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

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F a c u l t y  o f  G r a d u a t e  P R O G R A M M E  O F  S t u d i e s  T H E  FINAL ORAL EXAMINATION FOR THE DEGREE OF  DOCTOR O F PHILOSOPHY of  MAX  LYNN  SWANSON  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. COMMITTEE IN CHARGE D E A N G . M . SHRUM,  W. M. ARMSTRONG F. A. FORWARD E. TEGHTSOONIAN B. MOYLES  Chairman  B. SAVERY W. O. RICHMOND G. A. McDOWELL R. W. STEWART  External Examiner: B. N. BROCKHOUSE Chalk River Ont.  THE  CONSTRUCTION OF AN ADIABATIC CALORIMETER  AND  ITS USE IN MEASURING SPECIFIC HEATS ABSTRACT  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 surrounding 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 extreme 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 thermometerheater 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 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 anomalies at - 35°C, and at 65°C. This double specific heat anomaly indicates, in agreement with neutron diffraction results, a complex magnetic behavior for the alloy. Although the high temperature Curie point anomaly did not have a sharp peak the low temperature anomaly's shape approached that of the theoretical Weiss curve.  GRADUATE  STUDIES  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 Bonding in Metals . . . X-Ray Diffraction  ..  H. Meyer J. Halpern --- J. Parr  1.  Other Studies: Theory of Measurements Quantum  W.Opechowski  Mechanics  G. M. Volkoff  Electromagnetic Theory  J. Brown  Nuclear Physics  K.Mann  Chemical Physics Group Theory  C. Reid -  - 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 t h e s i s as conforming t o the standard required from candidates f o r the degree of DOCTOR OF PHILOSOPHY.  Members o f t h e Department o f Mining and Metallurgy.  The U n i v e r s i t y o f B r i t i s h Columbia  December 1957-  ABSTRACT  A f l u i d l e s s a d i a b a t i c calorimeter was constructed and was used to measure the s p e c i f i c heats o f manganese^aluminum-carbon and manganese-zinccarbon a l l o y s from -150° to 150°C. In an a d i a b a t i c calorimeter, the temperature of a s h i e l d the calorimeter v e s s e l i s kept at approximately  surrounding  the same temperature as that o f  the v e s s e l , so that the thermal leakage between t h e 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 c a l c u l a t e d , can be e l i m i n a t e d .  Since leakage modulus v a r i a -  t i o n s are reduced by the a d i a b a t i c method, i t can be used f o r large temperature r i s e s , r e s u l t i n g i n f a s t and accurate measurements.  The aneroid  (fluidless)  a d i a b a t i c calorimeter eliminates s t i r r i n g and evaporation e r r o r s , 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 v e s s e l surrounded by an e l e c t r i c a l l y heated a d i a b a t i c s h i e l d and an evacuated outer case.  A platinum r e s i s t a n c e thermometer-heater was used t o supply heat  to the calorimeter v e s s e l and t o measure the v e s s e l temperature.' The heat input and the thermometer r e s i s t a n c e were measured by using a potentiometer i n conjunction w i t h standard r e s i s t a n c e s . " The thermometer was c a l i b r a t e d by measuring i t s resistance at -183, -40, 0, and 100°C.  The calorimeter was c a l i b r a t e d from -150 t o 150°C. The  accuracy of the calorimeter was approximately  0.5%, the main e r r o r a r i s i n g from  the method of measuring the temperature of the calorimeter v e s s e l . The s p e c i f i c heat curves of t h e s i n g l e phase magnetic a l l o y s Mn AlC 3  and Mn ZnC were measured. 3  A second order s p e c i f i c heat anomaly was found, as  expected, f o r the ferromagnetic a l l o y Mn AlC at i t s Curie p o i n t , -10°C. Although 3  the anomaly was close t o the t h e o r e t i c a l shape, dropping to zero over only a 10°C range at the Curie point, i t s maximum height was less than saturation magnetizat i o n measurements would i n d i c a t e . The a l l o y Mn ZnC showed second order s p e c i f i c heat anomalies at -35°C, 3  and at 65°C.  This double s p e c i f i c heat anomaly i n d i c a t e s , i n agreement with  neutron d i f f r a c t i o n r e s u l t s , a complex magnetic behaviour f o r the a l l o y .  Although  the high temperature Curie point anomaly d i d not have a sharp peak, the low temperature anomaly's shape approached that of the t h e o r e t i c a l Weiss curve.  In p r e s e n t i n g the  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  r e q u i r e m e n t s f o r an advanced degree at the  University  of B r i t i s h Columbia', I agree t h a t 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 r e f e r e n c e and agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may  study.  I further  copying of t h i s  be g r a n t e d by the Head o f  Department o r by h i s r e p r e s e n t a t i v e .  be a l l o w e d w i t h o u t my w r i t t e n  Department o f  Metallurgy  _  The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver 3, Canada. Date  May 7 ,  1958  my  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 g a i n s h a l l not  thesis  financial  permission.  ACKNOWLEDGEMENTS  The author i s g r a t e f u l f o r the advice and t e c h n i c a l assistance given by h i s research d i r e c t o r , Dr. H.P. Myers, and by R. Butters and R. R i c h t e r . He would a l s o l i k e t o thank L. Howe f o r h i s p r e l i m i n a r y reading of the t h e s i s . The work was c a r r i e d out w i t h the help o f Research Grant 281 provided by the Defence Research Board, and a Studentship from the N a t i o n a l ' Research C o u n c i l .  TABLE OF CONTENTS Page  I * INTRODUCTION  »eoeoo»BeQoeo«»oa«»oaoo«a»»eo»e«*o«»oo*a <>»«••• I  1. S p e c i f i c Heat Theory  .  2. The A l l o y s t o be Investigated  ........ .....6  1 .  3« C a l o r i m e t r i c Theory II.  12  THE CONSTRUCTION OF AN ANEROID ADIABATIC CALORIMETER  2. M a t e r i a l s and Detailed Construction III.  IV©  V.  ....  17  ........."..... 19 34  CALIBRATION AND PERFORMANCE 1. C a l i b r a t i o n of Resistance Thermometer  34  2. Heat Capacity of the Calorimeter  i*2  3. Accuracy of S p e c i f i c Heat Measurements  42  f££SUl*TS  oooooooeooooooooO4OOOooooooOO0O«o*oeoe«eoo«  $2  1. Preparation and Properties o f A l l o y s  52  2. S p e c i f i c Heat Measurements  56  DISCUSSION AND CONCLUSIONS  68  1. Discussion of Results 2. Conclusions VI.  6  APPENDICES  V I I . BIBLIOGRAPHY  ...... ..o O  68 .6 . . . . . . . . . . < > . . . , . . . . . . . . . . . .  68k 69 84  FIGURE INDEX Page  1. Perovskite Structure 2„  «  0  o a  «  . . . o o . .  .  «  a  a  o o o  .  .  7  .  10  V a r i a t i o n of Saturation Magnetisation, <S", with Temperature, T , • f o r Mn—Zn—0 A l l o y s  o « « o o  .  .  .  o  .  .  .  . a . . . .  o  3o  Possible Magnetic Structure at 0 ° K of Mn ZnC . . . . . . . «  O 0  11  4o  Basic Calorimeter Design  o o  IB  5-  The Calorimeter  3  « o . .  . . . . . . . . . . . .  . . . . .  o » . o .  .  o .  » . « • «  «  •  * » o o  6. Adiabatic S h i e l d Control C i r c u i t . . « . . . • . • , ' • • •' . . . 7o  Suspension o f the Calorimeter . . « . . . . . .  8.  C i r c u i t Diagrams f o r Thermometer-Heaters  9o  C a l i b r a t i o n Apparatus .  ID.  o o  .  .  .  .  .  o  a o  24  . . . . . •  a 0  27  o . .e  . o  31  • «  37  . . . . . o  .  .  .  .  .  0  .  .  .  .  Heat Capacities of Calorimeter and of Calorimeter plus  OOOOQOOOOOOO 0 O « «  AXUlffi Tift  o o 0  « o « 0 «  e  47  12.  The V a r i a t i o n o f Saturation M a g n e t i z a t i o n , © ' , with Temperature, . . . . o . .  .. o  . . . . .  .  44  V a r i a t i b h of Thermal Leakage Modulus, k, with Pressure  . ., T , f o r Mn3AlC  .  o  11.  13*  21  .  . ' » . . . ..  .  54  The V a r i a t i o n o f Saturation Magnetization,©" , w i t h Temperature, T, f o r Mn ZnC 3  o . o . . .'. • . . . .  . a a • • . * .  o a  55  •  a  58  . . < >. « o .. o . o . a a • a . . . e .  .  59  a  60  14.  Capacity of Mn AlC plus Calorimeter . . . . . . . . . • •  15.  Capacity of Mn AlC  16.  Capacity of Mn AlC plus Calorimeter (using the s t r a i n - f r e e  3  3  .  3  thermometer)  . . . . .a a o . o o o . . . .  o •  a  62  17. 18.  Capacity of Mn ZnC ( a l l o y 2) plus Calorimeter  19.  Magnetic S p e c i f i c Heat of Mn AlC  20.  Magnetic S p e c i f i c Heat of Mn ZnC  21.  Magnetization Curves  3  p » .  6 ©  66  3  3  64  67  . . . . < . . . . . . . . . ' .  ', . . . . . . . . . . . . . . . . . .  .  .0  63b  TABLE INDEX  Page I.  Measured Capacities of Calorimeter plus Aluminum 43  Oxide  II.  Calorimeter Capacity  .  45  III.  Capacities of Mn AlC plus Calorimeter  57  IV.  Capacities of Mn ZnC plus Calorimeter  63  3  3  THE CONSTRUCTION OF AN ADIABATIC CALORIMETER AND ITS USE IN MEASURING SPECIFIC HEATS.  I.  1,  INTRODUCTION  S p e c i f i c Heat Theory a.  General Theory; The o b j e c t of the present research i s to construct a calorimeter i n  order t o 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 p r o p e r t i e s , and consequently the atomic structure of these a l l o y s 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 u n i t 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 c a l c u l a t e d by Debye,! the r e s u l t s agreeing w e l l w i t h experiment. where Q  b  I f the s p e c i f i c heat of one substance i s p l o t t e d against T/6^ 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 c a l c u l a t e d value of s i x c a l . per mole per degree centigrade, i f the e l e c t r o n i c s p e c i f i c h e a t , C , i s not considered.' However, although the s p e c i f i c heat e  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, o f t e n have a l a r g e r C  e  than simple metals because  of t h e high d e n s i t y of e l e c t r o n l e v e l s i n t h e i r u n f i l l e d *d' bands.  C  can be  e  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 w i t h experimental r e s u l t s , p r e d i c t i n g 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 w i t h 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 r a t h e r r a p i d l y . b.  S p e c i f i c Heat Anomalies; The present work i s devoted p r i m a r i l y to measurement of s p e c i f i c heat  anomalies.  They are caused by t r a n s i t i o n s .'.in a substance, which involve a  i  ;  '.  - 3 -  change of symmetry i n the s t r u c t u r e 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 d e r i v a t i v e s of the Gibbs f u n c t i o n , G, change discontinuously. G = U - TS + PV where  by d e f i n i t i o n  U = i n t e r n a l energy S = entropy T = absolute temperature P = pressure V = volume.  since  dG  = VdP - SdT  dU  - TdS - PdV -S  therefore  and  dS  since  -  CpdT T  Consequently, the entropy changes d i s c o n t i n u o u s l y f o r a f i r s t order t r a n s i t i o n (e.g.: melting, v a p o r i z a t i o n ) , and the s p e c i f i c heat increases t o an i n f i n i t e value at the t r a n s i t i o n  J  point (producing a l a t e n t heat).  For  a second order t r a n s i t i o n , the  f i r s t order d e r i v a t i v e s of the Gibbs f u n c t i o n change continuously and the second order d e r i v a t i v e s change d i s c o n t i n u o u s l y .  Thus a  second order t r a n s i t i o n i s accompanied  ideally  by a d i s c o n t i n u i t y 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 t r a n s i t i o n s from an ordered to a disordered s t a t e , from a superconductor t o an o r d i n a r y conductor, and from a magnetic t o an non-magnetic s t a t e .  These t r a n s i t i o n s 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 t h e present research i s the magnetic one. The s p e c i f i c heat anomaly r e s 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 u n i t mass  of a ferromagnetic substance i s  where C i s the magnetic moment per u n i t mass, and the e f f e c t i v e f i e l d H,  H +  NI  (2)  H i s the a p p l i e d f i e l d , I i s the magnetic moment per u n i t volume and N i s t h e Weiss intermolecular f i e l d constant, which i s very large f o r ferromagnetic materials. Since H i s n e g l i g i b l e i n comparison w i t h NI, and I =^<T,  U = "C Np c der  (3)  where J> i s the d e n s i t y of the substance. Thus  U  _ N/><5-  U)  and t h e 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 p e r f e c t experimental conditions can never be a t t a i n e d , the d i s c o n t i n u i t y experimentally.  However, i t has been approached.  has not been observed 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  S i m i l a r anomalies should occur f o r antiferromagnetic and ferrimagnetic substances, s i n c e large i n t e r n a l exchange f o r c e s must e x i s t 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 r e s u l t a n t magnetization.  In f e r r i -  magnetics, the vectors a l s o oppose, but do not completely cancel, so that a r e s u l t a n t 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 j u s t as large as those observed f o r ferromagnetic substances i f the exchange f o r c e s are as l a r g e , even though the r e s u l t a n t magnetization may be much smaller. 2.  The A l l o y s to be Investigated. One aim of the present research i s t o determine the type of magnetiza-  t i o n of c e r t a i n substances by i n v e s t i g a t i n g t h e i r s p e c i f i c heat anomalies. substances are s i n g l e phase a l l o y s of Mn-Al-C and Mn-Zn-C.  These  These a l l o y s have an  ordered face-centered cubic 'perovskite' s t r u c t u r e and e x h i b i t 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, r a t h e r than a normal s u p e r l a t t i c e a t t r a c t i o n , i s responsible f o r the o r d e r i n g . p a r t i c u l a r i n t e r e s t are the a l l o y s Mn AlC 3  Of  and Mn ZnC% whose s t r u c t u r e s have the 3  maximum ordering (see Figure 1 ) . Mn-rAl-C a l l o y s are s i n g l e phase f o r a region near the Mn AlC composition. 3  I f the carbon content i s twenty atomic percent the a l l o y s are s i n g l e 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 Mn AlC and v a r i e s l i t t l e w i t h composition, i n d i c a t i n g t h a t 3  the Mn and A l atoms have almost the same diameter.  In these a l l o y s the s a t u r a t i o n  magnetization below the Curie temperature v a r i e s w i t h tenroerature i n a normal ferromagnetic manner, but the paramagnetic behaviour above the Curie point i n d i c a t e s ferrimagnetism, or at l e a s t a departure from Curie-Weiss behaviour.  Neutron  Figure Is  Peroyskite S t r u c t u r e .  d i f f r a c t i o n r e s u l t s i n d i c a t e that Mn AlC i s ferromagnetic above l i q u i d helium 3  temperature. As the Mn content i s increased from 60 to 69 atomic percent, the Curie temperature increases from 0°C t o 300°Cj, while the s a t u r a t i o n decreases from 1.20 t o  0.6 Bohr magnetons per Mn atom.  magnetization  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 a d d i t i o n a l Mn atoms (which must replace A l atoms i n cube corner p o s i t i o n s ) i s a n t i p a r a l l e l to t h a t of those i n face-centered p o s i t i o n s .  The  magnitude of the decrease i n magnetization corresponds t o the e x t r a Mn atoms e f f e c t i v e Bohr magneton value of ^/t4S-4/<B(  having an approximate  (This, i n c i d e n t a l l y , i s the value of ^M,  see  Appendix I ) .  f o r manganese i n the Heusler a l l o y s ) .  This i n t e r p r e t a t i o n could p o s s i b l y 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  v a r i e s 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 a l l o y s were always ferromagnetic, the height of the anomaly and also i t s t o t a l s i z e would become smaller as the magnetization decreased.  But f o r  a f e r r i m a g n e t i c 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 w h i l e the magne-  t i z a t i o n decreased i f the extra manganese atoms had large negative hypothesized.  as  Moreover, other t h i n g s 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 t o 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 s i n g l e phase over the range of composition from Mn6oZn oC o t o M n 7 Z n C o 2  of Mn ZnC i s 3.925& 3  a n  2  G  to  20  The l a t t i c e parameter  d i t v a r i e s only s l i g h t l y w i t h composition.  The Curie  temperature v a r i e s from about 80°C f o r Mn ZnC t o ~500°C f o r Mn Zn C2o« 3  70  io  The  v a r i a t i o n o f the s a t u r a t i o n magnetization f o r t h e a l l o y s w i t h 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 t o the behaviour at low temperatures predicted by Neel (see Appendix I I ) f o r one type of ferrimagnetic substance. The paramagnetic behaviour above t h e Curie point also appears t o agree w i t h that predicted by N£el f o r f e r r i m a g n e t i c s .  He deduced that the inverse of the  sus c e p t i b i l i t y , , 1 X  >= T + C  1 X»  C T ™Q  (see Appendix I I )  The curvature o f the r e s u l t i n g (L, T) curve i s concave t o the temperature a x i s , rather than convex, as i s u s u a l f o r a ferromagnetic substance. However, neutron d i f f r a c t i o n experiments c a r r i e d out a t Chalk R i v e r by Dr. B„ Brockhouse  on the a l l o y of approximate composition Mn ZnG suggest a 3  d i f f e r e n t and e n t i r e l y new magnetic concept t o e x p l a i n the magnetic properties of t h i s a l l o y .  These experiments i n d i c a t e that between the t r a n s i t i o n temperature  of -AO°C and the Curie temperature,, Mn ZnC i s a normal ferromagnetic substance 3  w i t h magnetically equivalent manganese atoms. t u r e , a complex magnetic s t r u c t u r e e x i s t s .  But below the t r a n s i t i o n tempera-  The a c t u a l s t r u c t u r e i s unknown, but  ohe i n good agreement w i t h 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 o f zero, two and three Bohr magnetons are arranged as shown i n Figure 3 f o r the state at absolute zero temperature. manganese atoms l^B  The magnetic moments o f t h e 2/*g  are opposing, so that t h e a l g e b r a i c mean moment of the a l l o y i s  per manganese atom at 0°K, i n agreement w i t h magnetic measurements.  I f the  magnetic moments o f 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 a r i t h m e t i c mean moment i n the low temperature s t r u c t u r e , then the ferromagnetic moment extrapolated t o 0°K would be 1.66/*go  This i s i n good  Figure 2 s  V a r i a t i o n of Saturation Magnetization, P", w i t h Temperature, T, f o r Mn-Zn-C A l l o y s  Only one- h a l f of the l a r g e unit c e l l i s shown; the other h a l f i s s i m i l a r but displaced one-half u n i t along the x - a x i s . Only manganese atoms are shown. The moments are oriented along a [ i l l ] diagonal.  -  12  -  agreement w i t h 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 t o become equivalent.  This  rearrangement involves a change i n interatomic exchange f o r c e s , so that a second order s p e c i f i c heat anomaly comparable t o t h a t o c c u r r i n g at the Curie temperature should be observable. In order to understand the form of the experimental magnetization curve f o r Mn ZnC i n terms of the two magnetic 3  structures mentioned, one may consider that below the t r a n s i t i o n point an opposing s u b l a t t i c e of magnetic moments comes i n t o 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 Mn ZnC becomes s l i g h t l y d i s t o r t e d 3  i n t o a face-centered t e t r a g o n a l s t r u c t u r e (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 result.  Thus, i f the ferrimagnetic model f o r Mn ZnC were c o r r e c t , the only 3  anomaly which would occur at r40°C would be a small t h i r d order one. I t i s seen t h a t measurement of s p e c i f i c heat anomalies should prove very u s e f u l i n checking the v a l i d i t y of magnetic models and thus i n determining b a s i c atomic s t r u c t u r e s of a l l o y s . 3.  C a l o r i m e t r i c Theory' a. General Theory; The s p e c i f i c heat o f a substance i s measured by means of a calorimeter,  which i s a device i n t o which heat can be introduced and the r e s u l t i n g temperature change measured.  Part of the heat brought i n t o the calorimeter r a i s e s i t s  -  temperature and part i s l o s t to the surroundings.  13  -  In p r e c i s e c a l o r i m e t r i c  measurements, the temperature change of the calorimeter and i t s thermal leakage (the amount of heat l o s t from the c a l o r i m e t e r d u r i n g the measurement) must be a c c u r a t e l y determined.  Since the greatest c a l o r i m e t r i c error f r e q u e n t l y a r i s e s  from measurement of the temperature change, an accurate thermometer must be used. 8 For precise work,, a platinum r e s i s t a n c e thermometer *  9 i s most o f t e n used.  The  r e s i s t a n c e of pure platinum, varying almost l i n e a r l y w i t h temperature,follows a smooth curve which i s determined by c a l i b r a t i o n at I n t e r n a t i o n a l Temperature Scale f i x e d points.  Because of i t s s t a b i l i t y , s t r a i n - f r e e platinum does not  require r e c a l i b r a t i o n very o f t e n . A platinum r e s i s t a n c e thermometer, used w i t h a Mueller bridge, w i l l measure temperatures t o 0.001°C accuracy i n the intermediate range, For l e s s 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 s u i t a b l e f o r vacuum apparatus. Most of the types of calorimeter design are the r e s u l t of thermal leakage considerations. The leakage depends on the thermal head, $ , which i s the temperature d i f f e r e n c e between the calorimeter and i t s surroundings. formula u s u a l l y used t o c a l c u l a t e the leakage i s ^ = ^ k t .  The  i s the temperature  change i n the calorimeter caused by the leakage f o r an experimental time t .  k is  the thermal leakage modulus of the c a l o r i m e t e r . In the o r d i n a r y c a l o r i m e t r i c method, since the thermal leakage i s r e l a t i v e l y high, i t must be a c c u r a t e l y estimated,,  To do this,, k i s found by  measuring ty't and p f o r a r a t i n g period immediately a f t e r each experimental period, and i t i s assumed to be the same i n the experimental period.  In order to  - 14measure the thermal head, p , a c c u r a t e l y , the calorimeter v e s s e l surroundings must be at a constant uniform temperature; a water j a c k e t i s o f t e n used f o r t h i s purpose.  The temperature change of the r a t i n g period must also be a c c u r a t e l y  measured.  Because the e r r o r i n measuring t h i s temperature change must be added  to the temperature e r r o r of the experimental period, the t o t a l thermometric e r r o r i s doubled by the leakage c o r r e c t i o n . Since the thermal leakage modulus v a r i e s w i t h temperature and other conditions, i t must be measured a f t e r 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 g r e a t l y 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 t o a c c u r a t e l y estimate t\„  In  Joule's method, t w i n calorimeters are heated the same amount, so that the thermal leakage, temperature, and heat capacity determinations become comparative measurements, made w i t h d i f f e r e n t i a l apparatus.  Other methods, such as the a d i a b a t i c  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 a d i a b a t i c calorimeter, the thermal head i s reduced almost t o  zero by keeping the temperature of a jacket or s h i e l d almost equal to that o f the calorimeter v e s s e l .  Thus, since the thermal leakage i s g r e a t l y reduced, i t  i s u s u a l l y neglected. But because the e r r o r involved i n a d j u s t i n g the jacket temperature f o r a d i a b a t i c conditions i s g r e a t e r than t h a t i n measuring the thermal head, the r e s u l t i n g e r r o r i n the thermal head i s greater f o r the a d i a b a t i c method than f o r the o r d i n a r y method.  And so the leakage neglected i n the a d i a b a t i c  method may be greater than the e r r o r i n the leakage c a l c u l a t i o n f o r the o r d i n a r y method. One advantage of the a d i a b a t i c method i s that i t reduces the e f f e c t of v a r i a t i o n s i n the leakage modulus, which are a chief source of e r r o r i n non-  - 15 a d i a b a t i c methods.  The smallness of the thermal head reduces the e f f e c t o f  f l u c t u a t i o n s i n t h e leakage modulus i n two ways;  F i r s t , the f l u c t u a t i o n s d u r i n g  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 n e g l i g i b l e ^ so that any such change i n i t w i l l a l s o be n e g l i g i b l e . Because leakage modulus v a r i a t i o n s are reduced by the a d i a b a t i c method, i t i s e s p e c i a l l y u s e f u l f o r protracted experiments, f o r which the leakage modulus might o r d i n a r i l y change considerably.  A l s o , although large temperature r i s e s can  not be used w i t h accuracy f o r non-adiabatic methods, because of the convection e f f e c t s , they can be s u c c e s s f u l l y used w i t h the adiabatic method.  Consequently,  the important temperature measurement errors can be g r e a t l y reduced.  A tempera-  ture r i s e increase from two t o t e n degrees, f o r example, would reduce the temperature measurement e r r o r by four f i f t h s .  Moreover, l a r g e r temperature r i s e s  w i l l permit more r a p i d measurement o f a s p e c i f i c heat curve.  The a d i a b a t i c method  avoids the ordinary r a t i 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, o r f l u i d l e s s , a d i a b a t i c calorimeter, by e l i m i n a t i n g s t i r r i n g , may almost completely eliminate temperature d r i f t .  Thus no c o r r e c t i o n s  f o r leakage need be made, and considerable time and c a l c u l a t i o n s are avoided.  One  experimental period can f o l l o w another without pause f o r large temperature ranges. Extreme temperatures can be much more e a s i l y a t t a i n e d w i t h an aneroid calorimeter than w i t h one containing f l u i d s . Another advantage o f the aneroid method i s that small dimensions (and l e s s dead m a t e r i a l ) are possible f o r t h e c a l o r i m e t e r , 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 f a s t e r thermal e q u i l i b r i u m  and thus a shorter experimental time (producing l e s s t o t a l thermal leakage), which i s u s u a l l y an advantage,.  However, at very low temperature, since the main  heat loss i s by conduction along w i r e s , smaller dimensions are not so advantageous. The main disadvantage of the aneroid calorimeter i s t h a t the temperature e q u a l i z a t i o n must be by conduction alone.  In order t o speed the conduction:  (1)  Metal of high c o n d u c t i v i t y (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 t h a t temperature d i f f e r e n c e s are not produced by thermal leakage.  The a d i a b a t i c method i s  thus e s p e c i a l l y u s e f u l . Necessary to the aneroid adiabatic calorimeter i s the measurement of the thermal head between calorimeter and s h i e l d . t h i s i s w i t h thermocouples.  The most convenient way t o do  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 e q u i l i b r i u m i s v e r y good f o r both calorimeter and s h i e l d . Thus, pains should be taken to insure good thermal e q u i l i b r i u m . The aneroid a d i a b a t i c calorimeter was chosen as most s u i t a b l e f o r the present work, because of the f o l l o w i n g reasons! (1)  I t can give s u i t a b l e accuracy (approaching 0.2%)for the temperature range r e q u i r e d : -150  (2)  t o 150°C.  I t can give quick and simple measurements, e s p e c i a l l y i f the r a t i n g period i s e l i m i n a t e d .  - 17 (3)  I t can be constructed simply and economically,  even though  the a d i a b a t i c controls introduce some complexity.  II, 1,  THE CONSTRUCTION OF AN ANEROID ADIABATIC CALORIMETER  Design a.  General Aims The basic design required was one of s i m p l i c i t y coupled w i t h accuracy,  since the simplest construction which w i l l s t i l l give the required accuracy i s undoubtedly the most d e s i r a b l e .  The p r e c i s i o n desired f o r the calorimeter was  approximately 0.5%, w i t h a reduction t o 0.2% possible by means of m o d i f i c a t i o n . Fast operation of the calorimeter by one person was d e s i r a b l e , 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 t o 150°C, since the a l l o y s t o be measured required t h i s .  Also, a d a p t a b i l i t y t o samples of a d i f f e r e n t form was  advantageous, w i t h regard t o future use of the calorimeter. b.  Basic Design? The basic design"^ of the a d i a b a t i c calorimeter i s shown i n Figure 4.  The c y l i n d r i c a l calorimeter v e s s e l (2) contains the specimen whose s p e c i f i c heat i s t o be measured, and the platinum resistance thermometer-heater ( 1 ) . The v e s s e l i s f i l l e d w i t h helium f o r good conduction.  The thermometer-heater-is  used, i n conjunction w i t h standard resistances and a potentiometer, t o measure a c c u r a t e l y the heat i t has introduced t o the v e s s e l and the r e s u l t i n g temperature change. The a d i a b a t i c s h i e l d (3) i s heated e l e c t r i c a l l y so that i t s temperature remains close t o that of the. v e s s e l , and thermal leakage may be neglected.  The  temperature d i f f e r e n c e between v e s s e l and s h i e l d i s measured with thermocouples  1: 2: 3: Ui 5: 6s  Thermometer-Heater Calorimeter Vessel Adiabatic Shield Outer Case Brass Ring Evacuation Tube  Figure l+i  Basic Calorimeter Design  connected t o a galvanometer.,  The outer case (4) i s evacuated through the tube  (6) so that thermal leakage between the s h i e l d and the v e s s e l , and between the s h i e l d and the case i s reduced.  I n order t o 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.  M a t e r i a l s and D e t a i l e d Construction a.  s  The Calorimeter Vessels (i)  Theory:  To promote f a s t conduction, the calorimeter v e s s e l was made of copper. By p l a c i n g the thermometer-rheater i n the center of the v e s s e l , the heat was generated uniformly, and thermal e q u i l i b r i u m was speeded.  Also, a central  p o s i t i o n o f the heater made more gradual the temperature change of the v e s s e l surface, so that the a d i a b a t i c c o n t r o l was e a s i e r . The calorimeter v e s s e l was made an a i r - t i g h t c y l i n d r i c a l container, so t h a t i t could accommodate p r a c t i c a l l y any type of s o l i d m a t e r i a l , and i n p a r t i c u l a r the powdered, e a s i l y corroded substances which were t o be i n v e s t i g a t e d . The s i z e o f the calorimeter v e s s e l depends on s e v e r a l f a c t o r s .  As  mentioned e a r l i e r , smaller dimensions r e s u l t i n f a s t e r thermal e q u i l i b r i u m , which i s advantageous.  However, the leakage modulus w i l l increase as t h e dimensions  decrease, since i t varies d i r e c t l y as t h e surface area but i n v e r s e l y as the heat capacity or volumes, i f the dimensions decrease by *n» times, the leakage modulus i s n /n = n times as l a r g e  0  However, f o r a smaller calorimeter, the shortening  of the experimental time caused by f a s t e r e q u i l i b r i u m may more than compensate f o r the increased leakage modulus, and produce l e s s t o t a l thermal leakage. For very s m a l l dimensions, t e c h n i c a l d i f f i c u l t i e s are encountered. A l s o , the increase i n weight, of the calorimeter v e s s e l w i t h respect t o t h e  - 20 weight of the sample decreases the p r e c i s i o n . Thus f o r the calorimeter constructed here, a compromise s i z e was chosen, which enabled the sample weight to be l a r g e r than the v e s s e l weight, while s t i l l r e t a i n i n g quite r a p i d e q u i l i b r i u m conditions, (ii)  Constructions  (see Figure  5)  The calorimeter v e s s e l (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. t h i c k (.020 i n . ) . I t was made by e l e c t r o p l a t i n g copper onto a s t a i n l e s s s t e e l mold having a 1/4° p i t c h 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 s o l u t i o n was a  d i l u t e sulphuric a c i d s o l u t i o n of CuSOi,. containing a small amount of g e l a t i n .  A  p l a t i n g current of 0.3 amperes f o r 120 hours was used, corresponding t o a thickness of 0,044 i n . , and the excess copper was machined o f f t o the required thickness. The calorimeter v e s s e l was s i l v e r - p l a t e d i n s i d e and out (to reduce the r a d i a t i o n of heat) to a thickness of ~0.003 i n . The s o l u t i o n used contained? AgCN  36 g . / l i t e r  KCN  52 g . / l i t e r  K C0 2  A s m a l l amount of CS  2  3  38 g . / l i t e r  was added as a brightener.  A current of 1 ampere (current  d e n s i t y of 5 amp/sq.ft.) and a voltage of 1 t o 2 were used. The bottom of the calorimeter v e s s e l was s o f t soldered on. thermometer (1) was f i x e d w i t h i n the v e s s e l and i t s leads (3)  The  axited from the  v e s s e l by s o f t s o l d e r i n g a disk connected t o the thermometer onto the t o p 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 s e a l (4) (No. 96.1010 from the Stupakoff Ceramic  and Manufacturing Co.), soldered through the top of the v e s s e l , was used to admit  Figure 5 s  The Calorimeter  (Actual S i z e )  -  22  -  helium to about four f i f t h s atmospheric pressure, i n order t o speed thermal equilibrium.  A f t e r the helium was  introduced t o the v e s s e l , the glass tube of  the s e a l was drawn o f f i n an oxygen-gas flame.  In order to change samples to  be measured i n the v e s s e l , the kovar s e a l and thermometer were removed, and the sample was admitted through the v e s s e l t o p .  The t o t a l weight of the calorimeter  v e s s e l plus thermometer was about 43 g. Binding posts (5) were soldered t o the s i d e and top of the calorimeter v e s s e l f o r thermocouple connections.  Three wire hooks (6) were soldered to the  v e s s e l top f o r suspension of the calorimeter w i t h i n the s h i e l d . b.  The Adiabatic S h i e l d (I)  Theory?  The a d i a b a t i c s h i e l d must be heated so t h a t i t s temperature remains close to that of the calorimeter v e s s e l surface.  In order to keep the s h i e l d  surface temperature uniform, the s h i e l d must be a h i g h l y conducting m a t e r i a l . The u n i f o r m i t y of surface,temperature  i s e s p e c i a l l y d e s i r a b l e i f only a few  thermocouples are used to determine the temperature..  To lessen temperature l a g s ,  the heat capacity of the s h i e l d must be small, and thus the s h i e l d w a l l s must be thin.  The s h i e l d must not be a i r - t i g h t , since a vacuum between s h i e l d and  calorimeter v e s s e l i s necessary to reduce thermal (ii)  leakage.  Construction: (see Figure 5)  The s h i e l d (7) i s a 6 by 8 cm. (2.4 x 3.2 i n . ) c y l i n d r i c a l brass container, 0.07  cm.  (.030  i n . ) t h i c k , w i t h a removable bottom.  I t was made by  reducing brass tubing t o the d e s i r e d thickness i n an a c i d bath, and s i l v e r s o l d e r i n g on a brass top.  The bottom was machined i n t o a f r i c t i o n - f i t t i n g cap.  The s h i e l d was heated by means of a non-inductive uniform winding of  - 23  No. 3*2 gauge s i l k - c o v e r e d manganin w i r e .  -  The t o t a l r e s i s t a n c e of the winding was  Z*20 ohmss 300 ohms f o r the side and 60 ohms each f o r t h e t o p and bottom. was i n s u l a t e d and connected t o the s h i e l d w i t h Dow Corning 935 v a r n i s h .  The w i r e This i s a  s i l i c o n e e l e c t r i c a l i n s u l a t i n g v a r n i s h having good f l e x i b i l i t y from -55° t o 260°C. I t drys t a c k - f r e e i n three hours at 200°C.  The winding onto the s h i e l d was  accomplished i n stages, the v a r n i s h being baked a f t e r each few windings. In order t o 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 s h i e l d under the manganin winding. the  Aluminum f o i l was cemented onto  s h i e l d surfaces (with the same v a r n i s h ) i n order t o reduce r a d i a t i o n of heat. Three small screws (8) were fastened t o the s h i e l d s i d e , top and  bottom f o r thermocouple connections. D i f f e r e n t i a l copper-constantan thermocouples  (No, 30 B and S, i n s u l a t e d w i t h t h e varnish) were attached from the  s h i e l d side t o t h e s h i e l d top, the s h i e l d bottom, the calorimeter v e s s e l s i d e , and the calorimeter v e s s l t o p . s h i e l d side temperature.  Thus a l l temperatures were found r e l a t i v e t o the  Three small holes through the s h i e l d t o p permitted  suspension w i t h threads of the calorimeter v e s s e l w i t h i n the s M e l d and the s h i e l d w i t h i n the outer case. c.  The Adiabatic S h i e l d 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 s h i e l d side and s h i e l d top, s h i e l d bottom, calorimeter side and calorimeter top.  The thermocouples were connected t o 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., g i v i n g a d e f l e c t i o n 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 t o the galvanometer at one time, (see Figure 6b).  - 24 -  Figure 6a:  Power C o n t r o l  o Thermocouples  Figure 6b:  Figure 6:  Thermal Head Measurement  Adiabatic S h i e l d C o n t r o l C i r c u i t  Galvanometer  - 25  -  The manually operated power c o n t r o l c i r c u i t f o r the a d i a b a t i c s h i e l d The power was supplied by a 60 cycle 120 v o l t s t a b i l i z e r ,  i s shown i n Figure 6a. A 0-110  V powerstat permitted power c o n t r o l ; the f i n e c o n t r o l , obtained w i t h a  50 ohm v a r i a b l e r e s i s t a n c e , was not normally used.  The current was r a i s e d b r i e f l y  by means of a tapping key shunting out a 400 ohm r e s i s t a n c e , and the current was lowered b r i e f l y by switching a 350 ohm r e s i s t a n c e i n t o the c i r c u i t .  Part of the  current was shunted from the top s h i e l d heater through a v a r i a b l e 500 ohm r e s i s t a n c e 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 f o r a temperature d i f f e r e n c e of more than 50°C between the s h i e l d and outer case, the d i f f e r e n c e 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 p r i m a r i l y as an evacuation chamber to make temperature c o n t r o l e a s i e r . the  When i t i s evacuated, the thermal leakage between  a d i a b a t i c s h i e l d and the calorimeter v e s s e l i s decreased g r e a t l y , enabling  the leakage t o be neglected. E q u a l l y important, the leakage between the s h i e l d and the outer case -is reduced, so t h a t the s h i e l d temperature c o n t r o l i s simplified. By keeping the outer case temperature near the s h i e l d temperature, the s h i e l d c o n t r o l i s made s t i l l e a s i e r .  The case temperature was c o n t r o l l e d 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 w i t h a v a r i a b l e a l t e r n a t i n g voltage.  For low temperatures, down t o -80°C, a mixture of dry i c e  and acetone was used.  For temperatures from ~80°C t o -rl60°C, l i q u i d oxygen was  fed through a copper c o i l immersed i n g a s o l i n e ; the gasoline was f i r s t cooled to -80°C w i t h dry i c e .  - 26 (ii)  Construction; (see Figure 5 ) .  The outer case (9) i s a 10 by 15 cm. (k x 6 i n . ) copper c y l i n d e r , 0.2 cm, t h i c k (1/8 i n ) , w i t h a brass bottom s o f t soldered on, and a removable 0  brass cover,  A groove was machined i n t o the cover, so that i t could f i t onto t h e  case i n a vacuum grease s e a l (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"' t h i c k and 3/4 " long) soldered onto the i n s i d e o f the cover. 1  Three  holes were d r i l l e d through the r i n g f o r suspension of the s h i e l d . The case was evacuated through a t h i n walled 3/8 (12) , s i l v e r soldered t o the cover.  511  German s i l v e r tube  A glass T-tube was connected t o the German  s i l v e r tube, so that the leads might emerge from the vacuum system through one arm of t h e T-tube, which was sealed w i t h de Khotinsky wax.  F l e x i b l e leads were  soldered t o the No. 29B and S Cu formex leads where they emerged from the system, and the junctions were f i x e d w i t h de Khotinsky wax. The case was suspended w i t h i n the Dewar f l a s k on a p l a t e 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  A platinum r e s i s t a n c e thermometer-heater >  q 7  was used both t o supply  heat t o the calorimeter v e s s e l and t o measure the v e s s e l temperature.  This dual  purpose can be accomplished f o r an a d i a b a t i c calorimeter, since no temperature measurement need be made while energy i s being supplied t o the calorimeter v e s s e l . The double use of the thermometer enables a l l temperature and heat input measurements t o be made w i t h one potentiometer  0  The disadvantage o f a thermometer-heater  i s that the e r r o r i n heat input measurement i s increased, unless s p e c i a l precautions are  taken, because o f the v a r i a t i o n of r e s i s t a n c e of the heater w i t h temperature.  -  Figure 7:  Suspension of the Calorimeter  27  - 28 .... •(ii)  Constructions  Two d i f f e r e n t kinds of platinum r e s i s t a n c e  thermometers were used. (a)  Glass embedded c o i l s  A commercial 100 ohm platinum r e s i s t a n c e  thermometer (produced by Wheelco Co., now Barber-Colman Co.) was used f i r s t .  It  consists of a non-inductive winding of f i n e platinum ribbon embedded i n a s o f t glass tube (0.17 i n . x 1.75 i n . ) . 0.010  The  two  i n . platinum leads emerge from a  cup-shaped top. A platinum d i s k , 0.005 inches t h i c k  u  and 0.5 inches diameter, punched so that i t f i t t e d into the cup, was sealed t o the glass  Thermometer: Actual Size  by heating i t by i n d u c t i o n to a b r i g h t red color.  F l e x i b l e copper leads were s o f t  soldered to the platinum leads, doubled back Ft.disk  i n t o the cup, and f i x e d there w i t h a r a l d i t e cement.  araldite  Two leads were soldered t o each  f l e x i b l e lead to make the f o u r - l e a d type of thermometer.  (The r e s i s t a n c e of the two  f l e x i b l e leads -  0.01 ohm - was  negligible  i n comparison w i t h the platinum r e s i s t a n c e ) . The thermometer was f i x e d w i t h i h the c a l o r i m e t e r v e s s e l by s o f t s o l d e r i n g the platinum d i s k to the v e s s e l top. The disadvantage of t h i s type of thermometer i s t h a t i t s c a l i b r a t i o n v a r i e s because of s t r a i n s produced i n the platinum by expansion of the g l a s s . (b)  S t r a i n - f r e e thermometer;  In order t o avoid repeated thermometer  c a l i b r a t i o n , a s t r a i n - f r e e platinum r e s i s t a n c e thermometer was  constructed.  The  thermometer must be small, i n order to f i t i n t o the calorimeter and to have f a s t  - 29 thermal e q u i l i b r i u m , but the platinum wire must be f r e e t o expand and have no 13 p o s s i b i l i t y of short c i r c u i t s .  The u s u a l method  14 '  of s a t i s f y i n g these  conditions i s t o wind f i n e platinum wire into a small diameter c o i l and then wind the c o i l around an i n s u l a t i n g support (usually mica)„  The whole i s then  enclosed i n an a i r - t i g h t container, t o avoid contamination of the platinum and water condensation on i t , as w e l l as f o r mechanical p r o t e c t i o n . This was the method adopted. diameter was used.  Chemically pure platinum wire 0.003 i n .  The s p e c i f i c a t i o n s ensuring p u r i t y are t h a t the r a t i o of the  r e s i s t a n c e at 100°C to that at 0°C i s greater than 1.390  and the r a t i o of the  r e s i s t a n c e at -183°C to that at 0°C i s l e s s than O.25O.  From 100 to 200 cm of  the wire was wound under 10 g t e n s i o n 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 s l i p p e d o f f 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 t e n s i o n n o n - i n d u c t i v e l y 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  Of the many types of p r o t e c t i v e case m a t e r i a l s used f o r resistance thermometers, pyrex glass was chosen because i t i s easy to work and i s s u i t a b l e f o r the temperature  range d e s i r e d .  Thin-  Pt wire Tube  walled pyrex was used f o r the p r o t e c t i v e  Pt s t r i p  tube, and t h i n platinum s t r i p s were used  Plug  f o r leads i n order t o make a vacuum s e a l through the g l a s s . The s e a l i n g Is d e t a i l e d i n Appendix IV,  procedure  - 30 The platinum s t r i p s emerging from the s e a l were fused t o 0 020 i n c h o  platinum wire leads.  The wire was f l a t t e n e d at one end by r o l l i n g , was held i n  contact w i t h the platinum s t r i p by p i n c e r s , 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 t o 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 d i l u t e n i t r i c a c i d s o l u t i o n , annealed f o r 15 hours a t 490°C, and de-gassed f o r 3 hours a t 400°C. A f t e r the s e a l was l e a k - t e s t e d , the thermometer was f i l l e d w i t h d r i e d helium at two-thirds atmospheric pressure and the pyrex tube was sealed o f f . The a c t u a l s i z e of the thermometer i s shown i n the diagram.  Kovar seals were used t o attach the thermometer to the c a l o r i m e t e r v e s s e l and t o b r i n g i t s leads out of the v e s s e l . The two platinum leads were soldered t o kovar s e a l s , which were soldered to a s i l v e r e d phosphor bronze d i s k . This d i s k was soldered t o the calorimeter v e s s e l top. connected t o each kovar s e a l on the outside. (~ 0.001 ohm)  Two f i n e leads were  The r e s i s t a n c e of the kovar seals  i s n e g l i g i b l e i n comparison w i t h t h a t of the r e s i s t a n c e thermometer.  fo Energy Input and Temperature Measurement 1  The potentiometer method was used to measure both the thermometer r e s i s t a n c e , f o r temperature determinations, and the energy input to the thermometer when used as a heater. one c i r c u i t .  Both measurements were made w i t h one potentiometer, using  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 s t r a i n - f r e e 20 ohm thermometer, i s shown i n Figure 8.  - 31 -  Q  H  H 10  10,000  a.  5000  -\VWW  •  T  —-4r  10.  H  T  6V  Therm.  Dummy 5 ~=  *t  6V ="  Commercial 100 ohm Thermometer C i r c u i t  Q  T  x  H  H 10,000  20  AVW(/A—•  •AVIV" 1  20  Therm. R. 10,00C  b.  H  6V Dummy 25  S t r a i n - f r e e 2 0 ohm Thermometer C i r c u i t .  Figure 8. C i r c u i t Diagrams f o r Thermometer-Heaters.  6V  - 32 To produce a very stable d i r e c t current source, s i x 120 amp.-hour 2 v o l t storage b a t t e r i e s were used. sources i n p a r a l l e l .  Theywere arranged t o create two 6 v o l t  Before an energy input, the b a t t e r i e s were discharged  through a dummy r e s i s t a n c e , so that t h e i r voltage would be constant w h i l e measurements were being made. For temperature measurements, the switches were thrown to p o s i t i o n T i n Figure 8, and f o r heat input the switches were at p o s i t i o n H. The potentiometer connections are at P and Q. potentiometer was used, having ranges 0 t o 1.9 x 10 readable to 0.00$% or  1 19,000  of the maximum.  maximum, or 2/*- V f o r the lowest s c a l e .  A T i n s l e y type 3387B  2 -1 , 1.9 x 10 and 1.9 v o l t s ,  The accuracy i s 0.01% of the  A l l r e s i s t a n c e s , the p o t e n t i a l across  which was t o be measured, were standard r e s i s t a n c e s . .  A l l leads i n s i d e the  calorimeter were No. 29B and S copper, formex i n s u l a t e d , and outside were No. 18B and S f l e x i b l e (i)  copper.  Energy Input:  The p o t e n t i a l across the heater was found by using a p o t e n t i a l d i v i d e r , c o n s i s t i n g of a 10 and a 10,000 ohm r e s i s t a n c e .  The p o t e n t i a l across the 10 ohm  resistance was measured; i t was VQ  = 10,010  the p o t e n t i a l across the heater.  +  10 E  times l e a d s  Since the resistance of the leads was about  1 ohm, the p o t e n t i a l across the heater, V  =  1001.1 V  Q  ;  n e g l e c t i n g the lead resistance produced an i n s i g n i f i c a n t e r r o r of 0.01%. The current through the heater was found by measuring the p o t e n t i a l across a 1 ohm standard r e s i s t a n c e .  This p o t e n t i a l , V , was equivalent t o t h e  - t o t a l current through the c i r c u i t .  The current through the heater, I , was equal  t o the t o t a l current minus the current through the p o t e n t i a l d i v i d e r s I =  vp  ZQ  10  The energy input = V l t . The d u r a t i o n of energy i n p u t , t,, was measured . w i t h a stopwatch t o b e t t e r than 0.05% accuracy. I ** 0.05 amp. f o r the 100 ohm heaters, ohm heater.  At 0°C, V ~ 6 v o l t s and  V ^ 3 v o l t s and I s=* 0.12 amp. f o r the 20  Thus the power was approximately  0.35 watts f o r each thermometer at  0°C, producing a temperature r i s e of ~ 0.5°C per minute f o r the calorimeter v e s s e l when containing a sample; weighing 30 g. Since the heater resistance v a r i e d w i t h temperature, the power input v a r i e d a l s o , and had t o be averaged.  The v o l t a g e , 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 r e s i s t a n c e  changed b y 4 ohms o r one t w e n t y - f i f t h of i t s value. changed by approximately t h i s f r a c t i o n . V = 6 - Vp. Vp was only i n V was  1 2500  1 100  Thus the current a l s o  Since the b a t t e r y p o t e n t i a l = V + Vp = 6,  of V, and changed by only _1, so that the change 25  or 0.04% f o r one heat input.  For the s t r a i n - f r e e heater (Figure 8b), the c i r c u i t was arranged so that the power input remained approximately  constant.  As the temperature r o s e  the heater r e s i s t a n c e increased, and the current was lowered.  s  Thus the voltage  drop across the 20 ohm r e s i s t a n c e of the c i r c u i t decreased, and the p o t e n t i a l across the heater rose.  Since the heater r e s i s t a n c e was approximately the same  as the 20 ohm r e s i s t a n c e , the r i s e i n p o t e n t i a l across the heater compensated f o r t h e lowering of current through i t . change c a l c u l a t i o n s ) .  approximately  (See Appendix V f o r power  (ii)  Temperature Measurement; (Figure 8)  The temperature was c a l c u l a t e d from the r e s i s t a n c e of the thermometer, which was found by comparing w i t h a standard,,  The p o t e n t i a l s across the  thermometer, VQ, and across a standard 10 ohm r e s i s t a n c e , Vp, were measured f o r the 100 ohm thermometer.  Since t h e currents were the same through each  r e s i s t a n c e , the thermometer r e s i s t a n c e , R-  =  1  0  V  V  Q  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 f o u r 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 f o r the 20 ohm thermometer.  These r e l a t i v e l y high currents  reduced the e f f e c t o f p a r a s t i c thermal e.m.f."s i n the c i r c u i t , while apparently causing n e 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 v e s s e l . The resistance measurements were reproducible to 0,01%, which corresponded t o 0.02°C near 0°C. III. 1.  CALIBRATION AND PERFORMANCE  C a l i b r a t i o n of Resistance Thermometer a.  Fixed P o i n t s ; The r e s i s t a n c e thermometers were c a l i b r a t e d at I n t e r n a t i o n a l Temperature  - 35 -  Scale f i x e d p o i n t s . ^  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 e q u i l i b r i u m between l i q u i d and gaseous oxygen at the pressure of one standard atmosphere (760 mm. of Hg): Oxygen point  (li)  ~182 97 C o  0  The temperature of e q u i l i b r i u m between i c e and a i r saturated water at normal atmospheric pressure; Ice point  (iii)  0.000°C ,  The temperature of e q u i l i b r i u m 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 f i x e d point; (iv)  The temperature o f f r e e z i n g mercury at the pressure o f 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 a t t a i n i n g t h i s temperature. b.  C a l i b r a t i o n Apparatus^ (i)  Oxygen point: In order to standardize i t at the oxygen' p o i n t , the thermometer,  contained i n a glass tube, was immersed i n l i q u i d oxygen.  The formula used t o  f i n d the temperature of e q u i l i b r i u m between l i q u i d and gaseous oxygen at a pressure p (mm. Hg) i s s  t  =  -182.97 + 9.530 / p  - l\  - 36 Because of t h e commercial brand of l i q u i d oxygen used and because of the u n c e r t a i n t y i n p, the oxygen point was accurate t o o n l y " 0„l C»,. a  (ii)  Ice point, and mercury point (see Figure 9a)& To c a l i b r a t e at the i c e and mercury p o i n t s , the thermometer was  placed i n a glass tube c o n t a i n i n g acetone f o r f a s t thermal conduction.  The tube  was immersed i n d i s t i l l e d water o r 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 o f dry i c e and acetone. When  the temperature approached the t r a n s i t i o n p o i n t , the f l a s k was evacuated, so that the temperature remained constant f o r some time at the i c e or mercury p o i n t . (iii)  Steam points The steam point apparatus i s shown i n Figure 9b. Steam i n  e q u i l i b r i u m w i t h water r i s e s about the thermometer, and i n s u l a t e s 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 r e s i s t a n c e thermometer, c a l i b r a t e d by the National Bureau of Standards above 0°C, was used t o determine the mercury and..steam point temperatures. at the i c e p o i n t .  The c a l i b r a t i o n o f t h i s thermometer was checked  Since i t s r e s i s t a n c e at 0°C d i f f e r e d by only -* 0.0005 ohms  from i t s c a l i b r a t e d value of 2.5117 ohms, the change was assumed t o i n d i c a t e a l i n e a r s h i f t i n r e s i s t a n c e , w i t h no change i n the c a l i b r a t i o n R  100  ~ o R  a  n  d  constants,  5 o  The r e s i s t a n c e of t h i s thermometer was measured w i t h a Leeds and Northrup Mueller d i a l b r i d g e .  The lead r e s i s t a n c e s were eliminated by using a  mercury commutator t° switch leads.  For a balanced bridge (see diagram),  Evacuation Tube Cork  Glass Wool Evaenable F l a s k Water or mercury N.B.S. Thermometer Thermometer Dry Ice plus Acetone Dewar a.  Ice and Mercury Point Apparatus  Cork Glass Wool  -Glass Tube • Thermometer  'ASteam b. Figure 9.  Steam Point Apparatus C a l i b r a t i o n Apparatus.  - 37 -  - 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 r e s i s t a n c e s of the t  commutator also are eliminated by  t a k i n g measurements w i t h the thermometer r e s i s t a n c e , R^ , shunted out. e„  C a l i b r a t i o n Formulae The r e l a t i o n between resistance and temperature, t , of a platinum  r e s i s t a n c e thermometer f o r temperatures from 0 to 630°C i s s R R  0  =  +  8, 9  R . ( l + At + B t ) 2  (1)  i s the r e s i s t a n c e 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 p o i n t s . R  R [ l + At + Bt, + C(t - 1 0 0 ) t 3 J  =  t  For temperatures from -183 to 0°C„ 2  Q  (2)  The a d d i t i o n a l constant, C, i s determined by c a l i b r a t i o n at the oxygen p o i n t  0  For these ranges, the temperatures found from these formulae, using a standard platinum r e s i s t a n c e thermometer, represent the I n t e r n a t i o n a l Temperature Scale.  The f i r s t equation i s equivalent to Callendar's equations  t  = ^ R  where  8  =  100  ~ ° R  100~  -  R  o  10  4  B  A + 100B (See Appendix VI),  + 8 jjo.oi t )  2  - 0.01 t j  (3)  -  39 -  The second: equation i s equivalent tos t -  100 + 8 [(0.01 t ) S  100 •" o R  where D ° -  2  - 0.01 tl + D(t-100)t  L  (4)  3  J  C A + 100 B  (See Appendix ¥1). These formulae enable the temperature t o be e a s i l y c a l c u l a t e d from the r e s i s t a n c e by a successive approximation method. t  x  -  *t - o  100  R  R  The f i r s t approximation,  100 • o R  The second approximation i s a t t a i n e d by s u 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 ( t g ) , a n d so on.  In practice, t  2  can be  estimated, so that o n l y one c a l c u l a t i o n need be made t o f i n d the f i n a l approximation t3o >  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 o f the commercial 100 ohm thermometer gave  the  following results? Oxygen point  R  -LS2 9  =  24.740 abs. ohms  Mercury point  R_^g  " ^»725  Ice  R  = 100.24  point  Steam p o i n t  R  Q  100  8  ° ^9°°^  ^  - 40 -  The temperature o f the mercury point was found from the Leeds and Northrup thermometer r e s i s t a n c e a t t h a t p o i n t . The value Rioo  w  a  s  c a l c u l a t e d 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 p o i n t , the r e s i s t a n c e o f the Leeds and  Northrup thermometer was R^» = 3.4844.  Thus, using equation (3), the tempera-  ture, t - 3.4844 - 2.5117 , 0.9732  n o X U U  +(a n e g l i g i b l e quantity) = 99.95°G.  at which R^ = 139.00 f o r the commercial thermometer. Since  R 99.95 ~ o  E  E  = 100.24 ,  0  =  38.76  To f i n d R]_oo  !  8R f o r 0,05°C i s 38.76 99.95 where 0.985 i s the value o f  x  O o 0 5  .A(pt)  x  0,985 « 0.019,  (and pt =  At  R  R  t~ o 100 R  100)  near 100°C f o r a thermometer w i t h 8 = 1.50 . R  Therefore R  100 ~ o R  Q - 139.00 + 0.019 = 139.02  1 G  =  38.78 abs. ohms.  The r e s i s t a n c e s obtained by c a l i b r a t i o n (equation (5)) were s u b s t i t u t e d i n equation (2) t o f i n d the constants: A = 3.952 x 10"  3  B = -8.38 x 10~ C = -1.3  7  x 10"  1 2  - 41 Thus  8 = 2.17  and  D = 3.4 x  10"  10  Equation (4) becomes? ,  t = (R - 100.24) 2.5786 + 2.17|j0.01 t )  - 0.01 t ] + 3.4  2  t  x 10"  10  (t - 1 0 0 ) t  3  A f t e r t h i s commercial thermometer had been used f o r temperatures from -150 to 150°C, i t was r e c a l i b r a t e d .  R  =38.87  R R therefore and therefore  =  0  1 0 0  R i o o ~ ^o  8  4.65  =  100.06  »  138.91 36.85  =  8  The r e s u l t s were:  =  1.47  t = (R^ - 100.06) 2.5740 + 1.47 ["(0.01 t ) - 0.01 t ] 2  for  0 ^ t < 630°C.  The c a l i b r a t i o n of the thermometer had changed considerably. change caused an appreciable  .The'  e r r o r 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. (ii)  S t r a i n - f r e e thermometer?.,-. The c a l i b r a t i o n of the  s t r a i n - f r e e thermometer gave the f o l l o w i n g  results: R 183.0 R  -38.87  R R  c  1 0 0  " =  4  1  5  °  -  4 9 1  3  9  0  -• 18.218 =25.348  From these,  R 0 ~ o R  and Thus,  7° 130  =  1Q  5  =  D  =  ,1.46 1.14 x 1 0 ~  9  t = (Rt - 18.218) 14.025 + 1.46 [(.01 t ) - .01 t ^ 2  + 1.14 x 1 0 " ( t - 1 0 0 ) t 9  3  A f t e r the thermometer had been used from -100 t o 150°C, i t s zero p o i n t was checked and was found t o be the same w i t h i n 0.001 ohm. 2.  Heat Capacity of the Calorimeter The calorimeter was c a l i b r a t e d by measuring i t s capacity when i t  contained a standard substance.  This method was preferable t o c a l i b r a t i n g the  empty calorimeter, because conditions were c l o s e r t o 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 r e s u l t s f o r low temperatures, obtained u s i n g the commercial thermometer, are shown i n Tables I and I I and Figure 10.  It i s  seen t h a t , as expected from the behaviour of copper, the heat capacity o f the calorimeter decreased rather r a p i d l y below 0°C. Since the c a p a c i t y a t temperatures above 0°C was found from preliminary measurements t o be approximatel y constant, i t was assumed t o increase only from 15.8 j/°C at 0°C,(see Figure 10) to 15.9 j/°C at 100°C. 3.  Accuracy of S p e c i f i c Heat Measurements. ' The accuracy of r e s u l t s and consistency of performance of the  a d i a b a t i c calorimeter are l i m i t e d by c e r t a i n e r r o r s inherent i n the apparatus;  - m-  Table I  Measured Capacities of Calorimeter Plus Aluminum Oxide ( P l o t t e d i n Figure 10) Weight, of Calorimeter - 42.26: g.  .Weight of A 1 0 2  (Temperature i n t e r v a l s v a r i e d from 11 t o 15°C.)  Average Temperature (°c)  Capacity  (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  3  - 19.91 g.  44. -  -100  Figure 1Q,  -80  -60  -40 Temperature (°C.)  -20  0  Heat Capacities of Calorimeter and of Calorimeter plus Alumina.  Table I I  Calorimeter Capacity (Capacity of calorimeter plus A l 0 3 2  obtained from Figure 10) ( P l o t t e d i n Figure 10) Weight of Calorimeter = 42.26 g. Weight of A 1 0 2  T°K  = 19.91 g. = 0.1953 moles  3  A1 0 Cp 2  3  Cal. + A1 0 c 2  p  3  Cal. C (j/ C) P  c  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 E r r o r Since f o r any calorimeter heat i s t r a n s f e r r e d between the  calori-  meter v e s s e l and i t s surroundings, an e r r o r i n accounting f o r t h i s heat t r a n s f e r i s involved.  I n an a d i a b a t i c calorimeter, the heat t r a n s f e r i s not c a l c u l a t e d ,  but i s reduced t o a minimum. The r e s u l t i n g e r r o r i n temperature o f the calorimeter is s  where <P i s the thermal head or the difference i n temperature between the calorimeter v e s s e l and s h i e l d , k i s the leakage modulus and t i s the length of time f o r which the thermal head occurs. The maximum e r r o r ,  = ^  m  a  x  ^ * if ^  i s m  a  maximum thermal  t h e  x  head f o r the time t». For t h i s a d i a b a t i c 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 a d i a b a t i c s h i e l d c o n t r o l s .  The r e s u l t i n g values ( f o r a 30 g.  sample i n the calorimeter vessel) were? Temperature (°C.) Pressure (microns) k (°C/min,/°C head)  -70 0.2 .01  -20  -20  0,1  0.5  0.015 •  0.02  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  Temperature  (See Figure  11)  - 4^ " For a thermal head between calorimeter and s h i e l d bottom only; Temperature (° G.)  50  Pressure (>t)  0.2  k  0.004  b  (°C/min,/°C)  A P h i l i p s i o n i z a t i o n gauge, c a l i b r a t e d w i t h 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 w i t h temperature, but g r e a t l y w i t h pressure up t o about one micron of mercury. The lowest pressure obtained w i t h 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 i n s u l a t i n g v a r n i s h ) .  As the  thermal head between the s h i e l d and outer case increased, and thus the s h i e l d 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 w i t h t h i s thermal  head a minimum (less than 50°C), the l e a s t thermal leakage was obtained because of the low pressure, and a l s o because of the ease of c o n t r o l l i n g the s h i e l d temperature. The maximum e r r o r i n temperature a r i s i n g from thermal leakage can be c a l c u l a t e d from the formula noted previously;  *Wx  °  ^max  For a 10°C temperature r i s e , heat was supplied to the calorimeter v e s s e l f o r about 20 minutes, and an e q u i l i b r i u m period of 5 t o 10 minutes was required. t = 30 minutes.  If  0  ^  =  lf/io'c, and k = 0 03 C/min./ C, then \ ^ o  o  o  Thus  =~0.1°C.  Consequently, the maximum e r r o r i n temperature i n t e r v a l would be 1%. Since  (p  w i l l average much l e s s than i t s maximum, the error from t h i s source w i l l be much l e s s than 1%»  With c a r e f u l s h i e l d  control,0  could be kept w i t h i n l°/30 C, and  the r e s u l t i n g maximum possible e r r o r would be 0.3%. The probable thermal leakage  -  m -  e r r o r (the mean value of a random s e l e c t i o n of p o s i t i v e e r r o r s ) would then l i k e l y be l e s s than 0.1%. Besides the thermal leakage caused by a thermal head measurable w i t h the thermocouples, there can be leakage by other means, such as by conduction along leads and by r a d i a t i o n from hot spots. This leakage causes the calorimeter v e s s e l temperature t o d r i f t l i n e a r l y w i t h time.  However, t h i s e f f e c t was u s u a l l y  n e g l i g i b l e f o r t h i s calorimeter i f the s h i e l d temperature was w i t h i n 50°C. of the outer case temperature. b.  Energy Input E r r o r Since the r e s i s t a n c e of t h e thermometer-heater v a r i e d considerably  w i t h temperature, t h e power supplied by the heater also changed.  The averaging  of t h i s power introduced an e r r o r i n the energy input measurement. The voltage across the commercial 100 ohm thermometer remained almost constant during a heat i n p u t , but the current decreased by** 4% f o r a t e n degree temperature r i s e (see Section I I , 2 f ) , The decrease i n current was not uniform. I t was r a p i d as the power input was s t a r t e d and quick l o c a l heating occurred, but a f t e r one or two minutes i t became uniform as the heating rate became more constant. intervals.  I n order t o average the current, i t was measured at one t o two minute:, The averaging process introduced a maximum e r r o r 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 s t r a i n - f r e e 20 ohm thermometer was arranged so t h a t 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 t o the r e s i s t a n c e of the r e s t of the c i r c u i t (see Appendix V ) . In p r a c t i c e , the power change r e s 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 a d d i t i o n a l e r r o r i n power input measurement r e s u l t e d from inaccuracies i n potentiometer readings.  This e r r o r amounted t o ~ 0,05%. The  t o t a l maximum e r r o r i n measurement of the energy input was thus ~ 0,1%. c.  Temperature Measurement E r r o r .  *  The l a r g e s t error i n s p e c i f i c heat c a l c u l a t i o n s (as i s u s u a l l y 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 c a 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 e r r o r 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 l a r g e , since i t causes a 1% e r r o r i n s p e 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 c a l i b r a t i o n deviations tend, t o cancel out f o r temperature d i f f e r e n c e s , 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 l a r g e , because the  c a l i b r a t i o n must be very poor t o 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 w i t h temperature, so that t h e i r e f f e c t on s p e 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 s t r a i n s or i m p u r i t i e s i n the platinum, are more serious.  The e r r o r 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 f o r 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 l a r g e , i t s e f f e c t on the s p e c i f i c heat anomaly was reduced  because of i t s gradual change w i t h temperature. E r r o r s r e s u l t i n g from c a l i b r a t i o n deviations were g r e a t l y reduced by using the s t r a i n - f r e e thermometer, whose c a l i b r a t i o n constants remained very steady, (ii)  Inaccurate resistance measurement;  The greatest e r r o r i n temperature measurement f o r t h e s t r a i n - f r e e thermometer was caused by the e r r o r i n potentiometer readings. was  The potentiometer  accurate t o 0,01%, Since t h e thermometer resistance was found from the  quotient o f two potentiometer measurements (see S e c t i o n I I , 2 f ) , the maximum e r r o r i n the resistance was 0,02%, which corresponded t o a temperature e r r o r of 0,05°G, The maximum e r r o r i n the temperature i n t e r v a l was then 0,1°C, but the probable e r r o r was considerably l e s s , approximately 0,015°C,  I n 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 t h e maximum e r r o r was considerably greater than t h i s , a more accurate means of measuring the thermometer r e s i s t a n c e would have been d e s i r a b l e . A more accurate potentiometer could be used. c i r c u i t i s often usedj  Some form of Wheatstone bridge  a Mueller bridge i s perhaps t h e most accurate one a v a i l a b l e  g i v i n g p r e c i s i o n up t o 0.001°C. The t o t a l e r r o r i n s p e c i f i c heat measurement f o r t h i s  calorimeter,  under adequate working conditions and using the s t r a i n - f r e e thermometer, was thus made up of probable errors of a 0.1% thermal leakage e r r o r , a 0,05% power measurement e r r o r , and a 0.15% temperature measurement e r r o r . was  ~ 1 . 5 % and the probable e r r o r was ~ 0.2%,  The t o t a l maximum e r r o r  However, the accuracy of the  calorimeter i s perhaps best described by an e r r o r which i s not l i k e l y t o be exceeded.  I f an e r r o r i s used which only f i v e percent of a random sample o f errors  -  52  -  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 s u i t a b l e value t o use, so that the accuracy of the s p e c i f i c heat measurements was 0.5%. IV. 1.  RESULTS  Preparation and Properties of A l l o y s , a.  Preparation of A l l o y s ; A l l o y s close t o the compositions Mn AlC and Mn3ZnC were prepared. 3  The materials used were; Mn;  99.9% p u r i t y , donated by the Electromanganese Corp. of America.  Al:  99.99% p u r i t y , donated by the Aluminum Co. of Canada.  Zn:  99<i#9% p u r i t y , donated by the Consolidated Mining and Smelting Co. of Canada.  C : spectroscopic grade. The a l l o y Mn AlC was prepared by i n d u c t i o n melting under an argon 3  atmosphere i n an alumina c r u c i b l e .  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 Mn ZnC was sintered, using components Mn C and z i n c , f o r 3  3  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 r e s i n t e r e d f o r 12 days at 600°C. b.  Properties of the A l l o y s : (i)  Mn AlC: 3  The a l l o y Mn AlC was m e t a l l i c and b r i t t l e , but e a s i l y corroded 3  i n humid conditions.  X-ray powder photographs showed the a l l o y t o be about 9S%  the ordered face-centered cubic phase expected, w i t h parameter 3.876 A.  The s a t u r a t i o n 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 t h a t o f a ferromagnetic substance.  The Curie p o i n t ,  obtained by p l o t t i n g (f against temperature and e x t r a p o l a t i n g the s t r a i g h t l i n e x  to <f*> 0, was 6 = 286°K = 13°C.  The s a t u r a t i o n magnetization at 0°K,  obtained  squared against (T , and e x t r a p o l a t i n g to zero  by p l o t t i n g absolute temperature  temperature, was 0£ = 102 ergs/g/oersted, which corresponds t o 1.2 Bohr magnetons per manganese atom.  (The same value was obtained by previous  researchers ). 4  (ii)  Mn ZnCs 3  The a l l o y Mn ZnC was i n powdered, non-metallic,easily-corroded 3  form.  X-ray powder photographs showed the a l l o y t o be > 95% the ordered f a c e -  centered cubic s t r u c t u r e expected. A l l o y Is  Two a l l o y s were used?  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 i n d i c a t e d a small amount (~ 2%) o f a o  second phase. A l l o y 2:  The main phase had l a t t i c e parameter: 3.9228 A.  A large a l l o y o f 70 grams was s i n t e r e d i n order to make s p e c i f i c heat measurements more accurate. even a f t e r r e s i n t e r i n g .  About 5% o f a second phase was present  The l a t t i c e parameter o f the cubic s t r u c t u r e  o  was 3.9233 A. The v a r i a t i o n of s a t u r a t i o n magnetization w i t h temperature f o r the two a l l o y s i s shown i n Figure 13.  The maxima were 82.6 and 80.7 erg/g/oersted f o r  a l l o y s 1 and 2, o c c u r r i n g at -45 (*2)°C o r 228 (±2)°K. E x t r a p o l a t i n g the curve to 0°K f o r a l l o y 1 gave a magnetization at absolute zero which corresponded t o a Bohr magneton value o f one per Mn atom (see S e c t i o n T, 2 ) .  E x t r a p o l a t i n g the  part of the curve above -45°C t o 0°K gave a value of ~ 1.5 Bohr magnetons per Mn atom.  These r e s u l t s agree w i t h i n experimental e r r o r w i t h those obtained  - 55 -  ol  1 100  I  I  200  300  ;  L  '400  T(°K.) Figure 13.  The V a r i a t i o n o f S a t u r a t i o n Magnetization, f T , w i t h Temperature, T, f o r Mn-^ZnC.  -  previously^ (see Figure 2). 0, - 391°K = 118°C and 2.  56 -  The Curie temperatures of a l l o y s 1 and 2 were 368°K = 95°C i n a f i e l d of 16,000 oersted.  S p e c i f i c Heat Measurements a.  The S p e c i f i c Heat o f Mn AJC; 3  The s p e c i f i c heat of Mn AlC was measured from -140 t o 100°C, using 3  the commercial thermometer.  The c a p a c i t i e s of the a l l o y plus calorimeter are  p l o t t e d 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 p l o t t e d i n Figure 15. The s p e c i f i c heat curve of Mn AlC showed the expected 3  ferromagnetic  anomaly a t the Curie point, and i t s shape was close t o t h a t predicted by Weiss (Section I , 1).  This anomaly occurred at -10 ±2°C, which i s the Curie tempera-  t u r e i n zero magnetic f i e l d o f 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 o f course higher (13°C). The s i z e 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 o f the anomaly f o r a ferromagnetic substance of magnetization h^kg  per atom i s 3 h ;cal/°C/g atom.^" But the e  height of the Mn AlC anomaly was ~ 2.2 j/°C f o r 0.157 moles, or ~1.1 caL/°C/ 3  g atom of Mn, which would correspond t o only 0,37 Bohr magnetons per Mn atom. The greatest s p e c i f i c heat measured f o r the a l l o y (ignoring the anomaly) was 6.6 c a l / 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 f r e e thermometer, w i t h good general agreement i n r e s u l t s (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 C a p a c i t i e s of Mn AlC (31.97 g) 3  plus Calorimeter (42.18 g ) / as measured w i t h the Commercial Thermometer (Temp, i n t e r v a l s v a r i e d from 5 t o 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  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  -57.2  ;  36.2  I  Figure 15.  Capacity of Mn^AlG  - 60 -  3i|  i  i  -60  -40  Figure 16  •  i -20 T(°C.)  i  i  I  0  20  40  Capacity of Mn^AlC + Calorimeter, (using the s t r a i n - f r e e thermometer)  - 61 -  b.  The S p e c i f i c Heat of Mn^ZnC: (i)  A l l o y 1:  The heat capacity of the sample of Mn ZnC on which neutron d i f f r a c t i o n 3  measurements were made was found from -100 to 60°C, using the commercial thermometer.  The temperature i n t e r v a l s used were from 8 to 12°C, which were  large enough t o obtain accurate i n t e r v a l measurements, and s m a l l enough t o d i s c e r n sudden changes i n heat c a p a c i t y . The r e 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 Mn ZnC, or 1.7 cal/°C/g atom 3  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 q u i t e r a p i d l y . (ii)  A l l o y 2s  Because the sample of a l l o y 1 was rather s m a l l , another l a r g e r sample was made, i n order t o 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 s t r a i n - f r e e 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 Mn ZnC. 3  I t s rounded shape was probably due t o measurement  e r r o r s , and p o s s i b l y a l s o due t o inhomogeneity i n the a l l o y , since the Curie temperature i s v e r y dependent on composition. The l a r g e s t s p e c i f i c heat measured f o r 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 AlC. 3  - 63 -  Table IV. Capacities of Mn ZnC ( A l l o y 2:51.38g.) plus Calorimeter (43.66 g.) 3  as measured w i t h the Strain-Free Thermometer. (Temperature i n t e r v a l s v a r i e d from 5 t o 11°C.)  Average Temp.  (°C)  -69.2  -58.4 -56.7 -53.3 -48.0 -46.5  -45.7 -40.9  -38.3 -36.8 -36.1 -31.2 -28.3 -17.9 - 7.6 27.9 29.7  37.9  Capacity  Average Temp.  Capacity  40.2 42.0 42.1  39.5 47.5 47.8 49.3 56.5 59.8 66.0 69.3  46.3  (j./°c.)  42.6  44.0 44.3 44.6 46.1  47.0 46.5  47.3  42.7 42.6  43.0 43.3 45.6 45.3 45.8  (°C)  72.5  73.1 77.3 79.0 82.2 82.2 89.6 91.9 101.6 111.4  (j./°C) 47.1  46.4 46.8 46.9 46.8  47.0  46.9 46.2 46.0  45.7 45.2 45.3 45.6 45.2  46.0  45.8 45.8  c(j./°c.) p  Figure 18. Capacity o f Mn^ZnC ( a l l o y 2) plus Calorimeter.  ON  •p-  The low temperature anomaly had a very sharp peak, dropping t o zero over only about a 3°C i n t e r v a l .  This anomaly thus was very close t o t h e  t h e o r e t i c a l form predicted from the Weiss theory, which has a d i s c o n t i n u i t y at the Curie temperature. before.  Such an approach t o d i s c o n t i n u i t y  has not been observed  The height o f the anomaly was 5 . 5 j/°C f o r 0 . 2 1 2 moles, or 2 . 1  cal/°C/g atom of Mn. • * c  Magnetic S p e c i f i c 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 Co ° C Cq =  where C  p  - C  v  -  q  as:^  + (C - C ) + C i + S p  v  s p 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  Ci «  excess over the Debye value common to most metals,  S  excess caused by magnetic and other changes.  =  r  In order t o use t h i s procedure, however, measurements over a large temperature range must be made.  I n 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 t o c a l c u l a t e the e l e c t r o n i c  and t h e l a t t i c e s p e c i f i c heats.  Consequently, because of the l i m i t e d temperature range of these experiments, and because of f u r t h e r u n c e r t a i n t i e s a r i s i n g a t higher temperatures, t h i s method was not s u i t a b l e . The magnetic s p e c i f i c heats i n the present research were thus approximately determined by e x t r a p o l a t i n g the observed s p e c i f i c heat curves from above and below the anomalies. 20.  In Figure  19  The r e s u l t s are shown i n Figures 1 9 and  the magnetic s p e c i f i c heat  MCJI  o f MnaAlC (from Figure  15)  is  p l o t t e d , as w e l l as the values of -o'dcr/dH obtained from the magnetization  T(°C)  F i g . 19.  Magnetic S p e c i f i c Heat of Mn AlC. 3  200  220  240  260  280  300  T°K F i g . 20  Magnetic S p e c i f i c Heat of Mn ZnC ( A l l o y 2). 3  320  340  360  - 68 -  curve (Figure 12). (M = gram molecular weight, Cj/j = magnetic s p e c i f i c heat/gr.). A curve o f M C M / T versus T i s also p l o t t e d , which when i n t e g r a t e d gives the CM dT/T =  entropy change associated w i t h the magnetic change; A S = JUL  The magnetic energy, U = £MCjidT =  1.8 j/mole/°K = 0.15 caL/g. atom Mn/°K. 460 j/mole = 37 cal/€» atom Mn.  Figure 20 shows the magnetic s p e c i f i c heat o f Mn ZnC ( a l l o y 2,from 3  Figure the  18),  and a curve (M  CJJ/T,  T ) . The entropy changes corresponding t o  low and high temperature anomalies r e s p e c t i v e l y w e r e A S ^ = 1.7  0. 14 c a l / g . atom Mn and AS  2  ^/mole/°K =  » 1.3 j/mole°K = 0.11 c a l / g . atom Mn/°K.  magnetic energies, U]_ = 385 j/mole = 31 c a l / g . atom Mn  and U2  0  The  420 j/mole =  34 c a l / g . atom Mn.  V. 1.  DISCUSSION AND CONCLUSIONS  Discussion o f Results The s p e c i f i c heat curves of Mn AlC showed a second order anomaly at 3  i t s CurieI point (-9°C), as expected f o r t h i s ferromagnetic a l l o y .  The curve  f o r Mn ZnC showed second order anomalies at -35°C and a t 65°C, which supported 3  the  p r e v i o u s l y discussed (Section I , 2) concept o f the magnetic s t r u c t u r e of  the  alloy:  i t i s f e r r i m a g n e t i c 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 t h e o r i e s .  One o f the f i r s t ferromagnetic t h e o r i e s was the phenomeno-  l o g i c a l 'molecular f i e l d ' theory o f W e i s s , ^ the quantum m o d i f i c a t i o n o f which i s i n good agreement w i t h 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 e f f e c t i v e magnetic f i e l d , He, which i s t h e sum of the  applied f i e l d , H, and the molecular f i e l d , NI, which i s p r o p o r t i o n a l t o the magnetization, I . The p r o p o r t i o n a l i t y constant, N, c a l l e d the molecular f i e l d constant, i s a measure of the exchange forces' a c t i n g 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 where  J  A  <= J g ug = the r e s u l t a n t angular momentum quantum number of an atom = the sum of o r b i t a l (L) and s p i n (S) quantum numbers,  g  = gyromagnetic r a t i o = 2 f o r s p i n angular momentum only (« 2 f o r ferromagnetic materials experimentally)  Ug = the Bohr magneton, the magnetic moment of a s i n g l e spinning e l e c t r o n = eh/4lTmc = 9.27 x 10~ erg/gauss. 21  Let  a » u He/KT  where  K = Boltzmann"s constant and the e f f e c t i v e f i e l d ,  A  H = H + NI :. e  Then  the average magnetic moment of the substance, u, i s deduced from;  3L = L. U  A  I  -  d:  S  E  M  /  J  £  0  y~ ma/J m =' magnetic moment per u n i t volume at 0°K. e  where I Then  Q  I _ = 2J + 1 c o t h f e j + l ) a - 1_ I 2J 2J 2J Q  coth a_  2J  - 68 b „  68 c This i s the B r i l l o u i n f u n c t i o n . For j -» 1/2  B  0  (a s i n g l e e l e c t r o n s p i n ) ,  I = tanh u H Io KT A  I f H = 0,  The Curie temperature,fi«(J+l)gu NI /3K .  g  = tanh a  I _ - tanh I / I I V&  a  0  where  © = u NI /K A  0  This curve agrees w e l l with experiment.  17  The s a t u r a t i o n magnetization ( I / I ,  curve c a l c u l a t e d f o r  Q  J = 1/2, and the experimental curves f o r N i and Mn AlC are p l o t t e d i n 3  Figure 21.  The c a l c u l a t e d curve i s q u i t e close t o experimental r e s u l t s f o r  N i , but d i f f e r s considerably from the observed Mn AlC r e s u l t s . 3  I t should be  noted that i n general the curves observed f o r a l l o y s are l e s s concave t o the  T/Q a x i s than the curves f o r pure m e t a l s .  ( Stoner's c o l l e c t i v e e l e c t r o n  2 0  theory gives r e s u l t s close t o the Mn AlC curve). 3  UgNI /K = 264°K, the molecular f i e l d constant f o r  Using  0  Mn AlC, 3  N = 6600 gauss cc/erg. The molecular f i e l d , N I Q = 3.9 x 10 gauss. 6  For ferromagnetic Mn ZnC, 3  N = 7400 gauss cc/erg. NI = 5.1 x 10^ gauss, assuming the s a t u r a t i o n moment o at 0°K t o be 1.5 u per Mn atom. B  From the B r i l l o u i n f u n c t i o n , 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^Sk "TT  3K(T - © )  =  °M  fT©  - 68 d The Curie constant per mole,Cj/[ - (J+l)gug G£M  where C ^ i s the magnetic  3K moment per mole w i t h a l l elementary magnets aligned p a r a l l e l (at 0°K).  From  paramagnetic data f o r Mn ZnC, 3  N = 0K  = 2700 gauss  2  cc/erg, where  M = gram molecular weight. This r e s u l t 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 ( e s p e c i a l l y zinc evaporation).*  The value of C^  could not be determined f o r Mn AlC because of the curvature of the (1, T) curve, 3  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 c a l c u l a t i o n s 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 u n i t  mass, dT  . 1  Then i f the B r i l l o u i n f u n c t i o n 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 c o n t i n u a l l y w i t h i n c r e a s i n g temperature to the Curie p o i n t , where i t drops d i s c o n t i n u o u s l y to zero. i s shown i n S e c t i o n l i b .  The form of the curve  I f the gyromagnetic r a t i o , g = 2, the magnitude  of the d i s c o n t i n u i t y per gram-atom at the Curie point i s AA C^ = 3 Rn /2 f o r Q  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 w i t h these r e s u l t s , '  '  The d i f f e r e n c e s 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 c a l c u l a t e d from ferromagnetic and from paramagnetic data u s u a l l y do d i f f e r considerably.  - 68 e a t t r i b u t e d t o l o c a l ordering above t h e Curie temperature, and that the observed curves r i s e more r a p i d l y than the c a l c u l a t e d ones below t h e Curie point.  The decreases i n CJJ at the Curie temperature f o r Fe, Co and N i are  i n f a i r agreement w i t h theory f o r J = 1/2. Because of the d i f f e r e n c e between the t h e o r e t i c a l magnetization curve and t h a t observed f o r Mn AlC, the observed s p e c i f i c heat anomaly f o r 3  t h i s a l l o y can not be i d e n t i c a l w i t h the anomaly predicted f o r J «* 1/2. The values of Cdtf/dT taken from the magnetization curve f o r Mn AlC are p l o t t e d against T i n Figure 19. I t i s seen that the r e s u l t i n g 3  curve i s much f l a t t e r than the (Cjj^ T) curve. f i e l d constant N i n C M  D  Thus the value o f the molecular,  -UpoUtf/dl must be changing r a t h e r r a p i d l y over the  measured temperature range.  This i s t o be expected, since t h e v e r y s i m i l a r  a l l o y Mn ZhC experiences a t r a n s i t i o n from f e r r i - t o ferromagnetism i n t h i s 3  r e g i o n , which e n t a i l s a r e v e r s a l of s i g n i n the molecular f i e l d . The anomalies observed f o r Mn AlC and Mn ZnC ( F i g s . 15, 16, 18, 3  3  19, 20) were t y p i c a l o f 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 AlC and 0.2|R per Bohr 3  magneton f o r the ferromagnetic Mn ZnC anomaly, as compared w i t h the t h e o r e t i c a l 3  3 R/2jalso observed f o r pure metals.  The low temperature Mn ZnC anomaly was 3  q u i t e steep, i n d i c a t i n g a r a t h e r abrupt t r a n s i t i o n from ferrimagnetism t o ferromagnetism.  The u s u a l l y observed ' t a i l i n g o f f * above t h e 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 c a l c u l a t e d from the p a r t i t i o n f u n c t i o n , Q, evaluated above the Curie temperature,  - 68 f J using the p r e v i o u s l y mentioned notation ,  m=-J  The f r e e energy, F = -RTlnQ S = -$F  The entropy,  *T  Thus f o r J = 1/2 and the e f f e c t i v e magnetic f i e l d , H gram-atom.  •= 0 ( T > 0 ) ,  e  S « Rln2 per  In general, S = R l n ( 2 j + 1 ) . The ferromagnetic t r a n s i t i o n may also be considered i n terms o f a  simple model, i n which each atom has one e l e c t r o n spin ( S = 1/2) capable o f an o r i e n t a t i o n 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 , g i v i n g zero entropy. Above the Curie temperature, the e l e c t r o n s w i l l represent a paramagnetic gas, with h a l f oriented p a r a l l e l and h a l f a n t i p a r a l l e l . S  = Kin N i N i „N S. 2 2 e  e  therefore  Then the entropy i s .  e  f  o  r  j j electrons. e  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 q u i t e small.  The s p 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 t o the paramagnetic s t a t e of CuCl .2 H 0 has been found by 2  Friedberg  2  t o 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 o f f according t o 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 t o the t h e o r e t i c a l Rln2 = 0.69R = 1.4 cal/g» tom/deg. expected. a  Similar  22 r e s u l t s have been obtained f o r other paramagnetic  salts.  For ferromagnetic materials at elevated temperatures, the r e s u l t s are however f a r from s a t 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 e r r o r involved i n c a l c u l a t i n g the entropy change.  The change corresponding t o the magnetic  anomalies o f some ferromagnetic substances has been c a l c u l a t e d 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 s e v e r a l researchers t o 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 A S = 0.5 c a l /  g.atom/deg. ( A S = 0.3 f o r only the peak o f the curve, n e g l e c t i n g 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 r e s t have J = 0.  I f , however, as predicted by Mott and Jones '  J « 0 f o r 70 percent o f the atoms and J =» 1 f o r 30 percent o f 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 f r e e i n the paramagnetic s t a t e , and r e s u l t s from t i g h t bonding theory. deg.  The experimental r e s u l t f o r i r o n ^ was A S » 2 . 0 cal/g.atom/ 2  The expected value i s a t l e a s t 2.4 ( f o r J = 1 ) . These experimental  values are considerably l e s s than the t h e o r e t i c a l , but are r e c o n c i l a b l e w i t h i t i f a rather large degree of l o c a l ordering not observed i n the form o f 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 K a s t e l e i j n and Van Kranendonk short range order above the Curie p o i n t .  t o e x p l a i n the presence o f  The entropy above the Curie p o i n t  i s then c a l c u l a t e d 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 t h e above mentioned discrepancies.  - 68 h I t i s seen t h a t even a f t e r very c a r e f u l c a l c u l a t i o n o f 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 r e s u l t s f o r t h e entropy change can be obtained f o r t r a n s i t i o n s a t elevated temperatures. The r e s u l t s obtained f o r Mn AlC by the author ( A S = 0.15 ca]/g«atom/deg.) 3  are indeed very low and i n d i c a t e that f o r accurate r e s u l t s a l a r g e r temperature range,and i n p a r t i c u l a r measurements a t low temperatures to determine the Debye constant,are required. The entropy changes observed f o r each o f t h e anomalies of Mn ZnC 3  (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 f o r both anomalies should be 3/2 Rln2 per gram-atom o f manganese, i f J = 1/2 f o r the ferromagnetic s t a t e .  The entropy change f o r the upper t r a n s i -  t i o n from -33°C t o 100°C ( f o r T/&>0.70), assuming the t h e o r e t i c a l ( C , T) M  curve f o r J = 1/2, i s (0.4l)3/2R.„ Thus the entropy change f o r t h e low temperature anomaly i s 3/2 R(0.28) = 0.8 cal/g.atom/deg.  This i s o f course  only a crude estimate, since l o c a l ordering above the ferromagnetic Curie temperature, as w e l l as other f a c t o r s , were not taken i n t o consideration. Nevertheless, i t i s apparent t h a t the experimental r e s u l t s f o r Mn ZnC are not 3  of the correct order o f magnitude. I t i s f e a s i 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 o f the entropy change and consequently a v e r i f i c a t i o n of the low temperature s t r u c t u r e of Mn ZnC. 3  magnetic  To do t h i s , a d e t a i l e d theory o f the degree of disorder  i n the ferrimagnetic and ferromagnetic s t r u c t u r e s , and the degree o f 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 o f the two anomalies, and the r a p i d r i s e of the l a t t i c e and e l e c t r o n i c s p e c i f i c heats i n the anomalous region make an accurate separation of t h e magnetic s p e c i f i c heats d o u b t f u l .  The object o f  - 68 i the present research i n any case was merely to determine the presence of the two anomalies, and the small experimental temperature range permitted no accurate entropy change c a l c u l a t i o n .  However, the determination that  t r a n s i t i o n s from ferrimagnetism to ferromagnetism to paramagnetism occurred is in i t s e l f  significant.  Calculation of the magnetic anomaly of a ferromagnetic substance i s further complicated by the p o s s i b i l i t y that the drop i n the magnetic s p e c i f i c heat at the Curie point may be masked by a sudden increase i n the s p e c i f i c heat caused by electronic d i s t r i b u t i o n .  This r e s u l t i s produced by  I J^ie  c o l l e c t i v e 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 r e l a t i v e magnetization (as d i d Weiss), and that the p a r t i c l e s 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 s p e c i f i c heat of  ACM/R  °  -1.8  was compensated by a r i s e i n  e l e c t r o n i c s p e c i f i c heat of A C j / R ° 1 . 2 , so that the resultant was only A G / R = - 0 . 6 .  discontinuity  This i s not however i n agreement with experiment  many ferromagnetic substances.  for  By making the exchange energy also vary with  higher powers of the magnetization t h i s d i f f i c u l t y was overcome, and also the magnetization curves were i n better agreement with experiment. c.  28  Theories of exchange interactionsi In order, to f u l l y appreciate the phenomena of f e r r o - and f e r r i -  magnetism, the cause of the Weiss intermolecular f i e l d must be considered. 29  Heisenberg  o r i g i n a l l y explained i t  by exchange i n t e r a c t i o n between electrons  of neighbouring atoms i n terms of the Heitler-London method of l o c a l i z e d atomic wave f u n c t i o n s .  - 68 Slater  3 0  j -  extended t h i s theory t o apply t o ferromagnetic m a t e r i a l s •  He assumed t h a t a p o s i t i v e exchange i n t e g r a l (and thus ferromagnetism) r e s u l t e d from exchange between adjacent 3d s h e l l s when the r a t i o o f i n t e r nuclear distance to d s h e l l diameter was l a r g e r than a c e r t a i n value. Zener, ^ however, s t a t e d t h a t d-d coupling always gives a negative 3  exchange i n t e g r a l (producing a n t i p a r a l l e l s p i n s ) , and that ferromagnetism i s caused by a p o s i t i v e exchange i n t e g r a l between conduction electrons and incomplete d s h e l l s .  (In h i s c a l c u l a t i o n s 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 e l e c t r o n s and band wave f u n c t i o n s f o r the outer s e l e c t r o n s ) .  Zener's theory i s u s e f u l f o r Heusler a l l o y s and f e r r i t e s ,  but some disagreement w i t h neutron d i f f r a c t i o n data has been noted. Slater  3 2  has proposed t h a t t o overcome d i f f i c u l t i e s of non-  orthogonal wave f u n c t i o n s o f the Heisenberg method, determinantal wave f u n c t i o n s composed o f orthogonal energy-band o r b i t a l s should be used.. I f a s i n g l e d e t e r minantal wave f u n c t i o n i s used, the energy-band o r c o l l e c t i v e e l e c t r o n theory r e s u l t s , which i s q u i t e u s e f u l f o r s m a l l i n t e r n u c l e a r d i s t a n c e s . However, f o r complete accuracy a l l p o s s i b l e l i n e a r combinations o f p o s s i b l e determinantal wave f u n c t i o n s i£tj®t ^be made.  This however e n t a i l s an enormous amount o f  c a l c u l a t i o n s , and makes p r a c t i c a l a p p l i c a t i o n s d i f f i c u l t . The d i f f i c u l t y i n e x p l a i n i n g the presence of ferrimagnetism and antiferromagnetism i n terms of exchange energies i s more profound.  One approach  i s the c o n s i d e r a t i o n o f exchange i n t e r a c t i o n between e x c i t e d valence s t a t e s of 33 cations o f the same t r a n s i t i o n element (super exchange). general theory o f S l a t e r problem.  3 2  The completely  i s a l s o c e r t a i n l y t h e o r e t i c a l l y a p p l i c a b l e to t h i s  - 68 k 2„. Cone Ins i o n s , a.  The Calorimeter, AD> anssoid? a d j a b a t i c calorimeter was cbnstructed i n order t o measure  s p e c i f i c heat anomalies o f c e r t a i n 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 overcome by using a s t r a i n - f r e e platinum r e s i s t a n c e thermometer-heater.  The  accuracy o f the calorimeter was 0.5$$. the use of an accurate bridge f o r resistance measurements would increase i t t o 0.2%, b.  S p e c i f i c Heat  Measurements.  S p e c i f i c heat curves were s u c c e s s f u l l y measured f o r the a l l o y s Mn AlG and Mn ZnG. 3  3  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 p o i n t , -10°C, The presence of two second order anomalies on the s p e c i f i c heat curve of Mn ZnC supported 3  the  idea of a complex magnetic behaviour f o r the a l l o y s  i t i s ferrimagnetic  below «35°C, ferromagnetic between, -35 and 65°C and paramagnetic above 65°C  0  The observed anomalies were q u a l i t a t i v e l y i n agreement w i t h t h e theory o f Weiss.  Accurate separation of the magnetic s p e c i f i c heats,, which  would make a q u a n t i t a t i v e t h e o r e t i c a l i n t e r p r e t a t i o n o f the r e s u l t s f e a s i b l e , was impossible because o f the l i m i t a t i o n s of the experimental data*  - 69 -  VI.  APPENDICES  Appendix I  Calculation  of E f f e c t i v e Magnetization o f Mn Atoms.  The a l l o y M n A l C 6 0  The a l l o y Mng/ Al 6C Q f  a  2  2 0  has a magnetization ofy"^  2 o  = 1,22/tg per Mn atem..  , i n which 4 atomic percent of Mn atoms have replaced A l  atoms, has an average magnetization yu. 0  = 0.8°>*B  per Mn atom.  The decrease i s  assumed caused by the e x t r a c (=4) atomic percent Mn atoms, w i t h a magnetization ofyU. , which i s assumed independent o f 2  then  (c + 60)^u  ^u ^u  z  z  c  - 6 0 ^ + c/*  » -  0.89  2  + 60(0.89 -  -4.06/<  1.22)/4  B  Thus, i f r e p l a c i n g A l atoms by Mn atoms in.the a l l o y Mn AlC does not 3  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 e x t r a Mn atoms have a magnetization o f  -4/Cg*  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 f e r r i m a g n e t i c or antiferromagnetic substance c o n s i s t s o f two or more s u b l a t t i c e s w i t h a n t i p a r a l l e l s p i n systems (with opposing magnetization v e c t o r s ) . Consider two s u b l a t t i c e s , 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), I  A  The magnetizations of the s u b l a t t i c e atoms are  and I g , so t h a t the r e s u l t a n t magnetization i s I  - M  A  +>u.I  (!)  B  The i n t e r n a l W e i s s molecular f i e l d s , H atoms depend on I  A  and I g : H  -  A  thus  H  (2)  N(B>u.I - A I )  (3)  A  B  A  = N(aAl +x.I ) A  A  Hg =  A  ->«.1 B )  N(aAl  H*B -  since I  and Hg, a c t i n g on A and B  A  N(B>cI  B  B  + 7vI ) A  =|H |  (4)  .  (5)  A  i s i n the opposite d i r e c t i o n t o I g . N i s the Weiss i n t e r m o l e c u l a r  f i e l d constant and a and B  represent the s t r e n g t h o f the e f f e c t the s u b l a t t i c e s  have on t h e i r own molecular f i e l d s . And, according t o C u r i e , f o r paramagnetism: *  I and  A  -  C (H + H ) , T A  Ig =  C (H + Hg) T C i s the Curie constant and 5 i s the applied f i e l d . H  e  = H + H  A  or H + Hg  (6)  (7) The e f f e c t i v e f i e l d ,  S o l v i n g equations ( l ) , (2), (3), (6), and (7): H »  1  -  ix where  T + 1_ c  <f  Xo  T  (8)  -e  X = susceptibility, Xo "  c  -n-  *• e  cT = -v-eLn-t-e) c 6 and  V  NGx?v(2 + a + B)  =  -NC(/LB  -  N c2«A(aB - 1)  +>a)  2  This i s the equation f o r the paramagnetic behaviour of a f e r r i m a g n e t i c substance. I t i s a hyperbola w i t h curvature concave t o the temperature  axis,  Ne*el predicted s e v e r a l types of v a r i a t i o n o f s a t u r a t i o n magnetization w i t h temperature f o r f e r r i m a g n e t i c substances.  2,  Three o f these are as shown:  Energy o f Magnetization of. A n t i p a r a l l e l Spin Systems: The absolute value o f the e f f e c t i v e f i e l d a c t i n g on A o r B atoms i s : H  e  «. H  A  or  Hg  Therefore the energy o f magnetization per u n i t mass,  =  l  - ^ ^ B ^ B  +  A  d  l  A  ^  +  +  I  l  B  d  l  A  ADIBJ  from equations (4) and (5), For an antiferromagnetic substance, >I and thus  since 1 = 0,  = y t l g ,'  A  a =  B  ,  since H  Ill  U  =  -  U  =  - N I  A  = Hg  > ( a + B + .2) 2  > ( a + 1)  2  2  A  This value i s t h e same as t h a t of a ferromagnetic substance of magnetization I. n  (see equation (4) of I , 1) i f 2 ^ ( a + 1) = 1. 2  This would be the case i f  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 f o r a n t i p a r a l l e l s p i n s , a n d they occupied equivalent lattice sites. Assuming the f e r r i m a g n e t i c postulated behaviour of Mn-Al-C a l l o y s , the magnetization energy can be c a l c u l a t e d f o r an a l l o y Mn^AljeCap, i n which f o u r out of every 64 Mn *B* atoms have a spin of -4, a n t i p a r a l l e l t o the s p i n of 1.2 of the other 60 *A» atoms:  h  -  J L _  1.2  I  A  60 therefore  ^-In =  2 y^j 9  - i° i .  a  13"  3  A  .  -  - 73 :-hus  U«  H  » - S /  2  +  2  Vsia  81 L - 2 N j 2-s 2-v2 4 / * 2  J5"  U' ~  £* + §* + T  £ A £ 2L  N  If o - B - 1 ,  0  * 4B  +  fx ] 2  3ol J  A  Since Mn AlC i s ferromagnetic, then i t s magnetization energy, 3  U««  = -  N  P  j  (equation (4) of S e c t i o n I, l ) .  2  2  But/><r = ^ I , since the magnetization of the Mn atoms i n Mh AlC is* assumed equal A  3  to 1^, the magnetization of the Mn atoms of the A s u b l a t t i c e i n Mn64Al| C » X i s 6  2o  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»«  = _ NT I 2  2 A  2/  = 2 u» 3  The t o t a l magnetization energy, I P ' , i s only 2 as great as f o r the f e r r i m a g n e t i c state ( i f N i s the same i n both cases).  3  Appendix I I I  Assembly of the Thermometer Mica  Cross  The mica cross was assembled as f o l l o w s : mica sheet 0.003 i n c h  thick  was clamped between smooth s t e e l bars and cut with a razor blade i n t o 0.20 by 1.1 i n c h pieces.  Two pieces were s l o t t e d h a l f way so that they could f i t  together lengthwise at r i g h t angles, forming a cross. s m a l l s t e e l bars was used f o r the s l o t t i n g .  A d i e formed from four  Small holes were d r i l l e d through the  mica at one end t o permit anchoring of the leads. The pieces of mica were notched along each side by p l a c i n g them i n a grooved d i e and c u t t i n g out the mica w i t h 0.008 i n . piano wire held i n a metal bow.  The d i e was made by machining 0.30 i n . of 3/8'* brass rod down t o 0.2 i n .  diameter and then threading at 29 per inch. then s p l i t by c u t t i n g lengthwise.  The threaded part of the rod was  One h a l f of the s p l i t r o d was removed and  attached t o the other h a l f w i t h screws at e i t h e r end, so t h a t the mica could be held f i r m l y between the halves.  1.1"  Notching Die  Notched Mica  The mandrel used t o hold the assembled mica cross was constructed by s o l d e r i n g together f o u r bars of 3/16 i n c h brass and machining down t o a_ 0.150  7 5 -  diameter rod.  The platinum h e l i x was wound n o n - i n d u c t i v e l y 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 t o about 0,28 inch diameter, i n order t o obtain t h i n - w a l l e d tubing.  This t u b i n g was necked s l i g h t l y at the end  where the leads were t o emerge. In order t o make a l e a k - t i g h t s e a l of the platinum leads through pyrex g l a s s , a s p e c i a l technique 0.010  was u s e d . ^  Platinum s t r i p s , r o l l e d from  inch wire t o l e s s than 0,001 inch t h i c k , were used f o r the s e a l .  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  s e a l through  pyrex.  To fuse a s t r i p t o the 0,003 inch platinum thermometer w i r e , the s t r i p was f o l d e d over the end of the w i r e , and the wire was fused t o the corner of the f o l d w i t h a small oxy-gas flame. Torch  A  micro-burner i s not needed f o r t h i s flame i f pincers are used, as shown,, t o  Pincers  prevent melting of the f i n e platinum wire.  A f t e r the platinum s t r i p s were fused t o the leads, the thermometer c o i l was pushed i n t o 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 w i t h acetone, and f i r e polished at near the P! g U  melting point w i t h a gas flame.  A plug,  made from pyrex rod, was placed i n the tube end, and fused w i t h a s o f t flame t o uross p  platinum s t r i p s .  the tube and about 1/8 i n c h of the  The s e a l was cooled s l o w l y i n a yellow flame t o remove s t r a i n s .  The s e a l was t e s t e d i n a vacuum system, a f t e r being Immersed i n l i q u i d oxygen.  Appendix V  C a l c u l a t i o n of Power Change f o r the Thermometer-Heater (see Figure 8b) Let  R -  *ti  'v  .  R  t  *  + G  R  the i n i t i a l heater r e s i s t a n c e ,  to  t  the f i n a l heater r e s i s t a n c e ,  E  the constant voltage supply,  R  the r e s i s t a n c e o f the c i r c u i t minus  P  heater r e s i s t a n c e ,  V  The i n i t i a l c u r r e n t , I  V  Q  l  i n i t i a l , f i n a l voltages across heater.  =  E R  1^  The f i n a l c u r r e n t ,  t  +R  G  E  ,= R  Btx  +  R  V I 0  0  =  Rt *o  =  Q  +  V  Po » e i n i t i a l power through heaters t n  the f i n a l power,  E'  P^  t • R^r  £i .  »ti  P  R  0  tQ  / + t ^R R  *' [n  + R^j  R  Q  +  For R ^ R^, as w i t h the 100 ohm thermometer (Figure 8a): f o r a 10°C temperature P  n  R  tn  "  rise,  - 78 For R = R. , as w i t h the 20 ohm thermometer ( F i g . 8b): ^o PiP  UH  * t i R  0  t  ( 0  2  0  R  t  +  = R t  Q  l  4  R  t o  (R  (2 R t  ) 2  G  t o  *A  R  t )  +ARt)^  For a 10°C temperature r i s e , A R t * .04 R t therefore  Pi  h_ R t 26 o Q  25(2 R  Q  P  4 x 26  R t  =  T  £l  D  =  itl60  =  + .04 R t )  t(j  (2.04)* x 25  a  Q  =  0.9995  4162  0  The percent d i f f e r e n c e i s 0 . 0 5 % for  ?  „  1,  R  =  1.02  1  Rt  Q  if A R  t  =  0.04  R  to  That i s , f o r no power change, R must be h a l f way between Rt  and R^  - 79 -  Appendix VI  Equivalence of Resistance-Temperature Thermometer.  Equations f o r a Platinum Resistance  The general equation connecting r e s i s t a n c e and temperature f o r a platinum resistance thermometer i s s Rt for thus  -  R ( l + At + B t ) 2  0  0 < t < 630°C.  t -  % R  - Ro  100 =  t -  100 ' °  t -  ^  R  2  1 0 0  100A + 100 B  R  2  =  therefore  At + B t _  "* o 100= 100 - o  lOOBt - B t A + 100B 2  100 B A + 100B  R  2  R  100  |~ t 100  - / t f ( 100J  (lOOj  according t o Callendar's equation.  therefore  £ =  -  2 100 B A + 100B  » 1.5 f o r pure platinum For temperatures between -183 and 0°C, Rt t -  % - p l00" o  R | l + At + B t + C(t - 1 0 0 ) t ^ 2  -  R  R  1 0 Q  m  t  R  R  At + B t + C(t - 1 0 0 ) t - | 100A + 1002 B 1  ,R  •  t -  t - Rp  ioo - o R  =  2  _ 100 B A  3  lOOBt - B t - 0 ( t - 1 0 0 ) t A + 100B 2  1 0 Q  3  0  +  1 0 0 B  f /t \  Lv™i  2  --  t "J-  m  3  C ,  t  A + IOOB^  .  1 0 0 ) t L  U  O  ;  T  3  - 80 The l a s t term may be w r i t t e n : + ( - I00)t3 D  where  t  D  =  -  C A + 100B  Thus i n Callendar form, the equation i s : t  =  ^ ~ o 100 " o R  R  R  100  +  £ [ ( O . O l t ) - O.Olt] + D(t - 1 0 0 ) t 2  3  -  81-  Appendlx VII  E r r o r i n Temperature I n t e r v a l Caused by E r r o r i n C a l i b r a t i o n of Resistance Thermometer. Above 0°C, the temperature of a resistance thermometer i s : t -  - Ro R  100  1 0 Q  ^[(.oit)  +  - .oit]  1  ~ o R  The i n t e r v a l between two temperatures, t - t  x  = At =  *t ~ R  -  100~ A  °  R  2  2  + <r[(.oi)\t + t ) A t -  1 0 0  1 -£[].0l)  D  r  .OlT  $ft. l) (t + tl) 2  x  +  0  ^  0  (t + t i ) - , 0 l |  2  L  J 2  ....^  2  x  0 £  100,  < t <  .01]  [(.0l) (t + ti) -  2  .oiAt]  Q  + £ ^ . o i ) ( t + t ) - .oi| + for  + .Oltx - .OltJ  o  ioo - R  A St 100  i o o -. R  (.Oltx)  1_  100 - R  R  R  +cf[7.01t) -  1 0 0  100  ARt R  =  t  l  t  x  R  t h e r e f o r e At  R  2  2  < (,oi$) < 2  .0004  so that t h i s term i s n e g l i g i b l e i n comparison with unity. therefore A t  = A*t R  With a d i f f e r e n t  A  t»  -  100  100" o R  / l+^|T.0l) (t 2  I  + t) t  -  .0l1\  L  J  J  calibration,  A R t loo (Rioo - Ro)'  / i + (.  l~(.oi) (t + ) ' - .of! I L Jf 2  H  1  Thus t h e e r r o r i n temperature i n t e r v a l ,  iQQ -  (i) At - At« * A n ^ f  \ R i o o - Ro 5 = 8»  If  » AR.100f  \  1  1 Rioo - R - AR+ 102  /  Rioo- Ro  -  +  5  -  \  (Rioo- Ro)' J  Ro  _  O.OlS  Q  0  R  i  \,  (Rioo - Ro)'  J  f o r platinum •  ** ( Q . 4 + 6 ) R .  (Rioo - Ro)*  0  I n a 10°C. temperature i n t e r v a l , A t - At*  A R t ^ »04 R  - .04 R 1 0 2 /  1  a  (0.4H "  4.08  Q  o  1 (0.4 +€ ) R  \ 0  J  € 0.4(0.4 + € )  I f (r= 0.002 (eg.  i f^  1 0 Q  - R  Q  changed by 0.2 ohms f o r the 100 ohm  thermometer: a very large change),. At  - A t » " 4.08 x 0.002 0.4 x 0.402  = o.051°C.  Thus the e r r o r i n temperature i n t e r v a l would be: At  - At' At  ^  Rioo - o (Rioe-Ro)'j  - Ro)'  -  0.018  2 .  1*100" R cs 0.4 R  therefore  1 (Rioo  0  i  (Rioo-*  put  (Rioo-Ro)*  - 100 o'Z-T  and % = 1Q0°C.  A t - At»  if  + iQQSH  100  = 0.05 ioo = 0.5$ 10  - 83 -  This e r r o r i s considerably change i n R]_oo~ R  Q  of 6 R ^  l e s s than the e r r o r i n temperature caused by a i fR  Q  remains the same, R^OO Ganges by € R . Q  Thus the temperature e r r o r would be € R 0.4 R  .  83  Q  l o c  0.5°C a t 100°C f o r 6 = 0.002.  0  The e r r o r 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 f o r the commercial thermometer AR^  (see Section I I I , I d ) , f o r  = 4.00 a t 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 •» 1 0 ° , t h e e r r o r was 0.9$. temperature between 0 and 100°C.  This e r r o r would be l e s s f o r any  - 84 BIBLIOGRAPHY  1.  F. S e i t z , 'The Modern Theory of S o l i d s ' , McGraw-Hill, 1940.  2.  J . Roberts, 'Heat and Thermodynamics * ^ B l a c k i e .  '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,  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 C o n t r o l ' , BlakLston 1941.  132 (1955).  J . Am. Chem. Soc. ,55,  10.  Southard, Brickwedde,  4378.  11.  A. Reddoch, Master of Science Thesis, Queen's U n i v e r s i t y , Aug.  12.  J . Southard and D. Andrews, J . F r a n k l i n I n s t . 209.  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-  16.  K. Grew, P.R.S. 14j>,  17.  R. Bozorth, 'Ferromagnetism',  18.  L. Neel, Ann. Phys. 3_> 137 (1948).  509 (1934). Van Nostrand, 1951.  349 (1930).  (1953).  1954.  - 85  6, 661,  -  19.  P. Weiss, J. Phys. (4)  (1907).  20.  J . Went, Physica, 1£, 98, 596 (1951).  21.  S . Friedberg, Physica 18, 714 (1952).  22.  J, Daniels, F. Robinson, Phil. Mag. (7),  23.  E . Stoner, Phil. Mag. (7), 22, 81 (1936).  24.  N. Mott, H. Jones, 'Theory of Properties of Metals arid Alloys', Clarendon  4Jt, 630 (1953).  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 . 28.  E . Wohlfarth, Rev. Mod. Phys. 2j5, 211  (1939).  (1953).  29.  W. Heisenberg, Z. Physik i£, 6 1 9 (1928).  30.  J. Slater, Phys. Rev. 3j6, 57 (1930).  31.  C  32.  J . Slater, Rev. M o d . Phys. 2^, 1 9 9 ( 1 9 5 3 ) .  33.  J . Van VLeck, Grenoble Conference, 114 (1951).  Zener, Phys. Rev. 81, 440 (1951); 8 3 . , 299 (1951).  

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