Curie’s principle: Application to

condensed matter

Curie’s principleCurie’s principle

• Enunciated by Pierre Curie in 1894, after his study of piezoelectricity

Allows a qualitative analysis of a phenomena

• Simplify theoretical modelling andDesign of experiment

Cause : physical system and its environment• Physical system: atom, molecule, crystal, any sample• Environment : electric, magnetic, gravitational fields

incident wave, force or stress

• Effect : a physical property

« Les symétries des causes sont inclues dans celles des effets »

« L’effet est plus symétrique que la cause »

• J. Sivardière. « La symétrie en

mathématiques, physique et chimie »

PUG, Grenoble 1995.

’’The symmetry of a cause is always preserved in its effects’’

’’Effects are more symmetrical than their causes’’

Exact wording (1)

« Lorsque certaines causes produisent certains effets,

les éléments de symétrie des causesdoivent se retrouver dans les effets produits »

Pierre Curie, 1894

‘’If certain causes yield the known effects, the symmetry elements of the causes should be contained in the generated

effects’’

Exact wording (2)

« Lorsque certains effets révèlent une certaine dissymétrie,

cette dissymétrie doit se retrouver dans les causes

qui leur ont donné naissance »

Pierre Curie, 1894

‘’If the known effects manifest certain dissymmetry (absence of symmetry elements), this latter

should be contained in the causes which have generated those

effects’’

Relations between

symmetry groups K : symmetry group of the cause G : symmetry group of the effect

K is a sub-group of G

K G

Curie’s principle similar to Franz Neumann’s principle (1833) :

Crystal macroscopic physical properties have the samepoint group as this crystal.

Minnigerode B. (1884) : K (crystal) sub-group of G (property)

Curie well defined cause and effect concepts: principle is operational.

Examples

Cause: water moleculeK=2mm

Effect: polarisation G=mK G

Cause: crystal andX-ray beam

K=3/m m=3Effect: diffraction

patternG=6

K G

Other wording

« Les effets produits peuvent être

plus symétrique que les causes »

’’The produced effects can be more symmetrical than their causes’’

« La dissymétrie crée le phénomène »

‘’Dissymmetry creates the phenomenon’’

« Il n’y a pas degénération spontanée de dissymétrie »

‘’There is no spontaneous generation of dissymmetry’’

Obvious?

Curie’s principle is used naturallywithout naming it

• Electrostatics problems• Kinematics

Dissymmetrical phenomena makes us think of dissymmetrical origin

For symmetry reasons…

Curie’s mistake……symmetry breaking

• Buckling (?)

/mm

?

• Hydrodynamics

• Two baloons (?)

R<RcR>Rc

Figures of Chladni

Metastability……phase transition

• Effect is not unique

Pasteur and asymmetry • In 1844 Pasteur works

on tartric acid

Dans les champs de l’observation, le hasard ne favorise que les esprits préparés

Louis Pasteur1822-1895

L’asymétrie, c’est la vie

• Fermentation gives optically active molecules because only one form is active in the process.

• Fermentation is a process of life Study of bacteria, vaccines...

• Two forms : tartric acid, optically activeacide paratartric, racemic, inactive

Problem of isomers...• Pasteur works on racemic acid

and separate right and left species (1848).

Dextral positive

In the fields of observation chance favors only the prepared mind

Symmetry of the cause K

If there is only one effect, of symmetry G

If the produced effect is not unique, it breaks the K symmetry,

and forms with the other effectsa set of symmetry G’

Towards the Wigner theorem…

Generalized Curie’s principle

K G

K = G’

Symmetry of physical quantities

• Polar and axial quantities• Polar (m) : sign does not depend on the orientation convention

• F, E, D, v, m,

• Axial (/m) : depends on the space orientation• Torque, B, H, M...

•Mach’s paradox

S

N

• Lorentz force? : F = q v B (polar)

S

N OZMA problem…Dimensionality and chirality

Parity breaking

Desintegration of 60Co T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956)  

Wu et al. Phys. Rev. 105, 1413 (1957)

N

/m Symmetry CPT…

Charge Parity Time

60Co

e-

OZMA solution

M. Gardner, the ambidextrous universe

S

N

I (front)

Up

1-Definition of North magnetic pole

2-Definition of the left

South=left

60Co

Symmetry breakingin life

DNA is a dextral helixfor ALL living beings

Halibut and flatfish are born with eyes on each side…

Mollusks are dextral, very rarely senestral (left-handed)...

Fiddler crabflatfish

Importance of chirality

• Origine of life • Homochirality of amino acids and sugars• No explanation yet

• Hypothesis: Coriolis force, weak interaction.

• Extraterrestrial origin?• Meteorites, polarised radiations.

• Pharmacology• Enantiomers behave differently• Vitamin C, parfumes, thalidomide

Crystals of aspirin viewed under crossed polarizers

Cauchy stress tensorCauchy stress tensor

J.F. Nye : Physical properties of crystals

x3

x1

x2

11

13

12

• Stress: force/m2

• Homogenous• No force or torque ij=ji

22

32

23

33

31

21

Force /mmSymmetricalcompression

Shear mmmm

Ferroelectricity

Point groups compatibles with polarisation P

• The effect P has the point group of a cone: G= m• Point group K of cause verifying K m are :

1, 2, 3, 4, 6, 2mm, 3m, 4mm, 6mm, m

• Extending to aperiodicity: 5, 7, 8, 5m, 7m, 8mm

32

Pz Pz

Example of quartz• Space group P312, class 32

a b

c

Piézoélectricité• Polarisation électrique sous

contrainteG /mm m

Si4+O2-

P

P

32

=32/mm

32

=2/mm

P

32

=2/mm

P

ContrainteQuartz

A2

// A2

// A3

Tenseur piézoélectriqueModèle Meissner (1927)

FAUX !

Symmetry and order

Decrease of symmetry increase of order

Water is very symmetrical /m/mBut disordered

Ice is crystalline, less symmetricalmore ordered

Phase Transitions

•Landau theory :

• G1 and G2 have no relation group/sub-group : 1st order transition (soufre soufre )

• G1 sub-group of G2 (G1 G2)

An order parameter can be defined, zero in the symmetrical phase

Tc

Phase IG1

Phase IIG2 T

Tc T

Tc T• discontinuous• 1st order transition• Hysteresis, latent heat

• continuous• 2nd order transition• Coexistence at critical point

Liquid crystals

Liquid /m/m

T=236 °C

Nematic /mm

T=200 °C

Smectic A

T=175 °C

Smectic C

BaTiO3• Ferroelectric

• Perovskite ABO3

• T > 120 °C, Cubic Pm3m, paraelectric• 0°C < T < 120 °C, Tetragonal P4mm, ferroelectric

P4mm Pm3m, 1st order transition (domains).• -90°C < T < 0 °C, Orthorhombic Cmm2

Cmm2 P4mm, 1st order transition .• T < -90 °C, Rhombohedral R3m

R3m Cmm2, 1st order transition .

Ti

Ba

O4 Å

1er

1er 1er

Rhombohedral Orthorhombic Tetragonal

Ba2+, Ti4+, O2-