Lecture3
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OklahomaOklahomaState State
UniversityUniversity
Lecture 3:
Bonding, molecular and lattice vibrations:
http://physics.okstate.edu/jpw519/phys5110/index.htm
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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.
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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.
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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
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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
),,(
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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
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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≠≠≠
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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
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Point Coordinates
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Crystallographic Directions[u,v,w] (integers)
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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”
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X-Ray Diffraction sinhkldTQQSn 2
222 lkh
adhkl
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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
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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
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• 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
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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
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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
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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. )
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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
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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
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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
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Interatomic Forces
Net Forces
Potential Energy: E
FdrE
drdEFr /
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Potential Energy CurveE(r)
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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
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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
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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
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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
)()(
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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
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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