Chapter 3. Crystal Binding & Elastic...

Chapter 3. Crystal Binding & Elastic Constants

Solid State Physics by Heesang Kim at SSU

Chapter 3. Crystal Binding

Solid State Physics by Heesang Kim at SSU

Ionization energy ? Energy required in taking an electron away from an atom.

Cohesive energy? Energy required in taking a compound away from a crystal. Inert gas crystal < alkali metal < C, Si, Ge, … < transition metal

Melting temp. & bulk modulus vary roughly as cohesive energies.

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU

Crystals of inert gases

Simplest crystal, transparent insulator, weakly bound, low melting point, closed packed (fcc, hcp)

Inert gas atoms : completely filled shell, stable, very high ionization energy.

What holds them together? Van der Waals-London interaction

Induced dipole dipole interaction

atom1 atom2

+ +

weak bonding between neutral atoms, between molecules

Solid State Physics by Heesang Kim at SSU

Van der Waals-London interaction (attractive interaction)

Simple argument

Solid State Physics by Heesang Kim at SSU

When there is no interaction between the atoms,

)(2

1000 wwE Ground state energy

Solid State Physics by Heesang Kim at SSU

Now, turning on the coulomb interaction between them.

Assuming that

3

21

22

2

2

1

2

2

2

110

2

2

1

2

1

2

1

2

1

R

xxeCxCxP

mP

mHHH

This is nothing but a coupled oscillator problem. (mech. Chap. 12)

Solid State Physics by Heesang Kim at SSU

3

21

22

2

2

1

2

2

2

110

2

2

1

2

1

2

1

2

1

R

xxeCxCxP

mP

mHHH

One of the ways to solve it: normal mode transformation

)(2

11 as xxx )(

2

12 as xxx )(

2

11 as ppp )(

2

12 as ppp

2

3

222

3

22

)2

(2

1

2

1)

2(

2

1

2

1aass x

R

eCp

mx

R

eCp

mH

2

3

2

3

2

0

2

1

3

2

)2

(8

1)

2(

2

11/)

2(

CR

e

CR

ewm

R

eCw

Solid State Physics by Heesang Kim at SSU

6

2

3

2

00 )2

(8

1

R

A

CR

ewEEU

Therefore, Van der Waals interaction (London int., induced dipole dipole int.) lowers the ground state energy : quantum effects( ).

2

3

2

00

2

3

2

0 )2

(8

1))

2(

8

11()(

2

1

CR

ewE

CR

ewwwE

Ground state energy

Solid State Physics by Heesang Kim at SSU

Pauli exclusion principle (repulsive interaction)

Two electrons can not have their quantum numbers equal.

12/~ RBEmpirical form of the interaction

Solid State Physics by Heesang Kim at SSU

All together gives the Lennard-Jones potential :

612

4)(RR

RU

6

'

12

')4(

2

1

RPRPNU

ijj

ijj

total

N=# of atoms ; R=nn distance ; Pij R= distance btw I & j atoms

; 13188.1212'

jij

P 45392.146'

jij

PFor fcc

; 13229.1212'

jij

P 45489.146'

jij

PFor hcp

Cohesive energy (total)

Solid State Physics by Heesang Kim at SSU

0)45.14)(6()13.12)(12(20

0

0 7

6

13

12

RR

RRtot

RRN

dR

dU 09.10

R

6

'

12

')45.14()13.12(2

RRNU

jjtotal

Exp. values from gas phase

Ne Ar Kr Xe

R0/σ 1.14 1.11 1.10 1.09

)4)(15.2(09.1

1)45.14(

09.1

1)13.12(2)(

6

'

12

'

0 NNRUjj

total

for all inert gases

Solid State Physics by Heesang Kim at SSU

)4)(15.2()( 0 NRUtotal

Calculate Xe case as an example,……

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU

Hermite polynomial

Quantum harmonic oscillator problem

*** Helium, He ***

Ionic Crystals : ionic bond

1622 3221 : spssNa

52622 33221 : pspssCl

Energy 손익계산서: Na + Cl = NaCl + 6.4 eV

-5.14 + 3.61 + 7.9 = 6.4

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU

Electron density distribution By x-ray study

Solid State Physics by Heesang Kim at SSU

Interaction btw/ i-th & j-th ions (cgs unit)

Attractive interaction : Coulomb interaction Repulsive interaction : Pauli exclusion principle

jij

i UU'

ij

ijij r

qrU

2

exp

It is found that exponential form works better in ionic crystal case.

RqR

2

exp

ijU

R

q

Pij

21

nearest neighbors

otherwise

Solid State Physics by Heesang Kim at SSU

R

qezNNUU

R

itotal

2

Cohesive energy (total)

0)('

ijj P

Madelung constant

01

00

2

2

RR

R

RR

i

R

qez

dR

dU

R0

Notice that there are 2N ions.

0

2

0

2

0 1)(0

RR

Nq

R

qezNRU

o

R

total

Madelung (electrostatic) energy

Solid State Physics by Heesang Kim at SSU

How to evaluate Madelong constant

Let us consider a 1-dim. Ionic crystal as in the figure.

4

1

3

1

2

11

2

4

1

3

1

2

112

RRRRRR

432

)1ln(432 xxx

xx

2ln2

Solid State Physics by Heesang Kim at SSU

Covalent Crystals

312222 221221 PSSPSS

exchange interaction ←spin dependent Coulomb energy

Si, Ge, C Share electrons=> fill their shells Strong bond Directionality Tetrahedral bond

Solid State Physics by Heesang Kim at SSU

exchange interaction ←spin dependent Coulomb energy

a way to avoid Pauli repulsion

Solid State Physics by Heesang Kim at SSU

If the bonding is not symmetric, it looks ionic as well.

Solid State Physics by Heesang Kim at SSU

Metals

High electrical conductivity Valence e get delocalized into conduction e (e’s delocalization reduces K.E.

Think of uncertainty principle)

Sea of e

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU

Hydrogen Bonds

A type of bond formed when the partially positive hydrogen atom of a polar covalent bond in one molecule is attracted to the partially negative atom of a polar covalent bond in another.

Responsible for the strange behavior of water, and for the DNA double helix structure & reproduction

Solid State Physics by Heesang Kim at SSU

Elastic properties of Crystal

Assume a homogeneous, continuous medium; Valid for elastic waves ; Consider small strain so that Hooke’s law may apply.

Solid State Physics by Heesang Kim at SSU

Displacement of the vector

Solid State Physics by Heesang Kim at SSU

zzzyzx

yzyyyx

xzxyxx

eee

eee

eee

Strain

ze

ye

x

ue zzzzyyyyxxxx

;;

xz

uxze

yzzye

xy

uyxe

xzzxzx

yzzyyz

xyyxxy

Dilation : fractional increase of volume

zyxV zzyyxx eee 1

zzyyxx eeeV

VV

Dilation

Symmetric matrix : only 6 components.

Solid State Physics by Heesang Kim at SSU

Stress : force acting on a unit area (a bit different from pressure)

zyx

zyx

zyx

ZZZ

YYY

XXX Capital : direction of force Sub: direction of plane

Symmetric matrix w.r.t. the diagonal: Thus, there are only 6 components.

yzxzxy ZYZXYX ; ;

Solid State Physics by Heesang Kim at SSU

xy

zx

yz

zz

yy

xx

y

x

z

z

y

x

e

e

e

e

e

e

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

X

Z

Y

Z

Y

X

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

Hooke’s law gives

C : Elastic stiffness constants, moduli of elasticity (somewhat like spring constant)

(e) = (S)(X…) = (C)^{-1}(X…)

S : Elastic compliance constants, elastic constant

Symmetric matrix : only 21 components.

Solid State Physics by Heesang Kim at SSU

xy

zx

yz

zz

yy

xx

y

x

z

z

y

x

e

e

e

e

e

e

C

C

C

CCC

CCC

CCC

X

Z

Y

Z

Y

X

44

44

44

113212

121112

121211

00000

00000

00000

000

000

000

Cubic crystal case : symmetry consideration gives

There are only 4 Cs.

Solid State Physics by Heesang Kim at SSU

Elastic energy density

eeCU2

1

Bulk modulus, B

Compressibility, K

2

2

1BU

BK

1

dV

dpVB or equivalently

변형에 대해 얼마나 rigid 한가를 나타내는 수치

얼마나 쉽게 압축되는가를 나타내는 수치

Solid State Physics by Heesang Kim at SSU

Elastic energy density

eeCU2

1We might expect elastic waves, which we call phonon, i.e., quantum lattice vibration.

Solid State Physics by Heesang Kim at SSU

Solid State Physics by Heesang Kim at SSU