UBC Theses and Dissertations

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UBC Theses and Dissertations

³He melting curve thermometer Shinkoda, Ichiro 1983

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3HE MELTING CURVE THERMOMETER by ICHIRO SHINKODA B . S C , UNIVERSITY OF BRITISH COLUMBIA, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA DECEMBER 1983 © ICHIRO SHINKODA, 1983 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by t h e Head of my Depar tment or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f PHYSICS The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P lace Vancouve r , Canada V6T 1W5 D a t e : DECEMBER 30 , 1983 A b s t r a c t T h i s t h e s i s d e c r i b e s t h e c h o i c e , t h e c o n s t r u c t i o n , t h e o p e r a t i o n , and t h e t e s t i n g o f a 3He m e l t i n g c u r v e t h e r m o m e t e r . The m e l t i n g c u r v e thermometer s a t i s f i e s t h e need o f t h i s p a r t i c u l a r l a b o r a t o r y f o r a c o n v e n i e n t s t a n d a r d thermometer i n t h e t e m p e r a t u r e range o f 6 mK t o 600 mK. Over most o f i t s o p e r a t i n g r a n g e , t h e r e s o l u t i o n o f t h e t e m p e r a t u r e measured w i t h t h e 3He m e l t i n g c u r v e thermometer i s 0.03%, however , t h e a c c u r a c y i s o n l y 0.3%. The c o n s t r u c t i o n o f a p r e c i s i o n low t e m p e r a t u r e s t r a i n gauge i s d e s c r i b e d . A l s o a r a t i o t r a n s f o r m e r c a p a c i t a n c e b r i d g e w h i c h has a r e s o l u t i o n o f 1 ppm i s d e s c r i b e d . i i i T a b l e o f C o n t e n t s A b s t r a c t . . i i T a b l e of C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i Acknowledgements v i i Chapte r I Tempera tu re and Thermometry Below 1K 1 1.1 I n t r o d u c t i o n 1 1.2 Tempera tu re 4 1.3 The I n t e r n a t i o n a l P r a c t i c a l Tempera tu re Sca le 8 1.4 Thermal P r o p e r t i e s a t Low Tempera tu re 10 1.5 Review of P r a c t i c a l Thermometers Below 1 K 16 1 .5 .1 R e s i s t a n c e Thermometry 18 1 .5 .2 Paramagne t i c Thermometry 21 1 .5 .3 N u c l e a r M a g n e t i c Thermometry . . . . 2 5 1 .5 .4 Vapour P r e s s u r e Thermometry 32 1 .5 .5 F i x e d P o i n t Dev i ces 35 1 .5 .6 3He M e l t i n g Curve Thermometry 38 1 .5 .7 Osmot ic P r e s s u r e Thermometry 42 1 .5 .8 Mossbauer E f f e c t Thermometry 45 1 .5 .9 N u c l e a r O r i e n t a t i o n Thermometry . . . 4 8 1 .5 .10 Thermal No i se Thermometry 54 i v Chapte r I I 3 He M e l t i n g Curve 63 2 .1 I n t r o d u c t i o n 63 2 .2 M a g n e t i c F i e l d Dependence o f t h e M e l t i n g Curve 70 2 .3 I m p u r i t y E f f e c t s on t h e M e l t i n g Curve 73 2 .4 Accu racy and R e s o l u t i o n o f t h e MCT 75 2 .5 S e l f - h e a t i n g and R .F . S e n s i t i v i t y o f t h e MCT 78 2 .6 Thermal Time Response o f t h e MCT 79 Chapte r I I I 3He M e l t i n g Curve Thermometer 82 3.1 I n t r o d u c t i o n 82 3 .2 P r e s s u r e T r a n s d u c e r C e l l 84 3 .3 S i n t e r s 88 3 .4 P r e s s u r e System . . . 9 1 3 .5 C a p a c i t a n c e B r i d g e 99 3 .6 Choice of C o a x i a l Cab le f o r Low Tempera tu res 111 Chapter I V Thermometry w i t h t he M e l t i n g Curve Thermometer 115 4 .1 I n t r o d u c t i o n 115 4 . 2 O p e r a t i n g t h e MCT 116 4 .3 Compar ison o f t h e MCT w i t h a Germanium R e s i s t o r . . . . 1 2 2 4 . 4 C o n c l u s i o n 127 B i b l i o g r a p h y 129 Append ix A - O p e r a t i n g a MCT 133 V L i s t of Tables I Table of 3He M e l t i n g Curve Parameters 76 II Table of Noise C h a r a c t e r i s t i c s of C o a x i a l Cable at D i f f e r e n t Temperatures 113 I I I Table of Temperatures Measured by D i f f e r e n t Thermometers 124 v i L i s t o f F i g u r e s 1 . A SQUID Magnetometer No ise Thermometer 57 2 . 3He M e l t i n g Curve 64 3. The 3He S t r a i n Gauge 85 4 . The P r e s s u r e System f o r a MCT 92 5. The Flow C r y o s t a t Used t o Cool t h e C h a r c o a l Pump 94 6. C i r c u i t D iagram o f t h e Thermometer P r e s s u r e Gauge 97 7. Graph o f t h e C a l i b r a t i o n Curve o f t h e T h e r m i s t o r 98 8 . R a t i o T r a n s f o r m e r B r i d g e C i r c u i t 100 9 . S e n s i t i v e C a p a c i t a n c e B r i d g e 101 10. A Three T e r m i n a l C a p a c i t o r , 102 1 1 . Ground C a p a c i t a n c e s i n a R a t i o T r a n s f o r m e r B r i d g e 104 12. E q u i v a l e n t C i r c u i t o f a R a t i o T r a n f o r m e r 105 13. Schemat ic o f a H i g h I n p u t Impedance Low No ise P r e a m p l i f i e r 1 08 v i i ACKNOWLEDGEMENTS I would l i k e to acknowledge the support of Dr. W.N. Hardy i n the s u p e r v i s i o n of t h i s p r o j e c t . I have a l s o b e n e f i t e d from the numerous d i s c u s s i o n s with B.W. S t a t t . A l s o , I would l i k e to thank M. Hurlimann and M.W. Reynolds f o r t h e i r a s s i s t a n c e d u r i n g the running of the experiment. F i n a l l y I wish to thank the N a t i o n a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l of Canada fo r a Postgraduate S c h o l a r s h i p . 1 I. Tempera tu re and Thermometry Below 1K 1.1 I n t r o d u c t i o n A c c u r a t e measurement o f t e m p e r a t u r e s below 1 K i s a v e r y demanding t a s k ; as t h e number o f e x p e r i m e n t s c o n d u c t e d a t v e r y low t e m p e r a t u r e s i n d i f f e r e n t l a b o r a t o r i e s i n c r e a s e s , t h e demand f o r s t a n d a r d methods g r o w s . The i n t e r c o m p a r i s i o n o f e x p e r i m e n t a l r e s u l t s w h i c h depends on an a c c u r a t e measurement o f t h e t e m p e r a t u r e i s c o m p l i c a t e d by t h e f a c t t h a t an i n t e r n a t i o n a l l y a c c e p t e d p r a c t i c a l t e m p e r a t u r e s c a l e has y e t t o be d e f i n e d f o r t h e t e m p e r a t u r e range 1 mK t o .5 K ( t h e I n t e r n a t i o n a l P r a c t i c a l Tempera tu re Sca le ( M e t r o l o g i a , 1979) i s p r e s e n t l y d e f i n e d o n l y down t o .5 K } . I n a d d i t i o n , t h e p h y s i c a l p r o p e r t i e s of m a t e r i a l s a t low t e m p e r a t u r e s make a c c u r a t e t h e r m o m e t r y e x t r e m e l y d i f f i c u l t a t t h e s e t e m p e r a t u r e s . The pu rpose o f t h i s t h e s i s i s t o b u i l d a 3 He m e l t i n g c u r v e the rmomete r t o s e r v e as a l a b o r a t o r y s t a n d a r d i n a t e m p e r a t u r e range f r o m a few mK t o 400 mK(Greywa l l and Busch , 1 9 8 2 ) . The 3 He m e l t i n g c u r v e thermometer was chosen f o r a number o f r e a s o n s . F i r s t o f a l l i t has t h e advan tage t h a t i t i s i n s e n s i t i v e t o r f , i t i s i n s e n s i t i v e t o magne t i c f i e l d s as l a r g e as 5 kG, i t has a f a s t t h e r m a l response t i m e , and i t d i s s i p a t e s e s s e n t i a l l y z e r o p o w e r ( G r e y w a l l and Busch , 1 9 8 2 a ) . S e c o n d l y , 2 s i n c e the 3He m e l t i n g curve has a w e l l d e f i n e d minimum i n i t s thermodynamic P-T diagram, once a value has been chosed f o r the minimum pr e s s u r e p o i n t , every l a b o r a t o r y u s i n g a 3He m e l t i n g curve thermometer can compare the pressure standard used i n the l a b and apply necessary c o r r e c t i o n s . A l s o , the A - t r a n s i t i o n on the 3He m e l t i n g curve c o u l d i n p r i n c i p l e be used i n c o n j u n c t i o n with the minimum p o i n t to make the thermometer a s e l f - c a l i b r a t i n g system. However, s i n c e the A - t r a n s i t i o n occurs at 2.75 mK, a temperature below the lowest temperature a t t a i n a b l e with the present d i l u t i o n r e f r i g a t o r , i t i s of no immediate p r a c t i c a l i n t e r e s t to t h i s l a b o r a t o r y . The 3He m e l t i n g curve thermometer i s a temperature standard i n the same sense as the He vapour pressure s c a l e s a r e . I t i s not p o s s i b l e to d i r e c t l y measure the thermodynamic temperature by measuring the p r e s s u r e , but one r e l i e s on the P-T diagram having been p r e v i o u s l y measured using a primary thermometer. The r e s t of the chapter c l a r i f i e s what i s meant by the temperature of a system and d e s c r i b e s b r i e f l y the p h y s i c a l d i f f i c u l t i e s one has to master i n order to do meaningful thermometry at low temperatures. A b r i e f review of the other thermometers which c o u l d be used at temperatures below 1 K i s given with an emphasis on the thermal c h a r a c t e r i s t i c s , r a t h e r than the t e c h n i c a l a s p e c t s of each thermometer. Chapter two c o n s i d e r s the t h e o r i e s which are used to d e s c r i b e the m e l t i n g curve, and summarizes the p r e d i c t i o n s and measured p r o p e r t i e s of 3 the m e l t i n g curve. The t h i r d chapter d e s c r i b e s the c o n s t r u c t i o n of the thermometer, which i s borrowed d i r e c t l y from Greywall and Busch(1982a). The f i n a l chapter d e s c r i b e s i n d e t a i l how to u t i l i z e the 3He m e l t i n g curve thermometer to measure temperatures to b e t t e r than 0.1 mK accuracy at low temperatures. 4 1.2 Tempera tu re There e x i s t s a p h y s i c a l q u a n t i t y t h a t d e s c r i b e s t h e thermodynamic s t a t e o f systems w h i c h a re i n t h e r m a l e q u i l i b r i u m w i t h each o t h e r . The f i r s t p o s t u l a t e o f thermodynamics c o n c e r n s i t s e l f w i t h t h e e x i s t e n c e o f such a q u a n t i t y , c a l l e d t e m p e r a t u r e . The second law o f thermodynamics d e f i n e s t h e a b s o l u t e t e m p e r a t u r e s c a l e ( R e i f , l 9 6 5 ) o f a sys tem by T 3 E ( D is ' where S i s t h e e n t r o p y o f t h e system and E i s t he i n t e r n a l energy o f t h e s y s t e m . The p a r t i a l d i f f e r e n t i a t i o n i s e v a l u a t e d w i t h a l l o t h e r e x t r i n s i c v a r i a b l e s o f t h e sys tem h e l d c o n s t a n t . P r a c t i c a l l y , t h e t e m p e r a t u r e can be measured by o b s e r v i n g a p h y s i c a l p r o p e r t y of a system w h i c h v a r i e s r e p r o d u c i b l y and m o n o t o n i c a l l y w i t h t h e t e m p e r a t u r e . The t e m p e r a t u r e s c a l e d e f i n e d by such measurements i s an a r b i t r a r y e m p i r i c a l s c a l e . A r e v e r s i b l e hea t e n g i n e can be used t o r e l a t e t h e thermodymanic t e m p e r a t u r e s c a l e t o a p r a c t i c a l t e m p e r a t u r e s c a l e . I f a p e r f e c t gas i s used as t h e w o r k i n g subs tance i n a Carno t c y c l e , i t can be shown ( K i t t e l , 1969) t h a t (2 ) — = ~ 1 5 where T, and are the thermodynamic temperatures at the p o i n t s i n the Carnot c y c l e , and 0 V and 0*. are the corresponding temperatures d e f i n e d by the equation of s t a t e of the gas, where P i s the p r e s s u r e , V i s the volume occupied by the gas, R i s the molar gas c o n s t a n t , and n i s the number of moles of the gas. To prove equation (2) i t i s necessary to r e s t r i c t the energy by A c c o r d i n g to e q u a t i o n ( 2 ) , the thermodynamic temperature s c a l e and the p e r f e c t gas temperature s c a l e are i d e n t i c a l i f both have the same value at one p o i n t . In 1954, the 10th General Conference on Weights and Measures chose 273.16 K f o r the thermodynamic t r i p l e p o i n t of water. The value of the gas constant, depends d i r e c t l y on the a r b i t r a r y c h o i c e of the temperature s c a l e . The equation of s t a t e of a n o n - i d e a l gas has c o r r e c t i o n terms to that of the i d e a l gas, but i t may be w r i t t e n i n the form (3) (4) (5 ) R0 + BP + CP* + ••• > 6 where B, C,..., are f u n c t i o n s of temperature. Because the c o r r e c t i o n terms can be c a l c u l a t e d t o s u f f i c i e n t accuracy, a gas thermometer, where a p p l i c a b l e , i s used t o d e f i n e the thermodynamic temperature s c a l e . In the f o r m u l a t i o n of s t a t i s t i c a l mechanics of many p a r t i c l e systems, a parameter appears which has the same value fo r systems i n e q u i l i b r i u m with each o t h e r . The e n t i r e d e r i v a t i o n of thermal p h y s i c s can be accomplished without r e f e r e n c e to the thermodynamic t e m p e r a t u r e ( K i t t e l , 1 9 6 9 ) . If that parameter i s p o s t u l a t e d to have the r o l e of a temperature, the laws of thermodynamics can be d e r i v e d from s t a t i s t i c a l mechanics. A constant can be found that r e l a t e s the s t a t i s t i c a l mechanical temperature, , with the thermodynamic temperature, T, through, (6) where k i s Boltzmann's c o n s t a n t . L i k e the gas const a n t , Boltzmann's constant i s not a fundamental c o n s t a n t , as i t s value depends upon the ch o i c e of the temperature s c a l e . Systems which can be d e s c r i b e d by s t a t i s t i c a l mechanics can be used to f i n d the s t a t i s t i c a l mechanical temperature and by d e f i n i t i o n the thermodynamic temperature. Two such systems, which can be used f o r thermometry below 1 K, are the system of weakly i n t e r a c t i n g paramagnetics and the system of e l e c t r o n s i n 7 a conductor. 8 1.3 The I n t e r n a t i o n a l P r a c t i c a l Tempera tu re Sca le A gas thermometer measures t h e thermodynamic t e m p e r a t u r e , b u t i t i s a c o m p l i c a t e d s y s t e m ; t h e demands of t e c h n o l o g y and s c i e n c e r e q u i r e more c o n v e n i e n t t h e r m o m e t e r s . To t h i s end , t h e Bureau I n t e r n a t i o n a l des Po ids e t Measures c r e a t e d t h e IPTS, w h i c h has been r e v i s e d s e v e r a l t i m e s . The a im of t he IPTS i s t o f u r n i s h a t e m p e r a t u r e s c a l e w h i c h i s p r a c t i c a l and i s as c l o s e as p o s s i b l e t o t h e thermodynamic s c a l e . The l a t e s t r e v i s i o n of t h e IPTS o c c u r r e d i n 1968, and t h e t e m p e r a t u r e s c a l e t h e n d e f i n e d i s c a l l e d t h e I P T S - 6 8 ( M e t r o l o g i a , 1 9 6 9 , P r e s t o n , 1 9 7 6 ) . The s c a l e i s d e f i n e d by e l e v e n f i x e d p o i n t s , w h i c h were a s s i g n e d t h e b e s t v a l u e o f t h e thermodynamic t e m p e r a t u r e . The f i x e d p o i n t s range f rom t h e t r i p l e p o i n t o f h y d r o g e n ( 1 3 . 8 1 K) t o t he f r e e z i n g p o i n t o f g o l d ( 1 3 3 7 . 5 8 K ) . F u r t h e r m o r e , i n t e r p o l a t i o n f o r m u l a e a r e g i v e n f o r s t a r d a r d s p e c i f i e d i n s t r u m e n t s . The p l a t i n u m r e s i s t a n c e thermometer i s used i n t h e range 13.81 K t o 630.74 *C, t h e p l a t i n u m w i t h 10% rhod ium v e r s u s p l a t i n u m t h e r m o c o u p l e i s used f o r t h e r e g i o n 630.74*C t o 1064.3 "C, and f i n a l l y t h e h i g h e r t e m p e r a t u r e s a r e d e f i n e d by t h e P lanck law o f r a d i a t i o n w i t h 1337.58 K as t h e r e f e r e n c e t e m p e r a t u r e and a v a l u e o f 0.01388 mK f o r C a , t h e second r a d i a t i o n c o n s t a n t . The IPTS-68 d e v i a t e s f rom t h e thermodynamic t e m p e r a t u r e s c a l e i n s e v e r a l p l a c e s , a summary o f w h i c h i s g i v e n by 9 H u d s o n ( 1 9 8 0 ) . I n an e f f o r t t o c o r r e c t t h e s e d e v i a t i o n s and t o e x t e n d t h e t e m p e r a t u r e s c a l e below 30 K, t h e "1976 P r o v i s i o n a l 0 .5 K t o 30 . K Tempera tu re Sca le " was d e f i n e d ( M e t r o l o g i a , 1 9 7 9 ) . No t e m p e r a t u r e s c a l e has been d e f i n e d by t h e Bureau I n t e r n a t i o n a l des Po ids e t Measures f o r t e m p e r a t u r e s below .5 K. N . B . S . has d e f i n e d a s c a l e f rom 10 mK t o .5 K u s i n g a n u c l e a r o r i e n t a t i o n ( N O ) thermometer o f 6 0 C o i n a s i n g l e c r y s t a l o f 5 7 C o , a Josephson n o i s e t h e r m o m e t e r ( J N T ) , and a CMN s u s c e p t i b i t i t y thermometer c a l i b r a t e d by t he NO and JNT(Sou len and Marshak , 1 9 8 0 ) . U n t i l an i n t e r n a t i o n a l t e m p e r a t u r e s c a l e f o r t e m p e r a t u r e s below .5 K i s c r e a t e d , t h i s l a b o r a t o r y w i l l use t h e s c a l e d e f i n e d by N . B . S . 10 1.4 Thermal P r o p e r t i e s at Low Temperature Thermometry below .5K s u f f e r s not only from a lack of a w e l l d e f i n e d s c a l e , but a l s o from t e c h n i c a l problems due to p h y s i c a l p r o p e r t i e s common to a l l m a t e r i a l s at very low temperatures. At very low temperatures the s p e c i f i c heat of a l l m a t e r i a l s become very s m a l l , so that a very small heat input d r a s t i c a l l y changes the temperature of the sample. By arguments i d e n t i c a l to the K i n e t i c theory of gases the conductance, K, can be w r i t t e n a s ( K i t t e l , 1 9 6 9 ) where (C/V) i s the s p e c i f i c heat per u n i t volume, v i s the average speed of the heat c a r r i e r s , and 1. i s t h e i r mean f r e e path. At temperatures below 1 K, the mean f r e e path of the c a r r i e s i s determined by the impurity c o n c e n t r a t i o n or the independent of the temperature at these low temperatures. Thus, the thermal c o n d u c t i v i t y of m a t e r i a l s decreases with the temperature. A second source of thermal g r a d i e n t s i s the thermal boundary r e s i s t a n c e , R 6 , o f t e n c a l l e d the K a p i t z a r e s i s t a n c e , between two bodies i n thermal c o n t a c t . To perform a c c u r a t e and p r e c i s e temperature measurements on a system, the thermometer must be c a r e f u l l y designed with these o b s t a c l e s i n (7) sample s i z e . The speed of the c a r r i e r s i s e s s e n t i a l l y mind. A f i r m understanding of the thermal p r o p e r t i e s of m a t e r i a l s at temperatures below 1 K i s necessary i n order to design and i n s t a l l a u s e f u l thermometer i n t o a cryogenic system. Here a quick scan of general p r o p e r t i e s and some common techniques used to reduce thermal g r a d i e n t s are presented, f o r more d e t a i l e d i n f o r m a t i o n chapter 9 of the book by Lounasmaa i s recommended. In i n s u l a t o r s , the heat i s t r a n s p o r t e d by phonons. The dominant wavelength , \ , i s approximately given by ( 9 0 /T)d, where 0 O i s the Debye temperature and d i s the l a t t i c e s p a c i n g . For most elements the Debye temperature i s on the order of 100 K or m o r e ( K i t t e l ) . Thus below 1 K the phonon wavelength i s much longer than l a t t i c e i m p e r f e c t i o n s and the phonons are not s c a t t e r e d by the i m p e r f e c t i o n s . In the m i l l i k e l v i n s , the phonon mean f r e e path, i s equal to the s h o r t e s t d i s t a n c e between boundaries. Moreover, as the number of phonons i s small there are very few umklapp c o l l i s i o n s ( A h s h c r o f t and Mermin, 1976) and so there i s no impedance to flow from phonon-phonon i n t e r a c t i o n s . The s p e c i f i c heat of phonons i s p r o p o r t i o n a l to T 3 , the speed of sound i s r e l a t i v e l y independent of T, t h e r e f o r e by equation (7) the c o n d u c t i v i t y i s p r o p o r t i o n a l to 1,T3. In metals, there i s a d d i t i o n a l heat t r a n s p o r t by the conduction e l e c t r o n s . At low temperatures the mean f r e e path of e l e c t r o n s i s determined by the impurity c o n c e n t r a t i o n . The Fermi speed, v, i s independent of the temperature and the 12 s p e c i f i c heat of e l e c t r o n i s p r o p o r t i o n a l ( K i t t e l ) to T, so that equation (7) g i v e s R «t T. However, the phonons in a metal are s c a t t e r e d by the e l e c t r o n s whose en e r g i e s l i e w i t h i n kT of the Fermi s u r f a c e . The number of such e l e c t r o n s i s p r o p o r t i o n a l to T, so that the mean f r e e path of phonons i s p r o p o r t i o n a l to 1/T. So f i n a l l y , u s ing equation (7) again the t o t a l heat conductance i n a metal i s given by (8) K • o-T + b I j where a and b are c o n s t a n t s . At very low temperatures a l l the heat i s c a r r i e d by e l e c t o n s , thus K = aT. The Wiedemann-Franz law ( A s h c r o f t and Mermin) i £ - t T , where <r i s the e l e c t r i c a l c o n d u c t i v i t y , i s then v a l i d . The Lorenz constant JL = 25nW /K 2. Equation (9) i s o f t e n used to c a l c u l a t e the thermal c o n d u c t i v i t y from the more r e a d i l y measured e l e c t r i c a l c o n d u c t i v i t y . The metal with the h i g h e s t measured thermal c o n d u c t i v i t y at low temperature i s very pure copper(Lounasmaa). The c o n d u c t i v i t y of a l l metals can be improved by c a r e f u l l y a n n e a l i n g the sample. To o b t a i n the best r e s u l t s , the a n n e a l i n g process should be the l a s t step before mounting i n t o the (9) 13 apparatus because copper i s r e a d i l y work hardened. Good thermal c o n d u c t i v i t y not only i n the thermometer but a l s o i n the apparatus, whose temperature i s being measured, i s important i f a c c u r a t e thermometry i s d e s i r e d . The thermal conductance between two s u r f a c e s i s found to be p r o p o r t i o n a l to the f o r c e p r e s s i n g the s u r f a c e s together(Lounasmaa). The i r r e g u l a r i t i e s i n the s u r f a c e s allow the two s u r f a c e s to touch i n a s m a l l number of high spots and as the p r e s s u r e i s a p p l i e d these p o i n t s are deformed u n t i l the pressure drops to j u s t below the y i e l d p r e s s u r e . Thus the s u r f a c e area i n thermal c o n t a c t i s p r o p o r t i o n a l to the a p p l i e d f o r c e and t h e r e f o r e so i s the thermal conductance. The thermal r e s i s t a n c e between s o l i d s made of the same or s i m i l i a r m a t e r i a l s separated by a b a r r i e r which i s much t h i n n e r than the wavelength o f . the heat c a r r i e r s i s p r o p o r t i o n a l to 1/T f o r metals and 1/T3 f o r d i e l e c t r i c s . D i r t y c o n t a c t between metals which permit only phonons to pass through causes the thermal r e s i s t a n c e to be p r o p o r t i o n a l to 1/T 2, as would be expected from e q u a t i o n ( 8 ) . The thermal r e s i s t a n c e between a metal and a d i e l e c t r i c i s p r o p o r t i o n a l to 1/T 3. A l l these temperature dependencies have been observed as predicted(Suomi et a l , 1968) As the thermal conductance i s p r o p o r t i o n a l to the a p p l i e d f o r c e , two bodies h e l d together by b o l t s w i l l have a l a r g e thermal boundary r e s i s t a n c e i f the b o l t s are not t i g h t e n e d with a l a r g e torque. I f good metal to metal c o n t a c t cannot be made, 14 a l a r g e t h e r m a l boundary r e s i s t a n c e can be a v o i d e d by u s i n g a v e r y t h i n l a y e r o f b o n d i n g agent t o p r o v i d e more t h e r m a l c o n t a c t between t h e b o d i e s . Ap iezon N, G e n e r a l E l e c t r i c 7031 v a r n i s h , and Ep ibond 121 have been used f o r t h i s p u r p o s e ( A n d e r s o n and P e t e r s o n , 1 9 7 0 ) . The most e f f e c t i v e method f o r m i n i m i z i n g t h e t h e r m a l boundary r e s i s t a n c e i s t o s o l d e r t h e two p i e c e s t o g e t h e r w i t h a n o n s u p e r c o n d u c t i o n g s o l d e r . B u t , when t h i s i s no t p o s s i b l e and the p i e c e s must be r e a d i l y d e m o u n t a b l e , some e f f o r t s h o u l d be p u t i n t o g o l d p l a t i n g a l l t h e m e t a l p i e c e s . F u r t h e r m o r e , t h e t h e r m a l c o n t r a c t i o n o f t he a p p a r a t u s p i e c e s must be c a r e f u l l y t a k e n i n t o accoun t when u s i n g b o l t s . Somet imes, t h e d i f f e r e n t i a l t h e r m a l c o n t r a c t i o n can be used t o p r o v i d e a l a r g e f o r c e . The t e r m K a p i t z a r e s i s t a n c e i s o f t e n r e s t r i c t e d t o mean the t h e r m a l , boundary r e s i s t a n c e between l i q u i d h e l i u m and a s o l i d body . The n a t u r e o f t h e boundary r e s i s t a n c e i s no t t h e o r e t i c a l l y w e l l u n d e r s t o o d ( L o u n a s m a a ) , a l t h o u g h much s t u d y has been d o n e ( K e l l e r , 1 9 6 9 ) . A comprehens ive r e v i e w o f t he K a p i t z a r e s i s t a n c e of h e l i u m and v a r i o u s s o l i d s was c o m p i l e d by H a r r i s o n ( 1 9 7 9 ) . The K a p i t z a r e s i s t a n c e i s v e r y s e n s i t i v e t o t h e s u r f a c e c o n d i t i o n o f t h e s o l i d . A copper sample c a r e f u l l y a n n e a l e d i n a vacuum w i l l have a K a p i t z a r e s i s t a n c e an o r d e r o f magn i tude g r e a t e r t h a n a s i m i l i a r u n t r e a t e d p i e c e . F u r t h e r , t h e K a p i t z a r e s i s t a n c e o f t h e t r e a t e d sample can t h e n be d e c r e a s e d by 30% by 15 wiping the s u r f a c e with a tissue(Lounasmaa). The K a p i t z a r e s i s t i v i t y , d e f i n e d by r = RftA, where R 6 i s the K a p i t z a r e s i s t a n c e and A i s the s u r f a c e area i n thermal c o n t a c t , i s approximately the same w i t h i n a f a c t o r of three between l i q u i d 3He, l i q u i d "He, or d i l u t e mixtures of 3He i n "He, and a metal or an i n s u l a t o r . The product R a T 3 i s approximately constant below 0.1 K , i n c r e a s e s by an order of magnitude between 0.1 and 0.7 K, and then i s approximately constant above 0.7 K. A standard method of d e c r e a s i n g the K a p i t z a r e s i s t a n c e i s to i n c r e a s e the s u r f a c e area i n c o n t a c t with the l i q u i d u s i n g a s i n t e r made of copper or s i l v e r . P r o p e r l y s i n t e r e d s i l v e r powder of 700A diameter p a r t i c l e s has a s u r f a c e area of a meter squared per gram of s i n t e r . ( H a r r i s o n , 1979). More i n f o r m a t i o n about s i n t e r s w i l l be given i n chapter 3, when the c o n s t r u c t i o n of the 3He c e l l i s d e s c r i b e d . 16 1 .5 Review of P r a c t i c a l Thermometers Below j_ K Thermometers are c l a s s i f i e d a c c o r d i n g to the extent they are capable of measuring the thermodynamic temperature. The primary thermometers are the c l a s s of thermometers which can determine the thermodynamic temperature without any p r e v i o u s knowledge of the temperature i n the r e g i o n of i n t e r e s t . Secondary thermometers, however, r e q u i r e that they be c a l i b r a t e d a g a i n s t a primary thermometer before they can be used to measure the thermodynamic temperature. There e x i s t s a t h i r d c l a s s of thermometers, which w i l l be c a l l e d secondary standards, although most review a r t i c l e s ( L o u n a s m a a , Hudson et a l ) i n c l u d e them under the c l a s s i f i c a t i o n of primary thermometers. A secondary standard thermometers must be c a l i b r a t e d by a primary thermometer before i t can measure the a b s o l u t e temperature. Because the p h y s i c a l q u a n t i t y measured by the thermometer was chosen f o r the reason t h a t i t i s independent of the sample, once a p a r t i c u l a r secondary thermometer i s c a l i b r a t e d , a l l others of the same type are a l s o c a l i b r a t e d . Examples of secondary thermometers are the 3He vapour p r e s s u r e , the superconducting f i x e d p o i n t , and the 3He m e l t i n g curve thermometers. One may be tempted to use only primary thermometers i n an experiment to achieve a c c u r a t e thermometry, but such experiments would be very t e d i o u s . The disadvantage of primary thermometers i s t h a t they are c o m p l i c a t e d and r e q u i r e s u b s t a n t i a l time per 17 measurement. The secondary thermometers are simple to implement and r e q u i r e only a few seconds per temperature measurement. The major disadvantage with secondary thermometers i s the p o s s i b i l i t y that the c a l i b r a t i o n may change when the d e v i c e i s r e p e a t e d l y c o o l e d to low temperature. The secondary standards have the problem that they are d i f f i c u l t to implement and r e q u i r e s e v e r a l minutes per measurement, but they are more convenient to use than the primary thermometers. The r e s t of the chapter i s devoted to a review of the m e r i t s of v a r i o u s thermometers. The secondary thermometers which are d i s c u s s e d are r e s i s t a n c e , paramagnetic s u s c e p t i b i l i t y , and n u c l e a r magnetic resonance thermometers. The 3He and "He vapour p r e s s u r e , the 3He m e l t i n g curve, and the superconducting f i x e d p o i n t are secondary standards covered i n t h i s t h e s i s . F i n a l l y , the osmotic p r e s s u r e , the n o i s e , the nuclear o r i e n t a t i o n , and the Mossbauer thermometers are primary thermometers which are d i s c u s s e d . 18 1.5.1 R e s i s t a n c e Thermometry The most common thermometers at c r y o g e n i c temperatures are r e s i s t a n c e thermometers, because they are r e l a t i v e l y easy to o b t a i n , and are compact and easy to operate. Although these thermometers are the most v e r s a t t i l e temperature sensors at low temperature, only a very b r i e f d i s c u s s i o n i s given here. Phosphorus or a r s e n i c doped germanium r e s i s t o r s are used as thermometers because t h e i r c a l i b r a t i o n i s s t a b l e to b e t t e r than 1 mK over long p e r i o d s of time. The advantage of a germanium r e s i s t o r i s that once i t has been c a l i b r a t e d , i t may be used as a temperature standard. Commerical germanium r e i s i t o r s are a v a i l a b l e which can be used down to 50 mK. The most common r e s i s t a n c e thermometer sensors are carbon composition r e s i s t o r s . They have the advantage that they much l e s s expensive than the germanium r e s i s t o r s . Carbon r e s i s t o r s are i n s e n s i t i v e to r a d i a t i o n or e x t e r n a l p r e s s u r e , but are s e n s i t i v e to magnetic f i e l d s . The magnetoresistance e f f e c t i n c r e a s e s with the temperature s e n s i t i v i t y of the r e s i s t o r , so that the temperature e r r o r i s approximately constant or s l i g h t l y i n c r e a s e s at lower temperatures. The magnetoresistance of an i n d i v i d u a l r e s i s t o r i s d i f f i c u l t to a c c u r a t e l y p r e d i c t , but a t y p i c a l v alue i s 1% change i n the apparent temperature per 1 T(Lounasmaa). The r e s i s t o r s most commonly used f o r thermometry below 1 K 19 a r e Speer(Grade 1 0 0 1 ) ( E d e l s t e i n and Mess,1965) and M a t s u s h i t a ( E R C - 1 8 G J ) ( S a i t o and S a t o , 1 9 7 5 ) . The M a t s u s h i t a s a r e p r e f e r r e d a t t h e l ower t e m p e r a t u r e s because t h e y have a s m a l l e r s p e c i f i c h e a t and a re e a s i e r t o t h e r m a l l y g r o u n d t h a n S p e e r s . The one ma jo r d i s a d v a n t a g e w i t h r e s i s t a n c e t h e r m o m e t e r s i s t h a t t h e y a r e secondary the rmomete rs and t h u s need t o be c a l i b r a t e d . The c a l i b r a t i o n of a t y p i c a l r e s i s t o r can change a few p e r c e n t as i t i s r e p e a t e d l y c y c l e d . The a c c u r a t e c a l i b r a t i o n o f t he rmomete rs i s a d i f f i c u l t t a s k t h a t r e q u i r e s a g r e a t d e a l o f t h o u g h t and h a r d w o r k . I n t h i s l a b o r a t o r y i t has been f o u n d t h a t c o m m e r i c a l l y c a l i b r a t e d d e v i c e s canno t be c o m p l e t e l y t r u s t e d and s h o u l d be checked a g a i n s t o t h e r t e m p e r a t u r e s t a n d a r d s . When measu r ing t e m p e r a t u r e s below 100 mK, c a r e must be t a k e n t o ensure t h a t t h e r e s i s t o r i s i n good t h e r m a l c o n t a c t w i t h t h e t h e r m a l b a t h . U s u a l l y t h e r e s i s t o r s a r e g round t o a t h i n w a f f e r and then mounted i n a g o l d p l a t e d h o l d e r t o a c h i e v e good t h e r m a l c o n t a c t (Rob ichaux and A n d e r s o n , 1 9 6 9 ) . S i m i l i a r l y , t h e l e a d s t o t h e r e s i s t o r s h o u l d be c a r e f u l l y t h e r m a l l y a n c h o r e d . When a t t a c h i n g l e a d s t o a r e s i s t o r i t s h o u l d be remembered t h a t h e a t i n g a c a r b o n r e s i s t o r can change i t s c a l i b r a t i o n . Rf p i c k u p i n t he l e a d s a t t a c h e d t o r e s i s t o r s can cause h e a t i n g o f t h e r e s i s t o r ; t h i s can be q u i t e s u b s t a n t i a l below 50 mK. I f an e l e c t r i c a l l y s h i e l d e d room i s no t a v a i l a b l e , low 20 pass f i l t e r s to ground should be a t t a c h e d to a l l leads going i n t o the c r y o s t a t . Most r e s i s t a n c e b r i d g e s use low a.c. e x c i t a t i o n v o l t a g e s so that thermal emf and spurious emf induced by moving leads does not a f f e c t the measurement of the r e s i s t a n c e . When s u f f i c i e n t p r e c a u t i o n s are taken, the r e s i s t o r thermometer can be used to measure temperatures down to 3 mK(Lounasmaa) A u s e f u l " r u l e of thumb" i s that the power d i s s i p a t e d i n a carbon r e s i s t o r by the r e s i s t a n c e b r i d g e should be l e s s than Q = T 3nW/K 3, i f the accuracy,AT/T, of the temperature measured i s to be b e t t e r than 10"*(Lounasmaa). 21 1.5.2 Paramagnetic Thermometry The paramagnetic s u s c e p t i b i l i t y , % , of a system of n o n - i n t e r a c t i n g magnetic d i p o l e s i s d e s c r i b e d by the C u r i e law (10) * e "7" 1 where c i s the C u r i e c o n s t a n t . In p r i n c i p l e , the s u s c e p t i b i l i t y of a d i l u t e system of paramagnets c o u l d be used f o r a b s o l u t e thermometry, but because the measured s u s c e p t i b i l i t y depends on many f a c t o r s such as the m a t e r i a l , the shape of the sample, and the c o n t r i b u t i o n s from i m p u r i t i e s , i t i s i n s t e a d used as a good secondary thermometer. I f nearest neighbour d i p o l e - d i p o l e i n t e r a c t i o n s are taken i n t o account, equation (10) i s m o d i f i e d i n t o the C u r i e Weiss law ( I D t- * f r © » where © i s the Weiss c o n s t a n t . The C u r i e Weiss law has two c o n s t a n t s , thus only two f i x e d p o i n t s are necessary to c a l i b r a t e the thermometer, but f o r very p r e c i s e work s e v e r a l f i x e d p o i n t s are used to determine the c o n s t a n t s . The most s u i t a b l e paramagnetic s a l t f o r thermometry(Soulen, 1982) i s a s i n g l e c r y s t a l of cerium magnesium n i t r a t e (CMN). T h i s m a t e r i a l has been e x t e n s i v e l y studied(Zimmerman et a l , 22 1980) and shown t o obey t h e C u r i e law down t o 50 mK. A s i n g l e c r y s t a l o f CMN l o s e s t h e r m a l c o n t a c t w i t h t h e e n v i r o m e n t below 30 mK. The t h e r m a l boundary r e s i s t a n c e can be reduced by u s i n g powdered CMN i n a s l u r r y o f g rease and copper w i r e s or i n l i q u i d 3 H e ( W h e a t l y , 1975, R i c h a r d s o n , 1 9 7 7 ) . A l t h o u g h no c a r e f u l thermodynamic s t u d i e s have been done on powdered CMN, c o m p a r i s o n ( A b e s b o u r n e e t a l , 1973) o f a s i n g l e c r y s t a l and a powder o f CMN f i t t i n g i n a r i g h t c i r c u l a r c y l i n d e r w i t h t he d i a m e t e r e q u a l t o t h e h e i g h t have shown agreement w i t h i n 0 .05 mK a t 3 mK. The s u s c e p t i b i l i t y o f t he powder i s w e l l d e s c r i b e d ( W h e a t l y , 1975) down t o 2 mK by (12) X ~ J - 0 where B i s a c o n s t a n t . At v e r y low t e m p e r a t u r e s , t he t h e r m a l t i m e c o n s t a n t o f a s l u r r y o f CMN and g rease shows an i n c r e a s e as T " 3 and a t 8 mK i s 40 m i n u t e s ( G r e y w a l l and Busch , 1982a) . Below a few mK t h e d i p o l e - d i p o l e i n t e r a c t i o n s become l a r g e so t h a t t h e measured s u s c e p t i b i l i t y d e v i a t e s f rom the C u r i e Weiss l a w . These two f a c t s l i m i t t h e l o w e s t t e m p e r a t u r e t h a t can be measured w i t h a CMN t h e r m o m e t e r . A thermometer u s i n g l an thanum d i l u t e d CMN(LCMN) has a s m a l l e r mo la r hea t c a p a c i t y ( H u d s o n e t a l ) . Because the c o n c e n t r a t i o n o f t h e p a r a m a g n e t i c i o n i s l e s s i n t h e LCMN, t he d i p o l e i n t e r a c t i o n s a re l e s s s i g n i f i c a n t , so LCMN can be used t o 23 lower temperatures than CMN. The s u s c e p t i b i l i t y of the LCMN has been measured(Wheatly, 1975) and shown t o obey (12). The thermal time constant of LCMN has been measured f o r LCMN i n 3He and found to be roughly constant at 7 minutes f o r temperatures between 20 mK and 8 mK. The s u s c e p t i b i l i t y of the paramagnetic sensor i s u s u a l l y measured by p l a c i n g the s a l t i n one of a p a i r of a s t a t i c a l l y wound c o i l s and then measuring the mutual inductance of the system as a f u n c t i o n of the temperature. T r a d i t i o n a l l y , the mutual inductance was measured using a r a t i o transformer and a low audio frequency (20-300 Hz) e x c i t a t i o n v o l t a g e , but f o r low temperatures s q u i d magnetometers have almost completely r e p l a c e d the r a t i o transformer system. The advantage of a s q u i d i s that the system r e q u i r e s only 1/1000 of the sensor m a t e r i a l f o r the same s e n s i t i v i t y as the r a t i o transformer system. The disadvantages of the squ i d magnetometer thermometer are that a small d.c. f i e l d i s r e q u i r e d to b i a s the s a l t and the a s t a t i c c o i l s must be w e l l s h i e l d e d from ambient magnetic n o i s e . In summary, the s u s c e p t i b i l i t y thermometer using a s i n g l e c r y s t a l of CMN i s a good thermometer i n the range 30 mK to 30 K. If the thermometer i s to be used below 1 K, then the thermometer must be c a l i b r a t e d at p o i n t s below 1 K. A thermometer c a l i b r a t e d only at the higher temperatures i s not a r e l i a b l e thermometer f o r temperatures below 1 K. In the m i l l i k e l v i n range, the thermometer can e a s i l y achieve a r e s o l u t i o n of a few 24 m i c r o k e l v i n . At the very low mK range powdered CMN or LCMN i n a s l u r r y of grease of 3He can be used t o measure temperatures down to 2 mK. 25 1.5.3 Nuclear Magnetic Thermometry Nuclear paramagnetism can be used to measure the temperature by methods s i m i l i a r to that used f o r e l e c t r o n i c paramagnetic thermometers. The advantage of using nuclear magnetic moments i s that they obey the C u r i e law down to temperatures i n the m i c r o k e l v i n s . The Weiss constant can be of the order of a few m i c r o k e l v i n s . The preceding i s s t r i c t l y true f o r zero e x t e r n a l f i e l d s . In a f i n i t e e x t e r n a l f i e l d the c o r r e c t i o n s to the s e r i e s expansion of the B r i l l o u i n f u n c t i o n i s of the order B 2 T 2 . For copper at 1 mK and 0.2 T, the c o r r e c t i o n i s l e s s than 1%. The nuclear magnetic thermometer(NMT) would be a good primary thermometer except that the e f f e c t i v e C u r i e constant cannot be a c c u r a t e l y c a l c u l a t e d f o r the experimental s i t u a t i o n s , t h u s the, NMT i s a secondary thermometer. The disadvantage of using the nucl e a r magnetic moments i s that the magnetic moments are much sm a l l e r than the e l e c t r o n i c paramgnetic moments. The sensor m a t e r i a l must s a t i s f y s e v e r a l c o n d i t i o n s . The Weiss constant w i l l be of the order of m i c r o K e l v i n only i f two c o n s t r a i n t s are met. I f the nucl e a r s p i n , I, i s g r e a t e r than 1/2, then the s i t e symmetry of the nucleus must be cubic otherwise there w i l l be s p l i t t i n g of the nucl e a r s p i n energy l e v e l s due to the e l e c t r i c f i e l d g r a d i e n t s . T h i s s p l i t t i n g can be as l a r g e as 0.01 K. The second requirement i s that the 26 m a t e r i a l should not be m a g n e t i c a l l y ordered, as the l o c a l magnetic f i e l d would then be the sum of an a p p l i e d f i e l d and an i n t e r n a l magnetic f i e l d and thus C u r i e ' s law would not apply to the a p p l i e d magnetic f i e l d . A NMT measures the nuclear s p i n temperature which i s not n e c e s s a r i l y the same as the l a t t i c e temperature. The sensor must t h e r e f o r e posses a short r e l a x a t i o n time, T, , between the n u c l e a r s p i n and the l a t t i c e . < The T, f o r metals are orders of magnitude s h o r t e r than the T, of diamagnetic i n s u l a t o r s . The T, and the conduction e l e c t r o n temperature, which i s the l a t t i c e temperature of a metal, are r e l a t e d by the K o r r i n g a r e l a t i o n (13) T - -^r ) where b i s a c o n s t a n t . The constant has been measured(Hudson et a l ) f o r s e v e r a l metals and i s 1.27 Ksec(Anderson and R e d f i e l d , 1959) f o r copper and 0.0296 Ksec f o r p l a t i n i u m . The requirement there be good thermal c o n t a c t between the sensor and the temperature bath n e c e s s i t e s the use of a metal f o r the sensor element. The metal must be a v a i l a b l e i n very pure form and must not be superconducting d u r i n g the measurement. The n u c l e i of i n t e r e s t should have high n a t u r a l abundance and a l a r g e gyromagetic r a t i o so t h a t the C u r i e constant w i l l be l a r g e . The two most common metals which are used as the sensor elements of a NMT are copper and p l a t i n i u m ( R i c h a r d et a l , 1973, Dundon et 27 a l , 1973). Although the NMT i s a secondary thermometer, i t i s p o s s i b l e to s e l f - c a l i b r a t e the thermometer u s i n g the K o r r i n g a r e l a t i o n , but the accuracy i s u s u a l l y only of the order of 3%(Hudson et a l ) . The s u s c e p t i b i l i t y of the sensor can be measured by nonresonant or resonant t e c h n i q u e s . The s u s c e p t i b i l i t y of the sensor m a t e r i a l i n a small a p p l i e d magnetic f i e l d i s measured by methods i d e n t i c a l to that used i n e l e c t r o n i c paramagnetic thermometers and the temperature i n f e r r e d u s i n g C u r i e ' s law. The major d i f f i c u l t y i n implementing a s t a t i c NMT i s that the e l e c t r o n i c magnetization due to the impurties i n c o n c e n t r a t i o n s 1 ppm can be equal to the n u c l e a r c o n t r i b u t i o n to the measured ma g n e t i z a t i o n . A s o l u t i o n to t h i s problem i s to s a t u r a t e the e l e c t r o n i c c o n t r i b u t i o n by a p p l y i n g a l a r g e magnetic f i e l d or to c a l i b r a t e the system at a low enough temperature so t h a t the e l e c t r o n i c m a g e t i z a t i o n i s f u l l y s a t u r a t e d . A problem with a p p l y i n g a l a r g e magnetic f i e l d i s t h a t the l i f t i n g of the n u c l e a r s t a t e degeneracies i n c r e a s e s the n u c l e a r s p i n heat c a p a c i t y , thus i n c r e a s i n g the thermal e q u i l i b r i u m r e l a x a t i o n time. For example, at 10 mK the n u c l e a r and e l e c t r o n i c heat c a p a c i t i e s of copper are equal i n a magnetic f i e l d of 145 G(Hudson et a l ) . U n t i l the developement of the Squid magnetometer, the s e n s i t i v i t y of the s t a t i c measurement was l e s s than that a c h i e v a b l e by resonant methods. The Squid magnetometer i s 28 u s u a l l y used (Bubrman et a l , 1971) to measure the magnetization of the sample i n a small magnetic f i e l d ( 1 0 0 G). The r e s o l u t i o n of a t y p i c a l system i s given by AT S x 10" * • (14) — = b At T =l0mK and B = 10 G, i t i s p o s s i b l e to r e s o l v e 50 uK. The accuracy of the temperature i s determined by the c a l i b r a t i o n and i s u s u a l l y worse than 1% (Hudson et a l ) . The advantage of the s t a t i c method over the resonant techniques f o r measuring the magnetization of the sensor m a t e r i a l i s that very l i t t l e energy i s absorbed by the thermometer duri n g the measurement. The he a t i n g i n the nonresonant system i s due to noise generated eddy c u r r e n t s . I t has been e s t i m a t e d ( H i r s c h k o f f et a l , 1970) that the temperature r i s e w i l l be 1% at 1 mK i n copper wires of 2.3 x 10" 3 mm diameter assuming a R.F. squ i d o p e r t i n g at 190 MHz and a magnetic f l u x f l u c t u a t i o n of one f l u x o n . The n u c l e a r magnetization can a l s o be measured by two d i f f e r e n t resonant methods. The f i r s t method i s a pulsed n u c l e a r magnetic resonance(NMR) technique and the second i s a continuous wave(CW) resonance techuique. The advantage of a resonance measurement of the magnetization i s that the e l e c t r o n i c i m p u r i t i e s , with Lamor frequency a thousand times l a r g e r , do not c o n t r i b u t e to the measured m a g n e t i z a t i o n . They may, however, d i s t o r t the l o c a l f i e l d and broaden the l i n e width 29 of the s i g n a l . The NMR technique measures the magnetization of the n u c l e i by measuring the f r e e i n d u c t i o n decay (FID) of the sample. Because standard NMR techniques are used to measure the m a g n e t i z a t i o n , the NMR apparatus w i l l not be d e s c r i b e d here. The magnetizaion of a sample i n a steady f i e l d , B e, i n the z - d i r e c t i o n i s t i p p e d by a small angle by a p p l y i n g a magnetic p u l s e B c o s t a t ) f o r a time, , where w i s the Lamor frequency of the nucleus i n B e. Because the thermal c a p a c i t y of the nuclear s p i n system i s p r o p o r t i o n a l B, 2(Hudson et a l ) , i t i s d e s i r a b l e to keep B as s m a l l as p o s s i b l e ( u s u a l l y of the order of a 10 G). The magnetic component in the x - d i r e c t i o n i s MsinB, t h i s p recesses at the Lamor frequency around the z - a x i s . A v o l t a g e i s which i s p r o p o r t i o n a l to Msin© i s induced i n a loop p e r p e n d i c u l a r to both B 0and the x - a x i s . The induced v o l t a g e decays i n a time determined by the s p i n - s p i n r e l a x a t i o n time, j , and the inhomogeneity of the steady magnetic f i e l d . The induced v o l t a g e i s p r o p o r t i o n a l to 1/T through the m a g n e t i z a t i o n . Small t i p p i n g angles must be used because d e c r e a s i n g the magnetization along B 0 from M to Mcos© i n c r e a s e s the temperature of the s p i n s from T to T/cos6. T h i s reduces the induced v o l t a g e , but because at low temperature the FID s i g n a l s are l a r g e , s m a l l t i p p i n g angles are adequate. F u r t h e r , the p u l s e d NMR technique measures the n u c l e a r magnetization before the t i p p i n g p u l s e , thus the temperature of the nucleus before 30 the temperature i n c r e a s e i s measured. I f the s p i n - l a t t i c e r e l a x a t i o n time, T v , i s s h o r t , then the p u l s e d NMR method measures e q u i l i b r i u m temperature of the sample, which i s u n a f f e c t e d by the t r a n s i e n t s caused by the p u l s e . Because at temperatures below 1 K p l a t i n i u m has a of 1 msec compared to copper with a T A of 80 sec and a much s h o r t e r T , than the copper, p l a t i n i u m i s the best c h o i c e . Care must be taken when using p l a t i n i u m because i m p u r i t i e s can make C u r i e ' s law i n v a l i d f o r a p a r t i c u l a r specimen. The p u l s e d NMR thermometer can be made s e l f - c a l i b r a t i n g by f i r s t measuring T, by d e s t r o y i n g the magnetization i n 'the B d i r e c t i o n by a "T/2 pu l s e and observing the time f o r the magnet i z a t i o n to r e t u r n to e q u i l i b r i u m . Then K o r r i n g a ' s r e l a t i o n i s used to f i n d the e l e c t r o n i c temperature. The major source of s e l f - h e a t i n g i n the thermometer i s the eddy c u r r e n t h e a t i n g i n the sensor. For a c y l i n d r i c a l conductor of r a d i u s r , volume V, r e s i s t i v i t y R, and i n a changing magnetic f i e l d B along the d i r e c t i o n of i t s a x i s assuming f u l l p e n e t r a t i o n of the f i e l d , the eddy c u r r e n t heating(Lounasmaa) i s (15) Q In the range of a few mK to 4 K, the p r e c i s i o n of the temperature measured with the pulsed NMR i s about 0.5% with an 31 a b s o l u t e accuracy of 2% depending on the c a l i b r a t i o n u s e d ( A n u f r i e u and Peshor,1972). A NMR thermometer s e l f - c a l i b r a t e d u s ing the K o r r i n g a r e l a t i o n has an accuracy of about 3%(Hudson et a l ) . The second resonant method i s continuous wave NMR (CW). The sensor sample i s p l a c e d i n a high, steady magnetic f i e l d B i = Be and i s r a d i a t e d with a much smaller B^ = B s i n ^ t at r i g h t angle to B z. The t r a n s v e r s e s u s c e p t i b i l i t y , X = % - i% , of the sample i s d e s c r i b e d by the Bloch equation. E i t h e r the r e a l or imaginary p a r t may be measured by CW and used f o r thermometry. The imaginary p a r t of the s u s c e p t i b i l i t y , ~x", can be measured by d e t e c t i n g the energy absorbed f o r the magnetic f i e l d by the nucl e a r s p i n system, and the r e a l p a r t of the s u s c e p t i b i l i t y , X', can be measured by obs e r v i n g the vo l t a g e induced i n a c o i l p e r p e n d i c u l a r to both the z and y a x i s by the r o t a t i n g n u c l e a r magnetization induced by B^ . The power absorbed by the s p i n s from the radio f r e q u e n c y f i e l d i s given by Q = W J ^ ' B 2 / 2 and can be measured i n an NMR experiment. Because the CW method i s c o n s t a n t l y i n j e c t i n g energy i n t o the s p i n system, i f T , i s too long, the temperature of the s p i n system can be be higher than the temperature of the l a t t i c e . 32 1.5.3 Vapour Pressure Thermometry The vapour p r e s s u r e thermometer i s not a primary thermometer i n the sense that another primary thermometer i s r e q u i r e d to measure the l a t e n t heat of v a p o u r i z a t i o n and the b o i l i n g p o i n t or to measure the vapour pressure as a f u n c t i o n of the temperature. Once these v a l u e s are t a b u l a t e d the vapour p r e s s u r e can be used to f i n d the temperature of a system. When measuring temperatures to high accuracy the t a b u l a t e d v a l u e s of temperature versus the vapour p r e s s u r e are used, r a t h e r than the thermodynamic vapour pressure-temperature r e l a t i o n ( D i j k and Durieux, 1958) f o r a monatomic vapour gas given by (20) UU0= - * f ^ T > + £ e * * t T ) U ^ p rJL f s ^ T f where L e i s the heat of v a p o r i z a t i o n at T = 0, R i s the molar gas c o n s t a n t , and Le= l n [ ( 2 m ) 3 2 K 5 2 / h 3 ] i s the chemical constant of the gas. £(T) = InLPVa/R-rJ " " -y i s a c o r r e c t i o n term r e p r e s e n t i n g the n o n - i d e a l i t y of the vapour and the l a s t two terms c o n t a i n i n t e g r a l s of the molar entropy,s^, and the molar volume of the l i q u i d , v^, along the s a t u r a t i o n curve. The accuracy of the thermodynamic r e l a t i o n depends on how a c c u r a t e l y the l a s t three c o r r e c t i o n terms can be obtained from experimental data. The l a t e n t heat at T = 0 i s determined by f i t t i n g equation(20) to the experimental P-T d a t a . Because the 33 magnitude of the c o r r r e c t i o n terms decrease with deceasing temperature (20) may be used f o r low tempertures. For example, with "He equation (20) may be used f o r the c a l c u l a t i o n of T up to 1.5K with an accuracy of 0.2 mK(Durieux et a l , ). In the case of 3He, the l a r g e l i q u i d c o n t r i b u t i o n s makes i t p r e f e r a b l e to use t a b l e s to as low temperature as p o s s i b l e . For low temperatures 3He i s p r e f e r a b l e to "He because there i s no s u p e r f l u i d f i l m . Below 1.3 K f o r "He and 0.7 K f o r 3He c o r r e c t i o n f o r the pressure head i n the sensing tube and thermomolecular pressure e f f e c t (Greywall and Busch, 1980) must be made. The f i r s t c o r r e c t i o n i s of the order of 1 mK and can be e a s i l y c a l c u l a t e d or measured duri n g the experiment. The s i z e of the thermomolecular p r e s s u r e c o r r e c t i o n depends on the s i z e of the sensing tube down i n t o the c r y o s t a t and can be as high as a 17 mK c o r r e c t i o n at .5 K f o r a tube with a diameter of 2mm and the top end at 293 K. The c a p a c i t a n c e pressure gauge of Gonano and Adams(l970) measures the vapour pressure i n s i t u and avoids the n e c e s s i t y of making any c o r r e c t i o n to the measured p r e s s u r e . Such a thermometer with 3He can be used down to 0.3 K with an accuracy of 1 %(Lounasmaa). When using a vapour pressure thermometer i t i s important to remember that the vapour i s i n e q u i l i b r i u m with the l i q u i d when the system i s c o o l e d and the vapour pressure can be used to f i n d the temperature of the l i q u i d . But, when warming up, the 34 temperature of the vapour i s d i f f e r e n t from the temperature of the l i q u i d . Furthermore, the temperature of the v a p o u r - l i q u i d i n t e r f a c e i s not the temperature of the l i q u i d below the s u r f a c e . Due to the weight of the l i q u i d above i t , the temperature of the l i q u i d below the s u r f a c e i s higher than the temperature at the s u r f a c e . One note of warning, the "He vapour p r e s s u r e scale(T58) has been found to be i n c o r r e c t ( B r i c k l e d d e et a l , 1960) g i v i n g temperatures which are everywhere too low by 2%. S i m i l i a r l y the 3He scale(T62) i s i n c o r r e c t as i t had been f o r c e d to agree with the T58. The c o r r e c t e d vapour p r e s s u r e d v a l u e s are given by the T76 t a b l e s ( M e t r o l o g i a , 1979). 35 1.5.5 F i x e d Point Devices A c r i t i c a l p o i n t can be used as a r e f e r e n c e temperature d e v i c e once i t s behaviour i s w e l l understood. At temperatures l e s s than 0.5 K, the superconducting phase t r a n s i t i o n i n metals i s the most common c r i t i c a l phenomenon. Not a l l superconductors, however, are u s e f u l as f i x e d p o i n t d e v i c e s . Only metals with sharp t r a n s i t i o n s , very l i t t l e h y s t e r e s i s , and a minimun of s u p e r c o o l i n g are a c c e p t a b l e f o r thermometry purposes. The d i f f i c u l t y with u s i n g superconducting f i x e d p o i n t d e v i c e to d e f i n e the temperature s c a l e i s that the t r a n s i t i o n temperature, T c, of d i f f e r e n t samples of the same m a t e r i a l can vary as much as 1 mK. The p o s s i b i l i t y of d i f f e r e n t t r a n s i t i o n temperatures r e q u i r e s that one l a b o r a t o r y be chosen to c a l i b r a t e every superconducting d e v i c e . Furthermore, the r e p r o d u c i b i l i t y of the T i n a given sample i s o n l y +0.1 mK (NBS 260-62). An ambient magnetic f i e l d degrades the performance of the superconducting f i x e d p o i n t d e v i c e i n two ways. F i r s t , there i s the f a m i l i a r d e p r e s s i o n of the t r a n s i t i o n temperature by a magnetic f i e l d . The temperature at which the t r a n s i t i o n occurs i n an ambient magnetic f i e l d H ^ fT), i s r e l a t e d to the c r i t i c a l magneic f i e l d , H c ( 0 ) , and the zero magnetic f i e l d c r i t i c a l temperature, T c, by (21) HJJ) « HI*) C l - (.V) 1) ' 36 Equation (21) can be so l v e d f o r T. By a p p l y i n g a known magnetic f i e l d , temperatures which are a few mK lower than T c can be measured using the f i x e d p o i n t d e v i c e . The BCS theory g i v e s a more exact r e l a t i o n than equation (21) and c o u l d be used i f higher p r e c i s i o n i s r e q u i r e d . The second e f f e c t of the ambient magnetic f i e l d i s to cause the superconductor to s u p e r c o o l . When a superconductor su p e r c o o l s due to a magnetic f i e l d , H, the sample must be c o o l e d c o n s i d e r a b l y below the T C(H) before i t can become superconducting. The maximum depression of T c due to s u p e r c o o l i n g can be c a l c u l a t e d using the Ginzberg-Landau theory(NBS). P r a c t i c a l l y , the s u p e r c o o l i n g h y s t e r e s i s can be di m i n i s h e d by spotwelding m a t e r i a l s with higher Tc to the sensor m a t e r i a l s . The spotwelded m a t e r i a l a c t s as a n u c l e a t i o n s i t e and promotes the superconducting t r a n s i t i o n . The s u p e r c o o l i n g e f f e c t i s always g r e a t e r than the depr e s s i o n of T c due to the magnetic f i e l d . For a superconducting f i x e d p o i n t thermometer to perform s a t i s f a c t o r i l y , the maximum ambient magnetic f i e l d should be l e s s than 10 mG(NBS). When superconducting magnetic s h i e l d i n g i s employed care must be taken to ensure that l a r g e magnetic f i e l d s ( 20G) are not generated due to d i f f e r e n t i a l c o o l i n g of the s h i e l d through the t r a n s i t i o n temperature. The N a t i o n a l Bureau of Standards manufactures a 37 superconducting f i x e d p o i n t d e v i c e c a l l e d the SRM 768 f o r temperatures below 0.5 K. I t c o n s i s t s of f i v e s uperconductors(Auln , AuAl, In, Be, and W) which have t r a n s i t o n temperatures near 0.208 K, 0.161 K, 0.099 K, 0.024 K and 0.015 K r e s p e c t i v e l y , and an assembly of c o i l s . The mutual inductance between the primary c o i l and 5 counterwound secondary c o i l s are used to de t e c t the superconducting t r a n s i t i o n s . The SRM 768's are c a l i b r a t e d a g a i n s t a gamma-ray a n i s o t r o p y thermometer and a Josephson j u n c t i o n n o i s e thermometer. The temperature s c a l e used i s claimed to c o i n c i d e with the a b s o l u t e temperature s c a l e to w i t h i n a few tenths of a percent throughout the range(NBS). The accuracy of the temperature measured i s s e r i o u s l y a f f e c t e d by the ambient magnetic f i e l d as a l r e a d y d i s c u s s e d . I t i s a l s o a f f e c t e d by the s e l f - h e a t i n g of the superconducting f i x e d p o i n t d e v i c e . The a.c. e x c i t a t i o n v o l t a g e used to measure the mutual inductance of the c o i l s induces eddy c u r r e n t h e a t i n g i n the samples and the sample h o l d e r s . The requirement that the s e l f - h e a t i n g be small imposes a compromise i n the frequency and amplitude of the e x c i t a t i o n s i g n a l used, but t h i s r e s t r i c t i o n i s not a s e r i o u s o b s t a c l e i n the implementation. The estimated power d i s s i p a t i o n i n the SRM 768 i s 8x10" 1 1 watts, when an e x c i t a t i o n c u r r e n t of 29 uA at a frequency of 400 Hz i s used. 38 1.5.6 3He M e l t i n g Curve Thermometry The m e l t i n g curve of 3He has two d i s t i n c t f e a t u r e s : a minimum i n the pressure at T = .32 K and a d i s c o n t i n o u s change i n the slope at the s u p e r f l u i l d A t r a n s i t o n at a temperature of 2.75 mK(Greywall and Busch). These f e a t u r e s make the 3He m e l t i n g curve themometer(MCT) a very u s e f u l one f o r comparing l a b o r a t o r y temperature s c a l e s . Once the pre s s u r e of the two p o i n t s are ass i g n e d , any l i n e a r p r essure gauge may be c a l i b r a t e d a g a i n s t the m e l t i n g curve. The MCT would then be independent of the i n d i v i d u a l l a b pressure standard. Even i f only one p o i n t were to be used, s l i g h t c o r r e c t i o n s to the pr e s s u r e s c a l e of the l a b c o u l d be made. The C l a u s i u s - C l a p e y r o n equation f o r the e q u i l i b r i u m m e l t i n g pressure curve can be a p p l i e d to the 3He m e l t i n g curve (23) 1 ^ , where Pm i s the me l t i n g p r e s s u r e , S 4 and S s are the molar e n t r o p i e s of the l i q u i d and s o l i d , r e s p e c t i v e l y , and v t and v s are the molar volumes of the l i q u i d and the s o l i d at m e l t i n g . If t h e o r e t i c a l e x p r e s s i o n s f o r the e n t r o p i e s and the volumes c o u l d be d e r i v e d , then i n t e g r a t i o n of equation(23) would g i v e P m ( T ) . The MCT would then be a primary themometer. Some progress has been made i n the understanding of 3 H e ( K e l l e r ) , but 39 much s t i l l remains a mystery. Some of the t h e o r i e s and some of the p r e d i c t i o n s made about the behaviour of the melting curve w i l l be presented i n chapter 2. The a l t e r n a t i v e s are to measure the m e l t i n g curve using another primary thermometer or to d e r i v e the m e l t i n g curve from measured valu e s of the q u a n t i t i e s i n equation (23)(Hudson et a l ) . To date, the only method which a l l o w s the MCT to give temperature i n the range 0.3 K to 2 mK with an accuracy of 1%, i s to d i r e c t l y c a l i b r a t e the 3He m e l t i n g curve (Greywall and Busch,1982a) a g a i n s t other primary thermometers. The MCT measures the pressure of a constant number of 3He atoms i n a f i x e d volume as the temperature i s changed. A MCT c o n s i s t s of a low temperature pressure t r a n s d u c e r , with a constant volume c e l l f o r the 3He, a room temperature pressure standard, and a system f o r compressing the very pure 3He to the r e q u i r e d p r e s s u r e s . The 3He c e l l i s c l o s e d by a plug of 3He that forms i n a s e c t i o n of the c a p i l l a r y at a higher temperature than the c e l l . The b l o c k e d - c a p i l l a r y method(Webb et a l , l 9 5 2 ) , which w i l l be d i s c u s s e d f u r t h e r i n chapter f o u r . The trapped 3He then t r a c e s out the m e l t i n g curve as the temperature i s v a r i e d . Because the mixture f o l l o w s the m e l t i n g curve, the p r e s s u r e of the system can be used to f i n d the temperature. The p r e s s u r e must be measured in s i t u because the blocked c a p i l l a r y prevents the pressure v a r i a t i o n from being t r a n s m i t t e d up the c a p i l l a r y . 40 Although some commerical t r a n s d u c e r s can be used to measure the pressure(Weinstock et a l , 1962 ), they tend to d i s s i p a t e l a r g e amounts of heat. The most common method i s to use a c a p a c i t i v e p r e s s u r e transducer of the Straty-Adam t y p e ( S t r a t y and Adam, 1969). The c a p a c i t i v e transducer u l i t i l i z e s a w a l l or w a l l s of a c o n t a i n e r , u s u a l l y made of B e r y l c o 25 ( C o r r u c c i n i and M o u n t f i e l d , 1978), t o vary the s e p a r a t i o n of two w e l l d e f i n e d c a p a c i t o r p l a t e s . The change i n c a p a c i t a n c e i s measured by using the 3-wire guarded ground technique, which d e t e c t s only the c a p a c i t a n c e between the p l a t e s and i s only s l i g h t l y a f f e c t e d by the l e a d c a p a c i t a n c e . The diaphragm w a l l and the c a p a c i t o r s e p a r a t i o n are chosen to o p i m i t i z e the pr e s s u r e s e n s i t i v i t y of the transducer i n the range of i n t e r e s t ( S t r a t y and Adam, 1969). The 3He c e l l must be c a l i b r a t e d with a pressure standard, before i t can be used as a thermometer. As the a b s o l u t e accuracy of the MCT can be l i m i t e d by the u n c e r t a i n t y i n the c e l l c a l i b r a t i o n , the pressure standard f o r the system must be choosen with some c a r e . The c a p a c i t a n c e can be measured to AC/C = 10" 8 using a General Radio Co. Type 1620-A c a p a c i t a n c e b r i d g e . The corr e s p o n d i n g pressure r e s o l u t i o n can be about 5 x 10" 6 bar at 30 b a r s ( G r e y w a l l and Busch, 1982a). The 3He i s compressed to the r e q u i r e d p r e s s u r e s using a c h a r c o a l pump(Lounasmaa) or a t o e p l e r pump(Johnson and 41 Wheatley,1970). The impurity l e v e l of "He i n the 3He must be kept below 600 ppm ( S c r i b n e r et a l , 1969) and i s u s u a l l y < 20 ppm(Greywall and Busch,1982a). Many of the advantages of the MCT have been a l r e a d y l i s t e d and w i l l not be r e i t e r a t e d , but one f u r t h e r advantage of the MCT i s that when b e t t e r measurements of the m e l t i n g curve are made, the temperatures measured u s i n g the MCT can be e a s i l y a d j u s t e d to the new c a l i b r a t i o n . 42 1.5.7 Osmotic Pressure Thermometry The p r e s s u r e d i f f e r e n c e , P, between a d i l u t e mixture of 3He in "He and "He l i q u i d at T = 0 connected by a superleak i s given(Lounasmaa) by where x*0 i s the molar c o n c e n t r a t i o n of 3He = n 3 / ( n ^ + n^) , Vy; i s the molar volume of pure °He l i q u i d , A^.o i s the chemical p o t e n t i a l of 3He i n "He and s i s the molar entropy of pure "He l i q u i d . The f i r s t term of equation (22) i s the osmotic pressure of 3He i n "He, and the second term i s c a l l e d the f o u n t i a n pressure of "He. The behaviour of 3He molecules i n "He can be d e s c r i b e d a c c u r a t e l y by t r e a t i n g 3He as q u a s i - p a r t i c l e s with Fermi s t a t i s t i c s . ( L a n d a u et a l , 1970) Complete a n a l y t i c a l e x p r e s s i o n s are a v a i l a b l e f o r the osmotic pressure of 3He i n "He over the whole temperature range and c o n c e n t r a t i o n s . The osmotic pressure thermometer i s a primary themometer once the d e n s i t y of 3He i n "He i s known, otherwise the thermometer can be used to e x t r a p o l a t e the 3He temperature s c a l e down i n t o the m i l l i k e l v i n range. When the thermometer i s used as a t r a n s f e r d e v i c e , the c o n c e n t r a t i o n , x„, need only be known to w i t h i n 10%(Rosenbaum et a l , 1977). The osmotic pressure thermometer i s u s e f u l f o r experiments which study the p r o p e r t i e s (22) 43 of 3He i n "He. Furthermore, t h i s thermometer has the advantage that i t i s i n s e n s i t i v e to l a r g e magnetic f i e l d s . There has been no i n t e n s i v e i n t e r e s t i n r e f i n i n g the osmotic p r e s s u r e thermometer, but the thermometers that have been used have encouraging c h a r a c t e r i s t i c s . One thermometer has been developed(Bloyet et a l , 1975) f o r thermometry i n the range 50 mK to 500 mK with a r e s o l u t i o n of 10 uK and an accuracy of 2 mK. The thermometer c o n s i s t s of a c a p i l l a r y with one end i n the mixing chamber and a bulb on the other end. A heater on the bulb d r i v e s away the 3He and a t t r a c t s the s u p e r f u i l d "He. The heater s u p p l i e s a very small amount of power so that the temperature of the bulb i s independent of the heater power. The bulb i s above the mixing chamber so that the osmotic pressure t r i e s to push 3He i n t o the bulb, but the heater d r i v e s the 3He away, so temperature of the bulb i n c r e a s e s u n t i l osmotic pressure on the 3He i n the bulb i s equal to the osmotic pressure at the mixing chamber. The p r e s s u r e balance between the bulb and the chamber d e f i n e s a r e l a t i o n s h i p between the temperature of the two volumes. A thermometer i s used to measure the temperature of the bulb and i n f e r s the temperature of the chamber. Thus t h i s thermometer measures a low temperature by measuring a temperature an order of magnitude l a r g e r . The higher temperature can be measured by u s i n g a 3He vapour thermometer. A second thermometer(Lounasmaa) i n f e r s the temperature by 44 m e a s u r i n g t h e p r e s s u r e g i v e n by e q u a t i o n ( 2 2 ) . A s m a l l c y l i n d r i c a l r e s e r v o i r o f pu re "He i s c o n n e c t e d by a s u p e r l e a k t o a l a r g e chamber c o n t a i n i n g d i l u t e 3 H e ( x 0 = 0 .0004 - 0 .006) w h i c h i s a t t he same t e m p e r a t u r e as t h e "He c y l i n d e r . A c a p a c i t a n c e t e c h n i q u e i s used t o measure t h e h e i g h t o f t h e "He head i n t h e c y l i n d e r and t h u s t h e p r e s s u r e can be i n f e r r e d . The c r o s s - s e c t i o n a l a r e a o f t h e "He r e s e r v o i r i s made t o be much s m a l l e r t h a n t h a t o f t h e d i l u t e 3He r e s e r v o i r so t h a t t h e mo la r c o n c e n t r a t i o n o f t he 3He remains c o n s t a n t t o b e t t e r t h a n 1% as t h e f l i u d l e v e l s change. A thermometer w i t h x = 0.0004 i s u s e f u l i n t h e r e g i o n 20 t o 700 mK w i t h an a c c u r a c y o f 0 .2 mK. I n summary t h e o s m o t i c p r e s s u r e thermometer can be a v e r y u s e f u l thermometer i n e x p e r i m e n t s on p r o p e r t i e s o f d i l u t e 3He i n "He, bu t much more deve lopement i s necessa ry b e f o r e i t can be used as a r e l i a b l e g e n e r a l purpose p r i m a r y t h e r m o m e t e r . 45 1.5.8 Mossbauer E f f e c t Thermometry U n l i k e the vapour p r e s s r e thermometer, the Mossbauer e f f e c t thermometer does not r e q u i r e any d i r e c t knowledge of the a b s o l u t e temperature s c a l e t o i n i t i a l l y c a l i b r a t e the system. The Mossbauer e f f e c t i s the r e c o i l l e s s emission or a b s o r p t i o n of - r a d i a t i o n by the n u c l e i . The l a t t i c e i n which the n u c l e i i s embedded absorbs the momentum, but absorbs very l i t t l e energy s i n c e the e f f e c t i v e mass i s e s s e n t i a l l y i n f i n i t e . The Mossbauer themometer r e q u i r e s a source and an absorber to r a d i a t e and absorb the V - r a d i a t i o n r e s p e c t i v e l y . E i t h e r the source or the absorber can be at the temperature of i n t e r e s t . U s u a l l y the c o l d absorber thermometer i s used because t h i s arrangement i s s e l f - c a l i b r a t i n g . Here only the c o l d - a b s o r b e r thermometer i s c o n s i d e r e d . For d e t a i l s about source thermometry and Mossbauer thermometry i n g e n e r a l the paper by K a l i u s et a l ( l 9 6 9 ) i s recommended. The c o l d absorber Mossbauer thermometer measures the Boltzmann d i s t r i b u t i o n of the n u c l e a r h y p e r f i n e s t a t e s and thus measures the temperature of the s p i n system. The Tf-ray from an a p p r o p r i a t e source i s r e s o n a n t l y absorbed and e x c i t e s the absorbing n u c l e i from the ground s t a t e 1^, to the f i r s t e x c i t e d s t a t e I t . The h y p e r f i n e i n t e r a c t i o n s p l i t s the 1^ and I e i n t o h y p e r f i n e s u b l e v e l s , so that the t r a n s i t i o n s occur between two s u b l e v e l s m. and m t i n I - and I t r e s p e c t i v e l y . For t h i n 46 absorbers the r e l a t i v e t r a n s i t i o n s i n the Mossbauer spectrum i s given by (24) r \ U j - * ^ - < V C " V C U V " ^ * where oL i s a constant f o r a given experiment, C(m^m t) i s the square of the a p p r o p r i a t e l y normalized Clebsch-Gordon c o e f f i c i e n t , and /^m^) i s the p r o b a b i l i t y of the system e x i s t i n g i n a s u b l e v e l m^ c a l c u l a t e d assuming the n u c l e i are n o n - i n t e r a c t i n g s p i n p a r t i c l e s i n a magnetic f i e l d . As the host l a t t i c e i s doped with only 1% of the Mossbauer e f f e c t nucleus and the d i r e c t i n t e r a c t i o n s are n u c l e i - n u c l e i , t h i s i s a good approximation. The p r o b a b i l i t y i s c a l c u l a t e d as usual by B / k l l . (25) * *x } -7 where u n i s the nucl e a r magnetron, g n i s the nucl e a r g - f a c t o r , N i s the t o t a l number of Mossbauer n u c l e i i n the sample, and B i s the l o c a l magnetic f i e l d . There i s one unknown constant i n equation (24), but i f the n u c l e a r s t a t e s have a small number of s u b l e v e l s then ci need not be known. As an example, c o n s i d e r the case I,j = 3/2 + and I e = 1/2*. Then r e c a l l i n g the Clebsch-Gordan c o e f f i e c i e n t s have the pr o p e r t y that C(m^,mt) = C(-m^,-mfc) and then t a k i n g the r a t i o of the i n t e n s i t i e s of the two p o s s i b l e t r a n s i t i o n s one gets , 47 ( 2 6 ) R " R ( - V « ^ - « ) / r - % ) where = -/<,gjiB. By measuring the resonance l i n e s e p a r a t i o n , A£ can be independently determined. Once 4.6- i s measured the thermometer i s c a l i b r a t e d . The most common Mossbauer i s o t r o p e s 5 7 F e and 1 9 7 A u have A£/k of 4 mK and 14 mK r e s p e c t i v e l y , so that thermometers with sensory elements made from these are e f f e c t i v e thermometers from 2 mK to 20 mK with 5 7 F e and 7 mK to 50 mK with 1 9 7Au(Lounasmaa). The accuracy of the thermometer i s determined by the t o t a l number of events i n the Mossbauer spectrum. The major disadvantage of t h i s thermometer i s the l a r g e c o u n t i n g time needed f o r accurate temperature measurement. The Mossbauer thermometer r e q u i r e s the a b s o r p t i o n of t-rays and ,as such, the s e l f - h e a t i n g of the sensor becomes the l i m i t i n g f a c t o r i n the us e f u l n e s s at the lower temperatures. The t e c h n i c a l implementation of a Mossbauer thermometer i s a very d i f f i c u l t problem and the added f e a t u r e of very long measuring times s e v e r e l y l i m i t s the use of t h i s thermometer. 48 1.5.9 Nuclear O r i e n t a t i o n Thermometry If the p o l a r i z a t i o n of a nu c l e a r s p i n system can be measured, then the a b s o l u t e temperature can be deduced v i a the Boltzman f a c t o r . There are s e v e r a l methods f o r measuring the degree of o r d e r i n g i n the nu c l e a r s p i n s , but the only method e x t e n s i v e l y used f o r a b s o l u t e thermometry(Marshak, 1982) at low temperatures i s the measurement of the Y _ r a y a n i s o t r o p y from decaying n u c l e i o r i e n t e d i n a magnetic f i e l d . The a n i s o t r o p i c emission of K _ r a y s from o r i e n t e d r a d i o a c t i v e n u c l e i i s w e l l understood(Krane et a l , 1973), thus the n u c l e a r o r i e n t a t i o n thermometer(NOT) can be used as a primary thermometer(Berglund et a l , 1972) . O r i e n t e d nuclear spi n systems with a x i a l symmetry have a normalized s p a t i a l d i s t r i b u t i o n , W(0), of emi t t e d "V -rays given by (27) u)tri - * R K ^ P k t u ^ ) . where & i s the angle between the d i r e c t i o n the -rays are emitted and the o r i e n t a t i o n a x i s . The q u a n t i t i e s B K ( T ) d e s c r i b e s the i n i t i a l o r i e n t a t i o n of the n u c l e i and c o n t a i n s a l l the temperature dependence of W(0) . The Uk. are angular momentum r e o r i e n t a t i o n parameters, the R k are angular c o r r e l a t i o n c o e f f i c i e n t s , and the P (cos©) are the Legendre p o l y n o m i a l s . 49 Only even terms enter i n the summation of k because only the d i r e c t i o n a l d i s t r i b u t i o n of the r a d i a t i o n i s of i n t e r e s t and not the s t a t e of p o l a r i z a t i o n of the Y-rays. The summation over k goes from zero to the l e s s e r of 2L or 21, where L i s the m u l t i p o l a r i t y of the emitted r e d i a t i o n and I i s the spin of the o r i e n t e d n u c l e u s . For the purposes of t h i s t h e s i s i t i s s u f f i c i e n t to know that the sum can be c a l c u l a t e d to the necessary accuracy. C o n s i d e r a t i o n of W(T,©) shows that the l a r g e s t temperature v a r i a t i o n i n r a d i a t i o n f l u x with changes i n the temperature occurs at & = 0" or 180°. Furthermore, the r e l a t i o n between temperature and W i s unique at these p o i n t s , whereas at d i r e c t i o n s where P k(cos©) i s zero, t h i s i s not t r u e . The two most commonly used n u c l e i i n NOT are 6 0 C o and 5 0Mn, f o r which the decay processes are w e l l understood and the c o e f f i c i e n t s i n equation (27) can be c a l c u l a t e d . For 6 0 C o , W(T,0) i s given by (28) (oj(j.t^= u'los" + o . o i r f c f . m'ifCm) - 0.0 3 H f e i « V / W ' where ^(m), the p o p u l a t i o n of the m th s u b l e v e l as given by equation (25), i s the only temperature dependent q u a n t i t y . The index m i s summed from I to - I , where I i s the nuclear s p i n equal to 5 f o r 6 0 C o and 3 f o r 5"Mn. S i m i l i a r l y , f o r 5 aMn one has 50 (29) <uoU,0}= \ t O - ^ e & l - 0 . 0 0 3 ^ £ r * + / M ) The degree of alignment i s r e l a t e d to the r a t i o AE/kT, where AE = u ng*B, the h y p e r f i n e s p l i t t i n g . The r a d i o a c t i v e n u c l e i are o r i e n t e d by d i f f u s i n g them i n t o a host m a t e r i a l which i s magnetic. The i n t e r n a l l o c a l f i e l d can be of the order of 10 - 30 T i n the i r o n group metals. The domains i n the host are a l i g n e d by a p p l y i n g a magnetic f i e l d of 0.1 - 1 T. Some care must be taken to ensure that the a p p l i e d f i e l d i s s u f f i c i e n t to a l i g n a l l the domains. The NOT i s a p r a c t i c a l primary thermometer(Krane et a l , 1973), because the h y p e r f i n e s p l i t t i n g i n the host m a t e r i a l can be measured with h i g h p r e c e s i o n by NMR experiments. Measured E f o r 5*Mn i n i r o n i s AE/k = 9.11 - 0.01 mK, and f o r 6 0 C o i n i r o n AE/k = 7.945 i 0.003 mK(Templeton and S h i r l e y , 1967). The range of temperature over which the NOT can be used as a thermometer i s l i m i t e d because i t i s s e n s i t i v e only f o r temperature approximately AE/k. The most commonly used n u c l e i , 6 0 C o and 5 5Mn, are almost completely a l i g n e d at 30 mK. Recently 1 6 6 H o i n a s i n g l e c r y s t a l of Ho has been used(Marshak, 1982) to to cover the range 32 mK to 1.2 K, but AE has not yet been measured by NMR, t h e r e f o r e i t must be c o n s i d e r e d a secondary thermometer. The t h e o r e t i c a l accuracy of the NOT i s l i m i t e d by the f a c t t h a t W(T,0) a p p l i e s to a p o i n t source, although i n p r a c t i c e most experimental c o n f i g u r a t i o n s approximate the c o n d i t i o n q u i t e 51 w e l l . A l s o , W(T,0) a p p l i e s to a p o i n t d e t e c t o r , whereas a l l d e t e c t o r s have a f i n i t e s u r f a c e a r e a . As long as the d e t e c t o r s are c y l i n d r i c a l , the form of W(T,0) i s m o d i f i e d only by an a t t e n u a t i o n f a c t o r ,QK, i n the the sum(Rose, 1953). The t e c h n i c a l problem of a l i g n i n g the source with the de t e c t o r and measuring the s o l i d angle a c c u r a t e l y c o n t r i b u t e s to the u n c e r t a i n t y i n the measured temperature. The demagnetization e f f e c t s of the host m a t e r i a l must a l s o be taken i n t o account when c a l c u l a t i n g the e f f e c t i v e f i e l d at a s i t e , even i n a t h i n f o i l specimen. The major disadvantage of NOT i s the long c o u n t i n g times necessary f o r accurate measurements of the temperature. The s t a t i s t i c a l e r r o r , &n, i n a measured count i s An = n l / 2 = ( n b + n e W ( T , 0 ) ) 1 , where n e i s the number of p u l s e s i n the high temperature l i m i t and n b i s the background count. To keep the s e l f - h e a t i n g of the thermometer low, the f l u x has to be kept s m a l l , and t h i s i m p l i e s r e l a t i v e l y long counting i n t e r v a l s . Counting times longer than 30 minutes should be avoided as i t becomes too te d i o u s i n practice(Lounasmaa). Of the two, the 5ftMn thermometer i s more acc u r a t e than the 6 0 C o thermometer at temperatures l e s s than 20 mK because i t s e l f - h e a t s an order of magnitude l e s s than the 6 0 C o . I f the Y -ray a b s o r p t i o n i s i g n o r e d ( t h i s i s very small as long as the c r y o s t a t i s not too b u l k y ) , the r a d i o a c t i v e s e l f - h e a t i n g of 1 uCi of 6 0 C o i s 630 pW as opposed to 30 pW f o r 1 uCi of 5 aMn. 52 The 5flMn s e l f - h e a t s l e s s than 6 0 C o because i t decays through e l e c t r o n c a p t u r e , whereas 6 0 C o decays through emission of 110 Kev e l e c t r o n s . The most common host m a t e r i a l f o r 6 0 C o and 5ftMn i s i r o n ( S i t e s et a l , 1971, Johnson et a l , 1972), although n i c k e l and c o b a l t have been used(Cameron et a l , 1967). I f 6 0 C o i s d i s t r i b u t e d u n i f o r m l y throughout a hep 5 9 C o s i n g l e c r y s t a l ( T h o r p et a l , 1970), then no e x t e r n a l p o l a r i z i n g magnetic f i e l d i s necessary. 5"Mn w i l l not d i f f u s e i n t o hep c o b a l t c r y s t a l s . A f u r t h e r reason f o r p r e f e r r i n g 5 4Mn i n Fe i s that i t has a s p i n - l a t t i c e r e l a x a t i o n time of the order of 1 minute at 2 mK, whereas 6 0 C o i n Fe has a T, of 7 minutes at 2 mK. In Kondo m a t e r i a l s , below the Kondo temperature, the h y p e r f i n e f i e l d i s independent of the temperature and a very s t r o n g l y dependent upon the a p p l i e d f i e l d . I f the h y p e r f i n e f i e l d i s known as a f u n c t i o n of the a p p l i e d f i e l d , then the h y p e r f i n e s p l i t t i n g can be a d j u s t e d to gi v e the maximum s e n s i t i v i t y i n the reg i o n of i n t e r e s t . Two p o s s i b l e systems which have been s t u d i e d are 5"Mn doped to l e s s than 1 ppm i n pure c o p p e r ( P r a t t et a l , 1969), and 5ftMn doped to l e s s than 1 ppm i n zi n c ( M a r s h , 1970). Both themometers have been found to be most s e n s i t i v e i n the range 1-20 mK. The temperature range of 6 0 C o or s*Mn NOTs i s from a few mK to 30 or 40 mK with an accuracy of 1% when using a counting time of a few minutes per p o i n t ( S i t e s et a l , 1971). 53 Temperatures below 1 or 2 mK can be measured with a NOT using a Kondo m a t e r i a l as the host f i l m and c a r e f u l l y c o n t r o l l i n g the. a p p l i e d magnetic f i e l d . The "brute f o r c e technique" of alignment of the nuclear s p i n with a p p l i e d f i l d s of the order of s e v e r a l T e s l a can be used to extend the NOT down to the m i c r o K e l v i n range(Ono et al,l980)°, where i n t e r a c t i o n s between the n u c l e i become important. When lower temperatures are measured, i t must be remembered that the nuclear s p i n temperature need not be the same as the l a t t i c e . 54 1.5.10 Thermal Noise Thermometry Nyquist(1928) formulated a r e l a t i o n between the random v o l t a g e generated a c r o s s a r e s i s t o r and the Brownian motion of con d u c t i v e e l e c t r o n s i n the r e s i s t o r u s i n g the second law of thermodynamics and the law of e q u i p a r t i t i o n of energy. The most common form of the r e l a t i o n i s (30) <V a nW>= [ Y R ^ T d w i where < v 2 ( t ) > i s the time average of the thermal v o l t a g e f l u c t u c a t i o n s squared, R i s the r e s i s t a n c e of the element, and the i n t e g r a l i s over the frequency range over which the v o l t a g e i s measured. Nyquist g e n e r a l i z e d the f l u c t u a t i o n s to i n c l u d e quantum mechanical e f f e c t s , but the most ge n e r a l thermal n o i s e r e l a t i o n i n c l u d e s z e r o - p o i n t f l u c t u a t i o n s ( C a l l e n and Welton, 1951) and i s as f o l l o w s (31) < V , % > = ( % h v R ( v ) L ' / a « eL/ f cT , , ] 1 * » Furthermore, the r e s i s t a n c e R i n (30) has been g e n e r a l i z e d to a frequency dependent r e s i s t a n c e . In the l i m i t hv/kT —» 0, equation (31) reduces to Nyquist's o r i g i n a l e q u a t i o n . The e x p e r i m e n t a l l y measured thermal n o i s e has agreed very w e l l with the t h e o r e t i c a l p r e d i c t i o n s , thus i n p r i n c i p l e thermal n o i s e 55 thermomtry can be used to d e f i n e the temperature s c a l e ( J o h n s o n , 1928, Soulen and Marshak, 1980). The d i f f e r e n c e i n temperature found by using (30) i n s t e a d of (31) i s given by I t i s c a l c u l a t e d by expanding the i n t e g r a n d i n equation (31) i n powers of hv/kT, i n t e g r a t i n g the s e r i e s , and then e v a l u a t i n g the r e s u l t u s i n g a h i g h frequency c u t o f f at v-t. i f the c u t o f f frequency i s l e s s than 12.5 MHz and the temperature being measured i s more than 1 mK, then a c c u r a c i e s of b e t t e r than 1% can be achieved by using (30) i n s t e a d of the more complicated r e l a t i o n (31). The b a s i c components of a noise thermometer(NT) are a r e s i s t o r at the temperature of i n t e r e s t , an a m p l i f i e r , and a power meter. The amount of n o i s e an a m p l i f i e r adds to a s i g n a l can be d e s c r i b e d q u a n t i t a t i v e l y by a noise temperature. The noise temperature of the best room temperature a m p l i f i e r s i s 10 - 20 K. The noise temperature i s added l i n e a r l y to the n o i s e from the r e s i s t o r of i n t e r e s t , so that a very l a r g e c o r r e c t i o n must be s u b t r a c t e d from the output of the a m p l i f i e r i f the r e s i s t o r i s at a few mK. Other d i f f i c u l t i e s i n the implementation of the NT i s that the frquency response of the a m p l i f i e r must be w e l l known and the 1/f n o i s e spectrum from (32) 56 thermal emf's must be c a r e f u l l y accounted f o r . A d e s c r i p t i o n of the many techigues which have been developed to circumvent these problems i s given by Kamper(1973). At low temperatures, the use of Josephson j u n c t i o n s and superconductors permits the measurement of n o i s e power without adding e x c e s s i v e amounts of n o i s e to the s i g n a l . There are two ways to e x p l o i t the v a r i o u s c h a r a c t e r i s t i c s of the superconducting d e v i c e . One method i s to use a squid magnetometer to d i r e c t l y measure the n o i s e voltage(Webb et a l , 1973). The second method i s to apply the n o i s e v o l t a g e to a Josephen j u n c t i o n and to use the v o l t a g e to frequency c o n v e r s i o n p r o p e r t y of the junction(Webb et a l , 1973). The temperature i s i n f e r r e d from measurements of the frequency: the power output of the system i s not measured so the a m p l i f i e r gain c h a c a r t e r i s t i c need not be known. The s q u i d magnetometer NT achieves the very low a m p l i f i e r n o i s e ( 0.05 mK) (Webb et a l , 1973) by u s i n g the squid i n the f l u x locked mode to balance the n o i s e v o l t a g e generated i n the r e s i s t o r . The c i r c u i t c o n s i s t s of three p a r t s . The temperature sensing r e s i s t o r , R, i s connected to a superconducting i n d u c t o r , L, which i s f l u x coupled to a Squid. The s q u i d i s operated as a n u l l d e t e c t o r and i s connected to a feedback system, which maintains a f i x e d f l u x i n the s q u i d . The c u r r e n t needed to maintain a constant f l u x i n the squid i s processed through an a c t i v e bandpass f i l t e r and then an i n t e g r a t i n g mean square 57 r - > . Vnlll I (t) ^ ^ ^ ^ ^ o -L R L SQUID ^ V(0 •IVout (0* J •R, ACTIVE INTEGRATING | FILTER MEAN SQUARE VOLTMETER F i g u r e 1. A Squid magnetometer noise thermometer with low noise c h a r a c t e r i s t i c s as used by Webb et a l ( l 9 7 3 ) . 58 v o l t m e t e r . The system o p e r a t e s as f o l l o w , t h e n o i s e v o l t a g e , v ( t ) , i nduces a c u r r e n t , I ( t ) , i n t h e i n d u c t o r . The s q u i d magnetometer measures a f l u x <^(t) = L I ( t ) . The feedback system t h e n passes a c u r r e n t , I ' ( t ) , w h i c h g e n e r a t e s a f l u x e q u a l t o <Kt), i n t h e s q u i d . The c u r r e n t , I ' ( t ) , passes t h r o u g h a r e s i s t o r , R$, and the v o l t a g e v ' ( t ) i s t h e n p r o c e s s e d by a bandpass f i l t e r and measured i n a powerme te r . The n o i s e t e m p e r a t u r e of t h e sys tem i s found by p e r f o r m i n g a model c a l c u l a t i o n on t h e whole s y s t e m . By s u i t a b l y a d j u s t i n g t h e p a r a m e t e r s , such a s , R, L ,and r , t he n o i s e t e m p e r a t u r e of t h e sys tem can be dec reased t o 0 .05 mK. The t r a n s f e r f u n c t i o n o f t h e system must be known a c c u r a t e l y b e f o r e t h e NT can be used as an a b s o l u t e t h e r m o m e t e r . The a b s o l u t e a c c u r a c y o f a s q u i d magnetometer NT i s 3%, and i s m a i n l y due t o u n c e r t a i n t i e s i n L and R. One a l t e r n a t i v e i s t o use t h e NT as a secondary thermometer by c a l i b r a t i n g i t a t a s i n g l e f i x e d p o i n t . The components must be c a r e f u l l y compensated o r measured t o a c c o u n t f o r changes i n t h e pa ramete r v a l u e s as t h e t e m p e r a t u r e changes . The q u a n t i t y o f i n t e r e s t <v 2 ( t ) > i s a s t a t i s t i c a l q u a n t i t y s i n c e a f i n i t e i n t e g r a t i o n t i m e i s u s e d . There i s a c o n f l i c t i n t he c h o i c e of p a r a m e t e r s . I n o r d e r t o m i n i m i z e the n o i s e t e m p e r a t u r e o f t h e system and t o m i n i m i z e t h e s t a t i s t i c a l e r r o r i n a g i v e n measurement , a compromise of an a v e r a g i n g t i m e o f 2000 - 2500 sees and a p r e c i s i o n of 1% f o r 68 % o f t he 59 measurements of the temperatures between a few m i l l i K e l v i n and 20 K have been used(Webb et a l , l 9 7 3 ) . The second method u l i t i l i z e s the f a c t t h a t the c u r r e n t i n a weak l i n k which has an a p p l i e d d.c. b i a s v o l t a g e , V, o s c i l l a t e s at a frequency, f , given by where e i s the e l e c t r o n charge. The Josephson j u n c t i o n i s b i a s e d with a v o l t a g e , U, so that the n o i s e v o l t a g e , v ( t ) , modulates a frequency given by equation (33) with U as the v o l t a g e . The frequency i s u s u a l l y choosen to be 5KHz. The small amount of power r a d i a t e d by the j u n c t i o n ( of the order of 10" 1 5 W). i s a m p l i f i e d by parameteric up-conversion. T h i s i s accomplished by s u p p l y i n g a 30 MHz c a r r i e r s i g n a l to the j u n c t i o n . The s i g n a l i s f u r t h e r a m p l i f i e d and then demodulated by room temperature e l e c t r o n i c s . One way of i n t e r p r e t i n g the r e s u l t a n t power spectrum i s to r e a l i z e that the n o i s e v o l t a g e f l u c t u a t i o n s are compressed(Burgess, 1967) i n t o a L o r e n t z i a n curve with FWHM given by (33) (34) where <J>e i s the quantum of f l u x equal to h/2e. The method has been used to t e s t the noise thermometer to 10% i n the range 1 60 8 K ( S i l v e r et a l , 1967). The temperature can a l s o be i n f e r r e d from N frequency measurements of the system. The v a r i a n c e , o~ > ot the N measurements i s c a l c u l a t e d as (35) fj = 1 We - O 5 t-l 3. Is/ where V; i s the frequency measurement on the i t h t r i a l . The v a r i a n c e can be r e l a t e d to the temperature of the r e s i s t o r by(Kamper and Zimmerman,1971) (36) , ^ R f e T / t f * where *?" i s the gate time of the frequency counter. By t a k i n g a l a r g e number of frequency measurements, the v a r i a n c e can be determined to the d e s i r e d accuracy, thus the second method i s more u s e f u l than the f i r s t . In e i t h e r case, the equations c o n t a i n e a s l y measureable q u a n t i t i e s and c o n s t a n t s ; thus the thermometers are good primary thermometers. The r e s i s t o r of the system can be determined i n the same experiment by measuring the c u r r e n t i n the r e s i s t o r and averaging the frequency output of the system to get the v o l t a g e by equation (33). There are two unavoidable sources of u n c e r t a i n l y i n the thermometer. The number of frequency measurements that can be made i s f i n i t e , t h e r e f o r e , there i s a s t a t i s t i c a l e r r o r 61 a s s o c i a t e d with the c a l c u l a t e d v a r i a n c e . Simple c a l c u l a t i o n show that the rms s c a t t e r , AT R m s, i n the determined temperature i s given by There i s a count i m p r e c i s i o n of 1 i n the frequency measurement which adds an e x t r a c o n t r i b u t i o n to the v a r i a n c e . The e f f e c t can be c o n s i d e r e d to be the d e v i c e n o i s e , T 0 , and has been calculated(Kamper and Zimmerman, 1971) to be When the systematic c o r r e c t i o n i n the measured temperature i s equal to the u n c e r t a i n t y i n the measurement, then the adjustment of the measured temperature i s not u s e f u l . T h i s s e t s a l i m i t on the r e s o l u t i o n T of the thermometer. Combining (37) and (38), one f i n d s that the minimum r e s o l u t i o n of a NOT i s (37) (38) (39) 41 T To o b t a i n a u n c e r t a i n t l y of 1% i n the measured T, the number of frequency measurements r e q u i r e d i s 2x10". If the averaging time i s 1 sec then the t o t a l time necssary to measure 62 the temperature at one p o i n t i s 6 hours. The unavoidable systematic e r r o r s have beeen p r e d i c t e d to be i n the m i c r o k e l v i n range(Stephen, 1969). In summary, the noi s e thermometer i s the primary thermometer the l e a s t t r o u b l e d by c o r r e c t i o n s to the measured temperature. I t i s capable of measuring temperatures from a few mK to 0.5 K with 1% u n c e r t a i n t y . A major disadvantage of the NT i s that i t r e q u i r e s n e a r l y an hour to measure a s i n g l e temperature a c c u r a t e l y . A l s o , there always e x i s t s the p o s s i b i l i t y of an undetected extraneous noise source i n the system which i s undetected. 63 I I . 3He M e l t i n g Curve 2.1 I n t r o d u c t i o n The 3He m e l t i n g curve and the m e l t i n g curve thermometer(MCT) c h a r a c t e r i s t i c s w i l l now be d i s c u s s e d i n some d e t a i l , i n order to s u b s t a n t i a t e c l a i m s made e a r l i e r . Although both l i q u i d and s o l i d 3He are quantum systems, the thermodynamic C l a u s i u s - C l a p e y r o n equation <40> 1 7 ° - ^ J d e s c r i b e s the m e l t i n g curve. I f a l l the terms on the r i g h t hand s i d e of (40) were known, then the m e l t i n g curve c o u l d be c o n s t r u c t e d by i n t e g r a t i n g equation (40). The usual p r o c e d u r e ( S c r i b n e r et a l , 1969), i s to a c c u r a t e l y measure v^  - v 5 and use the va l u e s i n equation (40). Then the shape of the m e l t i n g curve i s determined by the molar entropy of the l i q u i d and s o l i d 3He. At pres s u r e s l e s s than 40 bar and high temperatures, the P-T diagram of 3He i s not p a r t i c u l a r l y i n t e r e s t i n g , but at low temperatures there are s e v e r a l phase t r a n s i t i o n s (Halper'in et a l , 1978). Along the m e l t i n g curve, there are three t r i p l e p o i n t s which correspond to the A - t r a n s i t i o n i n l i q u i d 3He, the B - t r a n s t i o n i n the l i q u i d , and 64 F i g u r e 2. The m e l t i n g curve of 3He using the r e s u l t s from Greywall and Busch (1982a) and G r i l l y ( 1 9 7 1 ) 65 the magnetic t r a n s i t i o n i n s o l i d 3He. Although very i n t e r e s t i n g and u s e f u l as thermometric f i x e d p o i n t s at very low temperatures(Roger et a l , l 9 8 3 ) , they w i l l not be c o n s i d e r e d i n the context of t h i s t h e s i s , s i n c e they occur below the base temperature of the r e f r i g e r a t o r s i n t h i s l a b . There are no accurate t h e o r i e s f o r the entropy of the v a r i o u s phases of 3He along the m e l t i n g curve. The 3He l i q u i d behaves l i k e a F e r m i - l i q u i d down to the A - t r a n s i t i o n (Legget, 1975), thus i t s entropy should be a l i n e a r f u n c t i o n of the temperature ( W i l k s ) . Experiments suggest, however, that there are higher order terms i n the e x p r e s s i o n f o r the entropy of the l i q u i d (Abel et a l , 1960). The l i q u i d commences to s p i n order at about 0.5 K ( S c r i b n e r , 1969), so that i t s entropy i s very small at the m i l l i k e l v i n temperatures. T h e o r e t i c a l l y , the entropy i s expected to be u n a f f e c t e d by magnetic f i e l d s which can be e x p e r i m e n t a l l y a p p l i e d ( G o l d s t e i n ) . The s e m i - c l a s s i c a l model of s o l i d 3He i s that of a system of n o n - i n t e r a c t i n g s p i n 1/2 p a r t i c l e s s i t t i n g at e q u i l i b r i u m s i t e s . The system has a molar entropy of R l n 2 ( K i t t e l ) . The l a t t i c e entroy c o n t r i b u t i o n s are very small as the Debye temperature of s o l i d bcc 3He i s 20 K ( H a l p e r i n ) . The entropy of the s o l i d would be expected to decreases as the s p i n s begin to order due to the n u c l e a r d i p o l e - d i p o l e i n t e r a c t i o n s , which have a c h a r a c t e r i s t i c energy of 10" 7 K. Above 50 mK, the measured s p i n entropy i s as p r e d i c t e d ( H a l p e r i n et a l ) , but near 1 mK the 66 entropy suddenly decreases to 0 .1Rln2-(0sheroff, 1972). The s p i n o r d e r i n g i n the s o l i d occurs at a higher temperature than expected because the zero p o i n t motion of the 3He, which i s aproximately 30% of the nearest neighbour s e p a r a t i o n , r e s u l t s i n l a r g e exchange o v e r l a p i n t e r a c t i o n s , of the order of 1 mK. I f only nearest neighbour exchange i s taken i n t o account, the s p i n H amiltonian of the system has the same form as the Heisenberg form(Thouless,1965) (*o H « - - i ^ ^ J > where J i s the exchange i n t e r a c t i o n between two neighouring atoms i and j , and I i s the n u c l e a r s p i n o p e r a t o r . S o l i d 3He i s a n t i - f e r r o m a g n e t i c , so the c o u p l i n g c o n s t a n t s J must be n e g a t i v e . High temperature s e r i e s expansion of the p a r t i t i o n have been found f o r s o l i d 3He by G o l d s t e i n ( 1 9 6 7 ) . At the low temperatures the q u a l i t a t i v e behaviour of the s o l i d i s not d e s c r i b e d by the Heisenberg model. The model p r e d i c t s a second order t r a n s i t i o n at about 2 mk, but experiments have shown a f i r s t order t r a n s i t i o n at about 1 mk. I t p r e d i c t s a decrease i n the s u s c e p t i b i l i t y r e l a t i v e to the Curie-Weiss law, but e x p e r i m e n t s ( B e r n i e r and D e l r i e u , 1977), shows that the s u s c e p t i b i l i t y i n c r e a s e s . A complete l i s t of the d e f f i c i e n c y of the Heisenberg model i s given by Roger et a l (1983). The l a t e s t model suggested to e x p l a i n the behaviour of s o l i d 3He below 30 67 mK i s that of a system of atoms which have m u l t i p l e exchange i n t e r a c t ions(Roger et a l ) . Although the very low temperature behaviour of the m e l t i n g curve i s d i f f i c u l t to e x p l a i n t h e o r e t i c a l l y , the higher temperature c h a r a c t e r i s t i c s are q u i t e w e l l understood. The shape of 3He m e l t i n g curve about the minimum can be e x p l a i n e d by a r e l a t i v e l y simple model. The entropy of the s o l i d , i n the range 10 mK to 500 mK, i s Rln2, because the 3He p a r t i c l e s are n o n - i n t e r a c t i n g s p i n 1/2. The entropy i s temperature independent u n t i l 10 mK, where due to o r d e r i n g of the s p i n s i t decreases r a p i d l y . The l i q u i d 3He i s a degenerate Fermi gas whose entropy decreses r a p i d l y below 0.3 K and i s equal to the s o l i d entropy at 0.32 K. The molar volume d i f f e r e n c e between the s o l i d and the l i q u i d i s approximately constant ( S c r i b n e r et al., 1969) from 0.33 K to zero K. At T =0 .32 K where the two e n t r o p i e s are e q u a l , the C l a u s i u s - C l a p e y r o n i m p l i e s that the m e l t i n g curve has an extrema. A f t e r i n s e r t i n g the q u a n t i t i e s i n t o equation(40) and i n t e g r a t i n g i t , one f i n d s a curve s i m i l i a r to the measured curve. More s o p h i s t i c a t e d models have t r i e d to p r e d i c t the e n t i r e m e l t i n g curve but no accurate m e l t i n g curves have been generated (Hudson et a l . ) . Given that i t i s not c e r t a i n that even the newer t h e o r i e s a c c u r a t e l y d e s c r i b e the behviour of the 3He along the m e l t i n g curve, an e m p r i c i a l approach w i l l be used to j u s t i f y the c l a i m s made about the 3He m e l t i n g curve thermometer. 68 Because there i s no complete theory f o r the 3He m e l t i n g curve, the MCT i s c a l i b r a t e d by comparing the m e l t i n g cruve with a primary thermometer. The most recent and p r e c i s e measurement of the 3He m e l t i n g curve i s by Greywall and Busch(1982a), who used the temperature s c a l e set by NBS through the SRM 768 f i x e d p o i n t d e v i c e s . They found the minimum pressure to be 29.316 0.003 bar. at a temperature of 0.318 0.001 K. T h e i r r e s u l t s agreed with accurate measurements made by H a l p e r i n et a l at the very low temperatures, and with the r e s u l t s of G r i l l y at the minimum i n the curve. Greywall and Busch f i t t e d the f u n c t i o n below to t h e i r data. (42) P - P, 5" T where 32.342 bar and T A = 2.752 mK, the A - t r a n s t i o n c o o r d i n a t e s found by H a l p e r i n et a l . The c o e f f i c i e n t s are a a a „ 8.406 x 10" 8 0.15435 157.616 624.330 - 1 a , = a , = a „ = -1.220 x 10-ft -44.4713 -365.784 -491.348 . The data of Greywall and Busch, and H a l p e r i n et al., d e v i a t e s 69 from the v a l u e s given by the e x p r e s s i o n (42) by l e s s than 2 mbar, which corresponds to 0.5% u n c e r t a i n t y i n the measured temperature over most of the temperature range of the thermometer. 70 2.2 Magnetic F i e l d Dependence of the M e l t i n g Curve The slope of the 3He m e l t i n g curve i s p r o p o r t i o n a l to the d i f f e r e n c e s - s , and s i n c e the entropy of a Fermi l i q u i d i s o n l y s l i g h t l y a f f e c t e d by the magnetic f i e l d ( G o l d s t e i n , 1968), the s l ope of the curve i s p r o p o r t i o n a l t o the entropy of the s o l i d 3He. In the Heisenberg approximation of the 3He s o l i d , the entropy i s given by ( G o l d s t e i n , 1967) (42) 5^ - U 1 _ + X s + --L<i\ \* ox * 5 t x * + where x = J/kT and y = Bu/kT. B i s the magnetic f i e l d , and u i s the magnetic d i p o l e moment of the 3He atom. The J i s the Heisenberg c o u p l i n g constant between nearest neighbours i n a 3He s o l i d and i s of the order of -1 mK(Trickey, 1972). In a l l experimental c o n d i t i o n s y i s very s m a l l , but even i f y 1, as long as the model i s c o n s i d e r e d i n the region T > 1 mK, where x < 1, the entropy i s e s s e n t i a l l y independent of the i n t e r a c t i o n c h a r a c t e r i s t i c s , because the c o e f f i c i e n t of y 2 i n (42) c o n t a i n s x as a second order term. Although at temperatures below 40 mK(Scribner) and high magnetic f i e l d s , the Heisenberg model i s incapable of d e s c r i b i n g the m e l t i n g curve, as even i t s q u a n t i t a t i v e p r e d i c t i o n s are i n c o r r e c t , the Heisenberg model can be used above t h i s temperature to d e s c r i b e the m e l t i n g curve respond to a a p p l i e d magnetic f i e l d . At the very low 71 temperatures, the multi-exchange i n t e r a c t i o n s must be i n c l u d e d i n the model to get s a t i s f a c t o r y agreement with the observed behaviour. I t has been e x p e r i m e n t a l l y found(Kummer et al., 1977) that f o r temperatures g r e a t e r than 1.35 mK and magnetic f i e l d s l e s s than 4.5 KG the d e p r e s s i o n of the m e l t i n g p r e s s u r e i s u n r e s o l v a b l y s m a l l . Even f o r f i e l d s as h i g h as 12 kG, as long as the temperature i s above 10 mK, the measured temperature w i l l be a c c u r a t e to b e t t e r than 1 % of the reading (Hummer et al., 1977). The m e l t i n g curve has been measured i n magnetic f i e l d s as high as 63 kG(Johnson et a l , 1971), but thermometry problems prevented the u s e f u l i n t e r p r e t a t i o n of the d a t a . That the m e l t i n g pressure i s depressed i n the presence of a magnetic f i e l d can be shown thermodynamically by c o n s i d e r i n g the Gibb's f r e e energy of two magnetized phases i n e q u i l i b r i u m . I t i s easy to show that at a constant temperature ( 4 3 ) Ai-r = M»- - ft* where and M s are the molar magnetization of the l i q u i d and the s o l i d . N e g l e c t i n g the m a g n e t i z a t i o n of the Fremi l i q u i d because i t i s s m a l l ( G o l d s t e i n , 1968), one gets 72 where v i s approximately constant and g r e a t e r than z e r o . As the magnetic f i e l d i s i n c r e a s e d , the magnetization i n c r e a s e s and the m e l t i n g pressure i s depressed. 73 2.3 Impurity E f f e c t s on the M e l t i n q Curve The m e l t i n g curve minimum occurs at higher temperature and at a depressed pressure as the "He c o n c e n t r a t i o n i n 3He i s in c r e a s e d (Weinstock et al., 1962). No major study has been made of the e f f e c t of i n c r e a s e d "He c o n c e n t r a t i o n s on the curve, but S c r i b n e r et al,(l969) found no s u b s t a n t i a l d i f f e r e n c e between the mel t i n g curve with 20 ppm "He and 600 ppm. But a small bump at a temperature of 0.1 K was n o t i c e d i n the m e l t i n g curve with 600 ppm "He. So as long the "He c o n c e n t r a t i o n i s smal l e r than 20 ppm then the bump should be u n r e s o l v a b l y s m a l l , the phase s e p a r a t i o n that occurs i n the "He 3He mixture probably i n s u r e s that the 3He i s very pure at the low temperatures. Furthermore, the "He w i l l tend to s o l i d i f y on the s i n t e r i n the 3He c e l l . An estimate of the amount of "He which can be fr o z e n out onto a s u r f a c e i s found by e s t i m a t i n g the area per atom and d i v i d i n g i n t o the t o t a l s u r f a c e area a v a i l a b l e . The nearest neighbour s e p a r a t i o n i n bcc s o l i d helium at 29 bars i s about 4 A ( K e l l e r ) . The atom diameter i s 2.2 A. The s u r f a c e area of the s i n t e r i n a c e l l of the Greywall and Busch type i s 0.5 m2. The t o t a l amount of 3He i s the c e l l and c a p i l l a r y below the plug i s about 4x10"* moles, thus, assuming that the spacing per atom i n a monlayer i s approximately equal to the l a t t i c e s e p a r a t i o n i n three dimensions, the "He c o n c e n t r a t i o n can be as hig h as 700 ppm and i t w i l l s t i l l f r e e z e out onto the s i n t e r s u r f a c e i n a monolayer. 74 A monolayer of "He w i l l not a f f e c t the K a p i t z a r e s i s t a n c e of the s i n t e r ( H a r r i s o n ) . Most 3He m e l t i n g curve thermometers have used 3He with "He i m p u r i t i e s l e s s than 20 ppm(Greywall and Busch, 1 982a). r 75 2.4 Accuracy and R e s o l u t i o n of the MCT The r e s o l u t i o n of the MCT depends on the r e s o l u t i o n of the s t r a i n gauge used to measure the pressure at m e l t i n g . Most pre s s u r e t r a n s d u c e r s used f o r thermometry(Greywall and Busch, 1982a, C o r r u c i n i et a l , 1978) have a pressure r e s o l u t i o n of b e t t e r than .1 ppm. The temperature s e n s i t i v i t l y of the m e l t i n g curve i n c r e a s e s as the temperature decreases down to about 3 mK where the curve beginning to l e v e l o f f as r e q u i r e d by the t h i r d law of thermodynamics. T h i s l e v e l i n g o f f can be seem i n the va l u e s of the ^ l i s t e d i n Table I, which are the r e s u l t s given by Greywall and Busch(1982a). At T = 300 mK, the temperature r e s o l u t i o n , given Ap/p = 10~ 7, i s 3 ju.K, and the r e s o l u t i o n i n c r e a s e s at the lower temperatures. I f a good dead weight t e s t e r i s used to c a l i b r a t e the s t r a i n gauge, then the pressure measured by the s t r a i n gauge w i l l have a u n c e r t a i n t y i n the accuracy of 0.003 ba r ( G r e y w a l l and Busch, 1982a) or b e t t e r ( G r i l l y , 1 9 7 1 ) . At T = 300 mK, 3 mbars corresponds to 3 mK u n c e r t a i n t y , at T = 200 mK, t h i s corresponds to 0.3 mK u n c e r t a i n t y , and f i n a l l y at T = 100 mK, 3 mbars corresponds to 0.1 mK u n c e r t a i n t y i n the measured temperature. Thus the accuracy of the MCT i n the range 3 mK < T < 300 mK i s b e t t e r than 1 %, i f the pressure i s measured to an accuracy of 3 mbar. The accuracy of any MCT depends on the accuracy to which the 3He m e l t i n g curve i s known t o . Greywall and Busch(1982a) 76 T A B L E I T, P, dP/dT bar K T, P, dP/dT. bar K mK bar mK bar 3 34.3330 -36.21 90 31.5407 -23.32 4 34.2957 -38.23 95 31.4262 -22.47 5 34.2569 -39.39 100 31.3159 -21.66 6 34.2171 -40.01 110 31.1072 -20.10 7 34.1769 -40.32 120 30.9136 -18.63 8 34.1365 -40.44 130 30.7343 -17.24 9 34.0961 -40.45 140 -.30.5685 -15.93 10 34.0557 -40.37 150 30.4154 -14.69 11 34.0154 -40.25 160 30.2745 -13.50 12 33.9752 -40.09 170 30.1452 -12.37 14 33.8954 -39.71 180 .30.0268 -11.30 16 33.8164 -39.26 190 29.9191 -10.26 18 33.7383 -38.79 299 ?9-82,U -9.27 20 33.6613 -38.30 210 29:7336 -8.31 25 33.4729 -37.05 220 29.6551 -7.39 30 33.2907 -35.81 230 29.5856 -6.51 35 33.1147 -34.58 240 29.5248 -5.65 40 32.9448 -33.39 250 29.4725 -4.82 45 32.7808 -32.23 260 29.4282 -4.03 50 32.6224 -31.11 270 29.3918 -3.26 55 32.4696 -30.02 280 29.3630 -2.52 60 32.3222 -28.97 290 29.3414 -1.81 65 32.1799 -27.95 300 29.3267 -1.14 70 32.0426 -26.96 310 29.3185 -0.50 75 31.9102 -26.00 320 29.3166 0.10 80 31.7826 -25.08 330 29.3204 0.66 85 31.6594 -24.18 Table of v a r i o u s parameters of the me l t i n g curve as c a l c u l a t e d by Greywall and Busch (1982a). 77 measured the m e l t i n g curve to a c c u r a c i e s of 3 mbar i n the p r e s s u r e and a few tenths of a percent i n the temperature. 78 2.5 S e l f - h e a t i n g and R.F. S e n s i t i v i t y of the MCT The s e l f - h e a t i n g of a MCT o r i g i n a t e s from the j o u l e h e a t i n g i n the c o a x i a l c a b l e s which are connected to the c a p a c i t o r of the MCT p r e s s u r e t r a n s d u c e r . The amplitude of the a.c. e x c i t a t i o n v o l t a g e must be kept to a minimum to reduce the j o u l e h e a t i n g i n the wires, but t h i s l i m i t s the measurable r e s o l u t i o n of the c a p a c i t o r . An e x c i t a t i o n v o l t a g e of 2 v o l t s RMS means that d u r i n g the n u l l i n g o p e r a t i o n , v o l t a g e s on the order of m i c r o v o l t s must be measured i f the c a p a c i t a n c e of the transducer i s t o be measured to 1 ppm. A l o c k - i n d e t e c t i o n system must be used to measure the v o l t a g e s because thermal emfs and s t r a y e l e c t r i c a l n o i s e would otherwise obscure the s i g n a l . The requirement that the c a p a c i t a n c e be r e s o l v a b l e to 1 ppm s e t s a lower l i m i t on the usable e x c i t a t i o n v o l t a g e . I f the e x c i t a t i o n v o l t a g e i n 2 v o l t s RMS, then with c a r e f u l thermal grounding of the c a b l e s , the power d i s s i p a t i o n i n the MCT can be l e s s than 1 0 - 1 4 watts. The MCT i s i n s e n s i t i v e to r . f . noise because the l a r g e impedance of the c a p a c i t o r i n the transducer a l l o w s only a very small c u r r e n t to flow i n the l e a d wires, thus there i s very l i t t l e j o u l e h e a t i n g . 79 2.6 Thermal Time Response of the MCT The thermal response time, t, of the thermometer a t t a c h e d to a thermal bath i s determined by the heat c a p a c i t y of the thermometer, C^, and the r e s i s t a n c e , R, to the heat flow to bath by (45) 2r - RC^p . There are r e a l l y two thermal systems that need to be c o n s i d e r e d fo r a p r e c i s e d e t e r m i n a t i o n of the thermal response time. The f i r s t system c o n s i s t s of the BeCu s t r a i n gauge. The s p e c i f i c heat of the BeCu i s about an order of magnitude s m a l l e r than that of 3He liquid(Lounasmaa) and the c o n d u c t i v i t y of the BeCu i s approximately the same order of magnitude as the 3He l i q u i d . Because the c e l l i s i n m e t a l l i c c o n t a c t with the heat bath, the thermal boundary r e s i s t a n c e i s much s m a l l e r than the K a p i t z a r e s i s t a n c e between the l i q u i d 3He i n the c e l l and the w a l l s of the c e l l . The second system c o n s i s t s of the s o l i d and l i q u i d 3He and the w a l l s of the c e l l . The thermal c o n d u c t i v i t y of the l i q u i d 3He i s f a i r l y l a r g e at the low temperatures being considered(Lounasmaa), t h e r e f o r e as long as there i s l i q u i d i n the c e l l , the 3He phases are i n e q u i l i b r i u m . T h i s i s not s t r i c t l y t r u e at the very low te m p e r a t u r e s ( S c r i b n e r et aL>, 1972), but the approximation w i l l s u f f i c e f o r these 80 c a l c u l a t i o n s . The thermal time constant of the 3He i s a good estimate of the thermal time constant of the thermometer. By c o n s i d e r i n g the m e l t i n g process as heat i s i n t r o d u c e d i n t o the 3He system, and using r e p r e s e n t a t i v e low temperature v a l u e s f o r the 3He parameters ( C o r r u c i n n i et a l , 1978), one gets that the e f f e c t i v e heat c a p a c i t y , f o r T < 100 mK, i s at most 5.6xl0 8nT erg/(mole K 2 ) , where n i s the number of moles of 3He. The measured K a p i t z a r e s i s t a n c e between 3He and s i l v e r s i n t e r i s R = 0.24T" 3A" 1K am 2/Watts, f o r T > 10 mK(Godfrin), where A i s the e f f e c t i v e s u r f a c e area of the s i n t e r i n square meters. Thus equation (45) g i v e s the thermal time to be ( 4 5 ) 2- = lU£_P- «**^*s«^ • AT* The s u r f a c e ares of 700 A s i l v e r s i n t e r i n a m e l t i n g curve thermometer of the Greywall and Busch(1982a) type i s 0.5 m2. The thermometer c e l l c o n t a i n s 0.09 cm 3 3He at 34 bars. Thus f o r T = 0.05 K, i s at most 40 sees. I f the temperature of the c e l l i s below 10 mK, then the K a p i t z a r e s i s t a n c e of the s i l v e r s i n t e r i s given b y ( H a r r i s o n , 1979) (47) R « U l o _ x _ l o - * w 4 K ^ , AT *'J thus ^= 450 sees. So below 10 mK, the thermal time constant of the thermometer i s at most 7 minutes. F o r t u n a t e l y , f o r the o p e r a t i n g c o n d i t i o n s of the c e l l , the thermal time constant about 30 sec. (Greywall and Busch, 1982a). 82 I I I . 3He M e l t i n g Curve Thermometer 3.1 I n t r o d u c t ion The 3He m e l t i n g curve thermometer c o n s i s t s of three p a r t s , a pressure transducer c e l l with a f i x e d volume, a c a p a c i t a n c e bridge c i r c u i t to measure the transducer response, and a pressure system to supply 3He at 35 bars and c a l i b r a t e the s t r a i n gauge. The p r e s s u r e transducer c e l l used at low tempertures must s a t i s f y s e v e r a l requirements; i t must be a b l e to withstand repeated c y c l i n g to low temperature without d e v e l o p i n g a leak; i t must have a low s p e c i f i c heat so that i t s thermal time, constant i s s h o r t ; i t must not be too bulky so that i t can be e a s i l y f i t t e d onto a d i l u t i o n r e f r i g e r a t o r ; and furthermore, the volume of the c e l l must remain a c o n s t a n t , r e g a r d l e s s of the pressure i n the c e l l . Because the s t r a i n gauge of Greywall and Busch(1982a) s a t i s f y a l l these requirements, i t as chosen as the c a p a c i t a n c e p r e s s u r e transducer to be used i n f o r t h i s MCT. The r e q u i r e d s e n s i t i v i t y i n the pressure measurements demands that the c a p a c i t a n c e of the transducer be measured to 1 ppm, with a d r i f t over a day of l e s s than 1 ppm. Because the c a p a c i t a n c e of the transducer i s approximately 10 pF and the c a p a c i t a n c e to ground of the c o a x i a l c a b l e s i s 300 pF, the guarded ground 83 3-wire technique must be used so that the s t r a y c a p a c i t a n c e does not completely mask the response of the t r a n s d u c e r . F o r t u n a t e l y , as the measurement of c a p a c i t a n c e i s a r e f i n e d technique, the r e q u i r e d r e s o l u t i o n and s t a b i l i t y can be achieved with mostly r e a d i l y a v a i l a b l e commerical instruments. The p r e s s u r e system of a MCT has many c o n s t r a i n t s , which makes i t d i f f i c u l t to c o n s t r u c t . The system must be 3He leak t i g h t over a p e r i o d of years so that no 3He i s l o s t , and no "He l e a k s i n t o the system. A l s o , the requirement t h a t the volume of the system be small so that only a small q u a n t i t y of h i g h l y p u r i f i e d 3He i s necessary to p r e s s u r i z e the whole system to 35 bars, n e c e s s i t a t e s the use of very t h i n c a p i l l a r y t u b i n g . The t h i n c a p i l l a r i e s i n c r e a s e s the time needed to pump down the system and thus hampers the leak t e s t i n g of the system. F i n a l l y , the 3He must be p r e s s u r e d to 35 bars without contaminating i t . These requirements f o r c e d many design d e c i s i o n s which makes the pressure system an awkward system to operate. 84 3.2 Pressure Transducer C e l l The c y l i n d r i c a l p r e s s u r e transducer of Greywall and Busch (1982a) i s shown i n F i g u r e 3. I t was c o n s t r u c t e d from 1/2 hardened b e r y l l i u m copper, type B e r y l c o 25. The p a r t s were hardened a f t e r machining by baking at 300 C f o r 2 hours i n a He atmosphere with 0.5 % to 1 % H^. The t r e a t e d BeCu has a t e n s i l e s t r e n g t h of 585 MPa, and an Young's modulus of 125x10 s Pa. The thermometer base was made of OFHC copper. U n l i k e many ca p a c i t a n c e gauges of the Straty-Adams t y p e ( l 9 6 9 ) , t h i s gauge does not have any f l a n g e s , and consequently, the spacing between the two c a p a c i t a n c e p l a t e s cannot be a d j u s t e d . That f e a t u r e makes t h i s gauge l e s s bulky and simpler to machine than the other d e s i g n s , but the assembly of the gauge i s c o r r e s p o n d i n g l y more d i f f i c u l t . The f i r s t s tep i n the assembly of the gauge was to s i l v e r s o l d e r a s t a i n l e s s s t e e l 0.56 mm O.D. f i l l l i n e c a p i l l a r y with 0.13 mm t h i c k w a l l s to the copper base of the thermometer. The bottom of the w e l l i n the base was then s i l v e r p l a t e d so that the s i l v e r s i n t e r would adhere to the copper base and form a m e t a l l i c c o n t a c t between the s i n t e r and the base. The s i n t e r i n g process w i l l be c o n s i d e r e d l a t e r . About 0.3 grams of s i n t e r was packed i n t o the w e l l ; t h i s corresponds to a s u r f a c e area of 0.8 m2. The s i n t e r o c c u p i e s about 45% of the volume of the c e l l . Next, the diaphragm was epoxied to the base using Emerson 85 F i g u r e 3. The 3He S t r a i n Gauge used by Greywall measure the 3He m e l t i n g curve. and Busch to 86 and Cumming, Type 1266, epoxy. Because the epoxy has a very low v i s o c i t y , the two p i e c e s were clamped t i g h t l y together i n a v i c e to prevent the epoxy from f l o w i n g i n t o the s i n t e r . The epoxy was cured f o r a day under a lamp, before the whole assembly was He leak t e s t e d and then p r e s s u r e t e s t e d to 40 bars with He from a high pressure c y l i n d e r . F u r t h e r , the c e l l was c o o l e d to l i q u i d n i t r o g e n temperature and p r e s s u r e d t e s t e d to t e s t that thermal c o n t r a c t i o n s d i d not crack the epoxy s e a l . The diaphragm has a diameter of 0.640 cm and i s 0.025 cm t h i c k . Using the formulas given by S t r a t y and Adams(l969), the optimum t h i c k n e s s of the diaphragm f o r maximum pressure s e n s i t i v i t y i s determined by the t e n s i l e s t r e n g t h of the BeCu and the p r e s s u r e s that are to be measured. The c a p a c i t o r p l a t e s s u r f a c e s were machined to a f i n e f i n i s h , sanded with very f i n e emery paper, and then lapped with 10 /cm diamond powder f o r a f l a t f i n i s h . Using i n i d i u m s o l d e r , #32 brass leads were a t t a c h e d to the p l a t e s . The s o l d e r f l u x was completely removed by washing with methanol. An i n s u l a t i n g l a y e r of Type 2850 FT epoxy was cured onto the back s i d e s of the c a p a c i t o r p l a t e s . Then the upper p l a t e was epoxied to the housing with 2850 FT and allowed to cure. A drop of 2850 FT was p l a c e d on the n i p p l e of the diaphragm, and the lower p l a t e was r e s t e d on i t . The housing was s l i p p e d onto the diaphragm, epoxied with 1266 epoxy, and allowed to r e s t on the lower p l a t e . The 3He chamber was then p r e s s u r i z e d to 34 bar and the whole 87 assembly cured f o r 2 days under a lamp. I t was found that two days are r e q u i r e d f o r the black epoxy to s e t , when under pr e s s u r e and i n a small c o n f i n e d volume. A Boonton Type 74C-S8 c a p a c i t a n c e bridge was atta c h e d to the leads and the pre s s u r e response of the transducer was measured. When the gauge was c o o l e d to LN a temperature a l a r g e decrease i n the pressure s e n s i t i v i t y was measured. Some e f f o r t was made to take t h i s s h i f t i n t o account, by d e c r e a s i n g the pres s u r e i n the 3He chamber while the epoxy was c u r i n g , but a f t e r s e v e r a l t r i a l s i t was found that the low temperature behaviour was not very p r e d i c a b l e . So, the epoxy was allowed to cure at 34 bars pressure which i n s u r e d there would- be no short between the p l a t e s w i t h i n the working pressure range of the MCT. The zero p r e s s u r e c a p a c i t a n c e of 3.15 pF at room temperature i m p l i e s a p l a t e s e p a r a t i o n of 3.7x10" 3 cm. The OFHC copper base was gold p l a t e d to ensure good m e t a l l i c c o n t a c t when the thermometer i s mounted on the mixing chamber of the d i l u t i o n r e f r i g e r a t o r . ( 88 3.3 S i n t e r s S i n t e r s are used i n low temperature apparatuses to decrease the K a p i t z a r e s i s t a n c e between l i q u i d helium and the w a l l s of i t s c o n t a i n e r . A summary of measured K a p i t z a r e s i s t a n c e s f o r v a r i o u s m a t e r i a l s i s given by H a r r i s o n ( 1979). Although copper s i n t e r s have been used(Lounasmaa), s i l v e r s i n t e r i s p r e f e r e d at the lower temperatures because i t s s p e c i f i c heat i s two times smaller than that of copper at 0.020 K(Lounasmaa). The s i l v e r powder commonly used to form the s i n t e r i s c a l l e d u l t r a f i n e Ag powder 700 A, Type II and was obtained from the f o l l o w i n g company: Vacuum M e t a l l u r g i c a l Comp. LTD. Shonan Bldg. n 14-10 1-Chome, Ginza, Chuo-ky, Tokyo. The v i r g i n powder has a s u r f a c e a r e s of 4.0 m2/g as measured by the BET method(Allen,1968) using Ar at LN a temperature and using a = .16 nm2 as the area covered by one Ar atom. A 400 A s i l v e r powder i s a l s o a v a i l a b l e from the same company, but because the s i l v e r powder tends to s e l f - s i n t e r (Roberston et al., 1983), i t i s not c l e a r t h at anything i s gained by using the sm a l l e r powder, unle s s very low temperatures w i l l be encountered. The K a p i t z a r e s i s t a n c e of the 400 A s i n t e r i s about one h a l f of the 89 r e s i s t a n c e of the 700 A s i n t e r ( G o d f r i n ) . The s i l v e r i s prepared f o r s i n t e r i n g by f i r s t p r e s i n t e r i n g the powder. The oxides are removed from the s u r f a c e s of the powder by baking the powder at 380*C f o r 10 minutes, while i t i s f l u s h e d with 1-2 t o r r of hydrogen. A f t e r baking i t i s allowed to c o o l i n a helium atmosphere. The s u r f a c e to r e c e i v e the s i l v e r s i n t e r must be p l a t e d with a t h i c k l a y e r of s i l v e r . The s i l v e r p l a t e i s p o l i s h e d with a f i n e diamond dust to remove any loose f l a k e s . Then the s i l v e r powder i s packed onto the s u r f a c e . In order f o r the s i n t e r to be s t r u c t u r a l l y s t r o n g , the powder must be packed with p r e s s u r e s around 4000 p s i . The moderately h i g h packing p r e s s u r e not only r e s u l t s i n a strong s i n t e r , but a l s o i n c r e a s e s the s u r f a c e area of the s i n t e r per u n i t area of the s i n t e r base. The s i n t e r i n g i s done i n a good vacuum or a helium atmosphere; hydrogen should be avoided as i t e m b r i t t l e s the copper base of the s i n t e r c o n t a i n e r . The s i n t e r must must be under p r e s s u r e d u r i n g h e a t i n g because of shrinkage of the s i n t e r . I f no pressure i s a p p l i e d , the r e s u l t i n g s i n t e r w i l l be f l a k e y . Greywall c l a i m s that 700 A powder packed to 3600 p s i p r e s s u r e and s i n t e r e d at a temperature of 200'C f o r 10 minutes w i l l r e s u l t i n a s i n t e r with a s u r f a c e area of 2.5 m2/g. The s i n t e r i n g parameters are not very c r i t i c a l . S i n t e r i n g temperatures can be anywhere from 100 *C to 400 "t and s i n t e r i n g times can be from 1 minute to 15 minutes. There i s a l o s s of 90 s u r f a c e a r e a o f t h e s i n t e r e d powder i f t h e h i g h e r t e m p e r a t u r e s or l o n g e r t i m e s a r e u s e d , bu t p resumab ly t h e s i n t e r w i l l be m e c h a n i c a l l y s t r o n g e r . S i l v e r s i n t e r , u n l i k e copper s i n t e r , c a n n o t be machined because i t t e n d s t o c h i p o f f a l o n g t h e p r e s s i n g l a y e r s . The s i n t e r i s b u i l t up by p r e s s i n g t h i n l a y e r s o f s i n t e r on t o p o f one a n o t h e r , i n o r d e r t o o b t a i n even p a c k i n g . O t h e r w i s e , t h e s i n t e r e d powder i s v e r y f r a g i l e . The s i n t e r i n t h e MCT c e l l was made by p a c k i n g t h e s i n t e r w i t h a v i s e and t h e n b a k i n g a t 90 "C f o r 20 m i n u t e s . W i t h t h i s c h o i c e o f s i n t e r i n g p a r a m e t e r s t h e s i n t e r was found t o adhere w e l l t o t h e b a s e . 91 3.4 Pressure System The p r e s s u r e system f o r a MCT must be ab l e to p r e s s u r i z e the 3He gas up to 35 bars without contaminating the gas and be able to c a l i b r a t e the 3He s t r a i n gauge with high accuracy. To minimize the contamination, a l l p a r t s of the system are made of s t a i n l e s s s t e e l and a l l connections are metal to metal s e a l s . Furthermore, an o i l - l e s s compressor was designed to prevent any p o s s i b i l i t y of pump o i l i n the system. The pressure system c o n s i s t s of three p a r t s , the c h a r c o a l pump, the v a l v e s and c o l d t r a p network, and the c a l i b r a t i o n gauge. The arrangement i s shown i n F i g u r e 4. The c h a r c o a l pump compresses the 3He by absorbing the helium-3 when the c h a r c o a l i s c o o l e d to 4 K and desorbing the gas when i t i s allowed to warm up to 30 K. As long as the p a r t i a l p r e s s u r e of the helium i s g r e a t e r than 10" 6 t o r r , the q u a n t i t y of helium gas that c h a r c o a l at 4 K w i l l absorb i s independent of the p r e s s s u r e . The c h a r c o a l used i n the pump has d e n s i t y of 0.67 g/cm3 and i s a b l e to absorb 0.43 l i t e r s STP of helium per gram when c o o l e d to 4 K. F i v e grams of c h a r c o a l were packed i n t o a 17 cm l e n g t h of 3/8" O.D. s t a i n l e s s s t e e l tube with 0.020" w a l l , which had end caps welded on. A f i l l l i n e of 1.22 mm O.D. and 0.20 mm w a l l s t a i n l e s s s t e e l was welded to one of the end caps. A small f i l t e r with pyrex g l a s s wool prevents c h a r c o a l dust from e n t e r i n g the f i l l l i n e . The other end of the 92 Charcool absorption Q°-2000psl pump Thermistor gauge (2)b P Transducer • Cryogenic Temperature • Cell Vacuum Cold trap F i g u r e 4. The pressure system f o r the MCT uses a c h a r c o a l pump to compress the 3He to 35 bars. A l l the v a l v e s are bellows v a l v e s manufactured by Nupro. Val v e s 1, 2, 3, and 4 are SS-4BG v a l v e s . V a l v e s 5 and 6 are SS-4BRG v a l v e s . Valve 0 i s a SS-4UG v a l v e . The v a l v e to the vacuum i s a brass bellows v a l v e . 93 f i l l l i n e i s connected to a 0-2000 p s i pre s s u r e gauge and a Nupro SS-4UG v a l v e . The assembly between the va l v e and pump w i l l operate s a f e t y with p r e s s u r e s up to 140 bars. A flow c r y o s t a t was chosen to c o o l the s o r p t i o n pump s i n c e there was i n s u f f i c i e n t space f o r a helium dewar near the r e f r i g e r a t o r . The flow c r y o s t a t , shown i n F i g . 5, c o o l s by flow i n g l i q u i d helium over the pump. The pump temperature i s monitored with a carbon r e s i s t o r mounted on the pump; i t has a room temperature r e s i s t a n c e of 10 k>* and a l i q u i d helium r e s i s t a n c e of 1140 k«v. The pump i s designed to be used i n the f o l l o w i n g manner: a f t e r the pump has absorbed a l l the helium from the r e s e r v o i r , i t i s slo w l y warmed up with the va l v e V© open. When the whole system has p r e s s u r i z e d to 25 bars, the va l v e i s c l o s e d . The pressure i n system i s now a d j u s t e d by c r a c k i n g open the v a l v e on the pump. The network of v a l v e s and c a p i l l a r i e s are arranged so that the helium can be f i l t e r e d with a l i q u i d n i t r o g e n t r a p before i t flows i n t o the small c a p i l l a r y tubes. A l s o , i n the event of the d i l u t i o n r e f r i g e r a t o r unexpectedly warming up, a r e l i e f v a l v e set at 600 p s i opens up to the r e s e r v o i r . A l l v a l v e s used i n the system are s t a i n l e s s s t e e l bellow v a l v e s with 1/4" swagelok conn e c t o r s . The network i s designed to operate with p r e s s u r e s l e s s than 40 bars (140 bars f o r the s o r p t i o n pump). When the MCT i s not i n use, the helium i s s t o r e d i n a 94 Bellows Gas return line 3 3He (il lint Einsu'laWn Charcoal pump .Uquid helium Copper thermal shield Vacuum line Super insulation Vacuum can F i g u r e 5. L i q u i d helium from a storage dewar i s f o r c e d to flow over the c h a r c o a l pump and c o o l the contents i n i t . 95 2570 cc volume at one bar p r e s s u r e . The L N a c o l d t r a p i s an 18 cm le n g t h of 1/8" O.D., 0.020" w a l l s t a i n l e s s s t e e l tube f i l l e d with crushed 14 A molecular s i e v e . Although there should not be any l a r g e amounts of i m p u r i t i e s i n the 3He, the t r a p helps to ensure t h a t no plugs of f o r e i g n gases w i l l be formed i n the narrow c a p i l l a r i e s which l e a d i n t o the r e f r i g e r a t o r . The f i l l l i n e i n t o the c r y o s t a t i s a c a p i l l a r y with a 1.22 mm O.D. and 0.20 mm w a l l . At the top of the c r y o s t a t the l i n e i s s i l v e r s o l d e r e d to a 0.71 mm O.D. and 0.15 mm w a l l c a p i l l a r y , which continues down to the vacuum can of the d i l u t i o n r e f r i g e r a t o r . There the l i n e becomes a 0.56 mm O.D. and 0.13 mm w a l l c a p i l l a r y tube. The t h i n c a p i l l a r y tube i s used to decrease the amount of 3He r e q u i r e d to p r e s s u r i z e the system, and ensure that the s o l i d 3He plug i n the tube i s f i x e d to a given p o i n t along the tube. Although i t r e q u i r e s s e v e r a l days to pump down the system to a pressure l e s s than 10 mT, the impedance of the system i s small enough so that the time constant f o r flow when the pr e s s u r e i s of the order of 10 bars i s a few seconds. A Sensotec Super TJE, 0-600 p s i a b s o l u t e pressure transducer with 0.05% accuracy, r e p e a t i b i l i t y , and d r i f t i s used to c a l i b r a t e the 3He c e l l . I t i s connected to the pressure network r i g h t a f t e r the l a s t v a l v e i n the system l e a d i n g to the c e l l . The power supply f o r the transducer i s a 10.000 v o l t s 96 supply with a d r i f t of l e s s than 10 ppm. -The output of transducer i s a 0-30 mV s i g n a l , which must be measured with a DVM with an input impedance of not l e s s than 1 0 1 o V / V . The sensor was c a l i b r a t e d with a dead weight t e s t e r by the manufacture. A l e a s t square f i t of a q u a d r a t i c to the c a l i b r a t i o n s u p p l i e d by the manufacture g i v e s (48) P- o.\o7C + \.zcv-fco v + a . f t - K , t « > x u > ~ % a , where the pr e s s u r e i s i n bars and the transducer reading i n m i l l i v o l t s . When using the gauge, the zero p r e s s u r e reading of the transducer must be s u b t r a c t e d from the v o l t a g e measurements to get the t r u e v o l t a g e output. The t h e r m i s t o r gauge measures the p r e s s u r e , by measuring the c o n d u c t i v i t y of the gas. A rather simple d e v i c e to c o n s t r u c t , i t s c i r c u i t diagram i s shown i n F i g u r e 6. The gauge was c a l i b r a t e d a g a i n s t a thermocouple gauge using a i r as the gas. The c a l i b r a t i o n graph i s shown i n F i g u r e 7. The gauge was designed because no commerical gauge c o u l d be found which can measure microbars and a l s o withstand o v e r p r e s s u r i n g t o 35 bars. 97 12 V 12V F i g u r e 6. One o f t h e t h e r m i s t o r s i s exposed t o t h e p r e s s u r e i s t o be measured . The second t h e r m i s t o r f o r t h e amb ien t t e m p e r a t u r e . gas whose compensates 98 E it 0 40 80 120 160 200 220 240 Thermistor Output (mV) F i g u r e 7. The Thermistor gauge was c a l i b r a t e d a g a i n s t a Thermocouple gauge using a i r as the gas. 99 3.5 Capacitance Bridge A c a p a c i t a n c e b r i d g e capable of measuring 10 pF with a r e s o l u t i o n of 10" 5 pF with a short term d r i f t of l e s s than 10" 5 pF per day i s d e s c r i b e d . The l e a d c a p a c i t a n c e of the c o a x i a l c a b l e s connecting the system together i s about 300 p f , so that s p e c i a l techniques must be used. In p a r t i c u l a r , the three wire c i r c u i t d e s c r i b e d by H i l l h o u s e and K l i n e (1960) i s f a i r l y i n s e n s i t i v e to the changes in the l e a d c a p a c i t a n c e . A s i m p l i f i e d schematic of the c i r c u i t used i s shown i n F i g u r e 8 and the p r a c t i c a l l a y o u t i s shown in F i g u r e 9. The c i r c u i t can be used to measure small c a p a c i t a n c e s i n the presence of l a r g e l e a d c a p a c i t a n c e s because i t makes use of- the p r o p e r t i e s of a three t e r m i n a l c a p a c i t o r . The three t e r m i n a l r e p r e s e n t a t i o n of a c a p a c i t o r e n c l o s e d with i n a metal s h i e l d i s shown i n f i g u r e 10. There are many bridge c i r c u i t s which can be used to measure the d i r e c t c a p a c i t a n c e of a 3-terminal c a p a c i t o r , but most r e q u i r e separate b r i d g e s to balance the ground c a p a c i t a n c e i n the c i r c u i t . The r a t i o transformer c a p a c i t a n c e bridge c i r c u i t i n F i g . 8 i s one c i r c u i t that does not r e q u i r e a secondary b r i d g e . Consider the c i r c u i t i n F i g u r e 11, where the ground c a p a c i t a n c e s The output impedance are shown. of the r a t i o transformer and the s i g n a l generator as seen by the b r i d g e c i r c u i t i s 5 *A-+ 5 0U I . S 2 , where S i s the d i a l s e t t i n g of the r a t i o 100 F i g u r e 8. Schematic of the r a t i o transformer c i r c u i t . When balanced, C X = 10S C, where s i s the d i a l r eading of the r a t i o t r a n s f o r m e r . Signal Generator Figures* Capacitance bridge for the MCT. Isolation Transformer Lock-in Detector Referenc 1 In o. D Ratio Transformer Reference Capacitor *>< DP DT Switch IK -» Preamp T-Uxxu) 1:1 r ZZ 3HeCell 102 B m on the l e f t - h a n d s i d e i s a c a p a c i t o r enclosed i n a s h K l S . On the r i g h t ' h a n d s i d e i s i t s schematic r e p r e s e n t a t . c n as a three t e r m i n a l c a p a c i t o r . 103 t r a n s f o r m e r . The output impedance of the b r i d g e c i r c u i t i s of the order of 1 Megohm, so that the ground c a p c a i t a n c e , C^, i s e f f e c t i v e l y shorted by the low impedance of the r a t i o t r a n s f o r m e r . The extent to which the c a p a c i t o r , , a f f e c t s the b r i d g e balance i s found by c o n s i d e r i n g the c i r c u i t i n f i g u r e 12(a). The coupled i n d u c t o r c i r c u i t i s e q u i v a l e n t to the c i r c u i t i n f i g u r e 12(b). The r a t i o of the v o l t a g e , E, , a c r o s s C^, to the v o l t a g e a p p l i e d , E, to the r a t i o transformer i s given by ( 4 9 ) E " aM-a.tU, - u,* U .L* -n*K<--v *<V> where u>= 2nf. L , and L x are r e l a t e d to the c o u p l i n g inductance M by (50) H •- v J T T L I where k i s a constant depending on the degree of the c o u p l i n g between L,and L x . I f the inductances are coupled together so that there are no reactance l o s s e s , as i n an i d e a l r a t i o t r a n s f o r m e r , k = 1, then r a t i o E t / E i s independent of the ground c a p a c i t a n c e s . In a p r a c t i c a l r a t i o transformer, the c o u p l i n g i s very h i g h ( K l i n e and H i l l h o u s e ) . An estimate of the e r r o r i n t r o d u c e d by the ground c a p a c i t a n c e i s found by s u b s i t u t i n g r e p r e s e n t a t i v e v a l u e s f o r the r a t i o transformer used i n the 104 F i g u r e 11. The r a t i o transformer bridge with three t e r m i n a l capac i t o r s . 105 (a) (b) F i g u r e 12. Schematic showing the ground c a p a c i t a n c e ( a ) and the e q u i v a l e n t c i r c u i t of the combination of r a t i o transformer and the ground c a p a c i t a n c e ( b ) . 106 b r i d g e : L| = L^= 500 H, k = .999, and the ground impedance i s 100 pF. E, /E i s found to d i f f e r from 1/2 by 10 ppm. Thus the e f f e c t of the ground c a p a c i t a n c e and k = 1, c o u l d be troublesome i f an a c c u r a t e measurement of a unknown ca p a c i t a n c e were d e s i r e d , but because the MCT r e q u i r e s only that the c a p a c i t a n c e be measurable to a high r e s o l u t i o n , t h i s i s of no r e a l concern. In f a c t , because only a p r e c i s e measurement i n an a r b i t r a r y c a p a c i t a n c e u n i t i s r e q u i r e d , the design of the bridge i s much simpler than that of a b r i d g e what must a c c u r a t e l y measure the c a p a c i t a n c e . The s t r a y c a p a c i t a n c e s and inductances i n the guard c i r c u i t s h i f t s the b a l a n c i n g c o n d i t i o n of the b r i d g e , and thus s h i f t s the measured c a p a c i t a n c e value, but t h i s s h i f t i s of no concern to a MCT, as long as the s e n s i t i v i t y i s not reduced by them, and the s h i f t s are r e p r o d u c i b l e . I t i s v i t a l that the leads to and from the r e f e r e n c e c a p a c i t o r and the 3He c e l l be w e l l guarded so that the c a p a c i t o r s approximate three t e r m i n a l c a p a c i t o r s . The leads between the d e t e c t o r and the c a p a c i t o r s are the most s e n s i t i v e to p i c k up and must be very c a r e f u l l y s h i e l d e d . The b r i d g e i s i n s e n s i t i v i t y to d r i f t s i n the ground c a p a c i t a n c e , but i s s e n s i t i v e to sudden f l u c t u a t i o n s i n the ground c a p a c i t a n c e , which may i n t r o d u c e s i g n a l s near the r e f e r e n c e frequency d r i v i n g the b r i d g e . The b r i d g e used in the MCT c i r c u i t i s d r i v e n by an HP 3330A S y n t h e s i z e r , which has an 50 oi. output impedance, connected to a 107 1:1 i s o l a t i o n transformer i n a sheet i r o n box, which i s a t t a c h e d to the case of the r a t i o t r a n s f o r m e r . The transformer ensures t h a t there i s only one ground i n the b r i d g e c i r c u i t . There are two secondary c o i l s on the i s o l a t i o n transformer, the s i g n a l from one i s used as the r e f e r e n c e s i g n a l f o r the HR8 PAR L o c k - i n a m p l i f i e r . The second winding s u p p l i e s the e x c i t a t i o n v o l t a g e to an E l e c t r o S c i e n t i f i c I n d u s t r i e s DT72A seven Decade R a t i o Transformer, which at a frequency of 1 kHz i s l i n e a r to 1 ppm. Most of the connecting c a b l e s i n the c i r c u i t are RG-172U c o a x i a l c a b l e s . However, the l e a d from the top of the vacuum can down to the 3He c e l l i s a Cooner AS636-1SSF coax, and the l e a d from the vacuum can to the tope of the c r y o s t a t i s a s e m i - r i g i d s t a i n l e s s s t e e l coax. I t has been found that the s e m i - r i g i d coax i s very n o i s y when c o o l e d to l i q u i d n i t r o g e n temperature ( i . e . v i b r a t i o n s induce e l e c t r i c a l s i g n a l s ) , whereas the RG-172U c a b l e i s almost u n a f f e c t e d by c o o l i n g . The drawback of the RG-172U i s that i t i s p o o r l y s h i e l d e d and, thus r e q u i r e s a d d i t i o n a l b r a i d e d s h i e l d s to reduce the 60 c y c l e pickup. The c i r c u i t f o r b a l a n c i n g the r e s i s t i v e component and the b a t t e r y powered p r e a m p l i f e r are housed in an i r o n box to s h i e l d the system from 60 Hz magnetic n o i s e . The p r e a m p l i f i e r uses a S i l i c o n i x U402 JFET, which has a very low gate leakage c u r r e n t and a reasonably low input noise v o l t a g e . The schematic diagram of the p r e a m p l i f i e r i s shown i n F i g u r e 13. The low leakage i s r e q u i r e d because at low 108 F i g u r e 13. The h i g h input impedance of the p r e a m p l i f i e r reduces the Johnson noi s e c u r r e n t seen by the b r i d g e . 109 f r e q u e n c i e s the c u r r e n t noise i s dominated by the shot n o i s e of the gate c u r r e n t . The source impedance i s e s s e n t i a l l y p u r e l y r e a c t i v e and of order 0.5 Meg. T h i s r e q u i r e s a very high value f o r the b i a s r e s i s t o r ( c h o s e n to be 800 MjO i f one i s to a v o i d i t s Johnson n o i s e . A NE5534 low noise o p e r a t i o n a l a m p l i f i e r which has a maximun input n o i s e v o l t a g e of 4.5nv/JnT lat 1 kHz i s used to a m p l i f y the s i g n a l by 228, before the s i g n a l i s fed i n t o the l o c k - i n d e t e c t o r . The p r e a m p l i f i e r i s b a t t e r y powered to decrease the number of ground loops i n the system. An 1000 */l 10 turn potentimeter can be switched i n s e r i e s with e i t h e r c a p a c i t o r , so that the quadrature s i g n a l can be balanced. The r e f e r e n c e c a p a c i t o r c o n s i s t s of a room temperature brass vacuum can which i s connected by three 3/16" O.D. t h i c k w a l l s t a i n l e s s s t e e l tubes to a lower brass vacuum can. The top can has a t t a c h e d to i t two h e r m e t i c a l l y s e a l e d BNC panel mounts and a 1.5 l b r e l i e f v a l v e . There are two separate compartments i n the top can so that the leads down to and from the c a p a c i t o r at the bottom are e l e c t r i c a l independent. The whole assembly i s evacuated. I t was found that the bubbling of l i q u i d n i t r o g e n i n c o n t a c t with the coax can cause v o l t a g e p u l s e s of s e v e r a l m i c r o v o l t s . The three s.s. tubes are s i l v e r s o l d e r e d to the top and bottom cans. Two of the tubes each c o n t a i n a RG-172U c a b l e . The c a b l e s are pressed f i r m l y a g a i n s t the inner w a l l s of the the tubes by a s t r i n g which i s wrapped around the c a b l e s . 110 The immoblized c a b l e s are then f a i r l y i n s e n s i t i v e to microphonic p i c k u p . A l l c a b l e s w i t h i n the r e f e r e n c e assembly are f i r m l y clamped to the body of the assembly or to s t a n d o f f s i n order to prevent v i b r a t i o n s causing motion in the c a b l e s and thus i n d u c i n g n o i s e . The t h i r d tube i s used as the pumping l i n e t o the bottom can. The bottom can c o n t a i n s a 31.90 pf s i l v e r e d mica c a p a c i t o r . When i t i s c o o l e d down to L N a temperature, i t s c a p a c i t a n c e decreases by 0.10 p f . Because the r e f e r e n c e c a p a c i t o r i s c o o l e d by an open LN^ bath, a s t a b i l i t y i n the c a p a c i t a n c e of 1 ppm, r e q u i r e s t h a t the atmospheric p r e s s u r e must not change by more than 5 t o r r d u r i n g the run. In f a i r l y s t a b l e weather, there i s a reasonable p r o b a b i l i t y t h a t the pressure w i l l not change by more than 5 t o r r over the course of 1 to 2 days. The change i n the d i e l e c t r i c constant of the i n s u l a t o r i n the coax, due to the change i n temperature as the l i q u i d n i t r o g e n l e v e l i n the bath f a l l s , i s not seen by the c a p a c i t a n c e b r i d g e because the ground c a p a c i t a n c e changes slowly and by f a i r l y s m a l l amounts. 111 3.6 Choice of C o a x i a l Cable f o r Low Temperature I t i s a f a c t , perhaps not g e n e r a l l y known, that e l e c t r i c a l n o i s e i s generated i n a c o a x i a l c a b l e when i t i s struck s h a r p l y . During the course of t e s t i n g the r e f e r e n c e c a p a c i t o r assembly, i t has a l s o been found that most c o a x i a l c a b l e s emit l a r g e amounts of e l e c t r i c a l n o i s e while they are being c o o l e d (to l i q u i d n i t r o g e n temperature, f o r example). In some cases, the c a b l e s a l s o became very microphonic. A b r i e f study of the noise c h a r a c t e r i s t i c s of c o a x i a l c a b l e s that were a v a i l a b l e i n the l a b was conducted. The s u r p r i s i n g r e s u l t of the study was that the cheapest coax was the q u i e t e s t and l e a s t microphonic of a l l the coax that were t e s t e d . The c a b l e s that were examined are the RG 172, RG 142, RG 58, s e m i - r i g i d coax, and an "homemade" a i r d i e l e c t r i c coax. The "homemade" coax c o n s i s t e d of a len g t h of 3/8" t h i c k w a l l s t a i n l e s s s t e e l tube with a s h o r t e r l e n g t h of 1/4" t h i n w a l l s t a i n l e s s s t e e l tube down the center of the l a r g e r tube. The inner tube was ce n t e r e d with four r i n g s of expended p l o y s t y r e n e i n s u l a t i o n which were epoxied to the inner tube. One end of the 3/8" tube had a copper end cap s o l d e r e d on. The other end had attac h e d to i t a BNC connector such that the cente r t e r m i n a l c o u l d be s o l d e r e d to the inner tube. A f t e r a rubber hose had been s l i p p e d over the BNC connector and the outer tube, a hypodermic needle was p i e c e d through the hose. The a i r i n s i d e 1 12 the tube was d i s p l a c e d with helium that was f o r c e d i n through the needle. The n o i s e i n d i f f e r e n t c a b l e s was s t u d i e d by a t t a c h i n g a l e n g t h of c a b l e to the input of the p r e a m p l i f i e r used i n the ca p a c i t a n c e bridge and then observing the output on a 5440 T e k t r o n i c scope, which had i t f i l t e r s set to permit o n l y D.C. to 10 KHz through. The op p o s i t e end of the coax was s h i e l d e d to prevent the inner conductor from p i c k i n g up the 60 Hz s i g n a l . Each p a r t i c u l a r c a b l e was i n t u r n tapped l i g h t l y with a p l a s t i c pen, w h i s t l e d a t , and f i n a l l y , t a l k e d to , while i t was at room temperature and then again at l i q u i d n i t r o g e n temperature. The r e s u l t s are summarized i n Table I I . The induced n o i s e was measured by observing the maximum peak to peak s i g n a l produced i n the c a b l e . Our c o n c l u s i o n s are as f o l l o w s . Coax c a b l e at low temperature should not be exposed to b o i l i n g l i q u i d s , because e l e c t r i c a l s p i k e s as l a r g e as 100 mV can be induced i n the c a b l e . Thick or r i g i d c a b l e s i n gen e r a l are n o i s i e r than p l i a b l e c a b l e s at low temperatures. T h i s may be due to the f a c t that the wires are more t i g h t l y wrapped i n the t h i c k e r c a b l e and t h e r e f o r e under more s t r e s s than the wires i n the more f l e x i b l e c a b l e s . A l s o , the b e t t e r q u a l i t y c a b l e s have t e f l o n as the d i e l e c t r i c , whereas RG 172 has a s o f t e r p l o y e t h y l e n e d i e l e c t r i c . When p i c k i n g a cable f o r low noise a p p l i c a t i o n s i n the audio frequency range, one should choose a f l e x i b l e c a b l e with a 113 Table II Peak to peak noise v o l t a g e i n coax c a b l e s at d i f f e r e n t temperatures. The ambient room temperature val u e s are a measure of the 60 Hz p i c k up of the c a b l e . The ambient noise at low temperature i s a measure of the ambient sound l e v e l i n the room. Coax Room Temperature Low Temperature ( 7 7 K) Ambient OxV) Tapping (mV) Microphonic (MV) Ambient (<tV) Tapping (mV) Microphonic UV) "homemade air core 18 0.4 7 6 0.13 13 RG 142 nil 9 n i l 8 6 4 44 RG 58 9 4 nil 87 0.4 nil Sem- rigid nil 0.2 nil 13 13 13 RG 172 80 0 . 3 nil 80 0.2 nil RG 172 shielded nil 0.9 nil nil 0.9 nil 1 1 4 n o n - t e f l o n d i e l e c t r i c ; one can reduce the e l e c t r i c p i c k up due to the poor outer b r a i d by running the c a b l e i n s i d e a conducting tube, f o r example, t h i n w a l l e d ..stainless s t e e l tube. 1 1 5 IV. Thermometry with the M e l t i n q Curve Thermometer 4.1 I n t r o d u c t i o n T h i s chapter d e a l s with the f i r s t use of the 3He m e l t i n g curve thermometer. As a t e s t i t was compared with a germanium r e s i s t o r which had p r e v i o u s l y been c a l i b r a t e d a g a i n s t an SRM 768 f i x e d p o i n t d e v i c e . The MCT was not d i r e c t l y compared with the SRM 768, because d u r i n g the same run an experiment i n v o l v i n g h i g h magnetic f i e l d s was a l s o conducted. I t was f e l t that exposing the SRM 768 to a 40 k i l o g a u s s magnetic f i e l d might a l t e r the c a l i b r a t i o n of the d e v i c e . A b r i e f d e s c r i p t i o n of the procedure f o r o p e r a t i o n the MCT i s given the next s e c t i o n , a more complete d e s c r i p t i o n can be found i n appendix A. The s e c t i o n f o l l o w i n g compares the temperature as measured by the MCT and the germanium d e v i c e , and d i s c u s s e s the d i s c r e p a n c y i n the measured temperatures. 1 16 4.2 Operating the MCT Before the d i l u t i o n r e f r i g e r a t o r was c o o l e d down, the 3He was c i r c u l a t e d through the c o l d t r a p to remove any i m p u r i t i e s i n the gas. The 3He was then returned to the r e s e r v o i r . The c e l l was pumped down with the c h a r c o a l pump so that there was only a few m i l l i t o r r of the 3He l e f t i n the c e l l . The 3He i n the c o l d t r a p was removed by pumping on the volume while the c o l d t r a p was s t i l l at l i q u i d n i t r o g e n temperature. The c o l d t r a p was then heated to 300 °C and pumped on with a d i f f u s i o n pump f o r one hour. An a n a l y s i s of the gas r e l e a s e d by the molecular s i e v e i n d i c a t e d t h at most of the contamination was due to water, with t r a c e s of argon and n i t r o g e n . The presence of argon gas was expected because the system had been pr e s s u r e t e s t e d with i t . The l a r g e c o n c e n t r a t i o n of water was a t t r i b u t e d to the outga s s i n g of the l a r g e 3He r e s e r v o i r . During the c o o l down of the r e f r i g e r a t o r , the c a p a c i t a n c e of the 3He tranducer was monitored with a Boonton c a p a c i t a n c e b r i d g e . At room temperature the c a p a c i t a n c e of the c e l l was 3.15 p f . When c o o l e d to l i q u i d n i t r o g e n temperature, the ca p a c i t a n c e decreased to 2.622 p f , and at l i q u i d helium temperature the c a p a c i t a n c e of the c e l l was 2.574 p f . At t h i s p o i n t the Sensotec transducer i s measuring e s s e n t i a l l y zero p r e s s u r e and the corresp o n d i n g v o l t a g e output of the transducer i s t h e r e f o r e the systematic zero s h i f t , which must be taken i n t o 117 account when using the c a l i b r a t i o n t a b l e s u p p l i e d by the manufacturer. When the r e f r i g e r a t o r was f u n c t i o n i n g and m a i n t a i n i n g a temperature of about 1 K, 3He was compressed by the c h a r c o a l pump and slowly l e t i n t o the c e l l through the c o l d t r a p at a ra t e of O.OlmV/sec on the Sensotec (about 0.2 p s i per s e c ) . To de t e c t the presence of plugs i n the c a p i l l a r i e s , the cap a c i t a n c e of the c e l l was c o n t i n u a l l y monitored with the Boonton during t h i s p r o c e s s . A f t e r a pressure of 28 bars was reached, the transformer bridge c a p a c i t a n c e was used i n s t e a d of the Boonton meter. The bridge was d r i v e n at 980 Hz with a 15 dBm s i g n a l . About f i v e d i g i t s c o u l d be u s e f u l l y read from the br i d g e , because the f l u c t u a t i o n s i n the room temperature a f f e c t e d the gas pressure and caused f l u c t u a t i o n s of 5 ppm per minute.. The pressure i n the c e l l was i n c r e a s e d to 34 bars and then decreased to 28 bars. The c e l l was e x e r c i s e d 5 times, d u r i n g each c y c l e the c a p a c i t a n c e of the c e l l was monitored as a f u n c t i o n of the p r e s s u r e . Between the t h i r d and f i f t h c y c l e a h y s t e a s i s of 0.1% was n o t i c e d i n the c a p a c i t a n c e at a given p r e s s u r e . The time r e q u i r e d f o r a pressure change to be t r a n s m i t t e d between the c e l l and the transducer was very s h o r t . However, i f l a r g e pressure changes ( 2 bar) were made, the r e l a x a t i o n time of the system at 30 bars pressure was about a minute. Between the f i f t h and s i x t h run there was no measurable h y s t e r e s i s , so c e l l was c a l i b r a t e d every 0.4 bars between 118 27.5 bars and 34.2 bars. In that r e g i o n the equation (5!) P = Vl.WtfM- 5-7 6*5-5 - S *, where P i s the pressure i n bars and S i s the d i a l reading of the r a t i o t r a n s f o r m e r , d e s c r i b e s the c a l i b r a t i o n p o i n t s with an e r r o r l e s s than the u n c e r t a i n t y i n the Sensotec c a l i b r a t i o n . Because the d i l u t i o n r e f r i g e r a t o r was running at a very high temperature, the temperature of each s e c t i o n was c a r e f u l l y monitored to ensure that the temperature at some p a r t of the c a p i l l a r y d i d not drop below the f r e e z i n g temperature d u r i n g the pressure c a l i b r a t i o n . The 3He pre s s u r e was decreased to 33.9 bars and a 3He plug formed i n the c a p i l l a r y near the s t i l l , by i n c r e a s i n g the pumping speed of the c i r c u l a t i n g system of the 3He-"He d i l u t i o n r e f r i g e r a t o r and so reducing the temperature at the s t i l l . The 3He f r e e z e s at a temperature of 0.75 K ( G r i l l y , 1871) at a pressure of 34 bars. When the s t i l l had c o o l e d below 0.7 K, the pressure on the plu g was i n c r e a s e d to 35 bars to promote the growth of the s o l i d p l u g i n t o r e g i o n s of higher temperature. The formation of the plug was v e r i f i e d by the f a c t that the 3He c e l l showed no i n c r e a s e i n pre s s u r e when the e x t r a pressure was s u p p l i e d . The best s t a r t i n g d e n s i t y corresponds to pr e s s u r e s between 32 and 34 b a r s . The p a r t i c u l a r c h o i c e of the s t a r t i n g p r essure 119 depends on the temperature range i n which the MCT i s to be used. To reach the lower temperatures, a higher s t a r t i n g p r e s s u r e i s r e q u i r e d ( G r e y w a l l and Busch,1982a). The l o c a t i o n of the minimum pr e s s u r e p o i n t i s independent of the i n i t i a l s t a r t i n g d e n s i t y f o r d e n s i t i e s c o r r e s p o n d i n g to p r e s s u r e s between 30.4 and 38 b a r s ( C o r r u c i n n i et al., 1978). The d i l u t i o n r e f r i g e r a t o r was then c o o l e d to about 0.55 K, where the temperature of the mixing chamber was r e g u l a t e d . The c a p a c i t a n c e b r i d g e was n u l l e d and the r e s i s t a n c e of v a r i o u s carbon and germanium r e s i s t o r s was measured. A second p o i n t was taken at 0.5 K. I t was then d e c i d e d to take the temperature measurements while the r e f r i g e r a t o r was warming up because i t proved to be very d i f f i c u l t to maintain r e g u l a t i o n at the higher temperatures while the r e f r i g e r a t o r was on the c o o l down c y c l e . The passage of the temperature through the minumum p o i n t on the m e l t i n g curve was very n o t i c a b l e . I t was p o s s i b l e , f o r example,to n u l l the brid g e to the 6th d i g i t while the r e f r i g e r a t o r was f r e e l y c o o l i n g . The r e f r i g e r a t o r would not c o o l down below 100 K, due to d i f f i c u l t i e s a s s o c i a t e d with the other experiment a t t a c h e d to the c r y o s t a t . There appeared to be a thermal short between the apparatus on the c o l d f i n g e r ( a ttached to the mixing chamber) and the thermal s h i e l d , but because the thermometers being compared were a l l s i t u a t e d on the massive mixing chamber and the temperature was about 0.100 K, the r e s u l t i n g thermal g r a d i e n t s 120 are estimated to be very s m a l l . From 0.1 K to the minimum pressure p o i n t , s i x readings were taken. About 15 minutes was r e q u i r e d to r e g u l a t e at a p a r t i c u l a r temperature. Because the s e n s i t i v i t y of the temperature r e g u l a t o r sensor i n c r e a s e s at low temperature, the temperature was r e g u l a t e d to 0.05 mK around 300 mK, and to b e t t e r than a few hundredth of m i l l i K e l v i n at the lower temperature. The thermal time constant of the thermometer at 300 mK was t e s t e d by measuring the pressure immediately a f t e r the temperature r e g u l a t i o n was achieved at a f i x e d p o i n t , and then 10 minutes l a t e r . The two pressure measurements agreed to b e t t e r than 4 ppm, i n d i c a t i n g a thermal time constant of l e s s than a minute f o r the system. The 4 ppm i s a measure of the t o t a l d r i f t i n the system due to the d r i f t i n the r e g u l a t e d temperature, the d r i f t i n the c a p a c i t a n c e bridge and the r e l a x a t i o n of the s t r a i n gauge. U n f o r t u n a t e l y the thermometer was never c o o l e d below 100 mK, so the thermal time constant at very low temperatures was not measured. A second measurement of the minimum p o i n t was taken as the mixing chamber was slowly warmed up. The two measurements d i f f e r e d by only one i n the s i x t h d i g i t , t h i s corresponds to a 0.03 mbar u n c e r t a i n t y i n the p r e c i s i o n to which the minimum was measured. The accuracy of t h i s measurement, however, « was only 0.05%, because the a b s o l u t e c e r t a i n t y i n the pressure 121 c a l i b r a t i o n of the 3He c e l l i s on l y 0.05%. Immediately a f t e r p a s s i n g through the minimum pr e s s u r e p o i n t the s o l i d 3He plug s l i p p e d and the c e l l was f i l l e d with s o l i d 3He. The plu g s l i p p e d because the d i f f e r e n c e i n pressure between the 3He c e l l and the Sensotec transducer became too great f o r the plug to h o l d : a f t e r the plug had been o r i g i n a l l y formed, the pressure had been i n c r e a s e d to 36 bars, but as the l i q u i d helium l e v e l i n the dewar f e l l , the pressure f u r t h e r i n c r e a s e d . The d i f f i c u l t y can be remedied by d e c r e a s i n g the pressure on the plug before attempting to warm up through the minimum. 122 4.3 Comparison of the MCT with a Germanium R e s i s t o r The MCT data c o n s i s t s of s e r i e s of d i a l readings from the r a t i o t r a n sformer. These numbers are converted to a pressure by u s i n g equation (50). The minumum pre s s u r e measured by t h i s MCT i s 29.3067 bar 0.05%. The a b s o l u t e u n c e r t a i n t y of 0.05% i n the measurement i s due to the Sensotec transducer and corresponds to 0.015 bar at t h i s p r e s s u r e . The e s t a b l i s h e d value of the minimum i s 29.313 0.003 b a r s ( G r e y w a l l and Busch, 1982a). The l a r g e number of f i g u r e s i n the MCT measurements was kept because the a b s o l u t e u n c e r t a i n t y i n the pressure measurements of the MCT can be decreased by s h i f t i n g the p r e s s u r e s c a l e so that the measured p r e s s u r e agrees with the accepted v a l u e . T h i s amounts to assuming that the Sensotec s u f f e r s only from a systematic zero o f f s e t , which can be c o r r e c t e d . Because the Sensotec was c a l i b r a t e d with a dead weight t e s t e r a c c u r a t e to 0.02%, the s i z e of pressure u n i t s are probably good to 0.02%. Furthermore, because the diaphragm transducer gauge i s u l i t i l i z e d over only 10% of i t s f u l l s c a l e c a p a c i t y , the gauge probably has a l i n e a r response curve over t h i s r e g i o n . A s l i g h t s h i f t i n the zero p r e s s u r e p o i n t , t h e r e f o r e , decreases the u n c e r t a i n t y a s s o c i a t e d with the p r e s s u r e c a l i b r a t i o n . The r e p r o d u c i b i l i t y of the s t r a i n gauge and the s t a b i l i t y of the c a p a c i t a n c e bridge over a p e r i o d of a day may be 123 q u e s t i o n e d . However, given that the minumum pressure measured d u r i n g the c o o l down c y c l e was w i t h i n 3 ppm of the minimum pres s u r e measured d u r i n g the warming up c y c l e , the d r i f t i n the system i s probably very small( on the order of 1 ppm per ho u r ) . T h e r e f o r e the d r i f t and r e p r o d u c i b i t i y of the pressure measurement may be ignored to the p r e c i s i o n of these measurements. Two thermometers i n a d d i t i o n to the MCT were a t t a c h e d to the mixing chamber. These were a S c i e n t i f i c Instruments germanium r e s i s t o r ( S I ) , Model 5-He3A, and a 400 -n. M a t s u s h i t a carbon r e s i s t o r (C). The carbon r e s i s t o r had p r e v i o u s l y been c a l i b r a t e d a g a i n s t the SI. The SI had been c a l i b r a t e d with the SRM 768 and, a l s o , with a Lakeshore germanium r e s i s t o r , which i n tur n had been c a l i b r a t e d with a CMN s u s c e p t i b i l i t y thermometer. Table III l i s t s the r e s u l t s of the c a l i b r a t i o n run. The m e l t i n g curve values given i n Table II of G r i l l y ( l 9 7 l ) was used as the c a l i b r a t i o n t a b l e f o r the MCT at temperatures above 0.318 K. A down s h i f t of 2 mK was made i n the temperatures i n t e r p o l a t e d from the graph drawn using the va l u e s given by G r i l l y to account f o r the 2 mK d e v i a t i o n of the T58 s c a l e from the thermodynamic s c a l e ( s e e sec 4.4). The temperatures below 0.318 K were found by using the valu e s given by equation (42). Where the carbon r e s i s t a n c e thermometer, C, i s not decoupled from the system, the SI and the C are c o n s i s t e n t i n the temperature reading, but both measure temperatures which are Table HI Temperatures measured by different thermometers, s Calculated Pressure (Bars) Corrected Pressure (Bars) MCT Temp. (mK) S 1 C Resistance (XI) Temp. (mK) Resistance (Kft) Temp. (mK) .41538 31.3947 31.4040 589 2.22.6 589 1.845 5 89 .434474 30.8561 30.86 45 550. 2 43.7 547 1.948 547 .467312 293067 29,3165 318 3.177 314 .434474 30.6315 30.6414 135.5 3.965 K 1 36 9.034 128 .441442 30.3518 30.3615 153.7 2.807 K 153 7.485 150 .446643 30.1425 30.1523 169.4 2.1 74 K 170 6.537 165 ,455294 29.4926 29.5024 202 I.4I7K 200 5.210 199 .466583 29.3363 29.3456 288 678.5 283 3.495 289 125 lower than the MCT measurements. Because the carbon r e s i s t o r i s l o c a t e d with a centimeter of the MCT, the p o s s i b i l i t y of thermal g r a d i e n t s i n the mixing chamber can be r u l e d out. The SI readings d e v i a t e s from the MCT temperatures by as much as 5 mK. I t was noted that the SI has s h i f t e d i t s temperature curve at the higher end, where the agreement i s the worst. The f i x e d c a l i b r a t i o n p o i n t d e v i c e c a l i b r a t i o n of the SI i n January 1983 show t h a t , although p r e v i o u s c a l i b r a t i o n curves of the SI s t i l l agrees with the SRM 768 temperature at 99.02 mK and 162.6 mK, the SI curve has s h i f t e d by 2 mK to a lower temperature at the 205.7 mK p o i n t . I f the SI c a l i b r a t i o n curve i s s h i f t e d to agree with the f i x e d p o i n t temperature, then the MCT and the SI agree w i t h i n a m i l l i K e l v i n . I t appears as i f the SI needs to be r e c a l i b r a t e d a g a i n s t the SRM 768, the Lakeshore, and the MCT. The only r e a l t e s t f o r the MCT i s to compare i t with the SRM 768 and p o s s i b l y the Lakeshore germanium r e s i s t o r . A s m a l l t e s t of the accuracy of the MCT was to c a l i b r a t e the C with the MCT and then the i n t e r p o l a t e d value of the r e s i s t a n c e at 0.318 K was compared to the taken value of C measured when the MCT was p a s s i n g through the p r e s s u r e minimum. The two v a l u e s d i f f e r d by 1%, which corresponds to a 1 mK d i f f e r e n t at that temperature. S i m i l i a r l y , i f the SI i s c a l i b r a t e d with the MCT, the f i x e d temperature at 205.7 mK from the January check i s only 0.5 mK from the new curve. Although these f a c t do not c o n f i r m the v a l i d i t y of r e l y i n g t o t a l l y on the 126 MCT, they do i n s t i l l some c o n f i d e n c e i n the MCT. As s t a t e d e a r l i e r , the only d e f i n i t i v e procedure f o r checking the MCT i s to compare i t s temperature s c a l e with the SRM 768, the Lakeshore germanium, and h o p e f u l l y a n u c l e a r o r i e n t a t i o n thermometer at the very low temperatures. 127 4 . 4 C o n c l u s i o n A 3 He m e l t i n g c u r v e the rmomete r sys tem has been b u i l t and t e s t e d . The p r e s s u r e r e s o l u t i o n o f t h i s p a r t i c u l a r MCT i s 0 .14 mbars . A t t e m p e r a t u r e s l e s s t h a n 100 mK, t h i s c o r r e s p o n d s t o an u n c e r t a i n t y o f 7 yu-K. At t h e h i g h e r t e m p e r a t u r e of 300 mK, t h e p r e c i s i o n i s o n l y 0 .1 mK. W i t h b e t t e r t e m p e r a t u r e r e g u l a t o r s and more c a l i b r a t i o n r u n s , i t may be p o s s i b l e t o d e c r e a s e t h e e s t i m a t e o f t h e u n c e r t a i n t y i n t h e p r e s s u r e measurement . The a b s o l u t e a c c a r a c y o f t h e p r e s s u r e measurement i s d e t e r m i n e d by t h e room t e m p e r a t u r e p r e s s u r e s t a n d a r d ( w h i c h i s 0 . 0 5 % ) . W i t h a u n c e r t a i n t y o f 0.05% i n t h e p r e s s u r e measurement , t h e MCT can d e t e r m i n e t h e t e m p e r a t u r e a c c u r a t e l y t o 0,5 mK a t 50 mK and 1 mK a t 260 mK. I n f a c t , t h e a c c u r a c y i n t h e p r e s s u r e i s somewhat b e t t e r t h a n 0.05%. The s h i f t i n g o f t h e measured minimum p r e s s u r e t o agree w i t h t h e a c c e p t e d v a l u e a t t h a t p o i n t , d e c r e a s e s t h e p r e s s u r e u n c e r t a i n t y i n t h e r e g i o n where t h e h i g h e s t a c c u r a c y i s n e c e s s a r y , s i n c e near t he p r e s s u r e minimum, t h e t e m p e r a t u r e measured by t h e MCT i s most s e n s i t i v e t o t h e p r e s s u r e . As l o n g as t h e p r e s s u r e s t a n d a r d i s r e a s o n a b l y l i n e a r a r o u n d 29 b a r s , t h e s h i f t i n t h e p r e s s u r e s c a l e w i l l d e c r e a s e t h e u n c e r t a i n t y i n t h e p r e s s u r e measurement below 0.05%. I f t h e r e were a l a r g e d i s c r e p a n c y between the measured minimum p r e s s u r e and t h e a c c e p t e d v a l u e , t h e r e m i g h t be some d o u b t as t o t h e v a l i d i t y o f r e s c a l i n g t h e p r e s s u r e . B u t , 128 because the determined minimum pressure i s with i t s u n c e r t a i n t y of the accepted v a l u e , there i s some co n f i d e n c e i n the r e s c a l i n g . J u s t how e f f e c t i v e the s h i f t i s i n reducing the u n c e r t a i n t y i n the pr e s s u r e i s hard to determine. 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Kammer, R.B.; M u e l l e r , R.M.; Adams, E.D., J . Low Temp. Phys. 27, 319 (1977). 30. Kamper, R.A., Temperature: I t s Measurement and C o n t r o l i n Science and In d u s t r y . 4, 349 (1973). 31. Kamper, R.A. and Zimmerman, J.E., J . A p p l . Phys. 43, 132 (1971). 32. K i r k , W.P.' and Adams, E.D., Phys. Rev. L e t t . 27, 392 (1971) 33. K i t t e l , C , Thermal Physic^. (John Wiley & Sons, Inc., New York, 1969). 34. Krane, K.S.; Murdock, B.T.; S t e y e r t , W.A., Phys. Rev. L e t t . 3_p_, 321 ( 1973). 35. Krane, K.S.; S t e f f e r , R.M.; Wheeler, R.M., N u c l . Data! Tables I I , 351 (1973). 131 36. Landau, J . ; Tough, J.T.; Brubaker, N.R.; Edwards, D.O., Phys. Rev. L e t t . 23, 283 (1969). 37. Landau, J . ; Tough, J.T.; Brubaker, N.R.; Edwards, D.B., Phys. Rev. A2, 2472 (1970). 38. Legget A.J., Rev. Mod. Phys. 47, 331 (1975). 39. Marsh, J.D., Phys. Rev. L e t t . 33A, 207 (1970). 40. M e t r o l o g i a 5_, 35 ( 1 969). 41. M e t r o l o g i a 1_5, 65 ( 1975). 42. O s h e r o f f , D.D.; Richardson, R.C.; Lee, D.M., Phys. Rev. L e t t . 28, 885 (1972). 43. P r a t l , W.P.; Schermer, R.I.; S t e y e r t , W.A., J . Low Temp. Phys. 1, 469 (1969). 44. Preston-Thomas, H., M e t e r o l o g i a 1_2, 7 ( 1966). 45. R i c h a r d s , M.G.; T o f f e , P.S.; Turner, P.R., Cryogenics 13, 182 (1973). 46. Richardson, R.C., Phys i c a 90B, 47 ( 1977). 47. Roberts, T.R. and Sydoriak, S.G., Phys. Rev. 93., 1418 (1954). 48. Robichaux, J.E. and Anderson, A.C., Rev. S c i . Instrum. 40, 1512 (1969). 49. Roger, M.; Heth e r i n g t o n , J.H.; Pelrium, J.M., Rev. Mod. Phys. 55, 1 (1983). 50. Rose, M.E., Phys. Rev. £1, 610 (1953). 51. Rosenbaum, R.L.; E c k s t e i n , Y.; Landau, J . , Cryogenics 1_4_, 21 (1977). 52. Rosenberg, H.M., J . Low Temp. Phys., Low Temp. S o l i d  S t a t e Phys. (Clarendon Press, Oxford, 1965). 53. S c r i b n e r , R.A.; Panczyk, M.F.; Adams, E.D., J . Low Temp. Phys. 1, 313 (1969). 54. S i l v e r , A.H.; Zimmerman, J.E.; Kamper, R.A., Appl. Phys. L e t t , i i , 209 (1967). 55. S i t e s , J.R.; Smith, H.A.; S t e y e r t , W.A., J . Low Temp. 132 Phys. 4, 605 (1971). 56. Sorelun, M.I., Noise, S.N.O. 57. Soulen, R.J. and Marshak, H., Cryogenics 20_, 408 ( 1980). 58. Soulen, R.J., Phys i c a 109-110, 2021 (1982). 59. Stephen, M.S., Phys. Rev. 1J82, 531 ( 1969). 60. Suomi, M.; Anderson, A.C.; Holmstein, B., Ph y s i c a 38, 67 (1968). 61. Sydoriak, S.G.; Roberts, T.R.; Sherman, R.H., J . Res.  N.B.S., G.B.A., 559 (1964). 62. Templeton, J.E. and S h i r l e y , D.A., Phys. Rev. L e t t . 1_8, 240 ( 1 967). 63. Thouless, P.J., Proc. Phys. Soc.(London) 86,, 893 (1965). 64. T r i c k l e y , S.B.; K i r k , W.P.;Adams, E.D., Rev. Mod. Phys. 44, 668 (1972). 65. van D i j k , H. and Durieux, M. , Physica 2_4_, ( 1959). 66. Webb, F.S.; W i l k i n s o n , K.R.; Wil k s , J . , Proc. Royal Soc. (London) 2J_4, 546 (1952). 67. Webb, R.A.; G i f f a r d , R.P.; Wheatby, J.C., J . Low Temp. Phys. 1_3, 383 (1973). 68. Weinstock, H.; L i p s c h u l t z , F.P.; K e l l e r s , C.F.; Tedrow, P.M.; Lee, D.M. , Phys. Rev. L e t t . S>, 193 ( 1962). 69. Wheatby, J.C., Rev. Mod. Phys. 47, 415 (1975). 70. Zimmerman, G.O.; Abeshouse, D.J.; Maxwell, E.; K e l l a n d , D.R., J . Low Temp. Phys. 41_, 79 ( 1980). 133 Apppendix A 2. Operating a MCT The MCT c o n s i s t s of three separate systems: these are the pr e s s u r e system, the c a p a c i t a n c e bridge c i r c u i t , and the c e l l i t s e l f . When using t h i s p a r t i c u l a r MCT, one must always remember a few important p o i n t s . F i r s t , the p r e s s u r e i n the system must not excede 40 bars, except i n the pump where 138 bars(2000 p s i ) can be s t o r e d s a f e l y . The c e l l should never be p r e s s u r i z e d above 35 bars. Second, the t h e r m i s t o r gauge has an epoxy pr e s s u r e s e a l , which should not be heated over 80 C. T h i r d , when the MCT i s o p e r a t i n g the va l v e to the 3He r e s e r v o i r should always be l e f t open so that the r e l i e f v a l v e can f u n c t i o n p r o p e r l y i n the event of the r e f r i g e r a t o r warming up. F i n a l l y , i n a p p r o p r i a t e opening of the v a l v e s to the vacuum l i n e c o u l d r e s u l t i n a s i z a b l e amount of 3He being l o s t . Because the c a p a c i t a n c e b r i d g e has very h i g h r e s o l u t i o n , i t i s s e n s i t i v e to many noise sources. F i r s t of a l l i t i s very s e n s i t i v e to microphonics, and t h i s i s mainly a s s o c i a t e d with the p r e a m p l i f i e r and the c a b l e s down i n t o the c r y o s t a t . The noise i n t r o d u c e d by microphonics can be reduced by damping the v i b r a t i o n s i n the c a b l e s by r i g i d i l y s u p p o r t i n g them. The guarding connections on the coax must be continuous or l a r g e 60 Hz s i g n a l s w i l l be p i c k e d up by the inner conductors. The l o c k - i n d e t e c t o r does not immediately f i l t e r out a l l of the beat f r q u e n c i e s between a m u l t i p l y of the 60 Hz and the 134 r e f e r e n c e frequency of the b r i d g e , because the bandwidth of the bandpass a m p l i f i e r i s 50 Hz. Thus i t i s important that the d r i v i n g frequency of the b r i d g e be chosen to minimize the p o s s i b i l i t y of very low frequency beats which cannot be f i l t e r e d by the output RC f i l t e r s . The ground of the b r i d g e i s through the c r y o s t a t . The l o c k - i n ground i s separate from that of the bridge to decrease the number of ground loops (This i s p o s s i b l e because of the transformer output of the p r e a m p l i f i e r ) . The o p e r a t i o n of the bridge i s s t r a i g h t forward. With the l o c k - i n d e t e c t o r on the l e a s t s e n s i t i v e s c a l e and u s i n g the l a r g e s t decade knobs of r a t i o t r ansformer, the output of the l o c k - i n i s zeroed. To converge to t h e ' c o r r e c t s e t t i n g , the zero i s always undershot. Then the quadrature s i g n a l i s n u l l e d by s w i t c h i n g the d i s s i p a t i o n r e s i s t o r i n t o the a p p r o r i a t e branch and a d j u s t i n g the r e s i s t a n c e to g i v e a zero output. A f t e r v e r i f y i n g t h a t the zero s e t t i n g on the s i g n a l has not changed, one can c o n t i n u e onto the l e s s s i g n i f i c a n t f i g u r e s of the r a t i o t r a n s f o r m e r . If a d j u s t i n g the quadrature has s h i f t e d the n u l l , then the procedure has to be repeated. As the s e t t i n g gets c l o s e r to the n u l l s e t t i n g , the g a i n of the l o c k - i n i s i n c r e a s e d a p p r o p r i a t e l y For the MCT, the d i a l r eading need not be converted to a c a p a c i t a n c e , r a t h e r , the pressure response of the transducer i s r e l a t e d d i r e c t l y to the d i a l s e t t i n g . 135 Before the i n i t i a l f i l l i n g of the system with 3He, the system i s pumped with an N^ trapped d i f f u s i o n pump f o r s e v e r a l days. Then the c h a r c o a l pump and the molecular s i e v e are cle a n e d by h e a t i n g the pump and the c o l d t r a p to high temperature while s t i l l pumping on them. The c o l d t r a p i s heated to 300°C with the wire heater that i s wrapped around the c o l d t r a p . The v a r i a c should not be turned aboue 45 v o l t s or the heater may melt. The c h a r c o a l pump i s warmed up by flo w i n g warm a i r through the flow c r y o s t a t . Because there are s o f t s o l d e r j o i n t s i n the flow c r y o s t a t the temperature should not be e l e v a t e d above 100*C. When the outgassing from the two c o n t a i n e r s has decreased to s u f f i c i e n t l y low l e v e l , the two assemblies are sealed from the r e s t of the system. A f t e r the r e s t of the system i s pumped down to l e s s than a few m i l l i t o r r , the vacuum l i n e i s shut. With the c o l d t r a p at LN^ temperature, 3He i s passed through the t r a p s and then l e t i n t o the r e s e r v o i r . The pre s s u r e i n the system i s monitored with the Sensotec pressure meter. Two l i t e r s of 3He at STP i s r e q u i r e d to p r e s s u r i z e the MCT. The main volume of the MCT i s the r e s e r v o i r with a 2.5 l i t e r volume. When there i s a s u f f i c i e n t q u a n t i t y of 3He i n the tank , the v a l v e s to the 3He r e s e r v o i r and the 3He supply are c l o s e d . When the p r e s s u r e i n the tubes i s l e s s than 10 m i l l i t o r r , the va l v e to the c h a r c o a l pump i s c l o s e d . The empty c o l d t r a p i s now 136 heated to 300 °C to d r i v e o f f the gasses on i t s s u r f a c e and then pumped with a d i f f u s i o n pump. I t i s then r e t u r n e d t o LN temperature. The MCT i s now ready to be used. The Boonton c a p a c i t a n c e meter i s a t t a c h e d to the 3He c e l l and the c a p a c i t a n c e of the c e l l i s monitored as i t c o o l s . The d i l u t i o n r e f r i g e r a t o r can now be c o o l e d down. With the r e f r i g e r a t o r at 1 K, the 3He i s slowly loaded i n t o the c e l l by slowly warming up the a d s o r p t i o n pump. The 3He should flow through the c o l d t r a p and i n t o the c e l l at a r a t e that i n c r e a s e s the Sensotec reading by 0.01 mV/sec. A l l the main l o c a t i o n s on the d i l u t i o n r e f r i g e r a t o r should be monitored to ensure that the r e f r i g e r a t o r does not c o o l below 0.9 K. The pressure i s i n c r e a s e d to 34.5 ba r s . A rec o r d of the ca p a c i t a n c e as a f u n c t i o n of the pressure s h o l d be kept so that the h y s t e r e s i s i n the c e l l can be observed. From 34.5 bars, the pressure i s slo w l y decreased to 28 bars. T h i s c y c l e i s repeated four or f i v e times. When the h y s t e r e s i s has decreased t o a value below the s e n s i t i v i t y of the Boonton meter, the ca p a c i t a n c e b r i d g e i s connected to the c e l l . The c e l l w i l l probably r e q u i r e about 5 c y c l e s before the h y s t e r s i s of the gauge i s i n s i g n i f i c a n t . The c a l i b r a t i o n run c o n s i s t s of p r e s s u r i z i n g the c e l l i n 0.3 or 0.4 bar increments and measuring the c e l l c a p a c i t a n c e at each p o i n t . The two v a l v e s l e a d i n g to the c e l l form metal to metal s e a l s at the seat and t h e r e f o r e , need to be f i r m l y c l o s e d 137 or they w i l l l e a k . The c e l l should be c a l i b r a t e d between 28 and 34.5 bars. The formula, P = a + a S + a S 3, i s f i t t e d to the c a l i b r a t i o n r e s u l t s . From 34.5 bars, the pre s s u r e i s decreased to the plug forming p r e s s u r e , and t h i s can be anywhere from 30.5 to 34 bar on t h i s MCT. The c i r c u l a t i o n of the 3He i n the r e f r i g e r a t o r i s now i n c r e a s e d to c o o l the r e f r i g e r a t o r below the f r e e z i n g p o i n t of the l i q u i d . When the temperature i s below f r e e z i n g p o i n t , an over p r e s s u r e of 0.5 bars i s a p p l i e d to promote the growth of the p l u g up the c a p i l l a r y . I f the pressure i n the c e l l i n c r e a s e s , the plug i s not completely formed. I f t h i s happens, the pressure i s r e l i e v e d to -the o r i g i n a l value and the r e f r i g e r a t o r c o o l e d f u r t h e r before the overp r e s s u r e i s a p p l i e d . When the plug i s completely formed, the MCT can be used as a thermometer. The p r e s s u r e on the top s i d e of the plu g i n c r e a s e s as the l i q u i d helium l e v e l i n the dewar drops. The pressure must not be allowed to b u i l d up because the over p r e s s u r e may cause the plug to s l i p . T h i s i s e s p e c i a l l y important when the MCT i s slow l y p a s s i n g through the minimum. At t h i s p o i n t , the pressure on the plug should be decreased to a few tenths of a bar above the p r e s s u r e used to form the o r i g i n a l p l u g and the temperature of the s t i l l checked. At the end of the c a l i b r a t i o n run, the e n t i r e r e f r i g e r a t o r should be warmed up to 1 K, so that the plug can melt and the 1 38 3He slowly b l e d out i n t o the r e s e r v o i r through the c o l d t r a p . And f i n a l l y , when the r e f r i g e r a t o r i s warming up to room temperature, the r e s e r v o i r should be opened to the c e l l so that the p r essure i n the c e l l does not b u i l d up. 

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