 - 11.00%, Mn 2.00% max, C 0.08%) used in the dewar has a similar composition, it was assumed that its thermal conductivity was close to the values given by the composite curve of Figure 6. Additional data on the Atlas steel may be found from Atlas Steels Limited, Technical Data Atlas Stainless Steels, Welland, Ontario, [l958*], pp. 10-11. 12. The following references discuss a few different methods of solving the heat flow problem. Further references may be obtained from their bibliographies. Car slaw and Jaeger, Conduction of Heat in Solids, George Merrick Dusinberre, Heat Transfer Calculation by Finite  Differences , Scranton, Pennsylvania, International Textbook Co. , 1961, Theodore R. Goodman, "Applications of Integral Methods to Transient Nonlinear Heat Transfer" , Advances in Heat Transfer, New York, Academic Press , 1964, vol. 1, pp, 52-122, Herbert Schenk, Fortran Methods in Heat Flow, New York, Ronald Press Co. , ^963^j , and 42 P. J. Schneider, Conduction Heat Transfer, Cambridge, Massachu-setts, Addison-Wesley, 1955. 13. Equation (6) also applies to the case in which K,p , and c are not constant. This is obtained by integrating twice the steady state from of Equation (2) i^.e. = ^ ) • T h e n t h e K o £ Equation (6) becomes the average value of K between the temperatures T Q and T. 14. The average thermal conductivity of the steel was calculated by numerically integrating the curve of Figure 6. The value for copper was estimated from data of Zemansky, Heat and Thermody- namics , p. 85. 15. "A Study of Heat Leaks into Cryostats Due to Electrical Leads", Cryogenics, vol. 5 (August 1965) , pp. 213-215. 16. Such new types as the p-n junction diode thermometer and the piezoelectric thermometer are discussed on pages 16 and 18 respectively and references are given in notes 22 to 24. 17. R. J. Corrucini, "Temperature Measurements in Cryogenics" , Advances in Cryogenic Engineering, ed. K. D. Timmerhaus, New York, Plenum Press, 1963, vol. 8, p. 316. 18. Temperature Its Measurement and Control in Science and Industry, Charles M. Herzfeld, ed. , New York, Rheinhold Publishing Corporation, [l962^ , vol. 3:1, pp. 329-380. These pages include several articles and references on platinum resistance thermometers. For information on copper resistance thermometers see T. M. 43 Dauphinee and H. Preston-Thomas, "A Copper Resistance Temperature Scale", Rev. Sci. Instr vol. 25 (September 1954), pp. 884-886. 19. Zemansky, Heat and Thermodynamics, p. 394. 20. S. A. Friedberg, "Semiconductors as Thermometers", Temperature  Its Measurement and Control in Science and Industry, H. C. Wolfe, ed. , New York, Rheinhold Publishing Corporation, [l955], vol. 2, cited in P. Lindenfeld, "Carbon and Semiconductor Thermometers for Low Temperatures", Temperature , Herzfeld, ed. , £l962], vol. 3:1, pp. 399-405. Friedberg has a good discussion on germanium thermometers up to 1954 with many references. Lindenfeld includes several other more recent references on the subject. 21. Minniapolis-Honeywell Regulator Co. , Pyrometer Sales Division, 151 East Hunting Park Ave. , Philadelphia, Pennsylvania, and Texas Instruments, Inc. , Recorder Products Dept. , P.O. box 6027, Houston 6, Texas, cited in Lindenfeld, "Carbon and Semiconductor Thermometers", Temperature, [l963] , vol. 3:1, p. 405. 22. "Measuring Temperature" , Electronics, vol. 35 (May 1962), pp. 38-40. 23. "GaAs p-n Junction Diodes for Wide Range Thermometery" , Rev. Sci. In str. , vol. 34 (October 1963), pp. 1091 -1093. 24. T. Gorini and S. Satori, "Quartz Thermometer", Rev. Sci. Instr. , vol. 33 (August 1963), pp. 883-884, and W. H. Wade and L. J. Slutsky, "Quartz Crystal Thermometer", Rev. Sci. Instr. , vol. 33 (August 1962), p. 212, cited in T . M. Flynn, et al. , "An Improved Cryogenic Thermometer", Advances in Cryogenic Engineering, New York, Plenum Press, 1963. 25. Temperature, vol. 3:2, pp. 3-294, has several papers on many applications of thermocouples to thermometry. 26. Guy Kendall White, Experimental Techniques in Low Temperature  Physics , London, Oxford Univer sity Pre ss , 1959 , p. 133. 27. "Low Temperature Thermocouples" , Cryogenics, vol. 1 (March 1961), pp. 139-150. 28. Ibid. p.150. 29. "The Thermoelectric Power of Copper", Philosophical Magazine, vol. 5 (1960), p. 765. 30 The constantan wire was from the Leeds and Northrup Co. spool No. 74718-9, Date 11-13-51, "Constantan for Copper-Constantan Thermocouples", 1938 calibration. The copper wire was "Polysol" insulated and was manufactured by The Canadian Wire and Cable Co. 31. Letter to the writer, 24 July, 1965. 32. Nine binding posts are provided on the outside of the dewar. Each is coloured in sequence to correspond to the first nine numbers of the standard resistor colour code numbers, 0 (black) through 8 (grey). An eight terminal feed through type covar seal is also mounted in the vacuum jacket to allow access with continuous leads. 33. Subsequent experiments showed that the calibration could be accomplished when the thermocouples were mounted directly onto 45 the sample holder as they are when in use. 34. White, Exp. Tech.in Low Temp. , p. 102. 35. Ibid., p. 205. 36. Ibid. , pp. 104-105. 37. "Low Temperature Thermocouples", Cryogenics, vol. 1 (March 1961) pp. 146-148. 38. This method of calibration is outlined in Handbook of Chemistry and  Physics, forty-fir st edition, Cleveland, 1954, Chemical Rubber Co., p. 2612. 39. PhD. candidate in the Department of Physics at the University of British Columbia. 40. Solomon Zwerdling, et. al. , "Internal Impurity Levels in Semiconductors: Experiments in p-Type Silicon'.', Phys. Rev. Letters, vol. 4 (February 15, I960), pp. 173-176. 41. The forthcoming PhD. thesis of Parsons will study the mechanisms involved in these transitions. 42. Ronald G. Ferrie, "Thermostat Operates with 0. 01°C Differential" , Electronics, vol. 37 (October 5, 1964), pp. 65-66. 43. Erik V. Bohn, The Transform Analysis of Linear Systems, Reading, Massachusetts, Addison- Wesley, [l963] , Chap.11. 44. Ibid, p. 246. 45 Ibid, p. 246. Equation (11-11). 46. Sokolnikoff and Redheffer , Math, of Phys. and Mod. Eng. , pp. 459-462. 46 47. These relations were obtained from Bohn, Transform Analysis , pp. 244-245, in the special case in which,L, the inductance per unit length and ,G, the leakage conductance per unit length are both zero. 48. Conduction Heat Transfer , p. 128. Equation 3.13 (7). 49. "Low Temperature Thermocouples", Cryogenics, vol. 1 (March 1961), pp. 146-148. 47 B I B L IOGRAPHY BOOKS Ame r i c a n Institute of P h y s i c s . Temperature Its Measurement and Control i n Science and Industry. New York, Rheinhold Publishing Corporation, vols. 2 [1955] , 3:1 and 3:2 1962 . (QC 271 A 6) Atlas Steels L i m i t e d . Technical Data A t l a s Stainless Steels. Welland, Ontario, [1958] . Bohn, E r i k V. The T r a n s f o r m A n a l y s i s of Linear. Systems. Reading, Massachusetts, Addison-Wesley, [1963] . (QA 401 B 6) Car slaw, H. S. and J . C. Jaeger. Conduction of Heat i n Solids. London, Oxford Un i v e r s i t y P r e s s , [1959] , Second Edition. (QC 321 C 28 1959) Dusinberre, George M e r r i c k . Heat Transfer Calculations by F i n i t e Differences. Scranton, Pennsylvania, Internation Textbook Co. , [1961] . (QC 320 D 79 1961) Handbook of Chemistry and P h y s i c s , f o r t y - f i r s t edition, Cleveland, 1954, Chemical Rubber Co. Irvine .Thomas F. and James P. Hartnett, eds. Advances i n Heat Tra n s f e r . New York, Academic P r e s s , 1964. v o l . 1. (QC 1 A 29) Organick, E l l i o t I. A F o r t r a n P r i m e r . Reading, Massachusetts , Addison-Wesley, [1963] . Schenk, H i l b e r t . F o r t r a n Methods i n Heat Flow. New York, Ronald P r e s s Co., [1963] . (QC 320 S 34 1963) Schneider, P . J . Conduction Heat T r a n s f e r . Cambridge, Massachusetts, Addison-Wesley, 1955. (QC 321 S 37 1955) Sokolnikoff, I. S. and R. M. Redheffer. Mathematics of P h y s i c s and  Modern Engineering. New York, M c G r a w - H i l l Book Co. , Inc. , 1958. (QA 401 S 64 1958) White, Guy Kendall. Experimental Techniques in Low Temperature P h y s i c s . Oxford, Claredon P r e s s , 1959. (QC 278 W 4 1959) Zemansky, Mar k W. Heat and Thermodynamics, 4th ed. New York, M c G r a w - H i l l Book Co. , Inc. , 1957. 48 A R T I C L E S Barton, L. E. "Measuring Temperature." E l e c t r o n i c s , v o l . 35 (May 1962), pp. 38-40. (TK 1 E7) Cohen, B. G.,.W. B. Snow, and A. R. T r e t o l a , GaAs p-n Junction Diodes for Wide Range Thermometry." Review of Scientific  Instruments, v o l . 34 (October 1963) pp. 1031-1093. (Q 184 R 5) Dauphinee , T. M. and H. Preston-Thomas. "A Copper Resistance Temperature Scale. " Review of Scientific Instruments, V o l . 25 (September 1954), pp. 884-886. (Q 184 R 5) F e r r i e , Ronald G. "Thermostat Operates with 0.01°C D i f f e r e n t i a l . " E l e c t r o n i c s , v o l . 37 (October 5, 1964), pp. 65-66. Gold, A. V., D. K. C. MacDonald, W. B. Pearson, and I. M. Templeton. "The Thermoelectric Power of Pure Copper." Philo s o p h i c a l Magazine . v o l . 5 (July I960), pp. 765-786. G o r i n i , T. and S. S a t o r i , "Quartz Thermometer." Review of Scientific  Instruments, v o l . 33 (August 1962), pp. 883-884. Matthews, P. W. , K. T. Khoo, and P. D. Neufeld. "A Study of Heat Leaks into Cryostats Due to E l e c t r i c a l Leads." Cryogenics, v o l . 5 (August 1965), pp. 213-215. P o w e l l , R. L. and W. A. Blanpied. "Thermal Conductivity of Metals and A l l o y s at Low Temperatures." National Bureau of Standards  C i r c u l a r 556; (September 1, 1954). (QC 100 U 555) P o w e l l , R. L., M. D. Bunch, and R. J . C o r r u c c i n i . VLow Temperature Thermocouples." Cryogenics, v o l . 1 (March 1961), pp. 139-150. (TP 490 C 78 v. 1-2) Timusk, Thomas. "Optical Absorption and Luminescence of the c=< - Centre i n K B r . " The Journal of the P h y s i c s and Chemistry  of Solids, v o l . 26 (May 1965), pp. 849-866. Timusk, Thomas. Letter to the w r i t e r , 24 J u l y , 1965. Wade, W. H. and L. S.Slutsky. "Quartz C r y s t a l Thermometer." Review of Scientific Instruments, v o l . 33 (February 1962), pp. 212-213. (Q 184 R 5) 49 Zwerdling , Solomon, Kenneth J . Button, Benjamin Lax, a n d L a u r a M. Roth. "Internal Impurity Levels in Semiconductors: Experiments in p-Type S i l i c o n . " P h y s i c a l Review L e t t e r s , v o l . 4 (February 15, I960), pp. 173-176. 50 A P P E N D I X A: HEAT F L O W IN T H E STAINLESS S T E E L T U B E B E T W E E N THE S A M P L E HOLDER AND H E L I U M CONTAINER ON COOLING. The following i s a l i s t of some of the symbols used throughout this appendix and Appendix B. A Cross sectional area of stainless steel tube. b = C^/C. This has the dimensions of length. C = cpA. Heat capacity per unit length of stainless steel tube. Cj_, Heat capacity of sample holder. c Specific heat capacity of the stainless steel tube. I Laplace transform of the heat current i n the steel. I Q = Q/s Laplace transform of Q. K Thermal conductivity of the stainless steel tube. Length of the stainless steel tube. T Temperature of the stainless steel tube at position x and time t. T Q Temperature of the helium container --usually 4°K. T j I n i t i a l temperature of the stainless steel tube and the sample holder. Q Constant heating power applied to sample holder at time t = 0. R = 1/KA Thermal resistance per unit length of the stainless steel tube, t Time. s Laplace transform parameter. V(x,s) Laplace transform of v(x,t). v(x,t) = T - T j 51 Z Q = R /Cs Transformed c h a r a c t e r i s t i c impedance of the stainless steel tube. Zg = 0. Transformed terminating impedance of the stainless steel at the helium container. Z = 1/CL S. Transformed terminating impedance of the steel tube at the sample holder. o< ^  - K/p c Thermal d i f f u s i t i v i t y of the stainless steel. It i s also equal to 1/RC. "X = N/RCi Assume the problem i s one dimensional and examine the following diagram, r STA/MSSS %re£L rose vfot). HELIUM CONTAINER SAMPLE Z3--O z - % c V / c . D£*/A R TAIL TRANSMISSION LlA/E ANALO& This problem i s equivalent to a tr a n s m i s s i o n line problem with a 43 capacitive load. Consider only the case i n which K, P , and c are constant; then equation (3) applies. V r d t = c x 2 -\z <)' T (3) Since T^ i s constant v(x,t) also satisfies Equation (3), v i z . , c) v c x c ) 2 v (9) 52 The Laplace transform of Equation (9) i s = y 2 V • o-2= s/o<2 = s p c / K = R C s (10) 44 r-. Bohn defines the ref l e c t i o n coeficients I 0 = (Z - Z ) / (Z_ + Z Q) and \^ = (Zg - Z Q)/(Z^-f Z Q) , which i n this case are VT= -1 and \~2 = (1 - b y ) / ( l f by ) . The s o l u t i o n 4 5 to Equation (10) i s V(x,s) = Z Q V g(s) ^ e + |jg e v ' ( H ) z o + z g 1 - r : ^ e - 2 ^ where V (s) = ( T Q - T.)/s i s the Laplace transform of the temperature step function applied at the helium container. Substitution of the values for Zn, Z „ , Z„, | , 1 „ , and V (s) into u JL g ° -C g Equation (11) gives V(x,s) = (T - T ) cosh - x) + b y sinh y ( - x) . — 2 — cosh yje f b y s i n h y i 1 ' s 2 The inverse Laplace transform of Equation (12) , with s = (X p, i s v(x,t) = T 0 - T j f e °< p t [ c o s h / p i / - x) + b ^ s i n h ^ ( ^ - x)] dp (13) 2 77-j J cosh^/p 1^ t bfp1 s i n h / p 1 ^ p~ The integral of Equation (13) most ea s i l y evaluated by the calculus of residues. The poles, p = p n , occur at the solution to ^ t a n ^ = l/b ; p n = - f i j ^ 2 ; n = 0 , l , 2 , (14) After some algebraic manipulation, the temperature i s found to be oo T(x,t) - T p + 2 ( 1 , - Tn ) y sin( & x / l ) exp - « X |3„/i ft (15) Z _ , ( ^ + B i n ^ c 0 s ^ ) 53 When no sample holder i s present, b = 0, Q - 2(n +• 1) ~JT , a oo » n 2 nd T(x,t) = T + 1 (T. - T o -rr 1 ^ V ^ s i n £(2n t- 1) 2n+ 1 TT x 2 X exp !0<(2n f 1)TT Equation (16) i s more readily obtained d i r e c t l y by the solution by 46 series method outlined i n Sololnikoff and Redheffer. At steady state conditions ( i . e . t—> oo ) , T approaches T q as expected. 5 4 A P P E N D I X B: HEAT F LOW IN THE STAINLESS S T E E L T U B E B E T W E E N THE S A M P L E HOLDER AND H E L I U M RESERVOIR ON APPLYING. A CONSTANT S U P P L Y OF HEAT AT THE S A M P L E HOLDER The same assumptions and parameters hold here as i n Appendix A except that v(x,t) = T - T . The following diagram i l l u s t r a t e s the trans m i s s i o n line analog of the problem. fi£LlUM CONTAINER STA'A/l£SS STEEL TU3£ -H5ATZR 2-= % $ Jrv(x^) I SAtAPLB HOLDER DEW A R TAIL 4 7 The relations between the parameters are - R I transmission LINE ANALOG? (a) V ^ x (b) B I •s C V ( 1 7 ) Both I and V satisfy the transformed diffusion equation ( i . e. Equation ( 1 0 ) ) The boundary conditions corresponding to the system i n i t i a l l y at a temperature T q suddenly having a heat input of Q applied at x = £ at a time t = 0 are (a) (b) V(JL.B) = d 0 + I ) / ( C L s) V ( 0,s) (t > 0 ) (for a l l t) ( 1 8 ) The solutions of the transformed diffusion equations are of the f o r m V = V e + v - Y x I = I e + + X ( 1 9 ) 55 Application of Equations (17) and (18b) shows that V = -V J I = I , V + = Z Q i + l and V_ := - Z Q i _ . Therefore, the solutions become V = 2V sinhyx and I = 21, cosh X x , + + The use of Equation (18a) gives I z V+ = 2 _ f . , (20) 2 ( Z Q C s sin l 6 l + cosh ~6l) so that I Z sinh if x V ( z , s ) = ° °- (21) Z C s sinh -yj? + c o s h y>£ o J_i JI T , \ L cosh a n d l ( x , s ) = ° . (22) Z0 C L s sinh 15 JL + c o s h y ^ Substitution of the values for I and Z Q and setting s = o< p i n Equation (22) give s V ( x . s ) = Q JjT1 sinh y/p~! x 4 2. <X C p (b^sinhv/p ,£ + cosh Vp1 X ) (23) The inverse Laplace transform of Equation (23) i s e** P t fp sinh f p 1 x ch> ( 24) (b / ^ s i n h /p~\£ + cosh P Cr-joo The calculus of residues i s used to evaluate the integral of Equation V(x,t) = Q (24). The poles, p = P n , occur at the solutions to Equation (14) and also at J> = 0. 56 The temperature i s then found to be sin pr\ sin ( x / l ) e  4ri ^ <P*+ 8 i n& c o>PJ T(x,t) = T + Qx o KA i. n * 0 A s i m i l a r result i s given without proof i n Car slaw and Jaeger. 48 With no sample holder present b = 0 , p = (2n + | ) TT 2 and T(x,t) oo T o + Q j i KA '—* (2n f l ) 2 r\.= o K A T T ' ot(2n+l)Tn 2£ 1 At steady state conditions ( i . e . t Oo ) , T approaches T +• Qx ° K A as expected. 57 A P P E N D I X C: I B M C O M P UTER P R O G R A M M E FOR THE T H E R M O C O U P L E C A L I B R A T I O N This programme was written i n the F 0 R T R A N IV language for use on the I B M 7040 computer at the U n i v e r s i t y of B r i t i s h Columbia. The F 0 R T R A N l i s t i s given i n F i g u r e 15. The execution of the programme is as follows: (a) The thermocouple-voltage data for 1°K to 100°K from the standard 49 thermocouple table of P o w e l l , et a l . i s read and stored under the variable name C T A B L E ( J ) , J = 1,100. (b) The reference temperature voltage and the helium point voltage for the par t i c u l a r experiment are read according to the format of statement 13. Alphanumeric messages pertaining to the experiment are also read at this point. F o r this reason two data cards should be supplied here even if the second one is blank. (c) The calculation of the ca l i b r a t i o n table i s performed according to Equation (6). The c a l i b r a t i o n table i s printed along with the alphanumeric message read i n step (b). (d) The experimental thermocouple voltages are read one at a time according to format statement 41. The voltages are transformed into temperatures by a linear interpolation of the c a l i b r a t i o n table. The thermocouple voltage and the corresponding temperature are then printed. When a - 1.0 value i s encountered for the thermocouple voltage," control is t r a n s f e r r e d to statement 29 which reads i n the reference voltage and helium point voltage for a new experiment. The programme automatically terminates when no more data i s input. 58 A B a m p l e of the input data a n d the c o r r e s p o n d i n g output i s g i v e n i n F i g u r e 16. The negative sign of the voltages i n the c a l i b r a t i o n table i n d i c a t e s that the c o r r e s p o n d i n g t e m p e r a t u r e i s b e l o w the r e f e r e n c e t e m p e r a t u r e . FIGURE 1 5 . FORTRAN SOURCE L I S T SOURCE STATEMENT $IBFTC MAIN DIMENSION T A B L f c ( 2 0 0 ) J E M P 1 ( 2 0 0 ) . C T A B L E ( 2 0 0 ) . D A T E ( 3 0 )  REAO 1 2 . N , ( C T A B L f c ( J ) , J = l , N ) 12 F O R M A T ( I 5 / ( 5 F 8 . ? ) ) 29 READ 1 3 , R E F V L T , H f c V 0 L T , ( D A T E ( J ) , J = l , 2 5 ) 13 F 0 R M A T ( 2 F 8 . 2 , 1 0 A 6 , A 4 / 1 3 A 6 , A 2 ) 14 DO 10 J = 1 , N T A B L E ( J ) = C T A B L E ( J l - R E F V L T  10 CONTINUE F U D G E = ( T A B L E ( 4 ) + H E V 0 L T ) / ( T A B L E ( 2 0 ) - T A B L E ( 4 ) ) CO 15 J = l , 2 0 T A B L E ( J ) = T A B L E ( J ) + F U D G E « ( T A B L E ( J ) - T A B L E ( 2 0 ) ) 15 CONTINUE PR INT _ 1 9 d C A T E ( J ) , J = l , 1 2 ) , ( D A T E ( J ) . J = 1 3 , 2 5 )  19 F0~RMAT(/ /6X,49HTABLE OF THERMOCOUPLE TEMPERATURE VS VOLTAGE FOR , 1 1 0 A 6 . A 4 / 6 X , 1 3 A 6 . A 2 / / 6 X . 4 ( 1 1 H T E M P E R A T U R E , 3 X , 7 H V 0 L T A G E , 8 X ) / 6 X , 4 ( 1 1 H ( D 2EGREES K ) , 2 X , 9 H ( M U V O L T S ) , 7 X ) / / ) DO 30 K = l , i 0 0 T E M P l ( K ) = K 30 CONTINUE '_ PRINT 2 5 , ( T E M P I ( J ) . T A B L E ( J ) . T E M P I ( J + 2 5 ) . T A B L E ( J + 2 5 ) , TEMP 1 ( J + 5 0 ) , T A 1 B L E ( J + 5 0 ) , T E M P I ( J + 7 5 ) , T A B L E U + 7 5 ) , J = 1 , 2 5 ) 25 F 0 R M A T ( 4 ( 8 X , F 6 . 1 , 5 X , F 8 . 2 , 2 X ) ) PRINT 57 57 FORMAT Kill) 40 READ 4 1 . V C L T S  4 1 FORMAT(F8 .2 ) I F ( V O L T S ) 2 9 , 4 2 , 4 2 42 M = N - l K=0 DO 52 J = 1 , M K = K+1  I F ( T A B L E ( J ) . L T . ( - V O L T S ) . A N D . ( - V O L T S ) . L E . T A B L E ( J + l ) ) GO TO 55 52 CONTINUE 55 TEMP2=K T E M P 3 " T E M P 2 - ( V C L T S + T A B L E ( K ) ) / ( T A B L E ( K + i J - T A B L E ( K ) ) PRINT 56 ,VOLTS,TEMP3 56 FORMAT(IX,23HP0TENTIOMETER VOLTAGE = . F 8 . 2 . 8 H MUVOLTS,IPX,20HSAMPLE I TEMPERATURE - , F 6 . l , 1 0 H DEGREES K) GO TO 40 99 STOP END 59 tb'NTRY 100 0.17 5.83 19.12 39.4: 1 '.6.SB <n.43 135.6 : s •». 3 ill. 460. i 552.3 6?a.i 707.3 7Sq.fi 3 7'. . 6 1057.3 949.0 964. 618. 619. 622. 627. • 608. ' 612. ' 609. | 549. i -1.0 J 0. 7. 22. 106. 147. 192. '* 3 • 29'j. 360. 494. 567. 6'»3. 723. 5 06. 393. 933. 1076. 964. 66 90 64 1.48 10.26 26.43 1 49.40 78.80 • I4 . 4 1 R5.;> ?02. I 254.7 311.3 '73.0 5 03.5 542.2 659.4 719 .9 323.7 91 0.9 1001.4 1095.2 JECEMBER 2.62 12.92 30.50 54.78 8^ .43 122.2 164.7 212.7 265.7 32 3.'. 335.5 4 5 2.4 5 2 ? . n' 59 /.3 675.3 756.4 84 0.8 928.7 1019.a 1114.4 4.07 15.88 34.84 60.40 92.31 1 '0.3 1 TX # a ?2?.9 2 76.8 3 35 .6 398 .8 466.2 5 3 7.5 612.7 691.2 773.0 858 .1 946.7 1033.5 1133.7 Thermocouple voltage -temperature data from Powell et a l . 4 9 1965 COUPLE 0 PRESS 755.8 RUN 5 (RRP SET 2) Thermocouple voltages for conversion into temperatures Reference voltage, helium point voltage, and alphanumeric message. INPUT DATA TABLE OF THERMOCOUPLE TEMPERATURE VS VOLTAGE FOR OECEMBER 6, 1965 COUPLE 0 PRESS 755.8 RUN 5 (RRP SET 2) TEMPERATURE VCLTAGb TEMPERATURE VCLTAGE TEMPERATURE VOLTAGE TEMPERATURE VOLTAGE ICEGREES K) (MUVCLTS) (CEGREES K) (HUVOLTS) (DEGREES K) (HUVOLTS) (DEGREES K) (MUVOLTS) 1.0 -967.20 26.0 -849.57 51.0 -601.10 76.0 -241.70 2.0 -966.56 27.0 -842.20 52.0 -588.70 77.0 -225.50 3.C -965.4* 28.0 -834.60 53.0 -576.00 78.0 -209.10 4.0 -964.00 29.0 -826.80 54.0 -563.20 79.0 -192.60 5*0 . U - 30.0 -818,70 55,0 -550.20 BO.O -176.00 6.0 -959.81 31.0 -810.40 56.0 -537.10 81.0 -159.20 7.0 -957.11 32.0 -801.90 57.0 -523.80 82.0 -142.40 8.0 -954.03 33.0 -793.20 58.0 -510.30 83.0 -125.30 9.0 -950.56 34.0 -784.30 59.0 -496.60 84.0 -108.20 10.0 -946.7C 35.0 -775.10 60.0 -482.80 85.0 -90.90 11.0 -942.47 36.0 -765.70 61.0 -46B.90 86.0 -73.40 12.0 "'-937.87 37.0 -756.10 62.0 -454.70 97.0 -55.90 13.0 -932.93 38.0 -746.30 63.0 -440.50 88.0 -38.10 14.0 -927.62 39.0 -736.30 64.0 -426.00 89.0 -20.30 15.0 -921.95 40.0 -726.10 65.0 -411.50 90.0 -2.30 16.0 -915.96 41.0 -715.70 66.0 -396.70 91.0 15.80 17.0 -909.62 42.0 -705.10 67.0 -381.80 92.0 v34.00 18.0 -902.95 43.0 -694.30 68.0 -366.80 93.0 52.40 19.0 -895.93 44.0 -683.30 69.0 -351.70 94.0 70.90 20.0 -888.60 45.0 -672.20 70.0 -336.30 95.0 89.50 21.0 -882.72 46.0 -660.80 71.0 -320.90 96.0 108.30 22.0 -876.58 47.0 -649.20 72.0 -305.30 97.0 127.20 23.0 -870.20 48.0 -637.50 73.0 -289.60 9B.0 146.20 24.0 -863.57 49.0 -625.50 74.0 -273.70 99.0 165.40 25.0 -856.69 50.0 -613.40 75.0 -257.80 100.0 184.70 POTENT IOMETER VCLTAGE = 964.00 MUVOLTS POTENTIOMETER VCLTAGE - 618.00 MUVOLTS POTENTIOMETER VCLTAGE = 619.00 MUVCLTS POTENTIOMETER VCLTAGE = 622.00 MUVCLTS . POTENTIflHEJER VCLIA&E..._62_7 ._QQ_MUJU1LIi. POTENTIOMETER VCLTAGE « 6C8.00 MUVOLTS POTENTIOMETER VOLTAGE » 612.00 MUVCLTS . POTENTIOMETER VCLTAGE - 609.00 MUVCLTS POTENTIOMETER VCLTAGE = 549.CO MUVOLTS SAMPLE TEMPERATURE = 4.0 DEGREES K SAMPLE TEMPERATURE = 49.6 DEGREES K SAMPLE TEMPERATURE - 49.5 DEGREES K SAMPLE TEMPERATURE - 49.3 DEGREES K SAMPLE TEMPERATURE - 48.9 DEGREES K SAMPLE TEMPERATURE • 50.4 DEGREES K SAMPLE TEMPERATURE « 50.1 DEGREES K SAMPLE TEMPERATURE = 50.4 DEGREES K SAMPLE TEMPERATURE = 55.1 DEGREES K OUT PUT. .DAT A FIGURE 16. SAMPLE COMPUTER INPUT AND OUTPUT. 

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