This differs from 03._CrystalBindingAndElasticConstants

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This differs from 03._CrystalBindingAndElasticConstants.ppt only in the section “Analysis of Elastic Strain” in which a modified version of the Kittel narrative is used.

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This differs from 03._CrystalBindingAndElasticConstants.ppt only in the section “Analysis of Elastic Strain” in which a modified version of the Kittel narrative is used. 3. Crystal Binding and Elastic Constants. Crystals of Inert Gases Ionic Crystals Covalent Crystals Metals - PowerPoint PPT Presentation

Transcript of This differs from 03._CrystalBindingAndElasticConstants

Page 1: This differs from  03._CrystalBindingAndElasticConstants

This differs from 03._CrystalBindingAndElasticConstants.ppt

only in the section “Analysis of Elastic Strain” in which a modified version of the Kittel narrative is used.

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3. Crystal Binding and Elastic Constants

• Crystals of Inert Gases

• Ionic Crystals

• Covalent Crystals

• Metals

• Hydrogen Bonds

• Atomic Radii

• Analysis of Elastic Strains

• Elastic Compliance and Stiffness Constants

• Elastic Waves in Cubic Crystals

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Introduction

Cohesive energy energy required to break up crystal into neutral free atoms.

Lattice energy (ionic crystals) energy required to break up crystal into free ions.

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Kcal/mol = 0.0434 eV/molecule KJ/mol = 0.0104 eV/molecule

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Crystals of Inert Gases

Atoms: •high ionization energy•outermost shell filled•charge distribution spherical

Crystal: •transparent insulators•weakly bonded•low melting point•closed packed (fcc, except He3 & He4).

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Van der Waals – London Interaction

Van der Waals forces = induced dipole – dipole interaction between neutral atoms/molecules.

Ref: A.Haug, “Theoretical Solid State Physics”, §30, Vol I, Pergamon Press (1972).

2 2 2 2

2 1 1 2

Q Q Q QV

R

R x x R x R x

Atom i charge +Q at Ri and charge –Q at Ri + xi.( center of charge distributions )

2 22

2 1 2 1 2 1ˆ ˆ ˆ ˆ ˆ2 R x x R x R x R x R x

2 1 R R R

1/22 212R a

R a

R a 2

2

2 2

ˆ3ˆ11

2 2

a

R R R R

R aR a ˆRR R

2 2 22 1 2 1 1 22 x x x x x x

2

2 1 1 23ˆ ˆ3

QV

R R x R x x x

2

1 2 1 2 1 232

Qx x y y z z

R ˆRR z

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0H H V H0 = sum of atomic hamiltonians

2

0

0 0 00 0

j

j j

VE E V

E E

0 = antisymmetrized product of ground state atomic functions

1st order term vanishes if overlap of atomic functions negligible.2nd order term is negative & R6 (van der Waals binding).

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Repulsive Interaction

Pauli exclusion principle (non-electrostatic) effective repulsion

Lennard-Jones potential: 12 6

4VR R

, determined from gas phase data

/Re

Alternative repulsive term:

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Equilibrium Lattice Constants

Neglecting K.E. 12 6

14

2toti j i j i j

E U Np R p R

For a fcc lattice:

12

12

112.13188

i j i jp

6

6

114.45392

i j i jp

For a hcp lattice:

12 12.13229 6 14.45489

R n.n. dist

At equilibrium:

0dE

dR

12 6

12 613 7

14 12 6

2N

R R

1/6

0 12

6

2R

1.09 for fcc lattices

Experiment (Table 4):

Error due to zero point motion

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Cohesive Energy

12 6

12 6

14

2totU R NR R

26

012

1

2totU R N

26

12

48

N

4 2.15N for fcc lattices

For low T, K.E. zero point motion.

For a particle bounded within length , p

2 2

2. .

2 2

pK E

m m

1

2

quantum correction is inversely proportional to the atomic mass:~ 28, 10, 6, & 4% for Ne, Ar, Kr, Xe.

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Ionic Crystals

ions: closed outermost shells ~ spherical charge distribution

Cohesive/Binding energy = 7.9+3.615.14 = 6.4 eV

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Electrostatic (Madelung) Energy

Interactions involving ith ion: i i jj i

U U

2/

2

. .R

i j

i j

qn ne

RU

qotherwise

p R

tot iU NUFor N pairs of ions:2

/R qN z e

R

z number of n.n.﹦ρ ~ .1 R0

j i i jp

﹦Madelung constant

At equilibrium: 0totdU

dR

2/

2Rz q

N eR

→ 0

2/2

0R q

R ez

2

00 0

1tot

N qU

R R

2

0

N qMadelung Energy

R

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Evaluation of Madelung Constant

App. B: Ewald’s method j i i jp

1 1 12 1

2 3 4

2ln 2

KCl

i fixed

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Kcal/mol = 0.0434 eV/molecule Prob 3.6

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Covalent Crystals

• Electron pair localized midway of bond.• Tetrahedral: diamond, zinc-blende structures.• Low filling: 0.34 vs 0.74 for closed-packed.

Pauli exclusion → exchange interaction

H2

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Ar : Filled outermost shell → van der Waal interaction (3.76A)Cl2 : Unfilled outermost shell → covalent bond (2A)

s2 p2 → s p3 → tetrahedral bonds

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Metals

Metallic bonding: • Non-directional, long-ranged.• Strength: vdW < metallic < ionic < covalent• Structure: closed packed (fcc, hcp, bcc)• Transition metals: extra binding of d-electrons.

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Hydrogen Bonds

• Energy ~ 0.1 eV• Largely ionic ( between most electronegative atoms like O & N ).• Responsible (together with the dipoles) for characteristics of H2O.• Important in ferroelectric crystals & DNA.

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Atomic Radii

Na+ = 0.97AF = 1.36ANaF = 2.33Aobs = 2.32A

Standard ionic radii ~ cubic (N=6)

Bond lengths:F2 = 1.417ANa –Na = 3.716A NaF = 2.57A

Tetrahedral:C = 0.77ASi = 1.17ASiC = 1.94AObs: 1.89A

Ref: CRC Handbook of Chemistry & Physics

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Ionic Crystal Radii

E.g. BaTiO3 : a = 4.004ABa++ – O– – : D12 = 1.35 + 1.40 + 0.19 = 2.94A → a = 4.16ATi++++ – O – – : D6 = 0.68 + 1.40 = 2.08A → a = 4.16ABonding has some covalent character.

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Analysis of Elastic Strains

ˆ ixLet be the Cartesian axes of the unstrained state

ix be the the axes of the stained state

Using Einstein’s summation notation, we have

ˆi j i j j xˆ ˆi i i j j x x x

1 1 11 1 11 2 k k x x 2 2 211 11 12 131 2

ˆi irr x

i ir r x

R r r ˆi i j jr x

Position of atom in unstrained lattice:

Its position in the strained lattice is defined as

Displacement due to deformation:

i j j iu r

iii ii

i

ue

x

jii j i j j i

j i

uue

x x

ˆi i ir x x ˆi iu x

Define ( Einstein notation suspended ):

i j

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Dilation

1 1 2 2 3 3ˆ ˆ ˆi i i j j j k k ka b c x x x

1 1 2 2 3 3i i j j k k i j kV

1 2 3ˆ ˆ ˆV a b c abc x x x

1 2 3V x x x

211 22 33

V VO

V

211 22 331V O

1 even permutation of 123

1 for odd permutation of 123

0 otherwisei j k ijk

2123 1 23 2 1 3 3i i j j k i jkV V O

2Tr O

where

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Stress Components

Xy = fx on plane normal to y-axis = σ12 .

(Static equilibrium → Torqueless) i j j i

y xX Y

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Elastic Compliance & Stiffness Constants

i j i j k l k le S

1 11 xX 1 1 1

2 2 2

3 3 3

2 3 4

3 1 5

1 2 6

i j

e S

4 23 32 z yY Z

C e

S = elastic compliance tensor

Contracted indices

C = elastic stiffness tensor

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Elastic Energy Density1

2U C e e

1

2C e C e U

e

C e 1

2C C e

C 1

2C C C

Let

then

1

2 i j k l i j k lU C u u

1

2C u u

Landau’s notations:

1

2ji

i jj i

uuu

x x

1

2

ii

i j

i jefor

ei j

1,2,3for

4,5,6ii

i j j i

uu

u u

u e

C

1

2C e e

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Elastic Stiffness Constants for Cubic Crystals

Invariance under reflections xi → –xi C with odd numbers of like indices vanishes

Invariance under C3 , i.e.,

1111iiiiC C

x y z x x z y x

x z y x x y z x

All C i j k l = 0 except for (summation notation suspended):

1122ii k kC C 1212i k ikC C

2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6

1 1

2 2U C e e e C e e e e e e C e e e

1 111 12 12

2 212 11 12

3 312 12 11

4 444

5 544

6 644

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

eC C C

eC C C

eC C C

eC

eC

eC

11C C 12C C 44C C , 1, 2,3 4,5,6

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1

11 12 12 11 12 12

12 11 12 12 11 12

12 12 11 12 12 11

44 44

44 44

44 44

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

C C C S S S

C C C S S S

C C C S S S

C S

C S

C S

where 11 12

1111 12 11 122

C CS

C C C C

12

1211 12 11 122

CS

C C C C

4444

1S

C

1

11 12 11 12S S C C 1

11 12 11 122 2S S C C

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Bulk Modulus & Compressibility

2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6

1 1

2 2U C e e e C e e e e e e C e e e

Uniform dilation:

211 12

12

6U C C

1 2 3 3e e e

4 5 6 0e e e

δ = Tr eik = fractional volume change

21

2B B = Bulk modulus

1 V

V p

11 12

12

3B C C = 1/κ κ = compressibility

See table 3 for values of B & κ .

Up

2

2

UB

p

pV

V

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Elastic Waves in Cubic Crystals

Newton’s 2nd law:2

2

i ki

k

u

t x

don’t confuse ui with uα

i k ik j l j lC u →2

2

jlii k j l

k

uuC

t x

221

2jl

i k j lk j k l

uuC

x x x x

2l

i k j lk j

uC

x x

2 22 2 2 2 2 23 31 1 2 2 1 1

1111 1122 1133 1212 1221 1313 13312 2 2 21 1 2 1 3 2 1 2 3 1 3

u uu u u u u uC C C C C C C

t x x x x x x x x x x x

2 22 2 2 2 23 31 2 2 1 1

1111 1122 12122 2 21 1 2 1 3 2 1 2 3 1 3

u uu u u u uC C C

x x x x x x x x x x x

22 2 2 2 2

31 1 2 1 111 12 44 442 2 2 2

1 1 2 1 3 2 3

uu u u u uC C C C

t x x x x x x x

Similarly 22 2 2 2 2

32 2 1 2 211 12 44 442 2 2 2

2 2 3 2 1 1 3

uu u u u uC C C C

t x x x x x x x

2 2 2 22 2

3 3 3 32 111 12 44 442 2 2 2

3 3 2 3 1 2 1

u u u uu uC C C C

t x x x x x x x

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Dispersion Equation2 2

2i l

i k j lk j

u uC

t x x

0

i ti iu u e k r

→2

0 0i i k j l k j lu C k k u

20 0il i k j l k j lC k k u

2 0i l i k j l k jC k k dispersion equation

2 0I kC i j imn j m nC k kkC

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Waves in the [100] direction

2 0I kC i j imn j m nC k kkC

1,0,0kk → 211i j i jC kkC

1111 1112 1113

22111 2112 2113

3111 3112 3113

C C C

k C C C

C C C

kC11

244

44

0 0

0 0

0 0

C

k C

C

11L

Ck

0 1,0,0u Longitudinal

44T

Ck

0 0,1,0uTransverse, degenerate 0 0,0,1u

11112

2112

3113

0 0

0 0

0 0

C

k C

C

11 16 152

61 66 65

51 56 55

C C C

k C C C

C C C

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Waves in the [110] direction

2 0I kC i j imn j m nC k kkC

1,1,02

kk →

2

11 12 21 22 2i j i j i j i j i j

kC C C C kC

1111 1221 1122 12122

2121 2211 2112 2222

3113 3223

0

02

0 0

C C C Ck

C C C C

C C

kC11 44 12 442

12 44 11 44

44

0

02

0 0 2

C C C Ck

C C C C

C

11 12 44

12

2L C C C k

0 1,1,0u Lonitudinal

442T

Ck

0 0,0,1u

Transverse 1 11 12

1

2T C C k

0 1, 1,0 u

Transverse

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Prob 3.10

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