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Oklahoma
  State
University


                  Lecture 3:
              Bonding, molecular
             and lattice vibrations:

http://physics.okstate.edu/jpw519/phys5110/index.htm
Lecture3
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.
Lecture3
Lecture3
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.
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
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
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
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°
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
Point Coordinates
Crystallographic Directions
     [u,v,w] (integers)
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
X-Ray Diffraction
 nλ = S Q + Q T = 2d hkl sin θ
             a
d hkl   = 2
          h +k +l
               2  2
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
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
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.
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
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
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. )
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
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
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
Interatomic
                Forces

Net Forces   Fr = − dE / dr
             E = ∫ Fdr

 Potential Energy: E
Potential Energy Curve
E(r)
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
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
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
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
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
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

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Lecture3

  • 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
  • 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
  • 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