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Thermal Expansion of Solids Mehdi Amara, Université Joseph-Fourier, Institut Néel, C.N.R.S., BP 166X, F-38042 Grenoble, France

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Thermal Expansion of Solids Mehdi Amara, Université Joseph-Fourier, Institut Néel, C.N.R.S., BP 166X, F-38042 Grenoble, France Cryocourse 2011 Thermal expansion of solids Mehdi Amara, Université Joseph-Fourier, Institut Néel, C.N.R.S., BP 166X, F-38042 Grenoble, France references : -"Thermal expansion", B. Yates, Plenum, 1972 - "Solid State Physics", N.W. Ashcroft, N.D. Mermin, Saunders, 1976 - "Introduction to solid state physics, C. Kittel, Wileys & sons, 1968 -"Magnetostriction", E. du Tremolet de Lacheisserie, CRC Press, 1993 ... www.neel.cnrs.fr Introduction Changes in size, volume while cooling/heating 1 "V # volume thermal expansion : ! = % V "T $ P 1 "L # 1 linear thermal expansion : ! = % = & L "T $ P 3 Orders of magnitude for α (room temperature and pressure) 1 Gases 10-3 – 10-2 ideal gaz : ! = 3T Liquids 10-4 – 10-3 insulators metals polymers Solids 10-6 < 10-5 < 10-4 Thermal expansion thermometry contrast between gas and solid Santorio Santorii (circa 1610) : liquid pushed by expansion of air in a glass tube. contrast between liquid and solid Galileo Galileis (1596) : water density change thermoscope. Ferdinand II, Grand Duke of Tuscany (1641): bulb alcohol thermometer. Gabriel Fahrenheit (circa 1710) : scaled mercury thermometer. Anders Celsius (circa 1710) : centigrade water thermometric scale. contrast between two solids John Harrison ( ! 1750): first bimetallic strip for temperature compensation latter used as thermometer and thermostat Technological issues Since early human industry: buildings, ceramics, metallurgy in case of thermal/material inhomogeneity Thermal compatibility (reinforced concrete...) Problem more acute with metals : in railways, heat engines, bridges, large buidings, transonic planes... and cryogenic equipements expansion joints or other... First systematic approaches : high precision mechanisms measurement of time John Harrison : gridiron pendulum (1726) L = constant Exercice : pendulum 1) Pendulum Clock: arm made of brass L0 Length at T = 20 °C, L0 = 50 cm #5 #1 !brass = 1.9 " 10 K L ! = 2" 0 = 2 s 0 g - compute τ at T = 15 °C - by how much is the clock shifted one day latter ? 2) Brass+Steel compensation: "gridiron" #5 #1 !steel = 1.3 " 10 K L l brass Lsteel - compute the T = 20°C ratio between Lbrass and Lsteel so that l is constant . Thermal expansion measurement Probing a small change in length Δl : sensitivities better than 10-4 - macroscopic optical lever : basic opto-mechanical amplifier 10-6 optical interferometric : Fizeau, fringes displacement 10-9 + radio-frequency resonance 10-11 capacitive : C(l) 10-9 electrical inductive : mutual inductance m(l) 10-7 resistive : strain gauge R(l) 10-6 - microscopic : diffraction of monochromatic photons, neutrons ! $l Bragg law : d = # $" = % tan" & reflections at large θ 2sin" l powder single crystal 10-5 10-6 A capacitance dilatometer Three-terminal capacitance method resolution up to 1 Angström sensitivity 10-6-10-8 Temperature range 2-300 K Magnetic Field range 0- 6.5 T Courtesy of Didier Dufeu Angular range 360 ° Institut Néel Some "reference" α curves Copper Silicon Thomas A. Hahn, J. Appl. Phys. 41 (1970) 5096 G.K. White, Journal of Physics D: Applied Physics 6 (1973), 2070 Graphite Fused Silica Expansion : microscopic interpretation Atomic : ! m thousands of K to change inflation ? ! (r ) = R (r).Y (",#) nl l radial electronic configuration ! "free" movement/kinetic energy ? gaz, liquids but not solids vibrations ? solids, liquids change in volume ? Electronic: Bonding electrons ? few levels well separated Free electrons in metals... but fermions Small amplitude: harmonic approximation V (X) ! e kT Crystals Glasses Liquids L Polymers oscillation 2 +" A X Liquids ! polymers # X e kT dX !" L ! L0 = 2 = 0 +" A X ! # e kT dX L !" libration No thermal expansion ! Failure of the harmonic approximation Zero thermal expansion No volume/Thermal dependence of elastic constants "high" temp. specific heat doesn't depart from Dulong and Petit infinite thermal conductivity ... Anharmonicity More realistic form of the interaction potential V(r) alkali halides strongly repulsive for L < L0 C e2 1 V(r) = m ! r 4"#0 r slowly varrying attractive for L > L0 X odd powers in the expansion of V(X) Classical anharmonic "description" 2 3 4 Expansion : V(X) = V0 + a X + b X + c X low temperature approximation +" +" 2 2 2 3 4 !x 4 !x 3 $ +" a X +b X +c X e dx = $ x e dx = ! # # 4 X e kT dX !" 2 3 4 !" 2 +" a X b X +c X +" a X # ! ! ! 3 4 !" 1 kT kT 1 kT b X + c X L ! L0 = 2 3 4 = X e e dX ! X e (1! )dX +" a X +b X +c X Z # Z # kT ! !" !" # e kT dX !" 2 2 2 +" a X +" a X +" a X 5/2 $ ! ! ( ! 1 & kT b 4 kT & 1 b 4 kT 1 b 3 + kT . L ! L0 ! % X e dX ! X e dX ) = ! X e dX = ! Z # kT # Z kT # Z kT 4 ,- a /0 &!" !" & !" ' * 2 2 2 $+" a X +" a X ( +" a X & ! b 3 ! & ! kT Z ! % e kT dX ! X e kT dX ) = e kT dX = + # kT # # a &!" !" & !" ' * 3 b d 3 b L ! L0 ! ! kT ! = L " L ! " k 4 2 0 2 a dT 4 a most common case (highly repulsive): b < 0 ! " > 0 but constant thermal expansion !!! Quantum description single particle Interactions Set of particles indistinguishable particles: -bosons : phonons... -fermions : electrons... Anharmonic spectra Thermal statistics E E 6 0 K Equilibrium 5 harmonic volume lattice 4 V(T) 3 2 ρ(Τ) 1 0 V0 V 0 ρ Phonons Solid = collection of quantum "harmonic" oscillators : Einstein, Debye etc. 1 Normal mode p, "phonon" : k , ω , energy ! ! ( V ), population np (T ) = p p p !! p 1 e kT " 1 Internal energy : U = U0 + ! !" p + !np !" p p 2 p T U dT T 1 "n $ Free energy: " $ # p F = U ! T S = U ! T ' & = U ! T ) !( p ' & dT # "T #%V T # T # "T # % 0 p 0 V 1 "(U0 + & !' p ) "F # p 2 "(!' p ) Phonons' pressure : P = ! % = ! ! np "V $ "V & "V T p !V " !V " dV = $ dP + $ dT = 0 $n $( % ) !P # !T # 1 "1 #V $ #P $ 1 p ! p T P ! = & ' & ! = " #T 3 V #P % #T % 3 & $T $V !V " !V " !P " T V p % $ = & $ = 3' V !T # !P # !T $ P T #V Κ isothermal compressibility T = 1/(bulk modulus) Grüneisen Parameter 1 $np $(!% p ) 1 "n ! = " # = ! C p T & & p VP = !! p p specific heat 3 p $T $V p V "T 1 $n $(!% ) 1 $n !% V $% 1 p p ( p p )( p ) C ! p = " #T = #T " = #T & Vp &' p 3 $T $V 3 $T V % p $V 3 V $# p $(ln# p ) ! p = " = " p Grüneisen factor # p $V $(lnV ) C # 1 1 $ Vp p 1 ! = " C # ! = "T % CV = "T %# %CV T $ Vp p 3 C $ p 3 3 p $ Vp p 3 Grüneisen Parameter ! = #$ "T #CV a k V $#D $(ln#D ) Debye simplification ! p = a.! D = ."D ! p =! = " = " ! #D $V $(lnV ) α 1 T << θ ! ! T 3 ! = " #$ #C D 3 T V "constants" T >> θD ! " const. θD 0 T Example: Alkali-Halides Grüneisen Parameter : 3 ! = #$ "T #CV of order of a few units not constant/ T ! (T ) stable at low T or high T at low T, ! "# = cst ! ! C T V G. K. White,Proceedings of the Royal Society of London. A .Vol. 286,(1965) 204 Conduction electrons in metals compressibility Ne 2 Ue 1 $Pe % 1 N 1 2 Drude model : Pe = kT = !e = "T # ' = "T k = "T CV e V 3 V 3 $T &V 3 V 3 3 Electrons pressure 2 Ue 1 2 Free electron gaz with Fermi statistics : Pe = ! = " C 3 V e 3 T 3 V e 1 ! = " (# C + # C ) γe (= 2/3) electronic Grüneisen Parameter 3 T e Ve l Vl electrons phonons ! 2 Pereira et al., J. of Appl. Physics, 41 (1970) C = k2g(E )T V e 2 F 3 at low temperature : ! ! aT + b T + .... Copper at low. temp. ! ! 1.3 " 10#10 T + 2.7 " 10#11 T 3 + .... γe = 0.57 Atomic configurations change in shape/size as function of T Orbital effects: L ≠ 0 ions Jahn Teller : symmetry lowering Crystal Field Splitting B. Lüthi, H.R. Ott, Solid State Comm., 33 (1980) 717 "Schottky type" thermal expansion E Change in valence B. Kindler et al., Phys. Rev. B, 50 (1994) 704 YbCu4In Phase transitions 2 m #V 1 1 % #V ( F(T,!,V ) = F(T,!,V0 ) " A! V0 + ' * V0 V0 2 $T & V0 ) Coupling between V and order parameter η Elastic energy "V !V = η V0 m m 1 #$V % 1 #" % !V = A"#T "$ !" = ' = A()T ( 3 #T & 3 #T ' P & P 1 ! = (T " T )# ! = # A$% $(& + m)$(T # T )&+m#1 C " 3 T C superconductivity in CeCu2Si2 Microscopic origin of A G. Bruls et al., Phys. Rev. Lett. 72, (1994) 1754 ordering = collective phenomena pair interactions ! ! example : localised magnetism H ij = !2Jij Si " S j !J A ij Jij depends on distance ! " j !V !Jij !Jij most commonly < 0 " < 0 " A < 0 " #$ > 0 !rij !V Example: Spontaneous magnetostriction and Invar effect Thermal expansion of Iron Thermal expansion of Nickel F. C. Nix and D. MacNair, Phys. Rev. (1941) TC Instability of iron magnetic moment !m !V < 0 " < 0 below T !V !T C In Invar the anomaly is shifted below room temp. 0 Fe65Ni35 !300K ! .
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