Lecture3

OklahomaOklahomaState State

UniversityUniversity

Lecture 3:

Bonding, molecular and lattice vibrations:

http://physics.okstate.edu/jpw519/phys5110/index.htm

Revisit 1-dim. caseRevisit 1-dim. caseLook at a 30 nm segment 0f a single walledcarbon nanotube (SWNT)Use STM noting that tunneling current is proportional toLocal density of states (higher conductance when near Molecular orbital.

Crystalline SolidsCrystalline Solids

Periodicity of crystal leads to the following properties of the wave function: 1-dim. (x+L)= (x); ‘(x+L)= ‘(x)

In 2-dim.

Periodic Boundary Conditions in a solid leads to traveling waves instead of standing waves

Excitations in Ideal Fermi Gas (2-dim.)

K-space

22 m

Eg Fd )(

Ground state: T=0 Particles and Holes: T>0

Distribution functions for T>0Distribution functions for T>0

•Particle-hole excitations are increased as T increases

•Particles are promoted from within kBT of EF to an unoccupied single particle state with E>EF

•Particles are not promoted from deep within Fermi Sea

Probability of finding a single-particle (orbital) state of particularspin with energy E is given by Fermi-Dirac distribution

-chemical potential

1

1

Tk

E

Be

TEf

),,(

Fermi-Dirac (FD) DistributionFermi-Dirac (FD) DistributionAs T 0, FD distribution approaches a step functionFermi gas described by a FD distribution that’s almost step like is termed degenerate

T=0

Crystal SystemsCubica=b=c°

Hexagonala=b≠c°°

Tetragonala=b≠c°

Rhombohedrala=b=c=≠90°

Orthorhombica≠b≠ca=b=g=90°

Monoclinica≠b≠c°≠

Triclinica≠b≠c≠≠≠

11

• ABCABC... Stacking Sequence• 2D Projection

A sites

B sites

C sitesB B

B

BB

B BC C

CA

A

• FCC Unit CellA

BC

FCC STACKING SEQUENCE

Point Coordinates

Crystallographic Directions[u,v,w] (integers)

d=n/2sinc

x-ray intensity (from detector)

c20

• Incoming X-rays diffract from crystal planes.

• Measurement of: Critical angles, c, for X-rays provide atomic spacing, d.

Adapted from Fig. 3.2W, Callister 6e.

X-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase to detect signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“2”

X-Ray Diffraction sinhkldTQQSn 2

222 lkh

adhkl

6

• Columns: Similar Valence Structure

Electropositive elements:Readily give up electronsto become + ions.

Electronegative elements:Readily acquire electronsto become - ions.

He

Ne

Ar

Kr

Xe

Rn

iner

t ga

ses

ac

cept

1e

ac

cept

2e

give

up

1e

give

up

2e

give

up

3e

F Li Be

Metal

Nonmetal

Intermediate

H

Na Cl

Br

I

At

O

S Mg

Ca

Sr

Ba

Ra

K

Rb

Cs

Fr

Sc

Y

Se

Te

Po

Adapted from Fig. 2.6, Callister 6e.

THE PERIODIC TABLE

Na (metal) unstable

Cl (nonmetal) unstable

electron

+ - Coulombic Attraction

Na (cation) stable

Cl (anion) stable

8

• Occurs between + and - ions.• Requires electron transfer.• Large difference in electronegativity required.• Example: NaCl

IONIC BONDING

• Requires shared electrons• Example: CH4

C: has 4 valence e, needs 4 more

H: has 1 valence e, needs 1 more

Electronegativities are comparable.

shared electrons from carbon atom

shared electrons from hydrogen atoms

H

H

H

H

C

CH4

Adapted from Fig. 2.10, Callister 6e.

COVALENT BONDING

12

• Arises from a sea of donated valence electrons (1, 2, or 3 from each atom).

• Primary bond for metals and their alloys

+ + +

+ + +

+ + +

METALLIC BONDING

Electrons are “delocalized”

•Electrical and thermal conductor

•Ductile

13

Arises from interaction between dipoles

• Permanent dipoles-molecule induced

• Fluctuating dipoles

+ - secondary bonding + -

H Cl H Clsecondary bonding

secondary bonding

HH HH

H2 H2

secondary bonding

ex: liquid H2asymmetric electron clouds

+ - + -secondary bonding

-general case:

-ex: liquid HCl

-ex: polymer

Adapted from Fig. 2.13, Callister 6e.

Adapted from Fig. 2.14, Callister 6e.

Adapted from Fig. 2.14, Callister 6e.

SECONDARY BONDING

Secondary bonding or physical bondsVan der Waals, Hydrogen bonding,

Hyrophobic bonding

• Self assembly – how biology builds…

• DNA hybridization

• Molecular recognition (immuno- processes, drug delivery etc. )

18

Ceramics(Ionic & covalent bonding):

Metals(Metallic bonding):

Polymers(Covalent & Secondary):

secondary bonding

Large bond energylarge Tm

large E

Variable bond energymoderate Tm

moderate E

Directional PropertiesSecondary bonding dominates

small Tsmall E

SUMMARY: PRIMARY BONDS

14

Type

Ionic

Covalent

Metallic

Secondary

Bond Energy

Large!3-5 eV/atom

Variablelarge-Diamondsmall-Bismuth1-7 ev/atom

Variablelarge-Tungstensmall-Mercury0.7-9 eV/atom

Smallest.05-0.5 ev/atom

Comments

Nondirectional (ceramics,NaCl, CsCl)

Directionalsemiconductors, ceramicsDiamond, polymer chains)

Nondirectional (metals)

Directionalinter-chain (polymer)

inter-molecular

SUMMARY: BONDING

Energy bands in crystalsMore on this next lecture!!

(Bloch function)

Ref: S.M. Sze: Semiconductor Devices Ref: M. Fukuda, Optical Semiconductor Devices

)()()(2

22

rErrVm kk

),()( rkUer n

rkjk

OklahomaOklahomaState State

UniversityUniversity

Interatomic Forces

Net Forces

Potential Energy: E

FdrE

drdEFr /

Potential Energy CurveE(r)

2

• Non dense, random packing

• Dense, regular packing

Dense, regular-packed structures tend to have lower energy.

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

ENERGY AND PACKING

15

• Bond length, r

• Bond energy, Eo

F F

r

• Melting Temperature, Tm

Eo=

“bond energy”

Energy (r)

ro r

unstretched length

r

larger Tm

smaller Tm

Energy (r)

ro

Tm is larger if Eo is larger.

PROPERTIES FROM BONDING: TM

16

• Elastic modulus, C

• C ~ curvature at ro

cross sectional area Ao

L

length, Lo

F

undeformed

deformed

L F Ao

= C Lo

Elastic modulus

r

larger Elastic Modulus

smaller Elastic Modulus

Energy

ro unstretched length

E is larger if Eo is larger.

PROPERTIES FROM BONDING: C

Vibrational frequencies of moleculesFor small vibrations, can use the Harmonic approximation:

where orr Represents small oscillations from ro

=(k/ )1/2 where k=

Oscillation frequency of two masses connected by spring

=m1m2/(m1+m2)-reduced mass

m11 m2

k

orrE

2

2

22

2

ooo rrrE

rErEor

)()(

Quantized total energy (kinetic + potential):

,...,, 21021

nwheren

C2H2 C~~H 8.64 1.53 450

C2D2 C~~D 6.42 2.85 463

12C16O C~~O 5.7 11.4 1460

13C18O C~~O 5.41 12.5 1444

C O

CH C

[1013 Hz] [10-27 kg] k [N/m]

H

Vibrational energies of molecules

k k k kk k k

un-1 un un+1

Lattice vibrations in Crystals•Equilibrium positions of atoms on lattice points (monatomic basis)•Small displacements from equilibrium positions•Harmonic Approximation•Vibrations of atoms slow compared to motion of electrons- Adiabatic Approximation•Waves of vibration in direction of high symmetry of crystal – q•Nearest neighbor interactions (Hooke’s Law)

n

nn uukPE 2

121

nnnn uuuk

dtud

M 2112

2

n

nuM

KE 2

2