The document provides an overview of bonding, molecular vibrations, and lattice vibrations in crystals. It discusses different types of bonding including ionic, covalent, metallic, and secondary bonding. It examines the periodic table and how elements form bonds. It also covers crystal structures, unit cells, X-ray diffraction, and how bonding influences material properties like melting temperature and elastic modulus. Finally, it summarizes vibrational frequencies of molecules and lattice vibrations in crystals using the harmonic approximation.

1. Oklahoma
State
University
Lecture 3:
Bonding, molecular
and lattice vibrations:
http://physics.okstate.edu/jpw519/phys5110/index.htm

3. Revisit 1-dim. case
Look at a 30 nm segment 0f a single walled
carbon nanotube (SWNT)
Use STM noting that tunneling current is proportional to
Local density of states (higher conductance when near
Molecular orbital.

6. Crystalline Solids
Periodicity of crystal leads to the following properties of the
wave function: 1-dim. ψ(x+L)= ψ(x); ψ‘(x+L)= ψ‘(x)
In 2-dim.

7. Periodic Boundary Conditions in a solid leads
to traveling waves instead of standing waves
Excitations in Ideal Fermi Gas (2-dim.)
Ground state: T=0 Particles and Holes: T>0
K-space
m
g 2d ( EF ) =
π 2

8. Distribution functions for T>0
•Particle-hole excitations are increased as T increases
•Particles are promoted from within k T of E to an unoccupied
B F
single particle state with E>EF
•Particles are not promoted from deep within Fermi Sea
Probability of finding a single-particle (orbital) state of particular
spin with energy E is given by Fermi-Dirac distribution
1
f ( E , µ ,T ) =  E −µ 
 
 k T 
e  B 
+1
µ-chemical potential

9. Fermi-Dirac (FD) Distribution
As T 0, FD distribution approaches a step function
Fermi gas described by a FD distribution that’s almost
step like is termed degenerate
T=0

10. Cubic
a=b=c Crystal Systems
α=β=γ=90°
Orthorhombic
a≠b≠c
Hexagonal a=b=g=90°
a=b≠c
α=β= 90° ; γ=120°
Monoclinic
a≠b≠c
Tetragonal α=γ=90°≠β
a=b≠c
α=β=γ=90°
Triclinic
Rhombohedral a≠b≠c
a=b=c= α≠β≠γ≠90°
α=β=γ≠90°

11. FCC STACKING SEQUENCE
• ABCABC... Stacking Sequence
• 2D Projection
A
B B
C
A
A sites B B B
C C
B sites B B
C sites
A
• FCC Unit Cell B
C
11

12. Point Coordinates

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

14. X-RAYS TO CONFIRM CRYSTAL
STRUCTURE
• Incoming X-rays diffract from crystal planes.
de
te
c
”
to
“1
in ra
r
ys
co ys
reflections must
X-
ra
m
be in phase to
X-
”
“2
in detect signal
“1
g
g in
λ Adapted from Fig.
”
extra o
g
“2
distance
θ ut θ
”
3.2W, Callister 6e.
travelled o
by wave “2” spacing
d between
planes
• Measurement of: x-ray
Critical angles, θc, intensity d=nλ/2sinθc
(from
for X-rays provide
detector)
atomic spacing, d.
θ
θc
20

15. X-Ray Diffraction
nλ = S Q + Q T = 2d hkl sin θ
a
d hkl = 2
h +k +l
2 2

16. THE PERIODIC TABLE
• Columns: Similar Valence Structure
inert gases
give up 1e
give up 2e
accept 2e
accept 1e
give up 3e Metal
Nonmetal
H He
Li Be Intermediate Ne
O F
Na Mg Adapted
S Cl Ar
from Fig. 2.6,
K Ca Sc Se Br Kr Callister 6e.
Rb Sr Y Te I Xe
Cs Ba Po At Rn
Fr Ra
Electropositive elements: Electronegative elements:
Readily give up electrons Readily acquire electrons
to become + ions. to become - ions.
6

17. IONIC BONDING
• Occurs between + and - ions.
• Requires electron transfer.
• Large difference in electronegativity required.
• Example: NaCl
Na (metal) Cl (nonmetal)
unstable unstable
electron
Na (cation) + - Cl (anion)
stable Coulombic stable
Attraction
8

18. COVALENT BONDING
• Requires shared electrons
• Example: CH4 shared electrons
H
C: has 4 valence e, from carbon atom
CH4
needs 4 more
H: has 1 valence e, H C H
needs 1 more
shared electrons
Electronegativities H from hydrogen
are comparable. atoms
Adapted from Fig. 2.10, Callister 6e.

19. METALLIC BONDING
• Arises from a sea of donated valence electrons
(1, 2, or 3 from each atom).
+ + + Electrons are
“delocalized”
+ + + •Electrical and thermal conductor
•Ductile
+ + +
• Primary bond for metals and their alloys
12

20. SECONDARY BONDING
Arises from interaction between dipoles
• Fluctuating dipoles
asymmetric electron ex: liquid H2
clouds H2 H2
+ - secondary + - H H H H
secondary
bonding Adapted from Fig. 2.13, Callister 6e. bonding
• Permanent dipoles-molecule induced
Adapted from Fig. 2.14,
secondary
-general case: + - + - Callister 6e.
bonding
secondary Adapted from Fig. 2.14,
-ex: liquid HCl H Cl bonding H Cl Callister 6e.
secon
-ex: polymer dary
bond
ing
13

21. Secondary bonding or physical bonds
Van der Waals, Hydrogen bonding,
Hyrophobic bonding
• Self assembly – how biology builds…
• DNA hybridization
• Molecular recognition (immuno- processes,
drug delivery etc. )

22. SUMMARY: PRIMARY BONDS
Ceramics Large bond energy
(Ionic & covalent bonding): large Tm
large E
Metals Variable bond energy
(Metallic bonding): moderate Tm
moderate E
Polymers Directional Properties
(Covalent & Secondary): Secondary bonding dominates
small T
secon
dary
bond
small E
ing
18

23. SUMMARY: BONDING
Type Bond Energy Comments
Ionic Large! Nondirectional (ceramics,
3-5 eV/atom NaCl, CsCl)
Variable Directional
Covalent large-Diamond semiconductors, ceramics
small-Bismuth Diamond, polymer chains)
1-7 ev/atom
Variable
Metallic large-Tungsten Nondirectional (metals)
small-Mercury
0.7-9 eV/atom Directional
Secondary Smallest inter-chain (polymer)
.05-0.5 ev/atom inter-molecular
14

24. Oklahoma
State Energy bands in crystals
University
More on this next lecture!!
 2 2      
jk ⋅r
 
−
 2m ∇ + V (r )φ k ( r ) = Eφ k ( r ) φ k (r ) = e U n (k , r ) (Bloch function)
 
Ref: S.M. Sze: Semiconductor Devices Ref: M. Fukuda, Optical Semiconductor Devices

25. Interatomic
Forces
Net Forces Fr = − dE / dr
E = ∫ Fdr
Potential Energy: E

26. Potential Energy Curve
E(r)

27. ENERGY AND PACKING
• Non dense, random packing Energy
typical neighbor
bond length
typical neighbor r
bond energy
• Dense, regular packing Energy
typical neighbor
bond length
typical neighbor r
bond energy
Dense, regular-packed structures tend to have
lower energy.
2

28. PROPERTIES FROM
BONDING: TM
• Bond length, r • Melting Temperature, Tm
F
F Energy (r)
r
• Bond energy, Eo ro
r
Energy (r)
smaller Tm
unstretched length
ro larger Tm
r
Eo= Tm is larger if Eo is larger.
“bond energy”
15

29. PROPERTIES FROM BONDING: C
• Elastic modulus, C cross
sectional
length, Lo Elastic modulus
area Ao
undeformed F ∆L
∆L =C
Ao Lo
deformed F
Energy
• C ~ curvature at ro E is larger if Eo is larger.
unstretched length
ro
r
smaller Elastic Modulus
larger Elastic Modulus
16

30. Vibrational frequencies of molecules
For small vibrations, can use the Harmonic approximation:
 ∂2E 
E (r ) = Eo (ro ) +  2  ( r − ro ) 2
 ∂r  r o
where ( r − ro ) Represents small oscillations from ro
Oscillation frequency of two k
masses connected by spring m11 m2
 ∂2E 
ω=(k/ µ)1/2 where k=  2 
 ∂r  ro
µ=m1m2/(m1+m2)-reduced mass

31. Quantized total energy (kinetic + potential):
 n + 1 ω where n = 0,1, 2,...
 
 2
Vibrational energies of molecules
ω[1013 Hz] µ[10-27 kg] k [N/m]
C2H2 C~~H 8.64 1.53 450 C C
H H
C2D2 C~~D 6.42 2.85 463
C16O
12
C~~O 5.7 11.4 1460 C O
C18O
13
C~~O 5.41 12.5 1444

32. 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)
1 M
PE = ∑ k ( un+1 − un )
2
KE = ∑ un
2
2n 2 n
d 2u n
M 2 = k ( un+1 + un−1 − 2un )
dt
un-1 un un+1
k k k k k k k