Short historical notes Dislocations: V. Volterra (1905) K. Kondo - - PowerPoint PPT Presentation

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Short historical notes

Dislocations: V. Volterra (1905)

• K. Kondo (1952)
• J. F. Nye (1953)
• B. Bilby, R. Bullough, E. Smith (1955)
• E. Kröner, I. Dzyaloshinskii, G. Volovik, M. Kléman, I. Kunin,
• J. Madore, A. Kadić, D. Edelen, H. Kleinert,
• E. Bezerra de Mello, F. Moraes, C. Malyshev, M. Lazar, ...

Disclinations: F. Frank (1958)

• I. Dzyaloshinskii, G. Volovik (1978)
• J. Hertz (1978)

Torsion: E. Cartan (1922) Torsion in gravity: A. Einstein, E. Schrödinger, H. Weyl,

• T. Kibble, D. Sciama, R. Finkelstein,

F.Hehl, P. von der Heyde, G. Kerlich, J. Nester,

• Y. Ne’eman, J. Nitsch, J. McCrea, J. Mielke, Yu. Obukhov
• M. Blagojecić, I. Nikolić, M. Vasilić, K. Hayashi, T. Shirafuji,
• E. Sezgin, P. van Nieuwenhuizen, I. Shapiro, ...

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• M. O. Katanaev, Steklov Mathematical Institute, Moscow
• J. Zanelli, Centro de Estudios Cienificos, Valdivia, Chile

Chern-Simons term in the Geometric Theory of Defects

Katanaev, Volovich Ann. Phys. 216(1992)1; ibid. 271(1999)203 Katanaev Theor.Math.Phys.135(2003)733; ibid. 138(2004)163 Phisics – Uspekhi 48(2005)675. Notation Elasticity theory of small deformations

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ij j i ij ij k ij k

f σ σ λδ ε µε ∂ + = = +

• Newton’s law
• Hooke’s law
• continuous elastic media = Euclidean three-dimensional space
• Cartesian coordinates
• Euclidean metric
• displacement vector field
• strain tensor
• stress tensor

( ) ,

i

f x λ µ

• density of nonelastic forces
• Lame coefficients

( 0)

i

f =

3 1 2

, 1,2,3 ( ) ( )

i i ij i ij i j j i ij

x y i u x u u δ ε σ = = ∂ + ∂ 

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Differential geometry of elastic deformations

( )

i i

y x y → ( ) 2

k l ij kl ij i j j i ij ij i j

y y g x u u x x δ δ δ ε ∂ ∂ = ≈ − ∂ − ∂ = − ∂ ∂

1 2 (

)

ijk i jk j ik k ij

g g g Γ = ∂ + ∂ − ∂ ≠  ( )

l l m l ijk i jk ik jm

R i j = ∂ Γ − Γ Γ − ↔ =    

i i j k jk

x x x = −Γ    

l ijk

R =

k k k ij ij ji

T = Γ − Γ =   x ( )

i i i

x y u x = +

• diffeomorphism:

3 3

→  

• induced metric

(*) (*)

• Christoffel’s symbols
• curvature tensor
• extremals (geodesics)
• Saint-Venant integrability conditions of
• torsion tensor

i

y

i

x

ij

δ

ij

g y 

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Dislocations

Edge dislocation Screw dislocation

b b

1

x

2

x

b

1

x

2

x

3

x

3

x

• Burgers vector

Vacancy Linear defects:

( )

i

u x   

is continuous = elastic deformations is not continuous = dislocations Point defects:

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Edge dislocation

C

b

1

x

2

x

i i i C C

dx u dx y b

µ µ µ µ

∂ = − ∂ = −

∫ ∫  

, 1,2,3 xµ µ = ( )

i

y x ( )

i i i i C S

b dx e dx dx e e

µ µ ν µ µ ν ν µ

= = ∧ ∂ − ∂

∫ ∫∫ 

( ) ( )

i i ij j ij ij ik j k

T e e R

µν µ ν µ ν µν µ ν µ ν

ω µ ν ω ω ω µ ν = ∂ − − ↔ = ∂ − − ↔

i i

b dx dx T

µ ν µν

= ∧

∫∫

• arbitrary curvilinear coordinates
• is not continuous !
• Burgers vector in elasticity
• torsion
• curvature

ij ji µ µ

ω ω = −

• definition of the Burgers vector

in the geometric theory

ij

Rµν =

ij µ

ω →

then Back to elasticity: if

( ) lim

i i i

y e x y

µ µ µ

∂  =  ∂  

• triad field

(continuous on the cut) SO(3)-connection

(*)

(*) ⇒

• outside the cut
• on the cut

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Disclinations

( )

i

n x

i

n ( )

i j i j

n n S ω = (3)

j i

S ∈ (3)

ij ji

ω ω = − ∈ so 1 2

jk i ijk

ω ε ω =

ijk

ε

123

( 1) ε =

• unit vector field
• fixed unit vector
• orthogonal matrix
• Lie algebra element (spin structure)
• rotational angle
• totally antisymmetric tensor

Examples

C

1

x

2

x

C

1

x

2

x

ij ij C

dxµ

µω

Ω = ∂

∫ 

jk i ijk

ε Θ = Ω

i i

Θ = Θ Θ 2π Θ = 4π Θ =

• Frank vector

(total angle of rotation) Ferromagnets

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More examples Nematic liquid crystals

C

1

x

2

x

C

1

x

2

x Model for a spin structure:

1

( )

j k j i i k

l S S

µ µ −

= ∂

• trivial SO(3)-connection (pure gauge)

π Θ = 3π Θ =

2

cos sin (1 cos ) (3),

k j j j j i ki i i i i

S ω ε ω ω δ ω ω ω ω ω ω ω ω = + + − ∈ = 

• principal chiral SO(3)-model

ij

l

µ µ

∂ =

i i

n n −  ( ) (3)

i x

ω ∈ so

• basic variable

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( )

ij x

ω ( ) lim

ij ij ij

x

µ µ µ

ω ω ω ∂  =  ∂  

• is not continuous !
• SO(3)-connection

(continuous on the cut)

( )

ij ij ij ij

dx dx dx

µ µ ν µ µ ν ν µ

ω ω ω Ω = = ∧ ∂ − ∂

∫ ∫∫ 

( )

ij ij ik j k

Rµν

µ ν µ ν

ω ω ω µ ν = ∂ − − ↔

ij ij

dx dx R

µ ν µν

Ω = ∧

∫∫

• curvature
• definition of the Frank vector

in the geometric theory Back to the spin structure: if

2

n∈

then

(3) (2) →  

• outside the cut
• on the cut
• the Frank vector

Frank vector

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Summary of the geometric approach (physical interpretation)

3



with a given Riemann-Cartan geometry Independent variables

i ij

eµ

µ

ω     

• triad field
• SO(3)-connection

( ) ( )

i i ij j ij ij ik j k

T e e R

µν µ ν µ ν µν µ ν µ ν

ω µ ν ω ω ω µ ν = ∂ − − ↔ = ∂ − − ↔

• torsion (surface density of

the Burgers vector)

• curvature (surface density of

the Frank vector)

0, 0, 0, 0,

ij i ij i ij i ij i

R T R T R T R T

µν µν µν µν µν µν µν µν

= = = ≠ ≠ = ≠ ≠

Elastic deformations: Dislocations: Disclinations: Dislocations and disclinations: Media with dislocations and disclinations =

=

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The free energy

[ ] [ ]

HE CS

S S e S ω = +

( )

i i

g e

µν µν µ µ

δ δ = =

( )

3

1 1 CS 2 3 j i i k i i ijk i

S d J ω ω ε ω ω ω ω

∧ ∧ ∧ ∧

= + −

∫

 1 2

, : ,

ij kij ij k k ijk µ µ µ µ

ω ω ε ω ω ε = = , ,... 1,2,3; , ,... 1,2,3 i j µ ν = =

( )

( ): 2

ij i j k ijk k k ijk

R R

µν µν µ ν ν µ µ ν

ω ε ω ω ω ω ε = = ∂ − ∂ +

No dislocations:

• SO(3) connection

(1-form, the only variable)

• free energy for disclinations
• the source term (2-form)
• equations of equilibrium
• the total free energy
• Euclidean metric

HE

S →

• indices

i

J

k k

R J

µν µν

=

• curvature

Three dimensions:

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One linear disclination

( )

2 int 3 i i

S q J q

µ µ

δ δ δω = −   x q

( ) ( )

int 3 3 3 2 3 i i i i i i i i

S dq J dtq J q dtd xq J x q d x J q

µ µ µ µ µ µ µ µ

ω ω ω δ ω δ = = = = − = −

∫ ∫ ∫ ∫

    x q

• the core of disclination
• the interaction term

Notation:

3 3

,

x y

ω ω

• the only nontrivial components

3

x

• the curvature tensor

1 3 3 3 2 2 i z x y

ω ω ω = −

• one complex component

One disclination along axis

( )

3,

q t t

µ

∈ ∈  

1 2

, , : x x x y z x iy = = = +

( )

3 3 3

2

zz z z z z

R ω ω = ∂ − ∂

• the source term

1 2

( , ) x x = x C

1

x

2

x

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One straight linear disclination

3 z

iA z ω = −

( )

1

z

z z πδ ∂ =

xy yx

ω ω ∂ = ∂ 2 arctan x A C y ω = − +

( )

3

4 ,

zz

R iA z A π δ = ∈

Fixing the source term:

• new kind of defect

The solution:

3 3 2 2 2 2

2 2 ,

x y

Ay Ax x y x y ω ω = − = + +

• real components

Rotational angle field The integrability conditions are fulflled

• a general solution

( )

ω x

• important formula

2 2 2 2

2 2 ,

x y

Ay Ax x y x y ω ω ∂ = − ∂ = + + tan = , : 2 x C A y A ω π ϕ ϕ = ⇒ =

• polar angle

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One straight linear disclination

( )

, x y ω

To make the rotational angle field well defined, we must impose the quantization condition:

, 2 n A n n ω ϕ = ∈ ⇒ =  ฀

12 2 2 12 2 2

sin , cos .

x y

ny n x y nx n x y ω ϕ ω ϕ = − = − + = = +

SO(3)-connection:

x y C

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Conclusion

1) The first example of disclination is described within the geometric theory of defects. 2) The Chern-Simons term is well suited for disclinations in the geometric theory of defects. 3) Linear disclinations correspond to a new type of defects of the SO(3) connection (the spin structure).