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Physical Properties

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(1)

Physical Properties

Reading Assignment:

1. J. F. Nye, Physical Properties of Crystals 2. R. E. Newnham, Properties of Materials 3. R. E. Newnham, Structure-property

Relations

(2)

Contents

First Rank Tensor First Rank Tensor

Second Rank Tensor Second Rank Tensor

Third Rank Tensor Third Rank Tensor 3

1 2

4

Physical Properties Physical Properties

5 Fourth Rank TensorFourth Rank Tensor

(3)

Physical Properties of Crystals

- first rank tensor

- pyroelectricity: develop an electric polarization when their temperature is changed

- hydrostatic pressure instead of temperature change - experiment- clamped- primary pyroelectricity

free - secondary pyroelectricity

( : pyroelectric coefficient)

i i i

P p T p

Δ = Δ

(4)

Pyroelectricity

- Neumann’s principle

in a crystal possessing a center of symmetry

inversion center

property is unchanged after applying the symmetry element

0 p = JG

1 1, 2 2, 3 3

p → − p p → − p p → − p

1 1 1 1 0

p → − =p p p =

centrosymmetric (11 groups) non-pyroelectric

2 2 2 2 2 4 4 2 2

1, , , 3, 3 , ,

m m m m m m m m m

6 6 2 2 2 4 2

, , 3, 3

m m m m m m m

(5)

Pyroelectricity

- a pyroelectric moment can only lie along a direction in a crystal which is unique, in the sense that it is not repeated by any symmetry element.

- polar direction: 이 축에 수직인 2회 회전축이나 거울면을 갖지 않는 극성을 갖는 축

ex) point group 32

a diad axis is a polar, but not unique direction

- all unique directions are polar, but only some polar directions are unique.

(6)

Pyroelectricity

(7)

Pyroelectricity

- polar class- non-zero pyroelectric coefficients 1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, 6mm

R. E. Newnham, Structure-Property Relations

(8)

Pyroelectricity

-Wurtzite-hexagonal ZnS-6mm

polar axis and P are parallel to c, six-fold axis

u=3/8

p=-quc p=+(1/2-u)c

R. E. Newnham, Structure-Property Relations

(9)

Pyroelectricity

R. E. Newnham, Properties of Materials

(10)

ZnS Wurtzite

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.html

(11)

•HCP S2- with Zn2+ in half Tetrahedral holes (only T+ {or T-}

filled)

•Lattice: Hexagonal - P

•a = b, c Å Ã(8/3)a

•Motif: 2S at (0,0,0) & (2/3,1/3,1/2); 2Zn at (2/3,1/3,1/8) & (0,0,5/8)

•2ZnS in unit cell

•Coordination: 4:4 (tetrahedral)

ZnS Wurtzite

(12)

Comparison of Wurtzite and Zinc Blende

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.html

(13)

Comparison of Wurtzite and Zinc Blende

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.html

(14)

Comparison of Wurtzite and Zinc Blende

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.html

(15)

Pyroelectricity

• ferroelectricity: spontaneous polarization reversability

hysteresis

*used as infrared radiation detectors

R. E. Newnham, Structure-Property Relations

(16)

Second Rank Tensor

- electrical conductivity - Ohm’s law

where : current density, : electrical conductivity tensor : potential, : electric field intensity

i ik ik k

k

i ik

k

i ik k

j E

x j

E

E j

σ φ σ

σ φ

ρ

= − ∂ =

=

J. F. Nye, Physical Properties of Crystals

(17)

J. F. Nye, Physical Properties of Crystals

(18)

Second Rank Tensor

- thermal conductivity

where : heat flow rate per unit area, : thermal conductivity tensor, : temperature gradient

i ij

j

i ij

T xj

h K T

x

h K

= − ∂

J. F. Nye, Physical Properties of Crystals

(19)

Second Rank Tensor

- electric polarization

where : electric displacement, : permittivity of vacuum P: polarization

( : dielectric susceptibility) , ( )

relative dielectric constant

o

o

i o ij j ij

i ij j ij o ij ij

r

D E P

D

P E

D E

ε

ε

ε χ χ

ε ε ε δ χ

ε

= +

=

= = +

JG JG JG

ij

= ε ε

ij

/

o

(20)

Second Rank Tensor

J. F. Nye, Physical Properties of Crystals

(21)

Second Rank Tensor

- magnetization

where : magnetic induction, : permeability of vacuum M: magnetization

( : magnetic susceptibility) , ( )

relative permeability

o o

o

i mij j mij

i ij j ij o ij mij

rij ij

B H M

D

M H

B H

μ μ

ε

χ χ

μ μ μ δ χ

μ μ

= +

=

= = +

=

JG JJG JJG

/ μ

o

(22)

Second Rank Tensor

J. F. Nye, Physical Properties of Crystals

(23)

Stress and Strain

11 21 31

21 22 23

31 32 33

σ σ σ

σ σ σ

σ σ σ

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

J. F. Nye, Physical Properties of Crystals

(24)

Third Rank Tensor

- piezoelectricity

- direct piezoelectric effect

stress Æ electric dipole moment

(piezoelectirc modulus) P = d σ

inversion center quartz

M. A. Omar, Elementary Solid State Physics

(25)

Third Rank Tensor

R. E. Newnham, Structure-Property Relations

(26)

Third Rank Tensor

1 111 11 112 12 113 13 121 21 122 22 123 23

131 31 132 32 133 33

- -

- : 27 coefficients - third rank tensor:

i ijk jk

ijk

ijk il jm kn lmn

P d

P d d d d d d

d d d

d

d a a a d

σ

σ σ σ σ σ σ

σ σ σ

=

= + + + + +

+ + +

=

(27)

Third Rank Tensor

111 112 113 211 212 213 311 312 313

1st layer 2nd layer 3rd layer 1 2 3

i i i

d d d d d d d d d

= = =

121 122 123 221 222 223 321 322 323

131 132 133 231 232 233 331 332 333

1 11

( ) ( ) ( )

( )( ) ( )( ) ( )( ) - (18 independent components)

- matrix notation

ijk ikj

d d d d d d d d d

d d d d d d d d d

d d

d

=

1 1 1 1 1

16 15 21 26 25 31 36 35

2 2 2 2 2 2

1 1 1

12 2 14 22 2 24 32 2 34

13 23

d d d d d d d d

d d d d d d

d d d

33

(28)

11 12 13 1 6 5

21 22 23 6 2 4

31 32 33 5 4 3

1 11

- tensor notation: 11 22 33 23,32 31,13, 12,21 matrix notation: 1 2 3 4 5 6

-

- P d

σ σ σ σ σ σ

σ σ σ σ σ σ

σ σ σ σ σ σ

σ

⎡ ⎤ ⎡ ⎤

⎢ ⎥ → ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

=

1 12 16 6 12 15 5 12 16 6 12 2 12 14 4

1 1

15 5 14 4 13 3

2 2

11 1 12 2 13 3 14 4 15 5 16 6

-

i ij j

( 1, 2, 3, 1, 2,.., 6)

d d d d d

d d d

d d d d d d

p d i j

σ σ σ σ σ

σ σ σ

σ σ σ σ σ σ

σ

+ + + + +

+ + +

= + + + + +

= = =

Third Rank Tensor

(29)

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

- array of

- the do not transform like the components of a second rank tensor

- converse piezoelectric effect electric

ij

ij

d

d d d d d d

d d d d d d

d d d d d d

d

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎝ ⎠

field strain

(matrix notation: ( 1, 2, 3, 1, 2,.., 6))

jk ijk i

j ij i

d E

d E i j

ε

ε

=

= = =

Third Rank Tensor

(30)

Third Rank Tensor

J. F. Nye, Physical Properties of Crystals

(31)

- reduction in the number of independent moduli by crystal symmetry

- a crystal with a center of symmetry cannot be piezoelectric

- modulus d36 in class 4

36

0

d = Third Rank Tensor

J. F. Nye, Physical Properties of Crystals

(32)

- piezoelectric: 20 point groups

( 32 groups- centro-symmetric groups (11)-432 ) - quartz (point group: 32)

1 1

1 11 1 2 3

2 2

1 11 1 2 3

- tensile stress parallel to

P , P 0, P 0

- tensile stress parallel to

P , P 0, P 0

x d

x d

σ σ

σ σ

= = =

= − = =

C/N

Third Rank Tensor

J. F. Nye, Physical Properties of Crystals

(33)

- zinc blende ZnS (point group: )

1 14 4

- P = d σ

43m

Third Rank Tensor

R. E. Newnham, Structure-Property Relations

(34)

J. F. Nye, Physical Properties of Crystals

(35)

J. F. Nye, Physical Properties of Crystals

(36)

J. F. Nye, Physical Properties of Crystals

(37)

Fourth Rank Tensor

- Hook's law

( : stiffness, Young modulus)

( : compliance)

- 81 components

, (36 components)

- matrix notation

( , 1, 2,.., 6)

ij ijkl kl ijkl

ij ijkl kl ijkl

ijkl ijlk ijkl jikl

i ij j

c c

s s

c c c c

c i j

σ ε

ε σ

σ ε

=

=

= =

= =

6x6 matrix

(38)

-orthorhombic class 222: s34

Fourth Rank Tensor

J. F. Nye, Physical Properties of Crystals

(39)

J. F. Nye, Physical Properties of Crystals

(40)

J. F. Nye, Physical Properties of Crystals

(41)

R. E. Newnham, Structure-Property Relations

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