www.neel.cnrs.fr

Thermal expansion of solidsMehdi Amara,Université Joseph-Fourier, Institut Néel, C.N.R.S., BP166X, F-38042 Grenoble, France

Cryocourse 2011

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...

Introduction

Changes in size, volume while cooling/heating

Solids

Liquids

Gases

! =1L"L"T

#$% P

=13&

! =1V"V"T

#$% P

volume thermal expansion :

linear thermal expansion :

Orders of magnitude for α(room temperature and pressure)

10-4 – 10-3

10-6 10-5

insulators

10-3 – 10-2

metals polymers10-4

ideal gaz : ! =13T

< <

Thermal expansion thermometry

Anders Celsius (circa 1710) : centigrade water thermometric scale.

Gabriel Fahrenheit (circa 1710) : scaled mercury thermometer.

contrast between liquid and solid

Galileo Galileis (1596) : water density change thermoscope.

contrast between two solidsJohn Harrison ( 1750): first bimetallic strip for temperature compensation

contrast between gas and solidSantorio Santorii (circa 1610) : liquid pushed by expansion of air in a glass tube.

!

Ferdinand II, Grand Duke of Tuscany (1641): bulb alcohol thermometer.

latter used as thermometer and thermostat

Technological issues

First systematic approaches : high precision mechanisms

Since early human industry: buildings, ceramics, metallurgy

Problem more acute with metals :

in case of thermal/material inhomogeneity

in railways, heat engines, bridges, large buidings,transonic planes... and cryogenic equipements

expansion joints or other...

John Harrison :measurement of time

gridiron pendulum (1726)

L = constant

Thermal compatibility (reinforced concrete...)

Exercice : pendulum

1) Pendulum Clock: arm made of brass

Length at T = 20 °C, L0 = 50 cm!brass = 1.9 "10

#5 K #1

2) Brass+Steel compensation: "gridiron"! steel = 1.3"10

#5 K #1

!0 = 2"L0g

= 2 s

- compute τ at T = 15 °C- by how much is the clock shifted one day latter ?

Lsteel

Lbrassl

- compute the T = 20°C ratio between Lbrass and Lsteel so that l is constant .

L0

Thermal expansion measurementProbing a small change in length Δl : sensitivities better than 10-4

- macroscopic

- microscopic : diffraction of monochromatic photons, neutrons

powder single crystal

optical

capacitive : C(l)inductive : mutual inductance m(l)

interferometric : Fizeau, fringes displacementoptical lever : basic opto-mechanical amplifier

electricalresistive : strain gauge R(l)

Bragg law : d =!

2sin"# $" = % tan" & $l

l

10-5 10-6

+ radio-frequency resonance

10-6

10-7

10-9

10-6

reflections at large θ

10-1110-9

A capacitance dilatometer

Courtesy of Didier DufeuInstitut Néel

Three-terminal capacitance method

resolution up to 1 Angströmsensitivity 10-6-10-8

Temperature range 2-300 K

Angular range 360 °Magnetic Field range 0- 6.5 T

Some "reference" α curves

Thomas A. Hahn, J. Appl. Phys. 41 (1970) 5096 G.K. White, Journal of Physics D: Applied Physics 6 (1973), 2070

Copper Silicon

Graphite Fused Silica

Expansion : microscopic interpretation

Atomic :inflation ?

vibrations ?

"free" movement/kinetic energy ? gaz, liquids but not solids

solids, liquids

Electronic:

Free electrons in metals... but fermions

radial electronic configuration !

! (! r ) = Rnl (r).Ylm(" ,#) thousands of K to change

change in volume ?

Bonding electrons ? few levels well separated

Small amplitude: harmonic approximation

L ! L0 =

X e!AX2

kT dX!"

+"

#

e!AX2

kT dX!"

+"

#= 0

Llibration

Loscillation

Liquidspolymers

CrystalsGlassesLiquidsPolymers

e!V (X )

kT

No thermal expansion !

Anharmonicity

"high" temp. specific heat doesn't depart from Dulong and Petit

No volume/Thermal dependence of elastic constants

Failure of the harmonic approximation

Zero thermal expansion

infinite thermal conductivity

More realistic form of the interaction potential

strongly repulsive for L < L0

...

slowly varrying attractive for L > L0

odd powers in the expansion of V(X)X

V(r)

V (r) = Crm

!e2

4"#0

1r

alkali halides

Classical anharmonic "description"

Expansion : V (X) = V0 + a X2 + b X 3 + c X 4

L ! L0 =

X e!a X2 +b X3+c X4

kT dX!"

+"

#

e!a X2 +b X3+c X4

kT dX!"

+"

#=1Z

X e!a X2

kT e!b X3+c X4

kT dX!"

+"

# !1Z

X e!a X2

kT (1! b X3 + c X 4

kT)dX

!"

+"

#

L ! L0 !1Z

X e!a X2

kT dX!"

+"

# !bkT

X 4e!a X2

kT dX!"

+"

#$%&

'&

()&

*&= !

1ZbkT

X 4e!a X2

kT dX!"

+"

# = !1ZbkT

34

kTa

+,-

./05/2

Z ! e!a X2

kT dX!"

+"

# !bkT

X 3e!a X2

kT dX!"

+"

#$%&

'&

()&

*&= e

!a X2

kT dX!"

+"

# = + kTa

L ! L0 ! !

34ba2

kT ! =

ddT

L " L0 ! "34ba2

k

most common case (highly repulsive): b < 0!" > 0

low temperature approximation

but constant thermal expansion !!!

e!x2dx

!"

+"

# = $ x4e!x2dx

!"

+"

# =3 $4

Quantum description single particle

Anharmonic spectra Thermal statisticsE

VV001

2

3

45

6

E

ρ0

ρ(Τ)

V(T)

Equilibriumvolume

indistinguishable particles:-bosons : phonons...-fermions : electrons...

Interactions Set of particles

0 Kharmonic

lattice

PhononsSolid = collection of quantum "harmonic" oscillators : Einstein, Debye etc.

Internal energy : U =U0 +12p

! !" p + np !" pp!

Phonons' pressure : P = !

"F"V

#$%T

= !

"(U0 +12p

& !' p )

"V! np

"(!' p )"Vp

&

Free energy: F =U ! T S =U ! T "U

" #T$%&V

d #T#T0

T

' =U ! T !( pp) 1

#T"np" #T

$%&Vd #T

0

T

'

np (T ) =1

e!! p

kT "1

! =13"1V

#V#P

$%&T

'#P#T

$%&V

! = "13#T

$np$T

$(!% p )$Vp

&

Normal mode p, "phonon" : kp, ωp , energy , population !! p (V )

dV =!V!P

"#$TdP +

!V!T

"#$ PdT = 0

%!V!T

"#$ P

= &!V!P

"#$T

!P!T

"#$V

= 3'VΚT isothermal compressibility

=1/(bulk modulus)

Grüneisen Parameter

! = "13#T

$np$T

$(!% p )$Vp

& = ! pp&

! =13"T CVp # p

p$

Debye simplification

! =13"T

CVp# p$CVp

p$

% CVp$ =13"T % # %CV

Grüneisen Parameter

! p = a.!D =

a k!."D ! p =! = "

V#D

$#D$V

= "$(ln#D )$(lnV )

! =13"T # $ #CV

"constants"

T << θD ! ! T3

T >> θD !" const.

! = 3"T #CV

#$

T θD

α

0

! p = "V# p

$# p

$V= "

$(ln# p )$(lnV )

CVP =

1V!! p

"np"T

! p = "

13#T

$np$T

$(!% p )$V

=13#T (

$np$T!% p

V)(" V

% p

$% p

$V)= 13#T &CVp & ' p

p specific heat

p Grüneisen factor

Example: Alkali-Halides

of order of a few units

Grüneisen Parameter :

not constant/ T ! (T )

G. K. White,Proceedings of the Royal Society of London. A .Vol. 286,(1965) 204

stable at low T or high T

at low T, ! ! CV!T " # = cst

! = 3"T #CV

#$

Conduction electrons in metals

Free electron gaz with Fermi statistics :

!e =13"T #

$Pe$T

%&'V

=13"T

NVk= 1

3"T

23CVeDrude model : Pe =

NeVkT =

23UeV

Pe =23UeV

!e =13"T

23CVe

CVe =! 2

2k2g(EF )T

! =13"T (# e CVe + # l CVl )

phononselectrons

γe (= 2/3) electronic Grüneisen Parameter

at low temperature : ! ! aT + b T 3 + ....

Copper at low. temp.

Pereira et al., J. of Appl. Physics, 41 (1970)

! ! 1.3"10#10 T + 2.7 "10#11 T 3 + ....

γe = 0.57

Electrons pressure

compressibility

Atomic configurations

Change in valence

Orbital effects:

B. Kindler et al., Phys. Rev. B, 50 (1994) 704

YbCu4In

Crystal Field Splitting

Jahn Teller : symmetry loweringchange in shape/size as function of T

"Schottky type" thermal expansion

B. Lüthi, H.R. Ott, Solid State Comm., 33 (1980) 717

L ≠ 0 ions

E

Coupling between V and order parameter η

G. Bruls et al., Phys. Rev. Lett. 72, (1994) 1754

Phase transitionsF(T ,!,V ) = F(T ,!,V0 ) " A!

m #VV0

V0 +121$T

#VV0

%&'

()*

2

V0

η!V ="VV0

Elastic energy

!V = A "#T "$m !" =13#$V#T

%&' P

=13A ()T (

#"m

#T%

&' P!" = #

13A $%T $ (& + m) $ (TC # T )&+m#1

superconductivity in CeCu2Si2

! = (TC " T )#

Microscopic origin of A

ordering = collective phenomenapair interactions

example : localised magnetism H ij = !2Jij

!Si "!Sj

depends on distanceJij A !

!Jij!Vj

"!Jij!rij

< 0 "!Jij!V

< 0" A < 0"#$ > 0most commonly

Example: Spontaneous magnetostriction and Invar effect

F. C. Nix and D. MacNair, Phys. Rev. (1941)Thermal expansion of Nickel

TC

Thermal expansion of Iron

Instability of iron magnetic moment!m!V

< 0 "!V!T

< 0 below TC

In Invar the anomaly is shifted below room temp.

Fe65Ni35 !300K ! 0