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 volume thermal expansion : linear thermal expansion :
1 "V # % V "T $ P 1 "L # 1 != % = & L "T $ P 3
!=
Orders of magnitude for α (room temperature and pressure) Gases
10-3 – 10-2
Liquids
10-4 – 10-3
Solids
ideal gaz : ! = 1
3T
polymers insulators metals 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 Length at T = 20 °C, L0 = 50 cm ! brass = 1.9 " 10 #5 K #1 ! 0 = 2"
L0
L0 = 2s g
- compute τ at T = 15 °C - by how much is the clock shifted one day latter ? 2) Brass+Steel compensation: "gridiron" ! steel = 1.3 " 10 #5 K #1
l
Lbrass 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
optical lever : basic opto-mechanical amplifier 10-6 interferometric : Fizeau, fringes displacement 10-9 + radio-frequency resonance 10-11 capacitive : C(l)
electrical
10-9 inductive : mutual inductance m(l)
10-7
resistive : strain gauge R(l)
10-6
- microscopic : diffraction of monochromatic photons, neutrons Bragg law : d =
! $l # $" = % tan " & 2 sin " l
powder 10-5
single crystal 10-6
reflections at large θ
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 Institut Néel
Angular range 360 °
Some "reference" α curves Copper
Thomas A. Hahn, J. Appl. Phys. 41 (1970) 5096
Graphite
Silicon
G.K. White, Journal of Physics D: Applied Physics 6 (1973), 2070
Fused Silica
Expansion : microscopic interpretation Atomic :
!
inflation ? ! (r ) = R nl (r).Y lm (" ,# )
thousands of K to change radial electronic configuration !
"free" movement/kinetic energy ? gaz, liquids but not solids vibrations ? solids, liquids
Electronic:
change in volume ?
Bonding electrons ? few levels well separated Free electrons in metals... but fermions
Small amplitude: harmonic approximation e
L oscillation
V (X) kT
Crystals Glasses Liquids Polymers
+"
Liquids polymers L ! L0 =
L libration
!
#
A X2 ! X e kT dX
!" +"
#
e
!
AX kT
=0
2
dX
!"
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)
strongly repulsive for L < L0 slowly varrying attractive for L > L0
alkali halides V (r) =
C rm
e2 1 ! 4"# 0 r X
odd powers in the expansion of V(X)
Classical anharmonic "description" Expansion :
V (X) = V0 + a X 2 + b X 3 + c X 4 +"
L ! L0
+"
3 $ # # 4 !" 2 3 4 !"a X 2 dX a X b X +c X # +" +" 3 4 ! ! ! 1 1 b X + c X !" kT = = # X e kT e dX ! # X e kT (1 ! )dX 2 3 4 a X +b X +c X +" Z Z kT ! !" !" kT e dX # +"
L ! L0
low temperature approximation
2
3
!
a X2 kT dX
a X +b X +c X ! kT Xe
!" $+"
1& ! % # Xe Z &!" '
e
4
b ! kT
+"
#
X e
dx = $
a X2 ( kT dX & =
) &*
!"
$+" ! a X 2 b & Z ! % # e kT dX ! kT &'!" L ! L0 ! !
4
!
! x2
+"
#
X 3e
!
a X2 ( +" kT dX & =
!"
3 b kT 4 a2
1 b ! Z kT
!=
) &*
#
e
+"
#
4
X e
!
a X2 kT dX
!" !
a X2 kT dX
= +
!"
d 3 b L " L0 ! " k 2 dT 4a
most common case (highly repulsive): b < 0 ! " > 0 but constant thermal expansion !!!
2
x 4 e! x dx =
1 b 3 + kT . =! 0 Z kT 4 , a / kT a
5/2
Quantum description Interactions
single particle indistinguishable particles: -bosons : phonons... -fermions : electrons...
Set of particles
Thermal statistics
Anharmonic spectra E
E 6
0K harmonic lattice
Equilibrium volume
5 4
V(T)
3 ρ(Τ)
2
V0
1 0
V
0
ρ
Phonons Solid = collection of quantum "harmonic" oscillators : Einstein, Debye etc. 1
Normal mode p, "phonon" : kp, ωp , energy !! p (V ), population n p (T ) = Internal energy :
e
1 U = U 0 + ! !" p + ! n p !" p p 2 p
Phonons' pressure : P = !
dV = %
kT
"1
"U $ dT # 1 "n p $ F =U !T S =U !T' = U ! T ) !( p ' &% &% dT # " T T T " T # # # # V p V 0 0 T
Free energy:
!! p
!V " !V " dP + $ $ dT = 0 !P # T !T # P
!V " !V " !P " = & = 3' V $# $# $ !T P !P T !T # V
"F # %$ = ! "V T
T
1 "(U 0 + & !' p ) 2 p "V
1 "1 #V $ #P $ != & ' & 3 V #P % T #T % V
! & np p
"(!' p ) "V
$ n p $(!% p ) 1 ! = " #T & 3 $V p $T
ΚT isothermal compressibility = 1/(bulk modulus)
Grüneisen Parameter $ n p $(!% p ) 1 ! = " #T & = &! p 3 $T $V p p
CVP =
" np 1 !! p V "T
p specific heat
$ n p $(!% p ) 1 $ n p !% p 1 V $% p 1 ! p = " #T = #T ( )(" )= # T & CVp & ' p 3 $T $V 3 $T V % p $V 3 $(ln # p ) V $# p !p=" =" # p $V $(lnV )
1 ! = " T $ CVp # p 3 p
1 ! = "T 3
$ CV # p % $ CV $ CV p
p
p
Grüneisen Parameter
Debye simplification 1 ! = " T # $ # CV 3 "constants"
! p = a. ! D =
T > θD
!=
ak ."D !
p
p Grüneisen factor
1 = " T % # % CV 3
3 #$ " T # CV ! p =! = "
α
V $# D $(ln # D ) =" # D $V $(lnV )
!!T3 ! " const. 0
θD
T
Example: Alkali-Halides Grüneisen Parameter : !=
3 #$ " T # CV
of order of a few units not constant/ T
! (T )
stable at low T or high T at low T, ! T " # = cst
G. K. White,Proceedings of the Royal Society of London. A .Vol. 286,(1965) 204
! ! CV
Conduction electrons in metals compressibility
Drude model : Pe = Electrons pressure
Ne 2 Ue kT = V 3V
1 $P % 1 N 1 2 ! e = " T # e ' = " T k = " T CV e 3 $T & V 3 V 3 3
Free electron gaz with Fermi statistics : Pe = 1 ! = " T (# e CVe + # l CVl ) 3 ! 2 = k g(EF )T 2 2
CV e
1 2 ! e = " T CV e 3 3
γe (= 2/3) electronic Grüneisen Parameter
electrons phonons
at low temperature : ! ! aT + b T 3 + .... Copper at low. temp. ! ! 1.3 " 10 #10 T + 2.7 " 10 #11 T 3 + .... γe = 0.57
2 Ue 3V
Pereira et al., J. of Appl. Physics, 41 (1970)
Atomic configurations change in shape/size as function of T Jahn Teller : symmetry lowering
Orbital effects: L ≠ 0 ions 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
#V 1 1 % #V ( F(T , !,V ) = F(T , !,V0 ) " A ! m V0 + V0 ' * V0 2 $ T & V0 )
Coupling between V and order parameter η
!V =
"V V0
!V = A" # T " $
Elastic energy
η 1 #$V % 1 #" m % !" = '& = A( ) T ( 3 #T P 3 #T '& P 1 !" = # A$ % T $ (& + m) $ (TC # T )& + m#1 3
m
! = (TC " T )#
superconductivity in CeCu Si
2 G. Bruls et al., Phys. Rev. Lett. 72, (1994)21754
Microscopic origin of A ordering = collective phenomena pair interactions
! ! = !2Jij Si " S j
example : localised magnetism H ij
Jij depends on distance A ! "
most commonly
!J ij !rij
