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The construction of an adiabatic calorimeter and its use in measuring specific heats Swanson, Max Lynn 1957-02-11

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F a c u l t y o f G r a d u a t e S t u d i e s P R O G R A M M E O F T H E FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of M A X L Y N N S W A N S O N B.A. University of British Columbia M. Sc. University of British Columbia IN ROOM 204. PHYSICAL METALLURGY BUILDING MONDAY, MAY 5, 1958. a t 10:30 a . m. C O M M I T T E E I N C H A R G E D E A N G . M . SHRUM, Chairman W. M. ARMSTRONG B. SAVERY F. A. FORWARD W. O. RICHMOND E. TEGHTSOONIAN G. A. McDOWELL B. MOYLES R. W. STEWART External Examiner: B. N. BROCKHOUSE Chalk River Ont. THE CONSTRUCTION OF AN ADIABATIC CALORIMETER AND ITS USE IN MEASURING SPECIFIC HEATS A B S T R A C T A fluidless adiabatic calorimeter was constructed and was used to measure the specific heats of manganese-aluminum-carbon and manganese-zinc-carbon alloys from -150° to 150°C. In an adiabatic calorimeter, the temperature of a shield sur rounding the calorimeter vessel is kept at approximately the same temperature as that of the vessel, so that the thermal leakage between the two is reduced to a negligible quantity. Thus the ordinary rating period, in which the thermal leakage modulus is calculated, can be eliminated. Since leakage modulus variations are reduced by the adiabatic method, it can be used for large temperature rises, resulting in fast and accurate measurements. The aneroid (fluidless) adiabatic calorimeter eliminates stirring and evaporation errors, and makes possible measurements at ex treme temperatures. The calorimeter consisted of a cylindrical silver-plated copper vessel surrounded by an electrically heated adiabatic shield and an evacuated outer case. A platinum resistance thermometer- heater was used to supply heat to the calorimeter vessel and to measure the vessel temperature. The heat input and the ther mometer resistance were measured by using a potentiometer in conjunction with standard resistances. The thermometer was calibrated by measuring its resistance at - 183, -40, 0, and 100°C. The calorimeter was calibrated from -150 to 150 .^ The accuracy of the calorimeter was approximately 0.5%, the main error arising from the method of measuring the temperature of the calorimeter vessel. The specific heat curves of the single phase magnetic alloys MnsAlC and MruZnC were measured. A second order specific heat anomaly was found , as expected, for the ferromagnetic alloy MmAlC at its Curie point, - 10°C. Although the anomaly was close to the theoretical shape, dropping to zero over only a 10°C range at the Curie point, its maximum height was less than saturation magnetization measurements would indicate. The alloy MmZnC showed second order specific heat anom alies at - 35°C, and at 65°C. This double specific heat anomaly indicates, in agreement with neutron diffraction results, a com plex magnetic behavior for the alloy. Although the high temper ature Curie point anomaly did not have a sharp peak the low temperature anomaly's shape approached that of the theoretical Weiss curve. G R A D U A T E S T U D I E S Field of Study: Metallurgy (Metal Physics) Structure of Metals W. Armstrong Phase Transformations in Metals . . . - W.Armstrong Ferromagnetism H.Meyers Plastic Deformations and Lattice Imperfections H. Meyer Bonding in Metals . . . J. Halpern X-Ray Diffraction . . 1. --- J. Parr Other Studies: Theory of Measurements W.Opechowski Quantum Mechanics G. M. Volkoff Electromagnetic Theory J. Brown Nuclear Physics K.Mann Chemical Physics C. Reid Group Theory - - B.Moyls THE CONSTRUCTION OF AN ADIABATIC CALORIMETER AND ITS USE IN MEASURING SPECIFIC HEATS by MAX SWANSON A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MINING AND METALLURGY We accept t h i s thesis as conforming to the standard required from candidates f o r the degree of DOCTOR OF PHILOSOPHY. Members of the Department of Mining and Metallurgy. The University of B r i t i s h Columbia December 1957-ABSTRACT A f l u i d l e s s adiabatic calorimeter was constructed and was used to measure the s p e c i f i c heats of manganese^aluminum-carbon and manganese-zinc- carbon alloys from -150° to 150°C. In an adiabatic calorimeter, the temperature of a shield surrounding the calorimeter vessel i s kept at approximately the same temperature as that of the vessel, so that the thermal leakage between the two i s reduced to a negligible quantity. Thus the ordinary r a t i n g period, i n which the thermal leakage modulus i s calculated, can be eliminated. Since leakage modulus v a r i a  tions are reduced by the adiabatic method, i t can be used f o r large temperature r i s e s , r esulting i n f a s t and accurate measurements. The aneroid ( f l u i d l e s s ) adiabatic calorimeter eliminates s t i r r i n g and evaporation errors, and makes possible measurements at extreme temperatures. The calorimeter consisted of a c y l i n d r i c a l s i l v e r - p l a t e d copper vessel surrounded by an e l e c t r i c a l l y heated adiabatic s h i e l d and an evacuated outer case. A platinum resistance thermometer-heater was used to supply heat to the calorimeter vessel and to measure the vessel temperature.' The heat input and the thermometer resistance were measured by using a potentiometer i n conjunction with standard resistances. " The thermometer was calibrated by measuring i t s resistance at -183, -40, 0, and 100°C. The calorimeter was calibrated from -150 to 150°C. The accuracy of the calorimeter was approximately 0.5%, the main error a r i s i n g from the method of measuring the temperature of the calorimeter vessel. The s p e c i f i c heat curves of the single phase magnetic alloys Mn 3AlC and Mn3ZnC were measured. A second order s p e c i f i c heat anomaly was found, as expected, for the ferromagnetic a l loy Mn3AlC at i t s Curie point, -10°C. Although the anomaly was close to the theoret ical shape, dropping to zero over only a 10°C range at the Curie point, i t s maximum height was less than saturation magnetiza t ion measurements would indicate. The a l loy Mn3ZnC showed second order speci f ic heat anomalies at -35°C, and at 65°C. This double speci f ic heat anomaly indicates, in agreement with neutron d i f f ract ion resul ts , a complex magnetic behaviour for the a l loy . Although the high temperature Curie point anomaly d id not have a sharp peak, the low temperature anomaly's shape approached that of the theoret ical Weiss curve. In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia', I agree that the 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of Metallurgy _ The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3 , Canada. Date May 7 , 1958 ACKNOWLEDGEMENTS The author i s g r a t e f u l f o r the advice and tec h n i c a l assistance given by his research d i r e c t o r , Dr. H.P. Myers, and by R. Butters and R. Richter. He would also l i k e to thank L. Howe for his preliminary reading of the thesis . The work was carried out with the help of Research Grant 281 provided by the Defence Research Board, and a Studentship from the National ' Research Council. TABLE OF CONTENTS Page I * INTRODUCTION »eoeoo»BeQoeo«»oa«»oaoo«a»»eo»e«*o«»oo*a <>»«••• I 1. Sp e c i f i c Heat Theory . ........ 1 2. The Alloys to be Investigated .....6 . 6 3« Calorimetric Theory 12 I I . THE CONSTRUCTION OF AN ANEROID ADIABATIC CALORIMETER .... 17 2. Materials and Detailed Construction ........."..... 19 I I I . CALIBRATION AND PERFORMANCE 34 1. Cal i b r a t i o n of Resistance Thermometer 34 2. Heat Capacity of the Calorimeter i*2 3. Accuracy of Spe c i f i c Heat Measurements 42 IV© f££SUl*TS o o o o o o o e o o o o o o o o O 4 O O O o o o o o o O O 0 O « o * o e o e « e o o « $2 1. Preparation and Properties of Alloys 52 2. Spe c i f i c Heat Measurements 56 V. DISCUSSION AND CONCLUSIONS 68 1. Discussion of Results 68 2. Conclusions . . . . . . O . . o .6 ..........<>...,............ 68k VI. APPENDICES 69 V I I . BIBLIOGRAPHY 84 FIGURE INDEX Page 1. Perovskite Structure « 0 o a « . . . o o . . . « a a o o o . . 7 2„ Variat ion of Saturation Magnetisation, <S", with Temperature, T , • for Mn—Zn—0 Al loys o « « o o . . . o . . . . a . . . . o . 10 3o Possible Magnetic Structure at 0 ° K of Mn3ZnC . . . . . . . « O 0 11 4o Basic Calorimeter Design . . . . . . . . . . . . . . . . . o o IB 5- The Calorimeter « o . . o » . o . o . » . « • « . « • * » o o 21 6. Adiabatic Shield Control C i r c u i t . . « . . . • . •, '• • '• . . . a o 24 7o Suspension of the Calorimeter . . « . . . . . . . . . . . • a 0 27 8. Circui t Diagrams for Thermometer-Heaters . . . . . 0 o . .e . o 31 9o Calibrat ion Apparatus . o o . . . . . o o . . . . . . . . . • « 37 I D . Heat Capacities of Calorimeter and of Calorimeter plus AXUlffi Tift OOOOQOOOOOOO 0 O « « o o 0 « o « 0 « o 44 11. Variatibh of Thermal Leakage Modulus, k, with Pressure . . . e 47 12. The Variation of Saturation Magnet izat ion,© ' , with Temperature, . ., T, for Mn3AlC . . . . o . . . . o . . . . . . ' » . . . . . . 54 1 3 * The Variat ion of Saturation Magnetization,©" , with Temperature, T, for Mn3ZnC o . o . . . ' . • . . . . . a a • • . * . o a 55 14. Capacity of Mn3AlC plus Calorimeter . . . . . . . . . • • . • a 58 1 5 . Capacity of Mn3AlC . . <> . « o . . o . o . a a • a . . . e . . 59 16. Capacity of Mn3AlC plus Calorimeter (using the stra in- f ree thermometer) . . . . . a a o . o o o . . . . o • a a 60 1 7 . 62 1 8 . Capacity of Mn3ZnC (al loy 2) plus Calorimeter p » . 6 © 64 19. Magnetic Speci f ic Heat of Mn3AlC 66 2 0 . Magnetic Specif ic Heat of Mn3ZnC . . . . < . . . . . . . . . ' . 67 2 1 . Magnetization Curves ', . . . . . . . . . . . . . . . . . . . . 0 63b TABLE INDEX Page I. Measured Capacities of Calorimeter plus Aluminum Oxide 43 I I . Calorimeter Capacity . 45 I I I . Capacities of Mn3AlC plus Calorimeter 57 IV. Capacities of Mn3ZnC plus Calorimeter 63 THE CONSTRUCTION OF AN ADIABATIC CALORIMETER AND ITS USE IN MEASURING SPECIFIC HEATS. I. INTRODUCTION 1, Sp e c i f i c Heat Theory a. General Theory; The object of the present research i s to construct a calorimeter i n order to measure the s p e c i f i c heats of c e r t a i n a l l o y s . From these s p e c i f i c heats, the basic magnetic properties, and consequently the atomic structure of these alloys might be explained. The s p e c i f i c heat of a substance i s defined as C = Sq/dT where Sq i s the amount of heat required to change the temperature of a unit mass of the substance by dT. The s p e c i f i c heat at constant volume f o r simple substances has been calculated by Debye,! the results agreeing w e l l with experiment. I f the s p e c i f i c heat of one substance i s plotted against T/6^ where Qb i s the Debye temperature, a constant dependent on that substance, the r e s u l t i n g curve i s the same as that f o r many other simple substances. / At high temperatures, the s p e c i f i c heat approaches the c l a s s i c a l l y calculated value of s i x c a l . per mole per degree centigrade, i f the electronic s p e c i f i c heat,C e , i s not considered.' However, although the s p e c i f i c heat produced by the electrons* i s n e g l i g i b l e at ordinary temperatures, i t becomes appreciable at high temperatures. Moreover, t r a n s i t i o n metals, which are involved i n the present work, often have a larger C e than simple metals because of the high density of electron levels i n t h e i r u n f i l l e d *d' bands. C e can be of the order of 6 cal./mole/°C. at very high temperatures f o r t r a n s i t i o n metals. At low temperatures the Debye theory agrees with experimental r e s u l t s , predicting that the s p e c i f i c heat w i l l vary as the cube of the temperature. The present research, however, i s concerned with values i n the intermediate temperature range (0.5< T/(=^  ^ 1.5), where the slope of the s p e c i f i c heat curve i s decreasing rather rapidly. b. Sp e c i f i c Heat Anomalies; The present work i s devoted primarily to measurement of s p e c i f i c heat anomalies. They are caused by tr a n s i t i o n s .'.in a substance, which involve a i ; ' . - 3 - change of symmetry i n the structure of the substance. For a f i r s t order t r a n s i  t i o n , the f i r s t order derivatives of the Gibbs function, G, change discontinuously. G = U - TS + PV where U = i n t e r n a l energy S = entropy T = absolute temperature P = pressure V = volume. by d e f i n i t i o n dG = VdP - SdT since dU - TdS - PdV therefore and since -S dS - CpdT T J Consequently, the entropy changes discontinu ously f o r a f i r s t order t r a n s i t i o n (e.g.: melting, vaporization), and the s p e c i f i c heat increases to an i n f i n i t e value at the t r a n s i t i o n point (producing a latent heat). For a second order t r a n s i t i o n , the f i r s t order derivatives of the Gibbs function change continuously and the second order derivatives change discontinuously. Thus a second order t r a n s i t i o n i s accompanied i d e a l l y by a discontinuity i n the s p e c i f i c heat curve. Some examples of second order t r a n s i t i o n are the trans i t i o n s from an ordered to a disordered state, from a superconductor to an ordinary conductor, and from a magnetic to an non-magnetic state. These transitions a l l produce s i m i l a r s p e c i f i c heat anomalies. The t r a n s i t i o n pertinent to the present research i s the magnetic one. The s p e c i f i c heat anomaly res u l t i n g from a ferromagnetic t r a n s i t i o n can e a s i l y be deduced from the theory of Weiss.^ The energy of magnetization per unit mass of a ferromagnetic substance i s where C i s the magnetic moment per unit mass, and the effe c t i v e f i e l d H, H + NI (2) H i s the applied f i e l d , I i s the magnetic moment per unit volume and N i s the Weiss intermolecular f i e l d constant, which i s very large f o r ferromagnetic materials. Since H i s negligible i n comparison with NI, and I =^<T, U = "C Np c der (3) where J> i s the density of the substance. Thus U _ N/><5- U) and the anomalous s p e c i f i c heat of magnetization i s Since t h i s i d e a l tf - T curve i s never achieved and since perfect experimental conditions can never be attained, the dis c o n t i n u i t y has not been observed experimentally. However, i t has been approached. The smallest discovered temperature range through which the s p e c i f i c heat drops near the Curie point 3 has been about seven centigrade degrees, observed by Lapp on n i c k e l . e T Similar anomalies should occur f o r antiferromagnetic and ferrimagnetic substances, since large i n t e r n a l exchange forces must exist i n them, l i k e i n ferromagnetic substances, and the energy of magnetization depends on these - 6 - forces between atoms. Whereas i n ferromagnetics the spontaneous magnetization vectors of a l l atoms l i n e up p a r a l l e l , i n antiferromagnetics the vectors of two or more types of atoms oppose, causing zero resultant magnetization. In f e r r i - magnetics, the vectors also oppose, but do not completely cancel, so that a resultant spontaneous magnetization e x i s t s . The anomalies observed f o r a n t i - ferromagnetic and ferrimagnetic substances at t h e i r Curie points should be just as large as those observed f o r ferromagnetic substances i f the exchange forces are as large, even though the resultant magnetization may be much smaller. 2. The Alloys to be Investigated. One aim of the present research i s to determine the type of magnetiza t i o n of certain substances by investigating t h e i r s p e c i f i c heat anomalies. These substances are single phase alloys of Mn-Al-C and Mn-Zn-C. These alloys have an ordered face-centered cubic 'perovskite' structure and exhibit spontaneous magnetization at room temperature. Because of the great s t a b i l i t y of the ordered structure over a wide range of composition, i t would seem that bond formation, rather than a normal superlattice a t t r a c t i o n , i s responsible f o r the ordering. Of p a r t i c u l a r interest are the a l l o y s Mn3AlC and Mn 3ZnC% whose structures have the maximum ordering (see Figure 1). Mn-rAl-C al l o y s are single phase f o r a region near the Mn 3AlC composition. I f the carbon content i s twenty atomic percent the a l l o y s are single phase f o r 60-69 atomic percent Mn and thus f o r 20-11 atomic percent A l . The l a t t i c e parameter i s 3.869 A f o r Mn3AlC and varies l i t t l e with composition, in d i c a t i n g that the Mn and A l atoms have almost the same diameter. In these alloys the saturation magnetization below the Curie temperature varies with tenroerature i n a normal ferromagnetic manner, but the paramagnetic behaviour above the Curie point indicates ferrimagnetism, or at least a departure from Curie-Weiss behaviour. Neutron Figure Is Peroyskite Structure. d i f f r a c t i o n results indicate that Mn3AlC i s ferromagnetic above l i q u i d helium temperature. As the Mn content i s increased from 60 to 69 atomic percent, the Curie temperature increases from 0°C to 300°Cj, while the saturation magnetization decreases from 1.20 to 0.6 Bohr magnetons per Mn atom. This decrease i n magnetization as the Mn content i s increased past the value f o r maximum ordering (60 atomic percent Mn) can be explained i n one way by assuming that the magnetiza t i o n of the additional Mn atoms (which must replace A l atoms i n cube corner positions) i s a n t i p a r a l l e l to that of those i n face-centered positions. The magnitude of the decrease i n magnetization corresponds to the extra Mn atoms having an approximate effective Bohr magneton value of ^/t4S-4/<B( s e e Appendix I ) . (This, i n c i d e n t a l l y , i s the value of ^M, for manganese i n the Heusler a l l o y s ) . This interpretation could possibly be v e r i f i e d by s p e c i f i c heat measurements. By simple Weiss theory, the height of the s p e c i f i c heat anomaly varies d i r e c t l y as the Bohr magneton number f o r a ferromagnetic substance. Thus, i f the Mn-Al-C alloys were always ferromagnetic, the height of the anomaly and also i t s t o t a l size would become smaller as the magnetization decreased. But f o r a ferrimagnetic a l l o y , the anomaly's height depends on the Bohr magneton number of the separate atoms. Thus, the anomaly height would increase while the magne t i z a t i o n decreased i f the extra manganese atoms had large negative as hypothesized. Moreover, other things being equal, the t o t a l s i z e of the anomaly would increase i f Mn atoms of large negative /J- vere added to the a l l o y (see Appendix I I ) 0 The Mn-Zn-C a l l o y s , l i k e the Mn-Al-C a l l o y s , are single phase over the range of composition from Mn6oZn2oC2o to Mn7 GZn t oC 2 0o The l a t t i c e parameter of Mn3ZnC i s 3.925& a n d i t varies only s l i g h t l y with composition. The Curie temperature varies from about 80°C f o r Mn3ZnC to ~500°C f o r Mn 7 0Zn i oC2o« The var i a t i o n of the saturation magnetization for the alloys with Mn content near 60 atomic percent i s unusual at low temperatures,^ (see Figure 2)„ The magnetiza t i o n has a maximum near -40°C, corresponding to the behaviour at low temperatures predicted by Neel (see Appendix II) for one type of ferrimagnetic substance. The paramagnetic behaviour above the Curie point also appears to agree with that predicted by N£el for ferrimagnetics. He deduced that the inverse of the sus c e p t i b i l i t y , , 1 >= T + 1 - C (see Appendix I I ) X C X » T ™Q The curvature of the resulting (L, T) curve i s concave to the temperature a x i s , rather than convex, as i s usual for a ferromagnetic substance. However, neutron d i f f r a c t i o n experiments carried out at Chalk River by Dr. B„ Brockhouse on the a l l o y of approximate composition Mn3ZnG suggest a different and e n t i r e l y new magnetic concept to explain the magnetic properties of t h i s a l l o y . These experiments indicate that between the t r a n s i t i o n temperature of -AO°C and the Curie temperature,, Mn3ZnC i s a normal ferromagnetic substance with magnetically equivalent manganese atoms. But below the t r a n s i t i o n tempera ture, a complex magnetic structure e x i s t s . The actual structure i s unknown, but ohe i n good agreement with neutron d i f f r a c t i o n and magnetic data i s that proposed by B.N. Brockhouse and H„P. Myers.^ Manganese atoms having magnetic moments of zero, two and three Bohr magnetons are arranged as shown i n Figure 3 fo r the state at absolute zero temperature. The magnetic moments of the 2/*g manganese atoms are opposing, so that the algebraic mean moment of the a l l o y i s l ^ B per manganese atom at 0°K, i n agreement with magnetic measurements. I f the magnetic moments of the manganese atoms above the t r a n s i t i o n temperature were the same as t h e i r arithmetic mean moment i n the low temperature structure, then the ferromagnetic moment extrapolated to 0°K would be 1.66/*go This i s i n good Figure 2 s Variation of Saturation Magnetization, P", with Temperature, T, f o r Mn-Zn-C Alloys Only one- half of the large unit c e l l i s shown; the other half i s s i m i l a r but displaced one-half unit along the x-axis. Only manganese atoms are shown. The moments are oriented along a [ i l l ] diagonal. - 12 - agreement with the experimental extrapolated value of l ,5^Ug. At the t r a n s i t i o n temperature the magnetic moments of the proposed complex l a t t i c e must rearrange themselves to become equivalent. This rearrangement involves a change i n interatomic exchange forces, so that a second order s p e c i f i c heat anomaly comparable to that occurring at the Curie temperature should be observable. In order to understand the form of the experimental magnetization curve f o r Mn3ZnC i n terms of the two magnetic structures mentioned, one may consider that below the t r a n s i t i o n point an opposing sublattice of magnetic moments comes into e f f e c t . The Curie tempera ture of t h i s f i c t i t i o u s l a t t i c e i s the t r a n s i t i o n temperature. Although the cubic structure of Mn3ZnC becomes s l i g h t l y distorted into a face-centered tetragonal structure (c/a = 0.995) near -40°C, t h i s phase change i s gradual, so that no appreciable f i r s t or second order anomaly should r e s u l t . Thus, i f the ferrimagnetic model for Mn3ZnC were correct, the only anomaly which would occur at r40°C would be a small t h i r d order one. I t i s seen that measurement of s p e c i f i c heat anomalies should prove very useful i n checking the v a l i d i t y of magnetic models and thus i n determining basic atomic structures of a l l o y s . 3. Calorimetric Theory' a. General Theory; The s p e c i f i c heat of a substance i s measured by means of a calorimeter, which i s a device into which heat can be introduced and the r e s u l t i n g temperature change measured. Part of the heat brought into the calorimeter raises i t s - 13 - temperature and part i s l o s t to the surroundings. In precise calorimetric measurements, the temperature change of the calorimeter and i t s thermal leakage (the amount of heat l o s t from the calorimeter during the measurement) must be accurately determined. Since the greatest calorimetric error frequently arises from measurement of the temperature change, an accurate thermometer must be used. 8 9 For precise work,, a platinum resistance thermometer * i s most often used. The resistance of pure platinum, varying almost l i n e a r l y with temperature,follows a smooth curve which i s determined by c a l i b r a t i o n at International Temperature Scale f i x e d points. Because of i t s s t a b i l i t y , s train-free platinum does not require r e c a l i b r a t i o n very often. A platinum resistance thermometer, used with a Mueller bridge, w i l l measure temperatures to 0.001°C accuracy i n the intermediate range, For less accurate work, thermocouples may be used; they have the advantage of having p r a c t i c a l l y no heat capacity, but are inaccurate because of thermal gradients. Mercury thermometers, besides having l i t t l e accuracy, can be used f o r only very r e s t r i c t e d temperature ranges, and are not suitable for vacuum apparatus. Most of the types of calorimeter design are the result of thermal leakage considerations. The leakage depends on the thermal head, $ , which i s the temperature difference between the calorimeter and i t s surroundings. The formula usually used to calculate the leakage i s ^  = ^ k t . i s the temperature change i n the calorimeter caused by the leakage f o r an experimental time t . k i s the thermal leakage modulus of the calorimeter. In the ordinary calorimetric method, since the thermal leakage i s r e l a t i v e l y high, i t must be accurately estimated,, To do this,, k i s found by measuring ty't and p f o r a rati n g period immediately after each experimental period, and i t i s assumed to be the same i n the experimental period. In order to - 1 4 - measure the thermal head, p , accurately, the calorimeter vessel surroundings must be at a constant uniform temperature; a water jacket i s often used for t h i s purpose. The temperature change of the r a t i n g period must also be accurately measured. Because the error i n measuring t h i s temperature change must be added to the temperature error of the experimental period, the t o t a l thermometric error i s doubled by the leakage correction. Since the thermal leakage modulus varies with temperature and other conditions, i t must be measured after each experimental period f o r accurate r e s u l t s , unless i t s v a r i a t i o n i s greatly reduced by some experimental arrangement. The vacuum-jacketed calorimeter invented by Dewar decreases the thermal leakage modulus, so that f need not be so precise i n order to accurately estimate t\„ In Joule's method, twin calorimeters are heated the same amount, so that the thermal leakage, temperature, and heat capacity determinations become comparative measure ments, made with d i f f e r e n t i a l apparatus. Other methods, such as the adiabatic method, i n d i r e c t l y reduce the errors involved i n estimating the thermal leakage. b. The Adiabatic Calorimeter In the adiabatic calorimeter, the thermal head i s reduced almost to zero by keeping the temperature of a jacket or shield almost equal to that of the calorimeter vessel. Thus, since the thermal leakage i s greatly reduced, i t i s usually neglected. But because the error involved i n adjusting the jacket temperature f o r adiabatic conditions i s greater than that i n measuring the thermal head, the r e s u l t i n g error i n the thermal head i s greater f o r the adiabatic method than f o r the ordinary method. And so the leakage neglected i n the adiabatic method may be greater than the error i n the leakage calculation f o r the ordinary method. One advantage of the adiabatic method i s that i t reduces the effect of variations i n the leakage modulus, which are a chief source of error i n non-- 15 - adiabatic methods. The smallness of the thermal head reduces the effect of fluctuations i n the leakage modulus i n two ways; F i r s t , the fluctuations during an experiment are caused mostly by convection, which i s d i r e c t l y dependent on the thermal head. Second, any change i n the leakage modulus w i l l cause a corresponding change i n the thermal leakage; but the thermal leakage i s already negligible^ so that any such change i n i t w i l l also be ne g l i g i b l e . Because leakage modulus variations are reduced by the adiabatic method, i t i s especially useful f o r protracted experiments, for which the leakage modulus might o r d i n a r i l y change considerably. Also, although large temperature ri s e s can not be used with accuracy f o r non-adiabatic methods, because of the convection eff e c t s , they can be successfully used with the adiabatic method. Consequently, the important temperature measurement errors can be greatly reduced. A tempera ture r i s e increase from two to ten degrees, f o r example, would reduce the temperature measurement error by four f i f t h s . Moreover, larger temperature rises w i l l permit more rapid measurement of a s p e c i f i c heat curve. The adiabatic method avoids the ordinary rati n g period, i n which the leakage modulus i s found, but a period i s often required f o r which any constant temperature d r i f t (such as caused by the heat of s t i r r i n g ) i s determined. The aneroid, or f l u i d l e s s , adiabatic calorimeter, by eliminating s t i r r i n g , may almost completely eliminate temperature d r i f t . Thus no corrections for leakage need be made, and considerable time and calculations are avoided. One experimental period can follow another without pause for large temperature ranges. Extreme temperatures can be much more e a s i l y attained with an aneroid calorimeter than with one containing f l u i d s . Another advantage of the aneroid method i s that small dimensions (and less dead material) are possible f o r the calorimeter, because of the absence of - 16 - f l u i d c i r c u l a t i o n problems. Smaller dimensions mean faster thermal equilibrium and thus a shorter experimental time (producing less t o t a l thermal leakage), which i s usually an advantage,. However, at very low temperature, since the main heat loss i s by conduction along wires, smaller dimensions are not so advantageous. The main disadvantage of the aneroid calorimeter i s that the temperature equalization must be by conduction alone. In order to speed the conduction: (1) Metal of high conductivity (copper or s i l v e r ) i s used, (2) The heat i s generated uniformly. (3) The thermal head i s kept small, so that temperature differences are not produced by thermal leakage. The adiabatic method i s thus especially useful. Necessary to the aneroid adiabatic calorimeter i s the measurement of the thermal head between calorimeter and s h i e l d . The most convenient way to do t h i s i s with thermocouples. The l a g of the thermocouples should present no problem, since i t i s small and probably constant. I f only a few couples are used, the average of the thermal heads measured may not be the correct average head, unless the thermal equilibrium i s very good f o r both calorimeter and shield. Thus, pains should be taken to insure good thermal equilibrium. The aneroid adiabatic calorimeter was chosen as most suitable for the present work, because of the following reasons! (1) I t can give suitable accuracy (approaching 0.2%)for the temperature range required: -150 to 150°C. (2) I t can give quick and simple measurements, especially i f the ra t i n g period i s eliminated. - 17 - (3) I t can be constructed simply and economically, even though the adiabatic controls introduce some complexity. II, THE CONSTRUCTION OF AN ANEROID ADIABATIC CALORIMETER 1, Design a. General Aims The basic design required was one of s i m p l i c i t y coupled with accuracy, since the simplest construction which w i l l s t i l l give the required accuracy i s undoubtedly the most desirable. The precision desired f o r the calorimeter was approximately 0.5%, with a reduction to 0.2% possible by means of modification. Fast operation of the calorimeter by one person was desirable, f o r convenience as w e l l as economy. The calorimeter must be operable f o r samples i n powder form over the temperature range -150 to 150°C, since the alloys to be measured required t h i s . Also, adaptability to samples of a different form was advantageous, with regard to future use of the calorimeter. b. Basic Design? The basic design"^ of the adiabatic calorimeter i s shown i n Figure 4. The c y l i n d r i c a l calorimeter vessel (2) contains the specimen whose s p e c i f i c heat i s to be measured, and the platinum resistance thermometer-heater (1). The vessel i s f i l l e d with helium f o r good conduction. The thermometer-heater-is used, i n conjunction with standard resistances and a potentiometer, to measure accurately the heat i t has introduced to the vessel and the re s u l t i n g temperature change. The adiabatic shield (3) i s heated e l e c t r i c a l l y so that i t s temperature remains close to that of the. vessel, and thermal leakage may be neglected. The temperature difference between vessel and shield i s measured with thermocouples 1: Thermometer-Heater 2: Calorimeter Vessel 3: Adiabatic Shield Ui Outer Case 5: Brass Ring 6s Evacuation Tube Figure l+i Basic Calorimeter Design connected to a galvanometer., The outer case (4) i s evacuated through the tube (6) so that thermal leakage between the shield and the vessel, and between the shiel d and the case i s reduced. In order to reduce thermal leakage through the leads, they are wrapped around the s h i e l d and the brass r i n g (5) before emerging' through the evacuation tube. 11, 12 2. Materials and Detailed Construction s a. The Calorimeter Vessels ( i ) Theory: To promote fast conduction, the calorimeter vessel was made of copper. By placing the thermometer-rheater i n the center of the vessel, the heat was generated uniformly, and thermal equilibrium was speeded. Also, a central position of the heater made more gradual the temperature change of the vessel surface, so that the adiabatic control was easier. The calorimeter vessel was made an a i r - t i g h t c y l i n d r i c a l container, so that i t could accommodate p r a c t i c a l l y any type of s o l i d material, and i n part i c u l a r the powdered, e a s i l y corroded substances which were to be investigated. The s i z e of the calorimeter vessel depends on several fa c t o r s . As mentioned e a r l i e r , smaller dimensions result i n f a s t e r thermal equilibrium, which i s advantageous. However, the leakage modulus w i l l increase as the dimensions decrease, since i t varies d i r e c t l y as the surface area but inversely as the heat capacity or volumes, i f the dimensions decrease by *n» times, the leakage modulus i s n /n = n times as large 0 However, for a smaller calorimeter, the shortening of the experimental time caused by fa s t e r equilibrium may more than compensate for the increased leakage modulus, and produce less t o t a l thermal leakage. For very small dimensions, technical d i f f i c u l t i e s are encountered. Also, the increase i n weight, of the calorimeter vessel with respect to the - 20 - weight of the sample decreases the precision. Thus f o r the calorimeter constructed here, a compromise s i z e was chosen, which enabled the sample weight to be larger than the vessel weight, while s t i l l retaining quite rapid equilibrium conditions, ( i i ) Constructions (see Figure 5) The calorimeter vessel (2) i s a s i l v e r - p l a t e d c y l i n d r i c a l copper container, approximately 4 by 5 cm. (1 .57 x 1 .97 i n . ) , and 0.05 cm. thick (.020 i n . ) . I t was made by electroplating copper onto a sta i n l e s s s t e e l mold having a 1/4° pitch and shouldered to f a c i l i t a t e removal. The e l e c t r o l y s i s solution was a d i l u t e sulphuric acid solution of CuSOi,. containing a small amount of ge l a t i n . A plating current of 0.3 amperes for 120 hours was used, corresponding to a thickness of 0,044 i n . , and the excess copper was machined off to the required thickness. The calorimeter vessel was si l v e r - p l a t e d inside and out (to reduce the radiation of heat) to a thickness of ~0.003 i n . The solution used contained? AgCN 36 g . / l i t e r KCN 52 g . / l i t e r K 2C0 3 38 g . / l i t e r A small amount of CS 2 was added as a brightener. A current of 1 ampere (current density of 5 amp/sq.ft.) and a voltage of 1 to 2 were used. The bottom of the calorimeter vessel was soft soldered on. The thermometer (1) was f i x e d within the vessel and i t s leads (3) axited from the vessel by soft soldering a disk connected to the thermometer onto the top of the vessel. (The thermometers used w i l l be described i n d e t a i l l a t e r ) . A kovar metal-glass seal (4) (No. 96.1010 from the Stupakoff Ceramic and Manufacturing Co.), soldered through the top of the vessel, was used to admit Figure 5 s The Calorimeter (Actual Size) - 22 - helium to about four f i f t h s atmospheric pressure, i n order to speed thermal equilibrium. After the helium was introduced to the vessel, the glass tube of the seal was drawn off i n an oxygen-gas flame. In order to change samples to be measured i n the vessel, the kovar s e a l and thermometer were removed, and the sample was admitted through the vessel top. The t o t a l weight of the calorimeter vessel plus thermometer was about 43 g. Binding posts (5) were soldered to the side and top of the calorimeter vessel f o r thermocouple connections. Three wire hooks (6) were soldered to the vessel top f o r suspension of the calorimeter w i t h i n the s h i e l d . b. The Adiabatic Shield (I) Theory? The adiabatic shield must be heated so that i t s temperature remains close to that of the calorimeter vessel surface. In order to keep the sh i e l d surface temperature uniform, the s h i e l d must be a highly conducting material. The uniformity of surface,temperature i s e s p e c i a l l y desirable i f only a few thermocouples are used to determine the temperature.. To lessen temperature lags, the heat capacity of the shield must be small, and thus the s h i e l d walls must be t h i n . The shield must not be a i r - t i g h t , since a vacuum between s h i e l d and calorimeter vessel i s necessary to reduce thermal leakage. ( i i ) Construction: (see Figure 5) The s h i e l d (7) i s a 6 by 8 cm. (2.4 x 3.2 in.) c y l i n d r i c a l brass container, 0.07 cm. (.030 in.) t h i c k , with a removable bottom. I t was made by reducing brass tubing to the desired thickness i n an acid bath, and s i l v e r soldering on a brass top. The bottom was machined into a f r i c t i o n - f i t t i n g cap. The shield was heated by means of a non-inductive uniform winding of - 23 - No. 3*2 gauge silk-covered manganin wire. The t o t a l resistance of the winding was Z*20 ohmss 300 ohms for the side and 60 ohms each f o r the top and bottom. The wire was insulated and connected to the shield with Dow Corning 935 varnish. This i s a si l i c o n e e l e c t r i c a l insulating varnish having good f l e x i b i l i t y from -55° to 260°C. It drys tack-free i n three hours at 200°C. The winding onto the shield was accomplished i n stages, the varnish being baked after each few windings. In order to prevent conduction along the thermocouple and thermometer leads (No. 29 B. and S. Cu formex), they were wrapped non-inductively around the outside of the shie l d under the manganin winding. Aluminum f o i l was cemented onto the shield surfaces (with the same varnish) i n order to reduce radiation of heat. Three small screws (8) were fastened to the shield side, top and bottom for thermocouple connections. D i f f e r e n t i a l copper-constantan thermo couples (No, 30 B and S, insulated with the varnish) were attached from the shield side to the shield top, the shield bottom, the calorimeter vessel side, and the calorimeter v e s s l top. Thus a l l temperatures were found r e l a t i v e to the shield side temperature. Three small holes through the shield top permitted suspension with threads of the calorimeter vessel within the sMeld and the sh i e l d within the outer case. c. The Adiabatic Shield Controls The d i f f e r e n t i a l copper-constantan thermocouples were used to f i n d the temperature difference between the sh i e l d side and shield top, shield bottom, calorimeter side and calorimeter top. The thermocouples were connected to a Leeds and Northrup galvanometer having a s e n s i t i v i t y of 0.6 A V per mm., giving a deflection of 60 mm per °C at room temperature. Any one of the four d i f f e r e n t i a l thermocouples was connected to the galvanometer at one time, (see Figure 6b). - 24 - Figure 6a: Power Control o Galvanometer Thermocouples Figure 6b: Figure 6: Thermal Head Measurement Adiabatic Shield Control C i r c u i t - 25 - The manually operated power control c i r c u i t for the adiabatic shield i s shown i n Figure 6a. The power was supplied by a 60 cycle 120 v o l t s t a b i l i z e r , A 0-110 V powerstat permitted power control; the f i n e control, obtained with a 50 ohm variable resistance, was not normally used. The current was raised b r i e f l y by means of a tapping key shunting out a 400 ohm resistance, and the current was lowered b r i e f l y by switching a 350 ohm resistance into the c i r c u i t . Part of the current was shunted from the top shield heater through a variable 500 ohm resistance i n order to equalize the s h i e l d top and bottom temperatures. Since the controls became d i f f i c u l t for a temperature difference of more than 50°C between the shield and outer case, the difference was kept smaller by c o n t r o l l i n g the case temperature. d. The Outer Case ( i ) Theory: The outer case serves primarily as an evacuation chamber to make temperature control easier. When i t i s evacuated, the thermal leakage between the adiabatic s h i e l d and the calorimeter vessel i s decreased greatly, enabling the leakage to be neglected. Equally important, the leakage between the shi e l d and the outer case -is reduced, so that the shi e l d temperature control i s s i m p l i f i e d . By keeping the outer case temperature near the shield temperature, the s h i e l d control i s made s t i l l easier. The case temperature was controlled by immersing i n a temperature bath i n a Dewar f l a s k . For temperatures up to 90°C, a water bath was used, containing a heating element supplied with a variable alternating voltage. For low temperatures, down to -80°C, a mixture of dry ice and acetone was used. For temperatures from ~80°C to -rl60°C, l i q u i d oxygen was fed through a copper c o i l immersed i n gasoline; the gasoline was f i r s t cooled to -80°C with dry i c e . - 26 - ( i i ) Construction; (see Figure 5). The outer case (9) i s a 10 by 15 cm. (k x 6 in.) copper cylinder, 0.2 cm, thick (1/8 i n 0 ) , with a brass bottom soft soldered on, and a removable brass cover, A groove was machined into the cover, so that i t could f i t onto the case i n a vacuum grease seal (10) (Dow Corning high vacuum grease was used). To reduce thermal conduction along the leads, they were wrapped around a brass r i n g (11) (1/8"' th i c k and 3/41" long) soldered onto the inside of the cover. Three holes were d r i l l e d through the ri n g for suspension of the s h i e l d . The case was evacuated through a t h i n walled 3/8511 German s i l v e r tube (12) , s i l v e r soldered to the cover. A glass T-tube was connected to the German s i l v e r tube, so that the leads might emerge from the vacuum system through one arm of the T-tube, which was sealed with de Khotinsky wax. F l e x i b l e leads were soldered to the No. 29B and S Cu formex leads where they emerged from the system, and the junctions were f i x e d with de Khotinsky wax. The case was suspended wit h i n the Dewar fl a s k on a plate held by three 1/8" threaded s t e e l rods, as shown i n Figure 7. e. The Thermometer-Heater ( i ) Theory; 8 q A platinum resistance thermometer-heater > 7 was used both to supply heat to the calorimeter vessel and to measure the vessel temperature. This dual purpose can be accomplished f o r an adiabatic calorimeter, since no temperature measurement need be made while energy i s being supplied to the calorimeter vessel. The double use of the thermometer enables a l l temperature and heat input measure ments to be made with one potentiometer 0 The disadvantage of a thermometer-heater i s that the error i n heat input measurement i s increased, unless spe c i a l precautions are taken, because of the v a r i a t i o n of resistance of the heater with temperature. - 27 Figure 7: Suspension of the Calorimeter - 28 .. • ( i i ) Constructions Two d i f f e r e n t kinds of platinum resistance thermometers were used. (a) Glass embedded coils A commercial 100 ohm platinum resistance thermometer (produced by Wheelco Co., now Barber-Colman Co.) was used f i r s t . I t consists of a non-inductive winding of f i n e platinum ribbon embedded i n a soft glass tube (0.17 i n . x 1.75 i n . ) . The two 0.010 i n . platinum leads emerge from a cup-shaped top. A platinum disk, 0.005 inches t h i c k and 0.5 inches diameter, punched so that i t f i t t e d into the cup, was sealed to the glass by heating i t by induction to a bright red color. F l e x i b l e copper leads were soft soldered to the platinum leads, doubled back into the cup, and f i x e d there with a r a l d i t e cement. Two leads were soldered to each f l e x i b l e lead to make the four-lead type of thermometer. (The resistance of the two f l e x i b l e leads - 0.01 ohm - was n e g l i g i b l e i n comparison with the platinum resistance). The thermometer was f i x e d withih the calorimeter vessel by soft soldering the platinum disk to the vessel top. u Thermometer: Actual Size Ft.disk a r a l d i t e The disadvantage of t h i s type of thermometer i s that i t s c a l i b r a t i o n varies because of strains produced i n the platinum by expansion of the glass. (b) Strain-free thermometer; In order to avoid repeated thermometer c a l i b r a t i o n , a strain-free platinum resistance thermometer was constructed. The thermometer must be small, i n order to f i t into the calorimeter and to have fast - 29 - thermal equilibrium, but the platinum wire must be free to expand and have no 13 14 p o s s i b i l i t y of short c i r c u i t s . The usual method ' of s a t i s f y i n g these conditions i s to wind fine platinum wire into a small diameter c o i l and then wind the c o i l around an insulating support (usually mica)„ The whole i s then enclosed i n an a i r - t i g h t container, to avoid contamination of the platinum and water condensation on i t , as w e l l as for mechanical protection. This was the method adopted. Chemically pure platinum wire 0.003 i n . diameter was used. The specifications ensuring purity are that the r a t i o of the resistance at 100°C to that at 0°C i s greater than 1.390 and the r a t i o of the resistance at -183°C to that at 0°C i s less than O.25O. From 100 to 200 cm of the wire was wound under 10 g tension on a stretched s t e e l wire mandrel 0.010 i n , diameter. The h e l i x , from 2,5 to 5 inches long, was slipped off the mandrel and stretched to about 10 inches, so that none of the c o i l s were i n contact. The platinum h e l i x was then wound under 1 g tension non-inductively around a notched mica cross held i n a mandrel. For assembly of the mica cross see Appendix I I I . One 20 ohm and one 50 ohm thermometer were constructed. Pincers Burner Pt wire Tube Pt s t r i p Plug Of the many types of protective case materials used f o r resistance thermometers, pyrex glass was chosen because i t i s easy to work and i s suitable for the temperature range desired. Thin- walled pyrex was used f o r the protective tube, and t h i n platinum s t r i p s were used fo r leads i n order to make a vacuum seal through the glass. The sealing procedure Is detailed i n Appendix IV, - 30 - The platinum s t r i p s emerging from the seal were fused to 0o020 inch platinum wire leads. The wire was flattened at one end by r o l l i n g , was held i n contact with the platinum s t r i p by pincers, and was fused with an oxy-gas flame* A pyrex tube was then f i t t e d over the leads and collapsed, so as to f i x the platinum s t r i p s and leave only the heavy platinum leads emerging from the / thermometer. The thermometer was then cleaned i n a di l u t e n i t r i c acid solution, annealed f o r 15 hours at 490°C, and de-gassed f o r 3 hours at 400°C. After the seal was leak-tested, the thermometer was f i l l e d with dried helium at two-thirds atmospheric pressure and the pyrex tube was sealed o f f . The actual size of the thermometer i s shown i n the diagram. Kovar seals were used to attach the thermometer to the calorimeter vessel and to bring i t s leads out of the vessel. The two platinum leads were soldered to kovar seals, which were soldered to a sil v e r e d phosphor bronze disk. This disk was soldered to the calorimeter vessel top. Two fin e leads were connected to each kovar seal on the outside. The resistance of the kovar seals (~ 0.001 ohm) i s negligible i n comparison with that of the resistance thermometer. fo 1 Energy Input and Temperature Measurement The potentiometer method was used to measure both the thermometer resistance, f o r temperature determinations, and the energy input to the thermometer when used as a heater. Both measurements were made with one potentiometer, using one c i r c u i t . The c i r c u i t , which was s l i g h t l y d i f f e r e n t f o r the commercial 100 ohm thermometer and the strain-free 20 ohm thermometer, i s shown i n Figure 8. - 31 - Q 10. H 10,000 H H T 10 5000 T -\VWW • — - 4 r Therm. * t 6V 6V Dummy 5 ~= =" a. Commercial 100 ohm Thermometer C i r c u i t Q T x H 10,00C H •AVIV" 1 20 10,000 A V W ( / A — • 20 H Therm. R. 6V 6V Dummy 25 b. Strain-free 20 ohm Thermometer C i r c u i t . Figure 8. C i r c u i t Diagrams f o r Thermometer-Heaters. - 32 - To produce a very stable direct current source, s i x 120 amp.-hour 2 v o l t storage batteries were used. Theywere arranged to create two 6 v o l t sources i n p a r a l l e l . Before an energy input, the batteries were discharged through a dummy resistance, so that t h e i r voltage would be constant while measurements were being made. For temperature measurements, the switches were thrown to position T i n Figure 8, and f o r heat input the switches were at position H. The potentiometer connections are at P and Q. A Tinsley type 3387B 2 -1 potentiometer was used, having ranges 0 to 1.9 x 10 , 1.9 x 10 and 1.9 v o l t s , readable to 0.00$% or 1 of the maximum. The accuracy i s 0.01% of the 19,000 maximum, or 2/*- V for the lowest scale. A l l resistances, the potential across which was to be measured, were standard resistances.. A l l leads inside the calorimeter were No. 29B and S copper, formex insulated, and outside were No. 18B and S f l e x i b l e copper. ( i ) Energy Input: The potential across the heater was found by using a potential d i v i d e r , consisting of a 10 and a 10,000 ohm resistance. The potential across the 10 ohm resistance was measured; i t was VQ = 10 times 10,010 + E l e a d s the potential across the heater. Since the resistance of the leads was about 1 ohm, the potential across the heater, V = 1001.1 V Q ; neglecting the lead resistance produced an i n s i g n i f i c a n t error of 0.01%. The current through the heater was found by measuring the potential across a 1 ohm standard resistance. This potential, V , was equivalent to the - t o t a l current through the c i r c u i t . The current through the heater, I, was equal to the t o t a l current minus the current through the potential dividers I = vp - ZQ 10 The energy input = V l t . The duration of energy input, t,, was measured . with a stopwatch to better than 0.05% accuracy. At 0°C, V ~ 6 v o l t s and I ** 0.05 amp. for the 100 ohm heaters, V ^  3 v o l t s and I s=* 0.12 amp. for the 20 ohm heater. Thus the power was approximately 0.35 watts for each thermometer at 0°C, producing a temperature r i s e of ~ 0.5°C per minute f o r the calorimeter vessel when containing a sample; weighing 30 g. Since the heater resistance varied with temperature, the power input varied also, and had to be averaged. The voltage, V, across the 100 ohm heater (Figure 8a) was v i r t u a l l y constant during one heat input, but the current changed considerably. For a 10°C temperature r i s e , the heater resistance changed by 4 ohms or one twenty-fifth of i t s value. Thus the current also changed by approximately t h i s f r a c t i o n . Since the battery potential = V + Vp = 6, V = 6 - Vp. Vp was only 1 of V, and changed by only _1, so that the change 100 25 i n V was 1 or 0.04% f o r one heat input. 2500 For the strain-free heater (Figure 8b), the c i r c u i t was arranged so that the power input remained approximately constant. As the temperature rose s the heater resistance increased, and the current was lowered. Thus the voltage drop across the 20 ohm resistance of the c i r c u i t decreased, and the potential across the heater rose. Since the heater resistance was approximately the same as the 20 ohm resistance, the r i s e i n potential across the heater approximately compensated f o r the lowering of current through i t . (See Appendix V for power change calculations). ( i i ) Temperature Measurement; (Figure 8) The temperature was calculated from the resistance of the thermometer, which was found by comparing with a standard,, The potentials across the thermometer, VQ, and across a standard 10 ohm resistance, Vp, were measured f o r the 100 ohm thermometer. Since the currents were the same through each resistance, the thermometer resistance, R- = 1 0 V Q V P W i t h the 20 ohm thermometer, a 20 ohm standard resistance was used, so that R t = 20 \ V P Because the four lead type of thermometer was used, the measurements were independent of lead wire resistances,, A current of 1.2 milliamperes was used f o r the 100 ohm thermometer and 0. 6 milliamperes for the 20 ohm thermometer. These r e l a t i v e l y high currents reduced the effect of parastic thermal e.m.f."s i n the c i r c u i t , while apparently causing ne g l i g i b l e heating. The power generated by the current f o r the 100 ohm thermometer was 1.4 x 10 ^  watts (7 x 10~^w f o r the 20 ohm thermometer). This was 3 x 10"^- as much power as used when heating the calorimeter vessel. The resistance measurements were reproducible to 0,01%, which corresponded to 0.02°C near 0°C. I I I . CALIBRATION AND PERFORMANCE 1. Calibration of Resistance Thermometer a. Fixed Points; The resistance thermometers were calibrated at International Temperature - 35 - Scale f i x e d points.^ The c a l i b r a t i o n points were; Basic f i x e d pointss ( i ) The temperature of equilibrium between l i q u i d and gaseous oxygen at the pressure of one standard atmosphere (760 mm. of Hg): Oxygen point ~182 097 oC ( l i ) The temperature of equilibrium between ice and a i r saturated water at normal atmospheric pressure; Ice point 0.000°C , ( i i i ) The temperature of equilibrium between l i q u i d water and i t s vapour at the pressure of one standard atmosphere: Steam point 100.000°C Secondary fix e d point; (iv) The temperature of freezing mercury at the pressure of one standard atmospheres Mercury point -38.87°C. Although the basic sulphur point (444.60°C) i s recommended f o r c a l i b r a t i o n , the secondary mercury point was used because of the r e l a t i v e ease of attaining t h i s temperature. b. Calibration Apparatus^ ( i ) Oxygen point: In order to standardize i t at the oxygen' point, the thermometer, contained i n a glass tube, was immersed i n l i q u i d oxygen. The formula used to f i n d the temperature of equilibrium between l i q u i d and gaseous oxygen at a pressure p (mm. Hg) iss t = -182.97 + 9.530 / p - l\ - 36 - Because of the commercial brand of l i q u i d oxygen used and because of the uncertainty i n p, the oxygen point was accurate to o n l y " 0„laC»,. ( i i ) Ice point, and mercury point (see Figure 9a)& To calibrate at the i c e and mercury points, the thermometer was placed i n a glass tube containing acetone f o r fast thermal conduction. The tube was immersed i n d i s t i l l e d water or mercury, which was contained i n an evacuable Dewar f l a s k . The f l a s k was surrounded by a mixture of dry ice and acetone. When the temperature approached the t r a n s i t i o n point, the f l a s k was evacuated, so that the temperature remained constant f o r some time at the ice or mercury point. ( i i i ) Steam points The steam point apparatus i s shown i n Figure 9b. Steam i n equilibrium with water r i s e s about the thermometer, and insulates i t s e l f from the surroundings by c i r c u l a t i n g down outside the inner tube. (iv) Determination of f i x e d pointss A Leeds and Northrup platinum resistance thermometer, calibrated by the National Bureau of Standards above 0°C, was used to determine the mercury and..steam point temperatures. The c a l i b r a t i o n of t h i s thermometer was checked at the ic e point. Since i t s resistance at 0°C di f f e r e d by only -* 0.0005 ohms from i t s calibrated value of 2.5117 ohms, the change was assumed to indicate a lin e a r s h i f t i n resistance, with no change i n the c a l i b r a t i o n constants, R100 ~ Ro a n d 5 o The resistance of t h i s thermometer was measured with a Leeds and Northrup Mueller d i a l bridge. The lead resistances were eliminated by using a mercury commutator t° switch leads. For a balanced bridge (see diagram), - 37 - Evacuation Tube Cork Glass Wool Evaenable Flask Water or mercury N.B.S. Thermometer Thermometer Dry Ice plus Acetone Dewar a. Ice and Mercury Point Apparatus 'A- Cork Glass Wool -Glass Tube • Thermometer Steam b. Steam Point Apparatus Figure 9. Calibration Apparatus. - 38 - R x + C = R t + T, where R l i s the bridge reading* With T and C and t and c switched, R 2 + T = Rt + C 2 The connecting resistances of the t commutator also are eliminated by taking measurements with the thermometer resistance, R^ , shunted out. e„ Calibration Formulae The r e l a t i o n between resistance and temperature, t , of a platinum resistance thermometer f o r temperatures from 0 to 630°C iss 8, 9 R+ = R . ( l + At + Bt 2) (1) R 0 i s the resistance at 0°C and A and B are constants determined by c a l i b r a t i o n at the steam and sulphur points. For temperatures from -183 to 0°C„ R t = R Q [ l + At + Bt,2 + C(t - 100)t3J (2) The additional constant, C, i s determined by c a l i b r a t i o n at the oxygen point 0 For these ranges, the temperatures found from these formulae, using a standard platinum resistance thermometer, represent the International Temperature Scale. The f i r s t equation i s equivalent to Callendar's equations t = ^ ~ R ° 100 + 8 j j o . o i t ) 2 - 0 . 0 1 t j (3) R 100~ R o where 8 = - 10 4 B A + 100B (See Appendix VI), - 39 - The second: equation i s equivalent tos t - 100 + 8 [(0.01 t ) 2 - 0.01 tl + D(t-100)t 3 (4) S100 •" R o L J where D ° - C  A + 100 B (See Appendix ¥1). These formulae enable the temperature to be e a s i l y calculated from the resistance by a successive approximation method. The f i r s t approximation, t x - * t - R o 100 R100 • R o The second approximation i s attained by su b s t i t u t i n g t ^ i n the r i g h t hand side of equation (4), t 2 - • % * . & | $ ^ D(tj, - 100)^3 • t-^ + f ( t j ) The t h i r d approximation, %$. m t ^ + f(tg),and so on. In prac t i c e , t 2 can be estimated, so that only one ca l c u l a t i o n need be made to f i n d the f i n a l approxima- t i o n > t 3 o d c C a l i b r a t i o n Results ( i ) Commercial thermometer: The f i r s t c a l i b r a t i o n of the commercial 100 ohm thermometer gave the following results? Oxygen point R-LS2 9 = 24.740 abs. ohms Mercury point R_^g " 8^»725 Ice point R Q = 100.24 ^ Steam point R100 ° ^9°°^ - 40 - The temperature of the mercury point was found from the Leeds and Northrup thermometer resistance at that point. The value Rioo w a s calculated using the Leeds and Northrup thermometer, whose c a l i b r a t i o n constants were Ro' = 2.5117, Rioo* - Ro* = 0.9732, and 8* = 1.502, At the steam point, the resistance of the Leeds and Northrup thermometer was R^ » = 3.4844. Thus, using equation (3), the tempera ture, t - 3.4844 - 2.5117 , n o +(a n e g l i g i b l e quantity) = 99.95°G. 0.9732 X U U at which R^ = 139.00 f o r the commercial thermometer. Since R 0 = 100.24 , E99.95 ~ E o = 38.76 To f i n d R]_oo! 8R f o r 0,05°C i s 38.76 x O o 0 5 x 0,985 « 0.019, 99.95 where 0.985 i s the value of .A ( p t ) (and pt = R t ~ R o 100) At R100 - near 100°C f o r a thermometer with 8 = 1.50 . Therefore R 1 G Q - 139.00 + 0.019 = 139.02 R100 ~ R o = 38.78 abs. ohms. The resistances obtained by c a l i b r a t i o n (equation (5)) were substituted i n equation (2) to f i n d the constants: A = 3.952 x 10" 3 B = -8.38 x 10~ 7 C = -1.3 x 10" 1 2 - 41 - Thus 8 = 2 .17 and D = 3.4 x 10" 1 0 Equation (4) becomes? , t = (R t - 100.24) 2.5786 + 2.17|j0.01 t ) 2 - 0.01 t ] + 3.4 x 10" 1 0 (t - 100)t 3 After this commercial thermometer had been used f o r temperatures from -150 to 150°C, i t was recalibrated. The results were: R=38.87 = 84.65 R 0 = 100.06 R 1 0 0 » 138.91 therefore R ioo ~ ^o = 36.85 and 8 = 1.47 therefore t = (R^ - 100.06) 2.5740 + 1.47 ["(0.01 t ) 2 - 0.01 t ] f o r 0 ^ t < 630°C. The c a l i b r a t i o n of the thermometer had changed considerably. .The' change caused an appreciable error i n temperature i n t e r v a l measurements, which increased as the temperature d i f f e r e d more from 0°C. ( i i ) Strain-free thermometer?.,-. The c a l i b r a t i o n of the strain-free thermometer gave the following r e s u l t s : R 183.0 " 4 ° 4 9 1 R-38.87 = 1 5 - 3 9 0 R c -• 18.218 R 1 0 0 =25.348 From these, R 1 Q0 ~ Ro = 7° 130 5 = ,1.46 and D = 1.14 x 10~ 9 Thus, t = (Rt - 18.218) 14.025 + 1.46 [(.01 t ) 2 - .01 t ^ + 1.14 x 10" 9(t - 100)t 3 After the thermometer had been used from -100 to 150°C, i t s zero point was checked and was found to be the same within 0.001 ohm. 2. Heat Capacity of the Calorimeter The calorimeter was calibrated by measuring i t s capacity when i t contained a standard substance. This method was preferable to c a l i b r a t i n g the empty calorimeter, because conditions were closer to those achieved when the calorimeter contained a specimen. Of the heat capacity standards recommended,-^ aluminum oxide was chosen because of i t s a v a i l a b i l i t y and s t a b i l i t y . E l e c t r i c a l l y fused c r y s t a l l i n e carbon-free alumina. (RR Alundum), produced by Norton Co., Mass., was used. The c a l i b r a t i o n results f o r low temperatures, obtained using the commercial thermometer, are shown i n Tables I and I I and Figure 10. I t i s seen that, as expected from the behaviour of copper, the heat capacity of the calorimeter decreased rather rapidly below 0°C. Since the capacity at temperatures above 0°C was found from preliminary measurements to be approximate l y constant, i t was assumed to increase only from 15.8 j/°C at 0°C,(see Figure 10) to 15.9 j/°C at 100°C. 3. Accuracy of Specific Heat Measurements. ' The accuracy of results and consistency of performance of the adiabatic calorimeter are limited by certain errors inherent i n the apparatus; - m- Table I Measured Capacities of Calorimeter  Plus Aluminum Oxide (Plotted i n Figure 10) Weight, of Calorimeter - 42.26: g. .Weight of A1 20 3 - 19.91 g. (Temperature int e r v a l s varied from 11 to 15°C.) Average Temperature Capacity (°c) ( j / ° c ) -92.8 22.9 -80.6 24.2 -60.4 26.3 -60.0 26.1 -46.1 , 27.2 I -33^3 28.3 -21.4 28.9 44. - -100 -80 -60 -40 -20 0 Temperature (°C.) Figure 1Q, Heat Capacities of Calorimeter and of Calorimeter plus Alumina. Table I I Calorimeter Capacity (Capacity of calorimeter plus Al 203 obtained from Figure 10) (Plotted i n Figure 10) Weight of Calorimeter = 42.26 g. Weight of A1 20 3 = 19.91 g. = 0.1953 moles T°K A1 20 3 Cal. + A1 20 3 Cal. Cp c p C P ( j/ cC) 180 8 .55 22.93 14.38 190 9.28 24.02 14.74 200 9.99 25.05 15.06 210 10 .66 25.95 15.29 220 11.31 26.75 15 .44 230 11 .93 27.50 15.57 240 12 .52 28.20 15.68 250 13.09 28.80 15.71 - 46 - a. Thermal Leakage Error Since f o r any calorimeter heat i s transferred between the c a l o r i  meter vessel and i t s surroundings, an error i n accounting for t h i s heat transfer i s involved. In an adiabatic calorimeter, the heat transfer i s not calculated, but i s reduced to a minimum. The resu l t i n g error i n temperature of the calorimeter i s s where <P i s the thermal head or the difference i n temperature between the calorimeter vessel and s h i e l d , k i s the leakage modulus and t i s the length of time for which the thermal head occurs. The maximum error, = ^ m a x ^ * i f ^ m a x i s t h e maximum thermal head f o r the time t». For t h i s adiabatic calorimeter, the thermal leakage modulus was measured f o r various temperatures and pressures by c o n t r o l l i n g the thermal head by means of the adiabatic s h i e l d controls. The re s u l t i n g values (for a 30 g. sample i n the calorimeter vessel) were? Temperature (°C.) -70 -20 -20 Pressure (microns) 0.2 0,1 0.5 k (°C/min,/°C head) .01 0.015 • 0.02 Temperature 20 20 40 40 100 100 140 Pressure 0.15 2 0.2 0.5 0.1 3 0,3 k 0.015 0.06 0.012 0.03 0.025 0.07 0.04 (See Figure 11) - 4^  " For a thermal head between calorimeter and shield bottom only; Temperature (° G.) 50 Pressure (>t) 0.2 k b (°C/min,/°C) 0.004 A P h i l i p s i o n i z a t i o n gauge, calibrated with a McLeod gauge, was used to measure the pressures. As can be seen by the Tables and Figure 11, the thermal leakage modulus increases only s l i g h t l y with temperature, but greatly with pressure up to about one micron of mercury. The lowest pressure obtained with t h i s calorimeter was about 0.1 micron, because of the v o l a t i l e s present (mainly the ins u l a t i n g varnish). As the thermal head between the shield and outer case increased, and thus the shield current increased, the pressure from the v o l a t i l e s m u l t i p l i e d . Thus with t h i s thermal head a minimum (less than 50°C), the least thermal leakage was obtained because of the low pressure, and also because of the ease of co n t r o l l i n g the sh i e l d temperature. The maximum error i n temperature a r i s i n g from thermal leakage can be calculated from the formula noted previously; *Wx ° ^max For a 10°C temperature r i s e , heat was supplied to the calorimeter vessel f o r about 20 minutes, and an equilibrium period of 5 to 10 minutes was required. Thus t = 30 minutes. I f 0 ^  = lf/io'c, and k = 0 o03 oC/min./ oC, then \ ^ =~0.1°C. Consequently, the maximum error i n temperature i n t e r v a l would be 1%. Since (p w i l l average much less than i t s maximum, the error from t h i s source w i l l be much less than 1%» With careful shield control,0 could be kept within l°/30 C, and the r e s u l t i n g maximum possible error would be 0.3%. The probable thermal leakage - m - error (the mean value of a random selection of positive errors) would then l i k e l y be less than 0.1%. Besides the thermal leakage caused by a thermal head measurable with the thermocouples, there can be leakage by other means, such as by conduction along leads and by radiation from hot spots. This leakage causes the calorimeter vessel temperature to d r i f t l i n e a r l y with time. However, t h i s effect was usually ne g l i g i b l e for t h i s calorimeter i f the s h i e l d temperature was within 50°C. of the outer case temperature. b. Energy Input Error Since the resistance of the thermometer-heater varied considerably with temperature, the power supplied by the heater also changed. The averaging of t h i s power introduced an error i n the energy input measurement. The voltage across the commercial 100 ohm thermometer remained almost constant during a heat input, but the current decreased by** 4% f o r a ten degree temperature r i s e (see Section I I , 2 f ) , The decrease i n current was not uniform. I t was rapid as the power input was started and quick l o c a l heating occurred, but after one or two minutes i t became uniform as the heating rate became more constant. In order to average the current, i t was measured at one to two minute:, i n t e r v a l s . The averaging process introduced a maximum error of 0.1% i n the t o t a l energy input measurement. The c i r c u i t f o r the strain-free 20 ohm thermometer was arranged so that the power remained almost constant (see Section I I , 2 f ) . The t h e o r e t i c a l power change f o r a 10°C temperature r i s e i s 0.05% i f the i n i t i a l heater resistance i s equal to the resistance of the rest of the c i r c u i t (see Appendix V). In practice, the power change res u l t i n g from a 10°C temperature r i s e was about 0.5% near 0°C. The error involved i n averaging the power was thus about 0.05%. An additional error i n power input measurement resulted from inaccuracies i n potentiometer readings. This error amounted to ~ 0,05%. The t o t a l maximum error i n measurement of the energy input was thus ~ 0,1%. c. Temperature Measurement Error. * The largest error i n s p e c i f i c heat calculations (as i s usually the case i n calorimetry) was caused by inaccurate temperature measurement. Errors i n temperature as measured by a platinum resistance thermometer may be caused by: ( i ) Inaccurate ca l i b r a t i o n or change i n c a l i b r a t i o n of the thermometer Uncertainties i n the thermometer c a l i b r a t i o n cause errors i n the temperature measurement, and consequently i n the temperature i n t e r v a l measurement. The l a t t e r must be the more accurate; although a temperature error of 0.5°C i s not serious, a 0.1°C error i n the temperature i n t e r v a l i s rather large, since i t causes a 1% error i n spe c i f i c heat i f the temperature i n t e r v a l i s 10°G. Fortunately, errors caused by ca l i b r a t i o n deviations tend, to cancel out for temperature differences, so that they are not so serious as might be supposed. Errors caused by inaccurate c a l i b r a t i o n are not large, because the ca l i b r a t i o n must be very poor to produce s i g n i f i c a n t errors (see Appendix V I I ) . Moreover, these errors w i l l not vary with time, and w i l l change gradually with temperature, so that t h e i r effect on spe c i f i c heat anomalies w i l l be small. Changes i n the c a l i b r a t i o n constants of the thermometer, caused by strains or impurities i n the platinum, are more serious. The error i n temperature i n t e r v a l caused by the measured changes i n c a l i b r a t i o n of the commercial thermometer was ~ 0,1°G for a 10°C i n t e r v a l near 100°C (see Appendix V I I ) . Although t h i s error was large, i t s effect on the s p e c i f i c heat anomaly was reduced because of i t s gradual change with temperature. Errors r e s u l t i n g from c a l i b r a t i o n deviations were greatly reduced by using the strain-free thermometer, whose c a l i b r a t i o n constants remained very steady, ( i i ) Inaccurate resistance measurement; The greatest error i n temperature measurement for the strain-free thermometer was caused by the error i n potentiometer readings. The potentiometer was accurate to 0,01%, Since the thermometer resistance was found from the quotient of two potentiometer measurements (see Section I I , 2 f ) , the maximum error i n the resistance was 0,02%, which corresponded to a temperature error of 0,05°G, The maximum error i n the temperature i n t e r v a l was then 0,1°C, but the probable error was considerably l e s s , approximately 0,015°C, In a 10°C tempera ture i n t e r v a l , t h i s produced an i n t e r v a l error of 0.15%. As the maximum error was considerably greater than t h i s , a more accurate means of measuring the thermometer resistance would have been desirable. A more accurate potentiometer could be used. Some form of Wheatstone bridge c i r c u i t i s often usedj a Mueller bridge i s perhaps the most accurate one available giving precision up to 0.001°C. The t o t a l error i n s p e c i f i c heat measurement for t h i s calorimeter, under adequate working conditions and using the strain-free thermometer, was thus made up of probable errors of a 0.1% thermal leakage error, a 0,05% power measure ment error, and a 0.15% temperature measurement error. The t o t a l maximum error was ~1.5% and the probable error was ~ 0.2%, However, the accuracy of the calorimeter i s perhaps best described by an error which i s not l i k e l y to be exceeded. I f an error i s used which only f i v e percent of a random sample of errors - 5 2 - exceeds, the accuracy was ~ 0.5%. For a l l p r a c t i c a l purposes, t h i s i s a suitable value to use, so that the accuracy of the sp e c i f i c heat measurements was 0.5%. IV. RESULTS 1. Preparation and Properties of Allo y s , a. Preparation of Alloys; Alloys close to the compositions Mn3AlC and Mn3ZnC were prepared. The materials used were; Mn; 99.9% purity, donated by the Electromanganese Corp. of America. A l : 99.99% purity, donated by the Aluminum Co. of Canada. Zn: 99<i#9% purity, donated by the Consolidated Mining and Smelting Co. of Canada. C : spectroscopic grade. The a l l o y Mn3AlC was prepared by induction melting under an argon atmosphere i n an alumina crucible. The c h i l l cast a l l o y was homogenized i n an evacuated s i l i c a tube f o r 72 hours at 1000°C. The a l l o y Mn3ZnC was sintered, using components Mn3C and zinc, f o r three days at 550°C i n an evacuated s i l i c a tube containing l i t t l e free' volume. I t was then ground and resintered f o r 12 days at 600°C. b. Properties of the Alloys: ( i ) Mn 3AlC: The a l l o y Mn 3AlC was me t a l l i c and b r i t t l e , but ea s i l y corroded i n humid conditions. X-ray powder photographs showed the a l l o y to be about 9S% the ordered face-centered cubic phase expected, with parameter 3.876 A. The saturation magnetization,<T, i n a magnetic f i e l d of 16,000 oersteds was measured from -160 to 20°C. (113 - 293°-^), using a Sucksmith r i n g balance. The behaviour (Figure 12) was that of a ferromagnetic substance. The Curie point, obtained by p l o t t i n g (fx against temperature and extrapolating the straight l i n e to <f*> 0, was 6 = 286°K = 13°C. The saturation magnetization at 0°K, obtained by p l o t t i n g absolute temperature squared against (T , and extrapolating to zero temperature, was 0£ = 102 ergs/g/oersted, which corresponds to 1.2 Bohr magnetons per manganese atom. (The same value was obtained by previous researchers 4). ( i i ) Mn3ZnCs The a l l o y Mn3ZnC was i n powdered, non-metallic,easily-corroded form. X-ray powder photographs showed the a l l o y to be > 95% the ordered face- centered cubic structure expected. Two alloys were used? All o y Is The f i r s t was the same a l l o y on which neutron d i f f r a c t i o n measurements were made. The X-ray photograph indicated a small amount (~ 2%) of a o second phase. The main phase had l a t t i c e parameter: 3.9228 A. Al l o y 2: A large a l l o y of 70 grams was sintered i n order to make s p e c i f i c heat measurements more accurate. About 5% of a second phase was present even a f t e r r e s i n t e r i n g . The l a t t i c e parameter of the cubic structure o was 3.9233 A. The variation of saturation magnetization with temperature f o r the two allo y s i s shown i n Figure 13. The maxima were 82.6 and 80.7 erg/g/oersted f o r alloys 1 and 2, occurring at -45 (*2)°C or 228 (±2)°K. Extrapolating the curve to 0°K f o r a l l o y 1 gave a magnetization at absolute zero which corresponded to a Bohr magneton value of one per Mn atom (see Section T, 2). Extrapolating the part of the curve above -45°C to 0°K gave a value of ~ 1.5 Bohr magnetons per Mn atom. These results agree within experimental error with those obtained - 55 - ol 1 I I ; L 100 200 300 '400 T(°K.) Figure 13. The Variation of Saturation Magnetization, fT , with  Temperature, T, f o r Mn-^ ZnC. - 56 - previously^ (see Figure 2). The Curie temperatures of alloys 1 and 2 were 0, - 391°K = 118°C and 368°K = 95°C i n a f i e l d of 16,000 oersted. 2. Specific Heat Measurements a. The Specific Heat of Mn3AJC; The s p e c i f i c heat of Mn 3AlC was measured from -140 to 100°C, using the commercial thermometer. The capacities of the a l l o y plus calorimeter are plotted i n Figure 14 and are l i s t e d i n Table I I I . The capacity of the a l l o y alone, obtained by using the calorimeter c a l i b r a t i o n curve of Figure 10 (Section I I I , 2), i s plotted i n Figure 15. The s p e c i f i c heat curve of Mn3AlC showed the expected ferromagnetic anomaly at the Curie point, and i t s shape was close to that predicted by Weiss (Section I, 1). This anomaly occurred at -10 ±2°C, which i s the Curie tempera ture i n zero magnetic f i e l d of the a l l o y - the Curie temperature obtained by magnetic measurements i n a f i e l d of 16,000 oersteds was of course higher (13°C). The size of the s p e c i f i c heat anomaly was considerably smaller than expected f o r an a l l o y of magnetization 1.2 Bohr magnetons per manganese atom. According"to the Weiss theory, the height of the anomaly f o r a ferromagnetic substance of magnetization h^kg per atom i s 3 h e ;cal/°C/g atom.^" But the height of the Mn 3AlC anomaly was ~ 2.2 j/°C for 0.157 moles, or ~1.1 caL/°C/ g atom of Mn, which would correspond to only 0,37 Bohr magnetons per Mn atom. The greatest s p e c i f i c heat measured for the a l l o y (ignoring the anomaly) was 6.6 cal/g atom/°C,at 100°C. The measurements were repeated f o r the same a l l o y , using the s t r a i n - free thermometer, with good general agreement i n results (see Figure 16). The Curie temperature was -9 i2°C and the anomaly height was ~ 1.9 j/°C. f o r 0.151 moles, or ~0.33 Bohr magnetons per Mn atom. Table I I I Capacities of Mn3AlC (31.97 g) plus Calorimeter (42.18 g ) / as measured with the Commercial Thermometer (Temp, inte r v a l s varied from 5 to 12°C). Uverage Temp. (°c) Capacity (J/°C) Average Temp. (°c) Capacity (j/°C) -138.6 25.3 -2.0 35.1 -136.8 25.4 -0.8 34.9 -124.8 27.0 4.9 34.8 -111.7 28.5 6.2 34.9 -102.2 29.7 17.1 35.2 -97.3 30.1 26.0 35.2 -89.2 31.1 31.5 35.5 -89.1 30.8 36.2 35.6 -76.8 31.7 41.7 35.3 -61.6 33.2 43.8 35.3 -57.2 ; 33.2 51.6 35.6 -43.4 34.1 52.8 35.5 -30.7 35.2 61.0 36.1 -17.4 36.1 62.3 35.9 -13.6 36.3 70.2 35.9 -11.0 36.3 74.2 36.0 -10.4 36.7 83.0 36.3 -8.8 36.6 91.7 35.9 -6.4 36.0 100.2 36.2 I Figure 15. Capacity of Mn^AlG - 60 - 3 i | i i • i i i I -60 -40 -20 0 20 40 T(°C.) Figure 16 Capacity of Mn^AlC + Calorimeter, (using the strain-free thermometer) - 61 - b. The Sp e c i f i c Heat of Mn^ZnC: ( i ) A l l o y 1: The heat capacity of the sample of Mn3ZnC on which neutron d i f f r a c t i o n measurements were made was found from -100 to 60°C, using the commercial thermometer. The temperature intervals used were from 8 to 12°C, which were large enough to obtain accurate i n t e r v a l measurements, and small enough to discern sudden changes i n heat capacity. The re s u l t s are shown i n Figure 17. The s p e c i f i c heat anomaly, occurring at -37°C, had the form of a normal ferromagnetic Curie point anomaly. The anomaly height was 1.8 j/°C f o r 0.083 moles of Mn3ZnC, or 1.7 cal/°C/g atom of Mn. The corresponding Bohr magneton number was 0.6 per Mn atom. The maximum value of the s p e c i f i c heat reached was 7.5 cal/g. atom/°C at 80°C, and at t h i s point i t was s t i l l r i s i n g quite rapidly. ( i i ) A l l o y 2s Because the sample of a l l o y 1 was rather small, another larger sample was made, i n order to increase the accuracy of the s p e c i f i c heat measurements. The r e s u l t s , obtained using the strain-free thermometer, are shown i n Table IV and Figure 18. The s p e c i f i c heat values show two anomalies, a very sharp one at -35°C, and a more rounded one at ~ 65°C. The high temperature anomaly occurred at the normal ferromagnetic Curie temperature of Mn3ZnC. I t s rounded shape was probably due to measurement errors, and possibly also due to inhomogeneity i n the a l l o y , since the Curie temperature i s very dependent on composition. The largest s p e c i f i c heat measured for the a l l o y (except f o r the anomaly) was 6.8 cal/g, atom/°C at 100°C, which was approximately the same as that f o r Mn 3AlC. - 63 - Table IV. Capacities of Mn3ZnC (Alloy 2:51.38g.) plus Calorimeter (43.66 g.) as measured with the Strain-Free Thermometer. (Temperature intervals varied from 5 to 11°C.) Average Temp. Capacity Average Temp. Capacity (°C) (j./°c.) (°C) (j./°C) -69.2 40.2 39.5 46.3 -58.4 42.0 47.5 47.1 -56.7 42.1 47.8 46.4 -53.3 42.6 49.3 46.8 -48.0 44.0 56.5 46.9 -46.5 44.3 59.8 46.8 -45.7 44.6 66.0 47.0 -40.9 46.1 69.3 46.9 -38.3 47.0 72.5 46.2 -36.8 46.5 73.1 46.0 -36.1 47.3 77.3 45.7 -31.2 42.7 79.0 45.2 -28.3 42.6 82.2 45.3 -17.9 43.0 82.2 45.6 - 7.6 43.3 89.6 45.2 27.9 45.6 91.9 46.0 29.7 45.3 101.6 45.8 37.9 45.8 111.4 45.8 cp(j./°c.) Figure 18. Capacity of Mn^ZnC (all o y 2) plus Calorimeter. ON •p-The low temperature anomaly had a very sharp peak, dropping to zero over only about a 3°C i n t e r v a l . This anomaly thus was very close to the theo r e t i c a l form predicted from the Weiss theory, which has a discontinuity at the Curie temperature. Such an approach to dis c o n t i n u i t y has not been observed before. The height of the anomaly was 5 . 5 j/°C for 0 . 2 1 2 moles, or 2 . 1 cal/°C/g atom of Mn. • c* Magnetic Specific Heats and Entropies; The anomalous s p e c i f i c heat caused by the magnetic change at the Curie point may be found by w r i t i n g the observed s p e c i f i c heat a s : ^ C q + (C p - C v) + C i + S sp e c i f i c heat at constant volume from Debye theory, correction f o r d i l a t a t i o n excess over the Debye value common to most metals, excess caused by magnetic and other changes. r In order to use t h i s procedure, however, measurements over a large temperature range must be made. In p a r t i c u l a r , s p e c i f i c heats near 0°K must be found i n order to calculate the electronic and the l a t t i c e s p e c i f i c heats. Consequently, because of the li m i t e d temperature range of these experiments, and because of further uncertainties a r i s i n g at higher temperatures, t h i s method was not suitabl e . The magnetic s p e c i f i c heats i n the present research were thus approximately determined by extrapolating the observed s p e c i f i c heat curves from above and below the anomalies. The results are shown i n Figures 1 9 and 20. In Figure 1 9 the magnetic s p e c i f i c heat MCJI of MnaAlC (from Figure 1 5 ) i s plotted, as w e l l as the values of -o'dcr/dH obtained from the magnetization Co ° where Cq = C p - C v - Ci « S = T(°C) F i g . 19. Magnetic Specific Heat of Mn3AlC. 200 220 240 260 280 300 320 340 360 T°K F i g . 20 Magnetic Specific Heat of Mn3ZnC (Alloy 2). - 68 - curve (Figure 12). (M = gram molecular weight, Cj/j = magnetic s p e c i f i c heat/gr.). A curve of M C M / T versus T i s also plotted, which when integrated gives the entropy change associated with the magnetic change; AS = JUL CM dT/T = 1.8 j/mole/°K = 0.15 caL/g. atom Mn/°K. The magnetic energy, U = £MCjidT = 460 j/mole = 37 cal/€» atom Mn. Figure 20 shows the magnetic s p e c i f i c heat of Mn3ZnC ( a l l o y 2,from Figure 1 8 ) , and a curve (M CJ J /T, T). The entropy changes corresponding to the low and high temperature anomalies respectively wereAS^ = 1.7 ^/mole/°K = 0. 14 cal/g. atom Mn and AS2 » 1.3 j/mole°K = 0.11 cal/g. atom Mn/°K. The magnetic energies, U]_ = 385 j/mole = 31 cal/g. atom Mn and U2 0 420 j/mole = 34 cal/g. atom Mn. V. DISCUSSION AND CONCLUSIONS 1. Discussion of Results The s p e c i f i c heat curves of Mn3AlC showed a second order anomaly at i t s CurieI point (-9°C), as expected for t h i s ferromagnetic a l l o y . The curve f o r Mn3ZnC showed second order anomalies at -35°C and at 65°C, which supported the previously discussed (Section I , 2) concept of the magnetic structure of the a l l o y : i t i s ferrimagnetic below -35°C, ferromagnetic between -35 and 65°C and paramagnetic above 65°C. a. Weiss Theory: The experimental r e s u l t s w i l l be discussed i n terms of e x i s t i n g magnetic theories. One of the f i r s t ferromagnetic theories was the phenomeno- l o g i c a l 'molecular f i e l d ' theory of Weiss,^ the quantum modification of which i s i n good agreement with experimental r e s u l t s . Weiss made the important assumption that the elementary magnets of a ferromagnetic substance are under - 68 a - the influence of an effective magnetic f i e l d , He, which i s the sum of the applied f i e l d , H, and the molecular f i e l d , NI, which i s proportional to the magnetization, I. The proport i o n a l i t y constant, N, called the molecular f i e l d constant, i s a measure of the exchange forces' acting between the atoms of the substance. The magnetization can then be evaluated by using Boltzmann s t a t i s t i c s i n a manner analogous to the Langevin treatment of a paramagnetic gas. The magnetic moment of an atom i n the quantum notation i s u A <= J g ug where J = the resultant angular momentum quantum number of an atom = the sum of o r b i t a l (L) and spin (S) quantum numbers, g = gyromagnetic r a t i o = 2 for spin angular momentum only (« 2 f o r ferromagnetic materials experimentally) Ug = the Bohr magneton, the magnetic moment of a single spinning electron = eh/4lTmc = 9.27 x 10~ 2 1erg/gauss. Let a » uAHe/KT where K = Boltzmann"s constant and the effec t i v e f i e l d , He= H + NI :. Then the average magnetic moment of the substance, u, i s deduced from; 3L = L. - d: S E M / J UA I 0 £ y~ ema/J m where I Q =' magnetic moment per unit volume at 0°K. Then I _ = 2J + 1 cothfej + l ) a - 1_ coth a_ I Q 2J 2J 2J 2J - 68 b „ 68 c This i s the B r i l l o u i n function. The Curie temperature, fi« (J+l)gu BNI 0/3K . For j -» 1/2 (a single electron spin), I = tanh u AH g = tanh a Io KT If H = 0, I _ - tanh I / I a I 0 V& where © = u ANI 0/K 17 This curve agrees w e l l with experiment. The saturation magnetization ( I / I Q , curve calculated for J = 1/2, and the experimental curves f o r Ni and Mn 3AlC are plotted i n Figure 21. The calculated curve i s quite close to experimental re s u l t s f o r Ni , but d i f f e r s considerably from the observed Mn3AlC r e s u l t s . I t should be noted that i n general the curves observed f o r alloys are less concave to the T/Q axis than the curves f o r pure m e t a l s . 2 0 ( Stoner's c o l l e c t i v e electron theory gives results close to the Mn 3AlC curve). Using UgNI0/K = 264°K, the molecular f i e l d constant f o r Mn 3AlC, N = 6600 gauss cc/erg. The molecular f i e l d , N I Q = 3.9 x 106 gauss. For ferromagnetic Mn3ZnC, N = 7400 gauss cc/erg. NI = 5.1 x 10^  gauss, assuming the saturation moment o at 0°K to be 1.5 u B per Mn atom. From the B r i l l o u i n function, the molar s u s c e p t i b i l i t y above the Curie point i s % = ° M = (J+l) g UB S^k = °M "TT 3K(T -©) f T © - 68 d The Curie constant per mole,Cj/[ - (J+l)gug G£M where C ^ i s the magnetic 3K moment per mole with a l l elementary magnets aligned p a r a l l e l (at 0°K). From paramagnetic data f o r Mn3ZnC, N = 0K = 2700 gauss 2 cc/erg, where M = gram molecular weight. This result i s not very r e l i a b l e because of the experimental d i f f i c u l t i e s of paramagnetic measurements (especially zinc evaporation).* The value of C^ could not be determined f o r Mn3AlC because of the curvature of the (1, T) curve, X For best o v e r a l l agreement with ferromagnetic and paramagnetic data, J i s between 1/2 and 1. The calculations of magnetic s p e c i f i c heat based on the Weiss theory have been summarized i n Section l i b . The magnetic s p e c i f i c heat per unit mass, . 1 dT Then i f the B r i l l o u i n function i s used to evaluate G^j, the r e s u l t i n g s p e c i f i c heat anomaly r i s e s continually with increasing temperature to the Curie point, where i t drops discontinuously to zero. The form of the curve i s shown i n Section l i b . I f the gyromagnetic r a t i o , g = 2, the magnitude of the discontinuity per gram-atom at the Curie point i s AA C^ = 3 RnQ/2 f o r J = 1/2 and 2 Rn Q f o r J = 1 (where n Q = the number of Bohr magnetons per atom and A = atomic weight). The general form of observed magnetic s p e c i f i c heat anomalies i s i n agreement with these r e s u l t s , ' ' The differences are that the observed drop at the Curie point extends over a range of temperature (from 5-100°C), A However, the values of N calculated from ferromagnetic and from para magnetic data usually do d i f f e r considerably. - 68 e - attributed to l o c a l ordering above the Curie temperature, and that the observed curves r i s e more rapi d l y than the calculated ones below the Curie point. The decreases i n CJJ at the Curie temperature f o r Fe, Co and Ni are i n f a i r agreement with theory f o r J = 1/2. Because of the difference between the t h e o r e t i c a l magnetization curve and that observed f o r Mn 3AlC, the observed s p e c i f i c heat anomaly f o r t h i s a l l o y can not be i d e n t i c a l with the anomaly predicted f o r J «* 1/2. The values of Cdtf/dT taken from the magnetization curve f o r Mn3AlC are plotted against T i n Figure 19. I t i s seen that the r e s u l t i n g curve i s much f l a t t e r than the (Cjj^ T) curve. Thus the value of the molecular, f i e l d constant N i n C M D -UpoUtf/dl must be changing rather r a p i d l y over the measured temperature range. This i s to be expected, since the very s i m i l a r a l l o y Mn3ZhC experiences a t r a n s i t i o n from f e r r i - to ferromagnetism i n t h i s region, which en t a i l s a reversal of sign i n the molecular f i e l d . The anomalies observed f o r Mn3AlC and Mn3ZnC (Figs. 15, 16, 18, 19, 20) were t y p i c a l of magnetic anomalies reported i n the l i t e r a t u r e . • > i 1 The anomaly heights were R/2 per Bohr magneton f o r Mn 3AlC and 0.2|R per Bohr magneton f o r the ferromagnetic Mn3ZnC anomaly, as compared with the t h e o r e t i c a l 3 R/2jalso observed for pure metals. The low temperature Mn3ZnC anomaly was quite steep, i n d i c a t i n g a rather abrupt t r a n s i t i o n from ferrimagnetism to ferromagnetism. The usually observed ' t a i l i n g off* above the t r a n s i t i o n point was absent, but was perhaps masked by the ferromagnetic anomaly. b. Entropy Changes; The t o t a l entropy associated with a ferromagnetic anomaly may be calculated from the p a r t i t i o n function, Q, evaluated above the Curie temperature, - 68 f - J using the previously mentioned notation , m=-J The free energy, F = -RTlnQ The entropy, S = -$F * T Thus for J = 1/2 and the e f f e c t i v e magnetic f i e l d , H e •= 0 (T> 0 ) , S « Rln2 per gram-atom. In general, S = Rln(2j +1). The ferromagnetic t r a n s i t i o n may also be considered i n terms of a simple model, i n which each atom has one electron spin (S = 1/2) capable of an orientation p a r a l l e l or a n t i p a r a l l e l to a given d i r e c t i o n . At absolute zero temperature, a l l spins are oriented p a r a l l e l , giving zero entropy. Above the Curie temperature, the electrons w i l l represent a paramagnetic gas, with half oriented p a r a l l e l and half a n t i p a r a l l e l . Then the entropy i s . S = Kin N e i N e i „NeS. f o r j j e electrons. 2 2 therefore S = Rln2 (to a good approximation f o r one gram-atom) At temperatures approaching absolute zero, the magnetic s p e c i f i c heats (and thus the entropy changes) involved i n a magnetic t r a n s i t i o n can be determined quite accurately, since the s p e c i f i c heats at constant pressure and at constant volume are almost i d e n t i c a l , and the l a t t i c e s p e c i f i c heat i s quite small. The sp e c i f i c heat curve f o r the t r a n s i t i o n from the a n t i f e r r o - magnetic to the paramagnetic state of CuCl 2.2 H 20 has been found by Friedberg to have the t y p i c a l sharp peak at the t r a n s i t i o n point (4.3°K). _2 But above t h i s temperature i t t a i l e d off according to a T low, probably because of short range order. The magnetic entropy change below the N l e l temperature was 0.45R and i n the t a i l was 0.20R, the sum of which i s close - 68 g - to the t h e o r e t i c a l Rln2 = 0.69R = 1.4 cal/g»atom/deg. expected. Similar 22 results have been obtained f o r other paramagnetic s a l t s . For ferromagnetic materials at elevated temperatures, the results are however f a r from sat i s f a c t o r y . The d i f f i c u l t y i n separating the magnetic part of the s p e c i f i c heat i s undoubtedly responsible f o r the error involved i n calculating the entropy change. The change corresponding to the magnetic anomalies of some ferromagnetic substances has been calculated by the author' from s p e c i f i c heat curves found i n the l i t e r a t u r e . For n i c k e l , S t o n e r 2 3 has used the data of several researchers to separate c a r e f u l l y the magnetic part of the s p e c i f i c heat. The corresponding entropy change was AS = 0.5 c a l / g.atom/deg. ( A S = 0.3 f o r only the peak of the curve, neglecting the long low temperature t a i l ) . The expected value i s 0.83 i f six-tenths of the atoms have J = 1/2 and the rest have J = 0. I f , however, as predicted by Mott and Jones ' J « 0 f o r 70 percent of the atoms and J =» 1 f o r 30 percent of them, the expected value i s 0.3 Rln3 = 0.66. This approach assumes that the i n d i v i d u a l electrons are not free i n the paramagnetic state, and results from t i g h t bonding theory. The experimental r e s u l t f o r i r o n 2 ^ was AS»2.0 cal/g.atom/ deg. The expected value i s at least 2.4 (for J = 1). These experimental values are considerably less than the t h e o r e t i c a l , but are reconcilable with i t i f a rather large degree of l o c a l ordering not observed i n the form of a magnetic s p e c i f i c heat i s assumed present above the Curie temperature. A theory of 'constant coupling' f o r Heisenberg ferromagnetism has 26 been proposed by Kasteleijn and Van Kranendonk to explain the presence of short range order above the Curie point. The entropy above the Curie point i s then calculated to be 0.31 Rln2 f o r coordination No.6 and 0.12 Rln2 f o r coordination Ho. 12. These values have the order of magnitude of the above mentioned discrepancies. - 68 h - It i s seen that even after very c a r e f u l calculation of the magnetic s p e c i f i c heat anomaly, only q u a l i t a t i v e l y correct results f o r the entropy change can be obtained f o r t r a n s i t i o n s at elevated temperatures. The results obtained f o r Mn 3AlC by the author ( A S = 0.15 ca]/g«atom/deg.) are indeed very low and indicate that f o r accurate results a larger tempera ture range,and i n p a r t i c u l a r measurements at low temperatures to determine the Debye constant,are required. The entropy changes observed for each of the anomalies of Mn3ZnC (A>S5» 0.15 cal/g.atom/deg.) are s i m i l a r l y very low. The t o t a l entropy ,change for both anomalies should be 3/2 Rln2 per gram-atom of manganese, i f J = 1/2 for the ferromagnetic state. The entropy change f o r the upper t r a n s i  t i o n from -33°C to 100°C (for T/&>0.70), assuming the t h e o r e t i c a l (C M, T) curve for J = 1/2, i s (0.4l)3/2R.„ Thus the entropy change f o r the low temperature anomaly i s 3/2 R(0.28) = 0.8 cal/g.atom/deg. This i s of course only a crude estimate, since l o c a l ordering above the ferromagnetic Curie temperature, as w e l l as other factors, were not taken into consideration. Nevertheless, i t i s apparent that the experimental results f o r Mn3ZnC are not of the correct order of magnitude. I t i s feasi b l e that accurate s p e c i f i c heat measurements over a large temperature range would make possible a precise determination of the entropy change and consequently a v e r i f i c a t i o n of the low temperature magnetic structure of Mn3ZnC. To do t h i s , a detailed theory of the degree of disorder i n the ferrimagnetic and ferromagnetic structures, and the degree of order i n the paramagnetic state would be necessary. But d i f f i c u l t i e s would c e r t a i n l y be encountered i n the overlap of the two anomalies, and the rapid r i s e of the l a t t i c e and electronic s p e c i f i c heats i n the anomalous region make an accurate separation of the magnetic s p e c i f i c heats doubtful. The object of - 68 i - the present research in any case was merely to determine the presence of the two anomalies, and the small experimental temperature range permitted no accurate entropy change calculat ion. However, the determination that transit ions from ferrimagnetism to ferromagnetism to paramagnetism occurred i s in i t s e l f s ign i f icant . Calculation of the magnetic anomaly of a ferromagnetic substance i s further complicated by the poss ib i l i ty that the drop in the magnetic spec i f i c heat at the Curie point may be masked by a sudden increase in the speci f ic heat caused by electronic d is t r ibut ion . This result is produced by I J^ie col lect ive electron theory of „ Stoner. 2 ^ He assumed that ferromagnetism was caused by holes i n the 3d band, which was assumed parabolic near the Fermi l i m i t . He also assumed that the exchange energy varied as the square of the relative magnetization (as did Weiss), and that the part ic les obeyed Fermi-Dirac s t a t i s t i c s . Then i t was found that at the Curie point a drop i n magnetic speci f ic heat of A C M / R ° - 1 . 8 was compensated by a r ise in electronic speci f ic heat of ACj /R ° 1 . 2 , so that the resultant discontinuity was only AG /R = - 0 . 6 . This is not however in agreement with experiment for many ferromagnetic substances. By making the exchange energy also vary with higher powers of the magnetization this d i f f i c u l t y was overcome, and also the 2 8 magnetization curves were in better agreement with experiment. c. Theories of exchange interactions i In order, to f u l l y appreciate the phenomena of ferro- and f e r r i  magnetism, the cause of the Weiss intermolecular f i e l d must be considered. 2 9 Heisenberg or ig ina l ly explained i t by exchange interaction between electrons of neighbouring atoms in terms of the Heitler-London method of local ized atomic wave functions. - 68 j - S l a t e r 3 0 extended t h i s theory to apply to ferromagnetic materials• He assumed that a positive exchange i n t e g r a l (and thus ferromagnetism) resulted from exchange between adjacent 3d shells when the r a t i o of i n t e r - nuclear distance to d s h e l l diameter was larger than a certain value. Zener, 3^ however, stated that d-d coupling always gives a negative exchange i n t e g r a l (producing a n t i p a r a l l e l spins), and that ferromagnetism i s caused by a positive exchange int e g r a l between conduction electrons and incomplete d s h e l l s . (In his calculations he used l o c a l i z e d atomic wave functions f o r the incomplete d s h e l l electrons and band wave functions f o r the outer s electrons). Zener's theory i s u s e f u l for Heusler alloys and f e r r i t e s , but some disagreement with neutron d i f f r a c t i o n data has been noted. S l a t e r 3 2 has proposed that to overcome d i f f i c u l t i e s of non- orthogonal wave functions of the Heisenberg method, determinantal wave functions composed of orthogonal energy-band o r b i t a l s should be used.. I f a si n g l e deter minantal wave function i s used, the energy-band or c o l l e c t i v e electron theory r e s u l t s , which i s quite useful f o r small internuclear distances. However, f o r complete accuracy a l l possible l i n e a r combinations of possible determinantal wave functions i£tj®t ^ be made. This however e n t a i l s an enormous amount of calculations, and makes p r a c t i c a l applications d i f f i c u l t . The d i f f i c u l t y i n explaining the presence of ferrimagnetism and antiferromagnetism i n terms of exchange energies i s more profound. One approach i s the consideration of exchange interaction between excited valence states of 33 cations of the same t r a n s i t i o n element (super exchange). The completely general theory of S l a t e r 3 2 i s also c e r t a i n l y t h e o r e t i c a l l y applicable to t h i s problem. - 68 k - 2„. Cone Ins ions, a. The Calorimeter, AD> anssoid? adjabatic calorimeter was cbnstructed i n order to measure s p e c i f i c heat anomalies of certain magnetic a l l o y s between -150 and 150°C, The d i f f i c u l t i e s of accurate temperature and heat input measurement were over come by using a strain-free platinum resistance thermometer-heater. The accuracy of the calorimeter was 0.5$$. the use of an accurate bridge f o r resistance measurements would increase i t to 0.2%, b. Sp e c i f i c Heat Measurements. Sp e c i f i c heat curves were successfully measured f o r the a l l o y s Mn3AlG and Mn3ZnG. The expected second order s p e c i f i c heat anomaly was observed f o r the former at i t s ferromagnetic Curie point, -10°C, The presence of two second order anomalies on the s p e c i f i c heat curve of Mn3ZnC supported the idea of a complex magnetic behaviour f o r the alloys i t i s ferrimagnetic below «35°C, ferromagnetic between, -35 and 65°C and paramagnetic above 65°C0 The observed anomalies were q u a l i t a t i v e l y i n agreement with the theory of Weiss. Accurate separation of the magnetic s p e c i f i c heats,, which would make a quantitative t h e o r e t i c a l interpretation of the r e s u l t s f e a s i b l e , was impossible because of the l i m i t a t i o n s of the experimental data* - 69 - VI. APPENDICES Appendix I Calculation of Effect i v e Magnetization of Mn Atoms. The a l l o y M n 6 0 A l 2 0 C 2 o has a magnetization ofy"^ = 1,22/tg per Mn atem.. The a l l o y Mng/fAla6C2Q , i n which 4 atomic percent of Mn atoms have replaced A l atoms, has an average magnetizationyu. 0 = 0.8°>*B per Mn atom. The decrease i s assumed caused by the extra c (=4) atomic percent Mn atoms, with a magnetization ofyU. 2, which i s assumed independent of then (c + 60)^uc - 6 0 ^ + c/* 2 ^uz » 0.89 + 6 0 ( 0 . 8 9 - 1.22)/4 ^uz - - 4 . 0 6 / < B Thus, i f replacing A l atoms by Mn atoms in.the a l l o y Mn 3AlC does not change the magnetization of those Mn atoms, already present, the r e s u l t i n g decrease i n magnetization i s explained by assuming the extra Mn atoms have a magnetization of - 4 / C g * Appendix I I A n t i p a r a l l e l Spin Systems 1» General Theory of A n t i p a r a l l e l Spin Systems : A ferrimagnetic or antiferromagnetic substance consists of two or more sublattices with a n t i p a r a l l e l spin systems (with opposing magnetization vectors). Consider two sublattices, A and B, and l e t the f r a c t i o n a l volume of A and B atoms be A and ju- (*+_/*.= 1), The magnetizations of the sublattice atoms are I A and Ig, so that the resultant magnetization i s I - M A +>u.IB (!) The i n t e r n a l W eiss molecular f i e l d s , H A and Hg, acting on A and B atoms depend on I A and Ig : H A - N ( a A l A ->«.1B) (2) H*B - N(B>u.I B - A I A ) (3) H A = N(aAl A + x.I B) =|H A| (4) Hg = N(B>cI B + 7vI A) . (5) since I A i s i n the opposite d i r e c t i o n to Ig . N i s the Weiss intermolecular f i e l d constant and a and B represent the strength of the effect the sublattices have on t h e i r own molecular f i e l d s . And, according to Curie, f o r paramagnetism: * I A - C (H + H A) , (6) T and Ig = C (H + Hg) (7) T C i s the Curie constant and 5 i s the applied f i e l d . The effective f i e l d , H e = H + H A or H + Hg thus Solving equations ( l ) , (2), (3), (6), and (7): H » 1 - T + 1_ - <f i x c Xo T -e (8) where X = s u s c e p t i b i l i t y , Xo " c -n- *• e cT = -v-eLn-t-e) c 6 - NGx?v(2 + a + B) and = - N C ( / L B +>a) V - N 2c2«A(aB - 1) This i s the equation f o r the paramagnetic behaviour of a ferrimagnetic substance. I t i s a hyperbola with curvature concave to the temperature axis, Ne*el predicted several types of v a r i a t i o n of saturation magnetization with temperature f o r ferrimagnetic substances. Three of these are as shown: 2, Energy of Magnetization of. A n t i p a r a l l e l Spin Systems: The absolute value of the e f f e c t i v e f i e l d acting on A or B atoms i s : H e «. H A or Hg Therefore the energy of magnetization per unit mass, = l A d l A + ^ l B d l A + - ^ ^ B ^ B + I A D I B J from equations (4) and (5) , For an antiferromagnetic substance, > I A = y t l g ,' since 1 = 0, and a = B , since H A = Hg thus U = - Ill > 2 ( a + B + .2) U = - N I A 2 > 2 ( a + 1) This value i s the same as that of a ferromagnetic substance of magnetization I. (see equation (4) of I, 1) i f 2 ^ 2 ( a + 1) = 1. This would be the case i f n a = 1 and 7* = 1 , which would occur f o r a simple l a t t i c e i n which A and B atoms 2' . . • . were i d e n t i c a l except for a n t i p a r a l l e l spins, a n d they occupied equivalent l a t t i c e s i t e s . Assuming the ferrimagnetic postulated behaviour of Mn-Al-C a l l o y s , the magnetization energy can be calculated f o r an a l l o y Mn^AljeCap, i n which four out of every 64 Mn *B* atoms have a spin of -4, a n t i p a r a l l e l to the spin of 1.2 of the other 60 *A» atoms: h - J L _ I a - i° i . 1.2 A - 3 A . - 60 1 3 " therefore ^ - I n = 2 y^j 9 - 73 - :-hus 0 U« » - S H / T £*2 + §*2 + + fx 2] If o - B - 1 , U' ~ - 2 N j 2-s 4 / A * N £A£ 2L Vsia * 4 B + 3ol J5" 2 81 L J 2-v2 Since Mn3AlC i s ferromagnetic, then i t s magnetization energy, U«« = - N P j 2 (equation (4) of Section I, l ) . 2 But/><r = ^ I A , since the magnetization of the Mn atoms i n Mh 3AlC is* assumed equal to 1^ , the magnetization of the Mn atoms of the A sublattice i n Mn64Al|6C2o» X i s the same f o r both a l l o y s , because the Mn and A l atoms have about the same diameter. Thus, U»« = _ NT2 I A 2 = 2 u» 2 / 3 The t o t a l magnetization energy, IP', i s only 2 as great as for the ferrimagnetic 3 state ( i f N i s the same i n both cases). Appendix I I I Assembly of the Thermometer Mica Cross The mica cross was assembled as follows: mica sheet 0.003 inch t h i c k was clamped between smooth s t e e l bars and cut with a razor blade into 0.20 by 1.1 inch pieces. Two pieces were slotted half way so that they could f i t together lengthwise at right angles, forming a cross. A die formed from four small s t e e l bars was used f o r the s l o t t i n g . Small holes were d r i l l e d through the mica at one end to permit anchoring of the leads. The pieces of mica were notched along each side by placing them i n a grooved die and cutting out the mica with 0.008 i n . piano wire held i n a metal bow. The die was made by machining 0.30 i n . of 3/8'* brass rod down to 0.2 i n . diameter and then threading at 29 per inch. The threaded part of the rod was then s p l i t by cutting lengthwise. One half of the s p l i t rod was removed and attached to the other half with screws at either end, so that the mica could be held f i r m l y between the halves. 1.1" Notching Die Notched Mica The mandrel used to hold the assembled mica cross was constructed by soldering together four bars of 3/16 inch brass and machining down to a_ 0.150 7 5 - diameter rod. The platinum h e l i x was wound non-inductively on the mica cross and the two leads were fastened through the holes d r i l l e d f o r the purpose. - 76* - Appendix IV Construction of the Thermometer Pyrex Case The thermometer case was made by f i r s t drawing out 1 inch diameter pyrex glass tubing i n a broad oxygen-gas f3ame to about 0,28 inch diameter, i n order to obtain thin-walled tubing. This tubing was necked s l i g h t l y at the end where the leads were to emerge. In order to make a leak-tight seal of the platinum leads through pyrex glass, a special technique was u s e d . ^ Platinum s t r i p s , r o l l e d from 0.010 inch wire to less than 0,001 inch t h i c k , were used f o r the seal. Because of t h e i r thinness and the r e s u l t i n g feathered edges, these s t r i p s give a vacuum-tight seal through pyrex. To fuse a s t r i p to the 0,003 inch platinum thermometer wire, the s t r i p was folded over the end of the wire, and the wire was fused to the corner of the f o l d with a small oxy-gas flame. A micro-burner i s not needed f o r t h i s flame i f pincers are used, as shown,, to prevent melting of the f i n e platinum wire. Torch Pincers After the platinum s t r i p s were fused to the leads, the thermometer c o i l was pushed into the pyrex tubing and the two s t r i p s were bent over the necked end. The s t r i p s were cleaned with acetone, and f i r e polished at near the P! Ug melting point with a gas flame. A plug, made from pyrex rod, was placed i n the tube end, and fused with a soft flame to p the tube and about 1/8 inch of the uross platinum s t r i p s . The seal was cooled slowly i n a yellow flame to remove s t r a i n s . The seal was tested i n a vacuum system, after being Immersed i n l i q u i d oxygen. Appendix V Calculation of Power Change for the Thermometer-Heater (see Figure 8b) Let R t o * t i - R t G + * R t E 'v R . P V V l the i n i t i a l heater resistance, the f i n a l heater resistance, the constant voltage supply, the resistance of the c i r c u i t minus heater resistance, i n i t i a l , f i n a l voltages across heater. The i n i t i a l current, I Q = The f i n a l current, 1^ ,= E R + R t G  E R + Btx R + V V 0 I 0 = Rt Q*o = Po » t n e i n i t i a l power through heaters the f i n a l power, P^ E' t • R ^ r *' [n + R ^ j £i . »ti / R + R t Q P 0 R t Q ^R + For R ^ R^, as with the 100 ohm thermometer (Figure 8a): f o r a 10°C temperature r i s e , P n R t n " - 78 - For R = R. , as with the 20 ohm thermometer (Fig. 8b): ^o P i - * t i UH02 = 4 R t o ( R t o *A R t ) P 0 R t 0 ( R t Q + R t l ) 2 (2 R t G + A R t ) ^ For a 10°C temperature r i s e , A R t * .04 R t D therefore P i = h_ R t Q 26 R t o = 4 x 26 TQ 25(2 R t ( j + .04 R t Q ) a (2.04)* x 25 £l = i t l 6 0 = 0.9995 P 0 4162 The percent difference i s 0.05% f o r ?1 „ 1, R = 1.02 R t Q i f A R t = 0.04 R t o That i s , f o r no power change, R must be half way between Rt and R^ - 79 - Appendix VI Equivalence of Resistance-Temperature Equations f o r a Platinum Resistance  Thermometer. The general equation connecting resistance and temperature f o r a platinum resistance thermometer i s s Rt - R 0 ( l + At + B t 2 ) fo r 0 < t < 630°C. thus t - % - Ro 100 = t - At + Bt 2_ 1 0 0  R100 ' R ° 100A + 1002B = lOOBt - B t 2 A + 100B therefore t - ^ "* Ro 100= R100 - R o 100 2 B |~ t - / t f A + 100B 100 ( 100J according t o Callendar's equation. 100 (lOOj 2 therefore £ = - 100 B A + 100B » 1.5 f o r pure platinum For temperatures between -183 and 0°C, Rt - R 0 | l + At + B t 2 + C(t - 1 0 0 ) t 3 ^ t - % - R p 1 0 Q m t - At + Bt 1 + C(t - 100)t 3-| m  R l 0 0 " , Ro 100A + 1002 B • lOOBt - B t 2 - 0(t - 100)t 3 A + 100B t - R t - Rp 1 0 Q = _ 100 2B f / t \ 2 -- t "J- C , t . 1 0 0 ) t 3 R i o o - Ro A + 1 0 0 B Lv™i A + I O O B ^ L U O ; T - 80 - The l a s t term may be written: + D ( t - I00)t3 where D = - C  A + 100B Thus i n Callendar form, the equation i s : t = ^ ~ Ro 100 + £ [(O.Olt) 2 - O.Olt] + D(t - 100)t 3 R100 " Ro - 8 1 - Appendlx VII Error i n Temperature Interval Caused by Error i n Calibration of Resistance  Thermometer. Above 0°C, the temperature of a resistance thermometer i s : t - - Ro 1 0 Q + ^ [ ( . o i t ) 1 - .oit] R100 ~ Ro The i n t e r v a l between two temperatures, t - t x = A t = * t ~ R t l 1 0 0 +cf[7.01t)2- (.Oltx) 2 + .Oltx - .OltJ R100~ Ro - A R t 1 0 0 + <r[(.oi)\t + t x ) A t - .oiAt] R i o o - R Q therefore At ° ARt 100 1 _ R 100 - R D 1 - £ [ ] . 0 l ) 2 (t + t i ) - , 0 l | = A St 100 rx + $ f t . 0 l ) 2 ( t + t l ) - .OlT R i o o -. R 0 ^ L J + £ ^ . o i ) 2 ( t + t x ) - .oi|2+ ....^ for 0 < t < 100, £ 2 [ ( . 0 l ) 2 ( t + t i ) - . 01 ] 2 < (,oi$) 2< .0004 so that this term is negligible i n comparison with unity. therefore A t = A*t 100 / l + ^ | T . 0 l ) 2 ( t + tt) - . 0 l 1 \ R 1 0 0 " R o I L J J With a different calibration, A t » - ARt loo / i + l~(.oi) 2(t + H ) ' - .of! I (Rioo - Ro)' (. L 1 Jf Thus the error i n temperature i n t e r v a l , (i) At - At« * A n ^ f iQQ - 100 + iQQSH - 100 o'Z-T \ \R ioo - Ro (Rioo-Ro)* Rioo- Ro (Rioo- Ro)' J I f 5 = 8» and % = 1Q0°C. A t - At» » AR . 1 0 0f 1 - 1 + 0.018 _ O.OlS ^ \ 1 Rioo - R 0 (Rioo - Ro)' Rioo - R o (Rioe-Ro)'j - AR+ 102 / i - i \ , (Rioo-* Ro (Rioo - Ro)' J i f 5 - 2 . 1*100" RQcs 0.4 R 0 f o r platinum • put (Rioo - Ro)* ** ( Q . 4 + 6 ) R 0 . In a 10°C. temperature i n t e r v a l , A R t ^ »04 R Q therefore A t - A t * - .04 R a 102/ 1 - 1 \ ( 0 . 4 H o (0.4 +€ )R 0 J " 4.08 € 0.4(0.4 + € ) I f (r= 0.002 (eg. i f ^ 1 0 Q - R Q changed by 0.2 ohms for the 100 ohm thermometer: a very large change),. A t - A t » " 4.08 x 0.002 = o.051°C. 0.4 x 0.402 Thus the error i n temperature i n t e r v a l would be: A t - A t ' = 0.05 ioo = 0.5$ A t 10 - 83 - This error i s considerably less than the error i n temperature caused by a change i n R]_oo~ RQ of 6 R^ i f R Q remains the same, R^ OO Ganges by € R Q. Thus the temperature error would be € R Q l o c. 8 3 0.5°C at 100°C for 6 = 0.002. 0.4 R 0 The error i n temperature i n t e r v a l caused by the change i n c a l i b r a  t i o n observed for the commercial thermometer (see Section I I I , Id), f o r AR^ = 4.00 at 100°C, was A t - A t * = 4.00(2.5786 - 2.5740 + 2.5786 x 2.17 x .01 - 2.5740 x 1.47x.0101) = 0.09°C (using Equation l ) Since A t •» 10 ° , the error was 0.9$. This error would be less f o r any temperature between 0 and 100°C. - 84 - BIBLIOGRAPHY 1. F. S e i t z , 'The Modern Theory of So l i d s ' , McGraw-Hill, 1940. 2. J . Roberts, 'Heat and Thermodynamics * ^  Blackie. ' 3 . E. Lapp, Ann. Physique 12, 442 (1929). 4. R. Butters, and H. Myers, P h i l . Mag. 46_, 895 (1955). 5. R. Butters, and H. Myers, P h i l . Mag. 46, 132 (1955). 6. B. Brockhouse and H. Myers, Canadian J. Phys. 3J>, 313 (1957) 7. W. White, 'The Modern Calorimeter', The Chemical Catalog Co. 8. E. G r i f f i t h s , 'Methods of Measuring Temperature*, G r i f f o n 1947. 9. R. Weber, 'Temperature Measurement and Control', BlakLston 1941. 10. Southard, Brickwedde, J . Am. Chem. Soc. ,55, 4378. 11. A. Reddoch, Master of Science Thesis, Queen's University, Aug. 1954. 12. J . Southard and D. Andrews, J . Franklin Inst. 209. 349 (1930). 13. C. Meyers, Bureau of Standards J . Research £, 807 (1932). 14. N. Rasor, Rev. S E L I n s t r . 2j>, No. 4, 316 (1954). 15. D. Ginnings.and G. Furukawa, J . Am. Chem. Soc. 75_, 522- (1953). 16. K. Grew, P.R.S. 14j>, 509 (1934). 17. R. Bozorth, 'Ferromagnetism', Van Nostrand, 1951. 18. L. Neel, Ann. Phys. 3_> 137 (1948). - 85 - 19. P. Weiss, J. Phys. (4) 6, 661, (1907). 20. J. Went, Physica, 1£, 98, 596 (1951). 21. S . Friedberg, Physica 18, 714 (1952). 22. J, Daniels, F. Robinson, Phil. Mag. (7), 4Jt, 630 (1953). 23. E. Stoner, Phil. Mag. (7), 22, 81 (1936). 24. N. Mott, H. Jones, 'Theory of Properties of Metals arid Alloys', Clarendon Press 1936. 25. J. Awberry, E . Griffiths, Proc. Roy. Soc. (London), 174A. 1. 26. P. Kasteleijn, J. VanKranendonk^ Physica 22, 317 27. El Stoner, Proc. Roy. Soc, (London) A169. 339 ( 1 9 3 9 ) . . 28. E . Wohlfarth, Rev. Mod. Phys. 2j5, 211 ( 1 9 5 3 ) . 29. W. Heisenberg, Z. Physik i£, 619 (1928). 30. J. Slater, Phys. Rev. 3j6, 57 (1930). 31. C Zener, Phys. Rev. 81, 440 (1951); 83. , 299 (1951). 32. J. Slater, Rev. M o d . Phys. 2^, 199 ( 1 9 5 3 ) . 33. J. Van VLeck, Grenoble Conference, 114 (1951). 

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