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Fluctuations in a conductor Kshatriya, Amarnath R. 1964

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FLUCTUATIONS IN A CONDUCTOR by AMARNATH.R. KSHATRIYA B.Sc.(Hons) Uni v e r s i t y of Bombay, 1953-M.Sc. Un i v e r s i t y of Bombay, 1956. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Department of PHYSICS We accept t h i s t h e s i s as.conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September., 1964 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 a d v a n c e d d e g r e e 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 a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y * I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s 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 n o 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 . . D e p a r t m e n t o f PHYSICS The 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 , V a n c o u v e r 8, C a n a d a D a t e September h, 196k. ABSTRACT Applying purely thermodynamic arguments i t has been shown that temperature f l u c t u a t i o n s i n a sample can be represented by introducing appropriate'series temperature generator 1 or .'shunt heat current generator'. The temperature f l u c t u a t i o n s lead to resistance f l u c t u a t i o n s i n the sample. These resistance f l u c t u a t i o n s due -to temperature f l u c t u a t i o n s can be detected as voltage-fluctuations (temperature noise") by using a sensing d i r e c t current through the sample. Sta t i s t i c a l - m e c h a n i c a l arguments are used to obtain t h e o r e t i c a l expressions f o r spectral density of heat current f l u c t u a t i o n s i n a m e t a l l i c conductor i n terms of the macroscopic properties of the conductor. 'Since the electrons are c a r r i e r s of heat and e l e c t r i c currents i n a metal., heat and e l e c t r i c current f l u c t u a t i o n s are correlated. Spectral density of c r o s s - c o r r e l a t i o n between e l e c t r i c and heat current i s derived. S t a t i s t i c a l considerations are extended to the c a l c u l a t i o n of the steady state spectral density of heat current f l u c t u a t i o n s between two black bodies i n r a d i a t i v e contact. Temperature noise i n a system i n which there i s only a p a r t i a l c o r r e l a t i o n between temperatures at d i f f e r e n t points along the length at any time (isothermal System) i s compared with a system i n which there i s a complete c o r r e l a t i o n between temperatures at a l l points along the length at -any time (Single Temperature System). Experimental r e s u l t s indicate that f o r the metal filament used Nyquist theorem can be applied-at the operating temperature. For frequencies close to the c h a r a c t e r i s t i c frequency of the system i t i s ' , observed that there i s an increase i n noise temperature of the filament due to temperature noise. v i . ACKNOWLEDGEMENTS I am indebted to .Professor-R.E. "Burgess f o r h i s guidance and •assistance during the preparation of the material f o r t h i s d i s s e r t a t i o n . I should also l i k e to thank the Defenpe Research Board of Canada f o r summer support. i i i . TABLE OF CONTENTS Page Chapter I. Introduction 1.1 Voltage f l u c t u a t i o n s i n thermodynamic equilibrium ... .... l 1.2 Temperature f l u c t u a t i o n s -in thermodynamic equilibrium . . . 2 1- 3 Object and scope of the t h e s i s .. ...... ... ... ....... k Chapter II. Spectral density of heat current f l u c t u a t i o n s from s t a t i s t i c a l considerations 2.1 Spectral density of e l e c t r i c current f l u c t u a t i o n s . . . . . 5 2.2 Heat and e l e c t r i c currents i n a conductor 5 2- 3 Spectral density of heat current f l u c t u a t i o n s . . 7 2- k Spectral density of radiant heat current f l u c t u a t i o n s between two black bodies . . . . .... . . . . . . ... 11 Chapter I I I . Temperature f l u c t u a t i o n s i n a conductor i n Thermodynamic Equilibrium. Isothermal Case. 3- 1 Equivalence between heat flow equations and transmission l i n e equations 15 3-2 Generalisation of Nyquist theorem .... . .... . . . . . 17 3-3 Spectral density of Resistance f l u c t u a t i o n s •. •• -. . . . . 17 Chapter TV Temperature f l u c t u a t i o n s i n a "'single temperature system' k.l Single temperature system . . . . . . . . . . . . 21 h.2 Temperature noise . . . . . . . 21 h.3 Discussion of temperature f l u c t u a t i o n s in• ' " ( l ) Isothermal System (Chapter III) and Single Temperature System 2k Chapter V Experimental Aspects 5.1 Selection of Sample ......... . ... . . . . . . . . . . . . . 27 5-2 Determination of parameters 28 5-3 Noise measurement . . . 29 i v . Page 5-4 Results . . . . . ... ... ... . ..... ... . 32 5.5 Noise Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I+ Chapter VI Conclusions and Recommendations . . . . . . 6.1 Conclusions . . . . . . . .... . 37 6.2 Recommendations f o r future project .... ... ... . . . . 37 Appendix l ( a ) Temperature f l u c t u a t i o n s i n thermodynamic equilibrium . ....... .... ... . . . . . . . . . . . . 39 Appendix 2(a) Relationship between Variance and Spectral density . . . .... . . . . . . 1+2 Appendix 3(a) D i s t r i b u t e d f l u c t u a t i o n generator approach fo r a transmission l i n e i n thermodynamic equilibrium . . . . . . . . ... . 1+5 Appendix 3(b) Properties of f(M) . . . . . . . . . . . . . . . . . . . 51 LIST OF FIGURES Figure Opposite Page 5.1+.1 I-V C h a r a c t e r i s t i c of the filament . . . . . . . . 32 5-4.2 A p l o t of R s(^) Vi iXs^ 'H/w at the operating point . ..... . .. . . . .32 5.4.3 Power dissipated-as a function of filament temperature . . . . . . . . . . . . . . 33 5.5.I Noise temperature T n of the filament as a function of frequency . . . .... . . .. .. .34 1. • CHAPTER I Introduction 1.1 Voltage f l u c t u a t i o n s i n thermodynamic equilibrium. By purely thermodynamic reasoning Nyquist (1928) proved that a r e s i s t o r R i n thermodynamic equilibrium with i t s surrounding at temperature T 0 e x h i b i t s spontaneous voltage fluctuations. Spectral density.of voltage f l u c t u a t i o n s i s given by Where k = Boltzmann constant. E i n s t e i n (191.0) has derived the following entropy p r o b a b i l i t y law f o r computing mean square f l u c t u a t i o n s ,of a macroscopic va r i a b l e x. P r o b a b i l i t y P(x) that the var i a b l e x l i e s between x and x+dx i s where A >S(x) = change i n entropy of the system as a r e s u l t of f l u c t u a t i o n i n x. Consider the following e l e c t r i c a l system i n equilibrium at a temperature T0.' Using (1.1.2) mean square voltage f l u c t u a t i o n i s obtained. Langevin equation f o r current i n the system gives voltage c o r r e l a t i o n function. (1.1.1) ASM fa (1.1.2) -0 F i g . (1.1.1) As f a r as the f l u c t u a t i o n s in,voltage V are concerned these are representable by a seri e s e-m.f generator or shunt current generator as i n F i g . ( l . l . l ) . The generators i n F i g . - ( l . l . l ) have following properties v 1 = kz/c 4 k r 0 fa (1-1.-3) 1.2 Temperature f l u c t u a t i o n s i n thermodynamic equilibrium. . Consider two bodies C-^  and C2 i n thermal contact at temperature "To. C-p C2 are the heat c a p a c i t i e s 1-^ and"T2 are instantaneous temperatures res p e c t i v e l y . T T 2 To -- < T > - <ir > Fig.(1.2.1) ^ H g = thermal conductance between C-j_ and C2 = "^ "rp where H = average heat flow between C i , C2 when there is.-a temperature difference T. Using E i n s t e i n Entropy-probability law i n Appendix l ( a ) following r e s u l t s f o r mean square temperature f l u c t u a t i o n s are derived. 2 ( i ) A T ; 2 = •/?Xcl/c,{c,+cx) (1.2.1) ( m ) A T A T Q = ~ ^ /fo^ j ^T-j_ and A T2 are f l u c t u a t i o n s i n temperatures of the bodies characterised by the heat c a p a c i t i e s C-j_ and C2 • I f C-j_/'c^ /V*/ ^' e* ^°0^ with heat cap-a c i t y C-j_ i s i n contact with an i n f i n i t e heat bath then AT, 1 - ti%l/c, (1.2.2) This i s consistent with Gibbs'1 result that i f i s the energy of a system i n thermodynamic equilibrium then energy fluctuations are: This result i s v a l i d no matter what s t a t i s t i c s apply to the process of heat conduction between and the i n f i n i t e heat bath. Equation (1/2.3) i s the well known energy fluctuation theorem. Temperature fluctuations i n C-^  when i n contact with an i n f i n i t e heat bath are derived i n Appendix l ( a ) . I t i s seen that as far as temperature fluctuations are concerned these are representable by a series'"temperature generator" or a shunt heat current generator analogous to the generators i n (1.-2-.-3) T Fig.(1.2.3) We have shown that ( i ) Spectral density of series "temperature generator'" Thermal impedance of the body C, to i t s surroundings ( i i ) Swt^  13 + j ^ l ) 1.3 Object and scope of the t h e s i s . G i l l (1958) claims that temperature f l u c t u a t i o n s and voltage f l u c t u a t i o n s i n a conductor are manifestations of the same underlying phenomena. One ,of the objects of t h i s i n v e s t i g a t i o n i s to show that such a claim i s erroneous. Temperature f l u c t u a t i o n i s a macroscopic e f f e c t and temperature noise can be detected i n the following way. A H - > A T - > A R - > A ^ { 1 , 3 m X ) where A^ H = f l u c t u a t i o n i n heat current C-j_ .= heat capacity of the system ^ T-^  = temperature f l u c t u a t i o n = i- ^ - 5 - = temperature c o e f f i c i e n t of resistance R dT I = b i a s i n g current V = voltage f l u c t u a t i o n due to temperature f l u c t u a t i o n , o This i n v e s t i g a t i o n i s concerned with t h e o r e t i c a l and experimental aspects of temperature f l u c t u a t i o n s . Theoretical aspects of temperature f l u c t u a t i o n deal with the foll o w i n g : ( i ) Thermodynamic Equilibrium General s t a t i s t i c a l considerations are used f o r obtaining the spectral d e n s i t i e s of heat current f l u c t u a t i o n s and c r o s s - c o r r e l a t i o n s between the heat and e l e c t r i c currents. A d i s t r i b u t e d system i n which the temperature at d i f f e r e n t points are p a r t i a l l y c o r r e l a t e d (isothermal System) i s investigated and compared with a system i n which there i s a complete c o r r e l a t i o n between temperatures at d i f f e r e n t points (Single Temperature System). ( i i ) Steady State Temperature noise i n the steady state f o r a single temperature system and spectral d e n s i t i e s of heat current f l u c t u a t i o n s between two black bodies are obtained. 5-CHAPTER I I Spectral density of heat current f l u c t u a t i o n s from s t a t i s t i c a l considerations 2.1 Spectral density of e l e c t r i c current f l u c t u a t i o n s . Employing general s t a t i s t i c a l considerations Bakker and H e l l e r (1939) have derived the s p e c t r a l density of e l e c t r i c current f l u c t u a t i o n s i n a conductor at temperature T5. . Applying Lorentz theory of the electron and assuming that e l e c t r i c currents T ( t ) and l ( t + ^ ) are corr e l a t e d only when \t!T\ i s small they e s t a b l i s h the following r e s u l t . temperature Ty. 2.2 Heat and e l e c t r i c currents i n a conductor. In a conductor the motion of electrons leads to a simultaneous flow of e l e c t r i c current and heat -energy. To the f i r s t approximation e l e c t r i c and heat current densities are dependent l i n e a r l y on both the 'forces' - e l e c t r i c and thermal. Accordingly i f J i s the e l e c t r i c current density and i s the heat current density i n the conductor when there i s an e l e c t r i c f i e l d F and (2.1.1) where QfJ = $e Yi|l"f) = Real part of the conductance of the conductor at frequency w and temperature gradient then quite generally, 1 = ^ ,U I + ^M, (2.2.1) where (a*t) = 1,2) are s p e c i f i c conductances and i s the ' e l e c t r i c force' ^2 i s the 'thermal force'. Using Boltzmann transport equatipn e x p l i c i t forms for U, and U t and the admittances M^. i s evaluated. For the sake of generality we s h a l l assume an 6. a l t e r n a t i n g e l e c t r i c f i e l d i s applied. f(|<) = Fermi Dirac D i s t r i b u t i o n function f o r the electrons i n the presence of applied f i e l d s (steady state conditions). In the steady state f ( K ) varies with the same frequency as the applied f i e l d To'. T 0 ax (2.2.2) where = Fermi Dirac D i s t r i b u t i o n function f o r the electrons i n thermodynamic equilibrium. si-- N F ex|={(E(^)-^)/r?T0\ -H ECk) = energy of an electron with wave vector K = Fermi energy relaxation time of-electrons with wave vec t o r s - i n the i n t e r v a l K and K+dK ^ = angular frequency of applied f i e l d 0 dl<x X-component of v e l o c i t y of energy transport of an electron. , Y\ - n/zTT h is phncts Ca»S^nt Following Mott and Jones (1936) pages 305-306 1 = ^ F c f t U x f c f t ^ K (2.2.3) 3 d°K t ^ an element of volume i n K space - £ = charge on an electron. Substituting f o r f(if) from (2.2.2) i n (2.2-3) leads to (2.2.1+) 470 7" x where K = - J_ j«a g f o ^ f 0 Comparing (2.2.4) with (2.2..1) leads to the following e q u a l i t i e s \ x ' ^ (2.2.5) 1 lo when i-j^ = o 0 - i<i <0> (2.2.6) where = = thermal conductivity of the specimen conductor when Ljj_ = o. 2.-3 Spectral density of heat current f l u c t u a t i o n s i n a conductor. . In t h i s section Bakker and H e l l e r type approach i s extended to the c a l c u l a t i o n of sp e c t r a l density of heat current f l u c t u a t i o n s i n a.conductor at temperature TQ.. Consider a conductor of length L and area of cross section A. 'N-£» (t) i s the number of electrons i n the conductor at a c e r t a i n time t with t h e i r wave vectors l y i n g between K and K+dK. P ( T;K) i s the p r o b a b i l i t y that an electron which has not suffered a c o l l i s i o n at t continues to move f r e e l y i n the time i n t e r v a l t and t + T . I t i s assumed that'P i s independent of t, i . e . the c o l l i s i o n process i s a Poisson Process. P(ttF) = e,Xp>(- Vfetft) ) (2.3-1) dH(t) = f l u c t u a t i o n of heat current along x d i r e c t i o n due to N^(t). = jEi !? l> x A>-^ (2-3-2) dH(t+7) = M ^ ^ f t + T j (2-3-3) (2.3-4) I f i s the occupancy number of the electrons then Mj? = ALYi^dJL (2.3.5) and VV/^ = M W O g A (2.3-6) ^ H t ) ^ ) , hi' Ety \£?(<j:m<r^A (2.3-7) Auto c o r r e l a t i o n function f o r the heat currents at (t) and (t-Hj) due to a l l e lectrons: Hit;v\(t+<r) - £ k ( E \ P J V £ ?(ie )Vn/ K ( 2- 3- 8 ) Using Wiener-Khinchin theorem, s p e c t r a l density of heat current f l u c t u a t j i s rx> Htfe) H(t-wr) Cos *r a<rp 0 (2.3.9) Substituting i n (2.3.9) from (2.3.8) and i n t e g r a t i n g w.r.t leads to whe whe re %Y ftg - _ fa iff V • = AL"' W 121 C7aa (Talman R.C. (19U6)) (2.3.II) Motion of electrons i n the conductor leads to a simultaneous flow of e l e c t r i c and heat currents. The spectral density of cross c o r r e l a t i o n between the e l e c t r i c and heat current i s c a l c u l a t e d as f o l l o w s : d Hit) = eg?> rvg,w (2.3.12) Here on following the procedures leading to equations (2.3.k) to (2.3.9) leads to where ^ © - A L r ' ^ t O = - K, 6 AL"' (2.3.I3) The negative sign i s expected as the heat current and e l e c t r i c current c a r r i e d by the electron are i n opposite d i r e c t i o n s . Equations (2.-2.1), (2.I.I), (2-3.ll) and (2.3.1k) suggest that f o r a conductor i n thermodynamic equilibrium at a temperature T Q, following k terminal network f u l l y describes i t s response to external forces and i t s properties noise-wise. Fig.(2.3.l) 10. ( i ) £ = 0 i . e . open c i r c u i t (2,2). Corresponding to t h i s s i t u a t i o n and the spectral density of current f l u c t u a t i o n s i s given by (Bakker and H e l l e r ) where \{{) « A L' ' ^ ( - 0 ( i i ) Open C i r c u i t ( l , l ) i . e . J=0 then from (2.2..1) 2. - ^ , M 2 Substitution f o r W Is made from (2.2-5). ^ ( k a . " * U ) (2.3-16) K = Conventional Thermal Conductivity of the Conductor I f H 0 i s the heat current flow when 1=0. Spectral density of H Q f l u c t u a t i o n s i s 5 0 1 , (2.3-17) where := Admittance between the terminals (2,2) when (l , . l ) are ' open c i r c u i t e d ' ^ = ( ^ £ ) = ^ ( , 3 . 1 8 , hence; r ,r\ it u-^z n a (2-3-19) ^ = AL 'K = Thermal Conductance of the Conductor at. T^. Equation (2.3-14) g i v i n g the spectral density of cross c o r r e l a t i o n between the currents H and I i s an example of.the generalised Nyquist theorem (Takahasi, 194l) i n which the elements of the •admittance matrix do not a l l have the same dimension. Spectral density of radiant heat current f l u c t u a t i o n s between -]_i n i two black bodies i n r a d i a t i v e contact 2 . 4 ;" .". '. . . „ — — — : : , — — — . . . .. . :... ;:. . . . . . . Two black bodies with surface areas A-j_ and A2 are maintained at temperatures T-j_ and T2 r e s p e c t i v e l y T-j_.^ T2 • There i s a net flow of radiant energy from A-^  to A2 which i s supplied by a power source i n A-j_ . Fig.(2 .4.1) F i r s t consider the radiant energy emitted by A^ . Let J^j/^i) be the occupancy of a c e l l i n bos.on phase-space. The number of photons i n the frequency range y and emitted from Aj to A2 i n time t i s (Planck, M. •1913')-where A = *S&. t y>} t r a n s i t time of photons from A^ to Ag C = speed of l i g h t V * * N y * ^ t (2.4.2) I f i s the energy flow from the surface of A^ i n time t due to photons i n the frequency rangeV and H+d'i) and 2, By {h) lAv/V, -(2.^-3) Using (2.U.2), (2.4.3) becomes w 2 (2 . 4 . 4 ) V«y = 2 Ah fe^Hj^W^ 12. spectral density of heat current f l u c t u a t i o n s f o r the radiant energy due to photons of frequency andV'+fl/v^ then °lfy " (Appendix 2a) (2,4..5) Substituting f o r Var Ey from(2.4.4) since the r a d i a t i o n of d i f f e r e n t frequencies are s t a t i s t i c a l l y independent, the spectral density of radiant heat current f l u c t u a t i o n s from A^ to Ag due to photons of a l l frequencies i s obtained by i n t e g r a t i n g (2.4.5). ^- 0 Any photon emitted by Aj and reabsorbed by i t w i l l not give r i s e to any f l u c t u a t i o n s . Hon '= heat current composed of r a d i a t i o n of frequency between)) and })+^}) radiated from A-j_ to Ag T o t a l heat current from A-j_ to Ag i s the i n t e g r a l o v e r a l l frequencies of the r a d i a t i o n i n equation (2.4.8) H z s *Lhk Uv (2.4.9) Average heat flow from A-j_ to Ag i s 0 2 • £i> f £ ( j ^ T/*^ * (2.4.10) 13-It i s assumed that the d i s t r i b u t i o n function f o r photons emitted by A-|_ i s one appropriate to i t s steady state temperature (Tolman R.C, 19k6, page 630). - r -! 4^ VW W» (h»lkT?J (2.4.11) AT, Substituting i n (2.4.10) from ( 2 - 4 . l l ) Comparing (2.4.7) with (2.4.12) one can e a s i l y write s H I,T ^ ' l (2.^.12) Since A i s symmetric i n su f f i x e s 1 and 2 introducing an appropriate geometrical f a c t o r which determines the f r a c t i o n of photons emitted by Ag and absorbed by Aj_ and following arguments s i m i l a r to the ones outlines above, one can show that J > "the spectral density of heat current f l u c t u a t i o n s from Ag to A-]_ i s given by <3S„ - W r i 4 J & (2.k:l3) 1. Spectral density of t o t a l heat'current f l u c t u a t i o n s , Is C, - 'K. 4-C Radiations from A-^  and Ag being independent (no r e f l e c t i o n s ) . < T J - P = * fe (TMBI* 4. Tv i ^ O (2 . 4 . 1 4 ) • I f the two bodies A]_ and A2 are i n thermodynamic equilibrium with each other then ( X ? * O i > * and C^-f) =" ^ f c T . ^ (2.4.15) where g = conventional thermal conductance between the two bodies "A-^  and Ag $ £ JllLJlil i's ^ l ^ - _ ? Hxj appendix (2b). T ~ 7 ^ c>T, ^ T a -Ik. Steady state c a l c u l a t i o n of heat current f l u c t u a t i o n s "becomes extremely complicated when the bodies and Ag are not black. R e f l e c t i o n from the bodies w i l l destroy the s t a t i s t i c a l independence of the radiant heat currents. 15-CHAPTER I I I Temperature f l u c t u a t i o n s i n a conductor i n Thermodynamic Equilibrium,Isothermal Case 3-1 Equivalence between heat flow equations and transmission-line equations. Consider a one dimensional non-inductive transmission l i n e . Let Re = Resistance per unit length Gic = Conductance per unit length Ce = Capacity per un i t length then shunt admittance per unit length and . = series Impedance per un i t length = Rc I f l ( x ) and V(x) are current and voltage re s p e c t i v e l y at some point x then and (3-1-1) Consider a thermal conductor of length L and area of cross-section A. The specimen i s i n thermodynamic equilibrium at temperature T Q . Microscopic temperature f l u c t u a t i o n s w i l l be present along the length of the specimen, except at the two ends. Two ends of the specimen are held constant at T Q . 16. Heat balance equation, neglecting l a t e r a l temperature gradient i n the specimen compared to the l o n g i t u d i n a l temperature gradient, i s given by: - A A 0 -_CAM (3.1.-2) where A = l a t e r a l conductivity K = l o n g i t u d i n a l conductivity C = heat capacity per u n i t volume Response of (3..I.2) to a sinusoidal s i g n a l at frequency i s where 9g = l a t e r a l heat conductance per unit length = AA R9 = l o n g i t u d i n a l thermal resistance per unit length =(I<A) CQ = heat capacity per unit length = C A If h i s heat current, Comparing (3.L3) and (3-1-4) with (3.I-I) i t i s seen that equations of voltage and current d i s t r i b u t i o n s i n a RCG transmission l i n e and those f o r temperature and heat current d i s t r i b u t i o n s i n a specimen with l a t e r a l and l o n g i t u d i n a l heat current flows are s i m i l a r , i . e . (3.1-3) and''(3.1-4) can be obtained by replacing ( i ) M b y e ( i i ) L by h ( i i i ) by R& ( i v ) G\t by 9e (v) C e by Ce i n (3-1.1). •17-Thus a conductor with heat flow described by (3.I.2) has an e l e c t r i c a l transmission l i n e as i t s analogue. 3-2 Generalisation of Nyquist theorem. Consider a h terminal network i n thermodynamic equilibrium at temperature T Q . Fig.(3-2.1) Impedance matrix i s defined as follows: (3.2.I) Reciprocal of the Impedance Matrix defines the Admittance Matrix: I , ^ Y„ + ^ V,x i r - \i,yw + vtv« (3.2.2) Twiss, R.-Q (1955) has shown that the s p e c t r a l density of cross c o r r e l a t i o n s between the voltages \|\ and and the currents I,.-and l a i s : M,,\l, ^ _ / w (3-2-3) The r e s u l t s (3-2-3) were f i r s t derived by Takahasi•(l9^l). •3-3 Spectral density of Resistance fluctuations,Isothermal System. For c a l c u l a t i n g the spectral density of resistance f l u c t u a t i o n s i n the specimen equivalent "temperature generators" i n series with thermal resistance and'"heat current generators" i n shunt with the thermal conductance are introduced. These generators are assumed to be uniformly d i s t r i b u t e d along 18. the length of the specimen, the generators are not c o r r e l a t e d and t h e i r s p ectral d e n s i t i e s of f l u c t u a t i o n s i n temperature and heat current are given hy (1.2.U). The generalisation of Nyquist theorem provides an elegant t o o l f o r computing temperature c o r r e l a t i o n s i n the specimen. I 1 r0 • "1 1 • I I 1 X= X, Fig.(3-3-l) Analogous to (3-2.3) s p e c t r a l density of cross c o r r e l a t i o n s i n temperature %ft " ^ e ' i x i j = Z e ' " ' 2 ( 3 " 3 ' i ) Appendix 3(a), equation (3-a.l3) from appendix "3a f o r the ends of the conductor clamped at T Q. , - z ^ ^ ( L - j ^ ) (3.3.2) 9 " &'C ^ K V „ L X K < / Z where H f t c C h a r a c t e r i s t i c thermal impedance' of the specimen (3-3-3) i f thermal propagation constant of the specimen D.C. e l e c t r i a c l resistance of the specimen •g- —rzz = temperature c o e f f i c i e n t of Resistance at T t0 dT0 * o e l e c t r i c a l r e s i s t i v i t y of the specimen at 'T . (3-3-h) 19-spectral density of resistance f l u c t u a t i o n s ^ ^ ^ r / J j /?« ^ . ^ x ^ x , (3.3.5) .Substituting f o r ^)Q^(U)^> f r o r a e ( l u a t i o n (3-3-1). F i n a l l y on inte g r a t i o n i n (3-3-5) 1 - _ / Putting SA(f) = ^ r 0 V > 0 V ^ ( 3 . 3 . 6 ) If .1 'is a sensing current through the specimen then the sp e c t r a l density of voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s : vSjH~9 is given by 6 ( / ) ^ 1 '2.. 2 (3.3.7) where \|0 .= D.C. voltage drop across the specimen Real part of fa) = [Co£$- y^ ')r £ ^ - f c J/l)} j (3-3-8) where o j (I 20. As shown i n Appendix 3(b) -f(A) has following p r o p e r t i e s : ( i ) i f \M « W ( T « | Re fa) 3 % ( i i ) i f \M\ » l •4 1 u- 1 , j U0«T #9 C3-3..2; 21. CHAPTER IV Temperature f l u c t u a t i o n s i n a "s i n g l e temperature" system 4.1 Single temperature system. In Chapter I I I an expression f o r spectral density of voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s was obtained. In the model considered there Macroscopic and Microscopic temperatures were not the same. This resulted i n temperature c o r r e l a t i o n s , spectral density of which i s given by (3-3-l)-In the 'single temperature' system i t i s assumed that both the temperatures, Microscopic and Macroscopic, of the specimen are the same.. 4-2 Temperature noise. Let V and I be the instantaneous values of current -and voltage applied to the system characterised by a single temperature. I f the rate of loss of heat i s determined only by the excess temperature, then heat balance equation i s c , | | + 3 e = p = i v CU.-2.-l) where (j = instantaneous excess of temperature ..= T-T c C ( = heat capacity of the system 3 -- 9-y~ = ' l o c a l ' thermal conductance ( i . e . at the operating point.) A.C. Impedance of the system under operating temperature T Q i s given by; Burgess, R.'E. (1955) ' ^+ 0 ( ? O -r JU)C,' (4.2.2) where i \ 0 = -r = D.C- resistance at the operating point. ^Q = \l0To = Steady state power di s s i p a t e d i n the system. o( = J_ = Temperature c o e f f i c i e n t of resistance at T Q. (4.2.3) • 1 - 9 - - ? . •22. The-non l i n e a r i t y of I-V c h a r a c t e r i s t i c of the specimen i s purely a thermal e f f e c t . I f o(.= 0 then ^ = K 0 . For frequencies ^ ^ £ M = R G ; due to i t s thermal i n e r t i a the specimen i s not able to follow rapid v a r i a t i o n s i n temperatures. A.C Resistance of the specimen at .angular frequency ^ i s P r s cxPo 1 RM = K«[l+(9-*P.)f l4Wx«r.*;J (a) (4.2.4) and the A.C. reactance i s given by (b) (4.2.4) where .(c) (4.2.4) Eliminating (14 W <07 / from (4.2.4) (a) and'(b), (Burgess, R.E.) (4.2.5) Thus f o r a system characterized by a single time constant a p l o t of R(w) and linear-. Intercept on the XW/w = O axis gives the D.C. resistance and slope of the l i n e a r p l o t gives the r e c i p r o c a l of time constant ^ . A.C. Responsivlty of the single temperature system: Consider the following set up: 4^  A V Fig.(4.2.1) 23-R i s the instantaneous resistance of the single temperature system. V, I are the voltage and current respectively. There i s a f l u c t u a t i o n i n heat current from the system to the ambient and vice-versa. R-^  = Load resistance. (?f^  and 6|_ .are the equivalent Johnson noise e.m.f. generators i n series with the specimen resistance R and the load resistance R^ • To a f i r s t approximation the A.G". components of the equation (4.2.l) .satisfy the following equation of heat balance. * e = v, i o - u ( V 2 - 7 ) further, from K i r c h o f f ' s law Substituting f o r 0 i n (4 . 2 .6) from (k. 2.7) and evaluating ^l/r,' from (4.2.8) leads to o<V0 H, f^-t^u) + ^c,-v<3-o<Pe (4.2.9) equation (4.2.9) states that the current f l u c t u a t i o n Xt i s the r e s u l t of three noise e.m.fs .acting i n the c i r c u i t . The e.m.f. generator due to temperature f l u c t u a t i o n s of the system has the value &\\ given by ek - - <Vo\\\ (u.2.10) Current f l u c t u a t i o n due to heat current f l u c t u a t i o n s only i s 1\Q where o(\l0 Hi (4.2.II) I, 24.. and the spectral density of voltage f l u c t u a t i o n s due to heat current f l u c t u a t i o n s S W JJL-(^-^P 8) l(KwVJ ( 1 1 . , 1 2 ) i? - Ru The term Yl r _ f i n (4.2.11) takes i n t o account the electro-thermal feedback. Electro-thermal i n t e r a c t i o n i s due to the f a c t that the power dis s i p a t e d i n the specimen i s a function of time because the resistance i s changing. The time constant °J^= ^ </($-+°<f%rlJ i s a function of operating condition. Under constant current conditions, i . e . ^ | ?(coJ| 4-. 3 Discussion of temperature f l u c t u a t i o n s i n ( l ) Isothermal System (Chapter III) and Single Temperature System For the sake of comparison we w i l l regard the Isothermal and the single temperature systems to be i n thermodynamic equilibrium at a temperature 'T . Isothermal System: Two ends of the specimen conductor, i . e . X = 0 and )( ..= L are •'clamped' at a temperature T Q. Temperature f l u c t u a t i o n ( s p e c t r a l density) at a given point on the conductor I s given by s e t t i n g X t - X| = X i n (3-3-2). At the given point and f o r f i x e d parameters of the system sp e c t r a l density of temperature f l u c t u a t i o n s i s a maximum when UJT^< I . With t h i s approximation 25-(4.3.1) 2 *2» where i -L96 = A L A . The expression (4.3.I) vanishes at the two ends y(. = 0 and X= 1— • This i s •expected "because the two ends are clamped at T Q . (4.-3.l) has a maximum at *--L/2-(4.3.2) .Single temperature system: The single temperature system i s characterised by a complete c o r r e l a t i o n between the temperatures at d i f f e r e n t points at any instant of time. i. This s i t u a t i o n represents a " t h e r m a l l y short" conduction. Spectral density of temperature f l u c t u a t i o n s at a l l points i s the same. In thermodynamic equilibrium ^ ^ / ^ - G . With t h i s value (4.2.10) becomes A p l o t of s p e c t r a l density of temperature f l u c t u a t i o n s i n the two systems as a function of distance looks l i k e Fig. ( 4.3 . 1 ) . T0 L F i g , ( 4 . 3 . i ) ' 26. Supposing a sensing current I Q i s passed through the two systems, spectral density of the voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s i n each case i s given by the follow i n g : Isothermal model 8«) (3-3-7) Single temperature model where »j" - S / ^ * Ce/^ c^d «P0/^«\ f o r LO <TJX< | (4.2.13) (4-3-4) For-a specimen of f i x e d length L increasing the l o n g i t u d i n a l conductivity K and or decreasing the l a t e r a l conductivity ^ reduces the voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s as compared to that i n a single temperature system. When W<T»I 9 t t f l ) ~ Temperature f l u c t u a t i o n s i n the Isothermal system are equivalent to fl u c t u a t i o n s i n temperature of a ' s t r i n g ' of single temperature systems. Temperature f l u c t u a t i o n s at points f a r t h e r from the thermally clamped ends of the system are not sens i t i v e to the boundary condition. Fig.(4-3-2) shows the spe c t r a l density of temperature f l u c t u a t i o n s i n the two systems. To Stogie "u.v i^j.tYftjMv'e-^S te.vw Ic Fig.(4.3-2) 27-CHAPTER V Experimental Aspects 5-1 S e l e c t i o n of Sample . In a metal current noise i s not detectable because the number of electrons i s fi x e d . Only two sources of noise are'present. They, are ( i ) Johnson Noise and ( i i ) Temperature Noise. The experimental i n v e s t i g a t i o n outlined below i s therefore concerned with -a m e t a l l i c conductor. The choice of sample i s dictated by the following requirements, ( i ) Material of the sample be known. ( i i ) Uniformly high temperature be attainable. This w i l l increase the heat current f l u c t u a t i o n s . ( i i i ) S u f f i c i e n t l y high d-c. resistance at the operating point. This helps i n improving the noise f i g u r e of the noise detecting c i r c u i t . ( i v ) Tod± should be s i g n i f i c a n t l y l a rger than unity so that a check can be made f o r the appropriate expression f o r Johnson noise. (v) S u f f i c i e n t l y small thermal time constant at the operating point. It i s the r e c i p r o c a l of the time constant that determines the frequency below which temperature noise increases. P i h l i t e s A x i a l leaded s t y l e (Type 13-7) manufactured by Kay E l e c t r i c Co. were found, acceptable. The lead material of the p i n l i t e i s platinum and the filament material i s tungsten. The lamps were examined with a microscope f o r uniformity of temperature at a suitable value of current. It was observed, i n case of some of the lamps, that at a c e r t a i n bias current the resistance dropped. This on v i s u a l check l e d to the observation that decrease i n resistance of the •lamp was due to shorting out of adjacent c o i l s of the filament.. This was the case with a non-uniformly c o i l e d lamp, Fig.(5-1-1). Fig.(5.1.1) 28. %2 Determination of Parameters In the measurements outlined below the sample was immersed i n an o i l -bath contained i n a m e t a l l i c container. This was done to assure that the ends of the filament were thermally clamped to the bath temperature and the metallic container provided an e l e c t r i c s h i e l d . ( i ) , Measurement of R 0, P Q, T Q and . D.'C. resistance R Q of the sample was measured using a standard wheatstone's bridge c i r c u i t f o r d i f f e r e n t currents through i t . Power dis s i p a t e d i n the lamp was noted. Knowing the r e s i s t i v i t y of tungsten as a function of temperature Smlthels..'. (1955) the temperature at the operating points was determined. Temperature c o e f f i c i e n t of resistance at the operating temperature was then derived from these tables f o r tungsten at the operating temperature. ( i i ) Measurement of A.C. Impedance Following bridge c u r c u i t was used f o r measuring the a . c resistance and reactance of the lamp at the operating point. A l l the bridge components were c a r e f u l l y shielded against extraneous pickup. 1-1 Fig.(5.2.1) E]_ i s a model 202C Hewlett-Packard low frequency o s c i l l a t o r C]_, C2 are f i x e d capacitors 29-R3 are standard capacitance box and General R'adio resistance box respectively. They constitute the variable arm of the bridge L-j_ i s a bridge transformer (?v is- the p i n l i t e lamp under I n v e s t i g a t i o n D i s the detecting c i r c u i t c o n s i s t i n g of Keithl e y Low Noise pre-amplifier connected to a General Radio Wave Analyser (Type No..736-A) B i s a variable output d.c. supply A i s a c a l i b r a t e d ammeter. Balance condition corresponds to An advantage of using the c i r c u i t of F i g . (5-2.-1) i s that no d.c. passes through the other arms of the bridge excepting,^ . Use of Wave -Analyser as the n u l l detector has the advantage of eliminating any possible harmonics at n u l l point. Band width of the Wave Analyser was Performance of the bridge was checked by replacing ^ by a f i x e d carbon r e s i s t o r -at room temperature and i t s a.c. impedance measured over the frequency range ^ =Zoc/$ to (• = 5"Kc/S . The a.c. bridge readings' agreed within 5$ with the value of the r e s i s t o r determined by a d.c. VJheatstone bridge. 5•3 Noi&e- Measurement 1-, ( i ) Assembly A pre a m p l i f i e r using E810F P h i l i p s tube was assembled. Choice of the tubes was d i c t a t e d by i t s high value of trans-conductance, which implies low noise tubes. Careful sh i e l d i n g of the input c i r c u i t from the 30. r e s t o f t h e c i r c u i t was n e c e s s a r y t o p r e v e n t o s c i l l a t i o n s . F o l l o w i n g i s a b l o c k d i a g r a m o f t h e n o i s e d e t e c t i n g c i r c u i t u sed. vi-S P(?E. A>v\|?. Amp; Fig.(5.3.1) W-A CR.O 1-1 i s t h e i n p u t t o n o i s e m e a s u r i n g c i r c u i t . W-S i s a w a f e r s w i t c h . T h i s c o n n e c t s t h e i n p u t o f t h e p r e - a m p l i f i e r t o t h e lamp ^ o r ^54^ • ' i s a G e n e r a l R a d i o Decade R e s i s t a n c e box k e p t a t room t e m p e r a t u r e . W-A i s a narrow b a n d (Af = 4 $ ) G e n e r a l R a d i o Wave A n a l y s e r (Type 736-A). Wirewound r e s i s t o r s a r e u s e d as s o u r c e r e s i s t a n c e t o en s u r e t h a t J o h nson n o i s e i s t h e o n l y . s o u r c e o f n o i s e i n t h e power s u p p l y . Source r e s i s t a n c e i s v e r y much g r e a t e r t h a n . T h i s i s done t o o b t a i n a c o n s t a n t c u r r e n t s o u r c e . ( i i ) Measurement o f E q u i v a l e n t N o i s e R e s i s t a n c e (Req). E q u i v a l e n t n o i s e r e s i s t a n c e o f t h e n o i s e m e a s u r i n g c i r c u i t a t a f r e q u e n c y i s o b t a i n e d as f o l l o w s . The i n p u t ( l , l ) i s s h o r t e d . The mean r.m.s. o u t p u t i s n o t e d i n W-A a t t h e chosen f r e q u e n c y . Then a v a r i a b l e R e s i s t a n c e S t a n d a r d , ^ s t d * room t e m p e r a t u r e i s i n t r o d u c e d a t t h e i n p u t ( l , l ) and i t s v a l u e a d j u s t e d t i l l t h e ave r a g e i n d i c a t i o n o f t h e r.m.s. o u t p u t V,,.^ i n W-A i s such t h a t V s t d / ^ s h = \J~2~. The v a l u e o f R s t a c o r r e s p o n d i n g t o t h i s s i t u a t i o n i s R g ^ ( f ) , o f t h e n o i s e m e a s u r i n g c i r c u i t . 31-Rg^ ( f ) of the noise detecting c i r c u i t was measured at f=40c/s, f=100e/s, f=1000c/s and f=5000c/s. The procedure used was to p l o t V g t d as a function of Rg-^ jL at each of these frequencies. It was observed that the p l o t was l i n e a r and therefore W-A readings were r.m.s. readings f o r the values of R s t d used-Values obtained f o r R e q ( f ) are as follo w s : R e q ( f ) = 35OO ohms -f=40c/s. R e q ( f ) = 85O ohms -f=UOc/s. R e q ( f ) .= 275 ohms f=lKc/s and 5Kc/s. These fi g u r e s show that l / f noise becomes apparent i n the system at frequencies below about 300c/s. ( i i i ) Measurement of noise i n the sample. (a) For checking the accuracy of noise measurements, the lamp i n F i g . ( 5 - 3 - l ) was replaced by a f i x e d r e s i s t o r at room temperature. W-S was switched to 3 and the mean output meter reading V n s i n W-A was noted. Then W-S was switched to 2 and the value of Rg^a adjusted t i l l the mean output meter reading Vs^.^ was equal to V n s . I t was noted that at f=40c/s the random error i n resistance measurement was about 20$> while at f=5Kc/s the error i n measurement was l e s s than 10$. 'This i s because the equivalent noise resistance of the noise detecting c i r c u i t Increases with decreasing frequency. (b) .Lamp current was adjusted to 6 .3 m.-a. The choice of b i a s i n g current i s dic t a t e d by the need to keep the evaporation rate small enough while obtaining a high enough uniform temperature. As outlined i n (a) above W-A was set to kOc/s. W-S was switched to 3 and the mean output meter reading V n was noted. Then W-S was switched to 2 and the value of R sta adjusted t i l l the mean out put meter reading V was equal to V n . Keeping the current constant the procedure was repeated f o r various frequencies up to 5Kc/s. ft$(U)) \'i |A^ wyw ,at the operating point 32. 5.k Results (i) D.C. and A.C Data R a = Resistance of the lamp at the ambient temperature . 296°K = 30 ohms. T = Current through the lamp at the operating point = 6 .3 m.-a. RQ = D.C. resistance at the operating = 206 ohm. _3 P q = D.C Power dissipated in the steady state •= 8.2 x 10 watt. RD/Ra = 6 . 8 Using Smithels (1955) page 638, from the resistivity-temperature relationship for tungsten the ratio RQ/Ra = 6 . 8 corresponds to the operating temperature T = 1440°K o From the I-V characteristic Fig . ( 5-4.l) 4-¥ at the operating point dl i s equal to kOO ohm. A.C •Impedance'measurements at low frequency f = 20c/s gives Re (Z) = 395 + 15 ohm. (jt\j Thus ^ obtained from I-V characteristic (at the operating point and the values of Re Z(f) for f=20c/s agree within the experimental error in measurement. A plot of Rg(^) and | X s ^ f / 6 0 " is linear, Fig. (5 .4.2). The slope of the graph gives the value of characteristic time •= 5xlO~3sec. This implies that the lamp at the operating point can be treated as a system having a single characteristic time-. Power d i s s i p a t e d as a f u n c t i o n of Temperature 33-( i i ) Discussion of D.C. and A.C Data For a single temperature system (4.2.3) S u b s t i t u t i n g the measured values of and R„ gives = 0.-32 • As a check the value of ^'0/^ c a n b e independently c a l c u l a t e d by determining the i n d i v i d u a l parameters involved. i s found from Smitheljs ( l o c c i t ) at the operating point. = 2-1 X lo4o/c '' Knowing P Q as a function of T Q Fig.(5 .4.3) Q - at T Q i s c a l c u l a t e d Using the i n d i v i d u a l values of <?(. , -P0 and ^ gives e<tfy = 0.37 The two values of obtained from independent means show f a i r l y close agreement providing a confirmation of the lamp at the operating point as a Single Temperature system. Thus by two d i f f e r e n t c r i t e r i a (plot of R^(^) ,vs I Yi^fur and the °{?O/Q comparison) we have corroborated t h e - e s s e n t i a l feature of the model used i n the theory. It i s also noted from Fig.(5 .-4.3) that the 4 power di s s i p a t e d tends to be proportional to T f o r temperatures i n the neighbourhood of T Q. Noise Temperature of filament as a fu n c t i o n of frequency Experimental Curve (Dotted l i n e ) T h e o r e t i c a l Curve ( s o l i d l i n e ) . - 2Z0OK N $4s 34. 5-5 Noise Data We define a noise temperature T n ( f ) of the lamp at the operating point as a function of frequency as follows: (5.5,2) where Q t o - t a l ( f ) i s mean square voltage f l u c t u a t i o n i n the lamp at frequency f.. k = Boltzmann constant o R Q - D.C. resistance of the lamp at T Q K, Experimental values of T n ( f ) as a function of frequency are p l o t t e d i n Fig.(5-5-l). Discussion of noise data-We have shown that the lamp at the operating point acts as a Single Temperature System. T o t a l noise i n the system i s composed of two components ( i ) Johnson noise, ( i i ) Temperature noise. We assume that as: f a r as temperature f l u c t u a t i o n s due to r a d i a l heat losses are concerned the two components are independent. In the t h e o r e t i c a l expression f o r noise temperature T n ( f ) we add the mean square voltage f l u c t u a t i o n s due to each CUB = Or*0 + ^ c " - * ) We adopt the view that f o r a metal (^(f) = Johnson noise of the conductor = 4kT 0R Q (5-5-3) Since the filament of the lamp i s i n r a d i a t i v e contact with the ambient at 296°K and t o t a l e m i s s i v i t y of tungsten at the operating temperature Is.0.-2 Smithels (1955) page 666, we assume that equation (2.4.14) i s v a l i d . At the operating temperature (T Q/ip )^ J .' Hence the dominating term i n S H ( f ) i n (2.4.14) i s S H *0 = 2kT.X% (5-5-4) 35-Temperature noise from (U.2.13) and su b s t i t u t i n g f o r S j f f ) from (5-5.4) i s (5-5-5) ve as before,' (5.5.6) mean value of Substituting f o r the parameters i n (5.5-6) at the operating point y i e l d s frequency. In passing, we note that Brophy (1963) has reported measurements of noise i n a si m i l a r sample of p i n l i t e operating under s i m i l a r conditions. He has used expressions f o r temperature f l u c t u a t i o n s and heat current f l u c t u a t i o n s that are not appropriate to the experimental conditions to which they are presumed to apply. For example, at the high temperature (l600°K) at which h i s noise measurements are made, contrary to h i s assumption, Newton's law of cooling does not apply. Electro-thermal i n t e r a c t i o n , our fa c t o r ( (|— ), at the operating point i n h i s experiments does not appear i n h i s expression f o r temperature noise. His expression f o r Nyquist noise i s erroneous f o r he has assumed that Nyquist noise i s given by ^~IAT> i n the metal. It i s worth while to r e c a l l at t h i s point that i t i s purely the thermal i n e r t i a , of the system at the operating point that gives r i s e to Z ( w ) d i f f e r e n t from RQ. His experimental r e s u l t In Fig.(5.5.I) t h e o r e t i c a l value of T n ( f ) i s p l o t t e d as a function of 36.. " ^ / f ( 0 ) = i S p r o l : ' a b l y d u e t o a component of l / f noise.. It i s noted that the product of mean square noise voltage and frequency i n the range f=0 to f=30c/s i s appreciably constant. Wow reverting to the discussion of our own data, i t i s observed that f o r frequencies much l e s s than the c h a r a c t e r i s t i c frequency ^ D - ^IJ3]^ , the upper l i m i t to the t h e o r e t i c a l T n i s ( Q = 2100°K and T n - T Q for frequencies i n the range 800 c/s to 5Kc/s. Comparing the experimental value of T n ( f ) with the t h e o r e t i c a l value i n F i g . ( 5 - 5 ' l ) w e observe that the experimental curve agrees with the t h e o r e t i c a l curve f o r frequencies i n the range 800c/s to 5Kc/s. In t h i s frequency range the experimental noise temperature T n of the filament i s observed to be 1550°K. This agrees with the conventional temperature (Thermodynamic) of the filament "at the operating point within the experimental error. This implies that the temperature noise i s absent at higher frequencies and the conventional Nyquist theorem ( ljkT0&o ) i s applicable to the filament at L450°K. There i s a d e f i n i t e increase i n the experimental noise temperature f o r frequencies l e s s than 800c/s. The r i s e i s of the type expected from the appearance of temperature f l u c t u a t i o n s at low frequencies. However, the frequency at which t h i s appears i s higher by a f a c t o r of about 5 than the c h a r a c t e r i s t i c frequency ( ^ f f j ^ ) derived from impedance data. This discrepancy remains unexplained. 37-CHAPTER VI Conclusions and Recommendations 6.1 Conclusions Experimental r e s u l t s indicate that f o r frequencies much greater than the c h a r a c t e r i s t i c frequency.the noise temperature of the p i n l i t e filament i s i n close agreement with the conventional (thermodynamic) temperature. This would imply that the electrons i n the conductor -are e s s e n t i a l l y i n equilibrium with the l a t t i c e and that temperature f l u c t u a t i o n s are n e g l i g i b l e at frequencies greater than the c h a r a c t e r i s t i c frequency. Results f o r noise temperature f o r frequencies of the order of c h a r a c t e r i s t i c frequency seem to indicate' a d e f i n i t e increase i n noise temperature. The r e s u l t s though not conclusive make a strong case f o r the existence of temperature fluctuations.. However, there remains an unexplained discrepancy between the frequency at which these f l u c t u a t i o n s ' begin to appear and the t h e o r e t i c a l c h a r a c t e r i s t i c frequency. 6.2 Recommendations f o r future p r o j e c t s P i n l i t e s being coiled.filaments with e s s e n t i a l non-uniformities are not n e c e s s a r i l y the most elegant smaples f o r measurement of temperature f l u c t u a t i o n s at frequencies f o r which small enough noise fi g u r e of the noise detecting c i r c u i t i s attainable. The only parameter an experimenter can hope to adjust i s the temperature of the filament. This again i s l i m i t e d by the need to keep the rate of evaporation small. Following experimental projects are expected to lead to conclusive experiments regarding temperature fluctuations.. ( i ) A.thin s t r a i g h t wire of a metal, with diameter of a few microns and length a few cms. To increase the c h a r a c t e r i s t i c frequency ^ w i l l have to be Increased. This can be done by choosing a material of high emissivity. •38. I t i s advisable to work under conditions of r a d i a t i v e cooling only. This provides the p o s s i b i l i t y of operating point at which -£s <^L i s much greater than unity. ( i i ) Using a se n s i t i v e i n f r a - r e d detector, spectral density of heat current f l u c t u a t i o n s from a black body can be measured. This experiment w i l l provide a check f o r the expression f o r non-thermodynamic equilibrium fluctuations i n heat current between two black bodies derived i n (2.k). Following t h e o r e t i c a l problems suggest themselves.. ( i ) Extension of general s t a t i s t i c a l considerations f o r obtaining an expression f o r heat current f l u c t u a t i o n s i n a conductor.with a temperature gradient and also when both electrons and phonons are the c a r r i e r s of heat energy. ( i i ) Extension of the arguments outlinec(in (2.U) to sp e c t r a l density of heat current f l u c t u a t i o n s between two non-black bodies i n r a d i a t i v e contact. APPENDIX 1(a) Temperature f l u c t u a t i o n s i n Thermodynamic Equilibrium 39-Two bodies with heat c a p a c i t i e s Cj_ and Cg and instantaneous .temperatures T-j_ and Tg r e s p e c t i v e l y are i n thermal contact. ^ is.the thermal conductance. T 0 i s the thermodynamic equilibrium temperature of the system. j c aJf H = nett heat flow between the two bodies 0 - difference i n temperature between the bodies leading a flow of heat H. 1> ^ F i g . ( l . a . l ) T o t a l change i n entropy of the system due to a spontaneous flow of heat from Tg to T-^  i s c a l c u l a t e d as follows X = T ° + ^ ( l . a . l ) Spontaneous flow of heat from Ag to A^ r e s u l t s i n no nett change i n energy AS= C ^ + ( ^ = t o t a l change i n entropy = - _L £*AJi (c^) (l.a.2) P r o b a b i l i t y of a f l u c t u a t i o n i n temperature ^ T-^  i n C-^  i s oC e 2 * T » c* (l-a-3) .Covariance (l.a.K) ko. Negative sign of the covariance implies that there Is a n t i c o r r e l a t i o n between the temperatures of the two bodies. This i s a oorsequenee of the f a c t that the -spontaneous flow of heat i s not accompanied by any nett -change i n energy of the system. Spectral density of temperature f l u c t u a t i o n s Consider a body of heat capacity C^ i n thermal contact with an i n f i n i t e heat bath at temperature T Q. Qft) i s an instantaneous temperature difference between the body CQ_ and the ambient. Heat balance equation i s C\$ + f6 - « W ( 1 . A . 5 ) r\(i) = f l u c t u a t i n g heat current with a sharp c o r r e l a t i o n function and = O R f Solution of (l.a.5) i s Q (t) = Q (o) £ + ^  6 J e ' hi^ij^t, - £ 0*0 ( i . a . 6 ) Spectral density of temperature f l u c t u a t i o n s i s (Wiener-Khinchin theorem) W i (l.a-7J where £ f e •= /<! f rom = A T( and =. 0« i n (l.a-3) As f a r as f l u c t u a t i o n s i n temperature i n are concerned analogous to e l e c t r i c a l case ( l . - l ) , we introduce series temperature generators and shunt heat current generators to give (l.a-7) I ' " —(3 T r i ^ Fig.(1.a.2) :kl. •Spectral d e n s i t i e s of the series temperature generator and shunt heat corrent generator are given by *lu> = ^ + > C ' J" ' (l.a.8) ,Thus i n thermodynamic equilibrium we have -an analogue of Wyquist theorem f o r heat current and temperature f l u c t u a t i o n s . k2. APPENDIX 2(a) Relationship between variance and spe c t r a l density Consider the following system. It has per f e c t memory up to a c e r t a i n time i n t e r v a l T and no memory l a t e r . T i s the sampling time. Xft) A(0 l ( t ) i s the frequency response of the system to ,a un i t d e l t a function. 0 e (2.a.l) Let ^ ( t ; - J t - T Then = s P e G t r a l density of f l u c t u a t i o n s i n ^  = l A C f ; | * ^ ( ^ (2.a,2.) where ^(f).= s p e c t r a l density of f l u c t u a t i o n s i n X • (Var^) = Variance of ^ , as measured i n the sampling time T. = W^/^-f from(2.a.2). o For sampling time T ^ longest c h a r a c t e r i s t i c time J n ^ y of the system ( i . e . i n the case treated i n (2.h) • .T max t r a n s i t time of photons between A-j_.• and :Ag) S^(f) i s uniform (y^jd) - S p ( ^ ) i ^ - f - (2,.a..3) This s i m p l i f i e s to . r ( O - d ^ ) T ( 2.a.,) 43-(2..a. 3) i s quite a general statement. I f has the following form ••Fig-. (2,a , 1) then f o r (.2. a. 3) to be v a l i d sampling time T should s a t i s f y the following i n e q u a l i t y : APPENDIX 2(b) Thermal Conductance i n Equilibrium . Consider the two bodies A-|_ and Ag in. thermodynamic equilibrium. H 0 i s the heat flow i n each d i r e c t i o n . Temperature of Aj_ i s ra i s e d so that ^ = ^ . ( - ^ > T > (2.b.l) S i m i l a r l y i f "17"""X r K i _ ~ Hi-i (thermodynamic equilibrium) ^ = ^ ( 2 . b . 2 ) from (2.b..l) and (2.b.2) r 1 s the conventional thermal conductance. 45. . APPENDIX 3(a) D i s t r i b u t e d f l u c t u a t i o n generator approach f o r a transmission l i n e i n thermodynamic equilibrium. Method: E.m.f. generators i n series with e l e c t r i c a l resistance and current generators i n p a r a l l e l with e l e c t r i c a l conductance are introduced uniformly along the l i n e . Spectral density of these generators are given by 1.1.5- Voltage and current components due to the two types of generators at given points are obtained. Replacing voltage by temperature current by heat current equivalent expressions f o r temperature c o r r e l a t i o n s i s written down. Appropriate value f o r sp e c t r a l density of the.series temperature generators and shunt heat current generators, proved i n 1.2.3, i s used. Outline of procedure. Fig.(3.a.l) * *' Assumptions: ( i ) The g e n e r a t o r s ^ / , ) , , 6(xV) and are completely uncorrelated. ( i i ) Response of the system i s l i n e a r . 1(X) = amplitude of shunt ( e l e c t r i c ) current generator \ = t o t a l ( e l e c t r i c ) current at a point X\ 1% = t o t a l ( e l e c t r i c ) current at a point Xa. V-j_ and Vg are voltages at the points %\ and X-Lrespectively £lXj= amplitude of series e.m.f generator YjtP) = admittance per unit length 'i^i = impedance per un i t length ^ = propagation constant = Y??e = c h a r a c t e r i s t i c impedance = e^/Y^  46. Series e.m.f generators and shunt current generators contribute to current and voltage at any given point on the l i n e . These contributions are c a l c u l a t e d from the following' formulae. -4^ Y e ^ V -V- l_(x Boundary ; condition i s = 0 (3-a.l) Part I: Current and voltage.at two points due to an e.m.f generator. 'Switch o f f the shunt current generators. Superscript .'g. ' r e f e r s to component due to s e r i e s e.m.f generator and 's' r e f e r s to the somponent due to shunt current generators. '6-g- V'i^) = current at a point / = due to a shunt current source at x' With t h i s notation: tlx'; <•*«•»' , (3-a.2) ;Part I I Currents and voltages at X, and Xi- due to a shunt current generator. t K^Ki X-U Fig,(3.a.3) T o t a l voltage at y and /T,. i s obtained by adding the components due to shunt current generators and serie s e.m.f generators and then integr a t i n g over )(' . Thus taking into account a l l the generators. (3-a.U) *• S i m i l a r l y an expression f o r • (complex conjugate of VJX) ) i s obtained. Integral expressions f o r V|QO and V^)- are m u l t i p l i e d and bearing i n mind the assumption ( i ) i s a d e l t a function. (a) (3-a <A/|\jt^  -is obtained. (3,. 2?I2 f o r a RCG transmission l i n e shorted at y T^ B and y = L where 2 l 2 - ( 2 U - 2I6C2)H; ^ fat, Sfyvh y^ .v-:|r() Hi] -(3-« 2;a v * c (3, k9. A c a r e f u l algebra then leads to the following < V , V * > - S£ = 4*7.fc:2J . (3.1 .8) Spectral density of current c o r r e l a t i o n s . < Total current at X\ and yC^_ > as i - n the preceeding case i s obtained by adding the components due to serie s e.m.f. generators and shunt current generators and then i n t e g r a t i n g over From (3.a.2) and (3-a.3) ; L * -f I (3-a.9) S i m i l a r l y an i n t e g r a l expression f o r ^2 ^ i s obtained. As before, bearing i n mind the r e l a t i o n s (3-a.5) an expression f o r l | L i / "is obtained. f o r a shorted transmission l i n e . Admittance matrix i s defined by: ' ( 3.a,10) 50. Zc^svWvjrL Xi>y, (3-a.ll) Careful algebra leads to the following < T , r * > - - * * T . & ^ ( 3 . a . l 2 ) Temperature Correlations . For calculating the spectral density of temperature correlations following replacements are made i n the preceeding analysis, hy © 0 0 TV) by H60 v, *y 0 , L, by H i by Zevw) The heat and current generator and temperature generator at any given point are not correlated; also the heat current generators and temperature generators at different points are not correlated with themselves:;Jnor with each other; (3-a.13) For spectral density of temperature correlations where 2^ * **c ^ 51-APPENDIX 3(b) Properties of f(p) \/U\« i CO 01« \ expanding £, and £.. <7 3 • up to,terms i n 1 0 fefw - ^ - f ; [ ( i - s*- . )+ u V( J s| ' - . - ' ) f o r W J has.an upper bound.= ly H » tQ fa) ^ J u \ (3-b.l) (3-V2) 52. BIBLIOGRAPHY Bakker, C. and H e l l e r , G. -Physica 6, 262 (1939). Brophy, J . J. Appl. Phys. 3_4, 1890 (1963). Burgess, R. Proc. Phys. Soc. B. 68, 766 (.1955). Einstein,A. Ann. Physik. 33, -1275 (l910). G i l l , T. Nature 182, 239; (1958). Mott, N. and Jones, H. The Theory of the properties of Metals and Al l o y s . Dover Publications Inc., New York. Nyquist,, H. Phys. Rev 32, ; 110 (1928). Planck., M. Waermestrahlung (1913).• Tr a n s l a t i o n : The Theory.of Heat  Radiation. Dover Publications Inc., New York. Smithels, C. . Metal Reference Book, Vol.2,-(Second E d i t i o n ) . Butterworths S c i e n t i f i c P u b l i c a t i o n s , London (1955)-Tak-ahasi,-H. Proc. The Physico-Mathematical Society of Japan, ser. 3, 23, 5^ 8 (1941). Tdlman, R. ;The P r i n c i p l e s of S t a t i s t i c a l Mechanics. ( F i r s t E d i t i o n ) Oxford U n i v e r s i t y Press (1946). Twiss, R. J . Appl. Phys. 26, 599 (1955). 

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