Transcript 投影片 1 - National Cheng Kung University

This differs from
03._CrystalBindingAndElasticConstants.ppt
only in the section “Analysis of Elastic Strain” in which a modified
version of the Kittel narrative is used.
3. Crystal Binding and Elastic Constants
•
•
•
•
•
•
•
•
•
Crystals of Inert Gases
Ionic Crystals
Covalent Crystals
Metals
Hydrogen Bonds
Atomic Radii
Analysis of Elastic Strains
Elastic Compliance and Stiffness Constants
Elastic Waves in Cubic Crystals
Introduction
Cohesive energy  energy
required to break up crystal
into neutral free atoms.
Lattice energy (ionic crystals)
 energy required to break
up crystal into free ions.
Kcal/mol = 0.0434 eV/molecule
KJ/mol = 0.0104 eV/molecule
Crystals of Inert Gases
Atoms:
• high ionization energy
• outermost shell filled
• charge distribution spherical
Crystal:
•transparent insulators
•weakly bonded
•low melting point
•closed packed (fcc, except He3 & He4).
Van der Waals – London Interaction
Ref: A.Haug, “Theoretical Solid State Physics”, §30, Vol I, Pergamon Press (1972).
Van der Waals forces ＝ induced dipole – dipole interaction between neutral atoms/molecules.
Atom i  charge +Q at Ri and charge –Q at Ri + xi.
( center of charge distributions )
Q2
Q2
Q2
Q2
V



R
R  x2  x1
R  x1
R  x2

ˆ a 3
1
1
R
2

  R  2R  a  a 
1


Ra
R
R

2
2
2
ˆ   x  x   R
ˆ x  R
ˆ  x  2 R
ˆ x
R
2
1 
2
1
2

2 1/2

 x2  x1 

2
V
 



ˆ a
R
2R

2
2
 Rˆ  x 
a2
 2
2R




R  R 2  R1
ˆ
RRR
1
 x 2 2  x12  2x1  x 2



Q2 
ˆ x R
ˆ x  x x 
3 R
2
1
1
2
3 

R
Q2
 3  x1 x2  y1 y2  2 z1 z2 
R
R  R zˆ
H  H0  V
H0 = sum of atomic hamiltonians
E  E0   0 V  0  
j0
 j V 0
E0  E j
2

0 = antisymmetrized product of
ground state atomic functions
1st order term vanishes if overlap of atomic functions negligible.
2nd order term is negative &  R6 (van der Waals binding).
Repulsive Interaction
Pauli exclusion principle 
(non-electrostatic) effective repulsion
Alternative
repulsive term:
Lennard-Jones potential:
  12   6 
V  4      
 R  
 R 
,  determined from gas phase data
 e R / 
Equilibrium Lattice Constants
Neglecting K.E. 
12
6






1


 N  4   
  
 

2
p
R
p
R
i  j  i j

 i j  

E U tot
12
For a fcc lattice:
At equilibrium:

6
 1 
12   
  12.13188

i  j  pi j 
12  12.13229
For a hcp lattice:
dE
0
dR
 1 
6   
  14.45392

i  j  pi j 
6  14.45489

  12 
  6 
1
N  4   1212  13   6  6  7  
2
R 
 R 

1/6
 2 
  12 
  6 
R0
1.09 for fcc lattices
Experiment (Table 4):
Error due to zero point motion
R  n.n. dist
Cohesive Energy
6
   12
1
  
U tot  R   N  4  12     6   
2
 R  
  R 
 62
1
Utot  R0    N
2
12
 62
  N  4 
812
N  4  2.15
for fcc lattices
For low T, K.E.  zero point motion.
For a particle bounded within length ,
p


p2
K .E. 
2m
 quantum correction is inversely proportional to the atomic mass:
~ 28, 10, 6, & 4% for Ne, Ar, Kr, Xe.
2
2m
2
1

2
Ionic Crystals
ions: closed outermost shells
~ spherical charge distribution
Cohesive/Binding energy
= 7.9+3.615.14 = 6.4 eV
Electrostatic (Madelung) Energy
Interactions involving ith ion:
 R/  q2

 e
R

Ui j  
2
  q

pi j R
ji
At equilibrium:
j i
n.n.
otherwise
Utot  NUi
For N pairs of ions:
 
Ui  Ui j

pi j

 q2 
R/ 
 N  z e


R 

﹦Madelung constant
 z  R /   q 2 
dU tot
 2 
 0  N  e

R 
dR

U tot 0
z ﹦number of n.n.
ρ ~ .1 R0
N q 2 
 

1 

R 0  R 0 
→
2
0
R e
 R0 / 
  q2

z
N q 2
Madelung Energy  
R0
Evaluation of Madelung Constant
App. B: Ewald’s method
 
ji



pi j
1
i fixed
1 1
  2 1    
2 3 4
KCl
  2 ln 2


Kcal/mol = 0.0434 eV/molecule
Prob 3.6
Covalent Crystals
H2
• Electron pair localized midway of bond.
• Tetrahedral: diamond, zinc-blende structures.
• Low filling: 0.34 vs 0.74 for closed-packed.
Pauli exclusion →
exchange interaction
Ar : Filled outermost shell → van der Waal interaction (3.76A)
Cl2 : Unfilled outermost shell → covalent bond (2A)
s2 p2 → s p3 → tetrahedral bonds
Metals
Metallic bonding:
• Non-directional, long-ranged.
• Strength: vdW < metallic < ionic < covalent
• Structure: closed packed (fcc, hcp, bcc)
• Transition metals: extra binding of d-electrons.
Hydrogen Bonds
•
•
•
•
Energy ~ 0.1 eV
Largely ionic ( between most electronegative atoms like O & N ).
Responsible (together with the dipoles) for characteristics of H2O.
Important in ferroelectric crystals & DNA.
Atomic Radii
Standard ionic radii
~ cubic (N=6)
Na+ = 0.97A
F = 1.36A
NaF = 2.33A
obs = 2.32A
Bond lengths:
F2 = 1.417A
Na –Na = 3.716A
 NaF = 2.57A
Tetrahedral:
C = 0.77A
Si = 1.17A
SiC = 1.94A
Obs: 1.89A
Ref: CRC Handbook of Chemistry & Physics
Ionic Crystal Radii
E.g. BaTiO3 : a = 4.004A
Ba++ – O– – : D12 = 1.35 + 1.40 + 0.19 = 2.94A → a = 4.16A
Ti++++ – O – – : D6 = 0.68 + 1.40 = 2.08A → a = 4.16A
Bonding has some covalent character.
Analysis of Elastic Strains
Let
xˆ i
be the Cartesian axes of the unstrained state
xi
be the the axes of the stained state
Using Einstein’s summation notation, we have
xi  xˆ i   i j xˆ j    i j   i j  xˆ j
x1  x1  1  211  1k 1k  1  211  112  122  132
Position of atom in unstrained lattice: r  ri xˆ i
Its position in the strained lattice is defined as r  ri xi
Displacement due to deformation:
R  r  r  r  x  xˆ   ri  i j xˆ j  ui xˆ i
i
i
i
ui  rj ji
Define ( Einstein notation suspended ):
eii   ii 
 ui
 xi
ei j   i j   ji 
 ui  u j

 x j  xi
i  j 
Dilation
V  axˆ 1  bxˆ 2  cxˆ 3   abc
V   x1   x2  x3   1i  1i  axˆ i   2 j   2 j  bxˆ j   3k   3k  cxˆ k 


 1i  1i  2 j   2 j  3k   3k V  i j k
where
i jk
even permutation of 123
 1

 1 for ijk  odd permutation of 123
0
otherwise

V   V 123  1i i 23   2 j 1 j 3   3k  i jk   O  2 
 V 1  11   22   33   O   2 
V  V
 11   22   33  O   2   Tr  O   2 
V
Stress Components
Xy = fx on plane normal to y-axis = σ12 .
(Static equilibrium → Torqueless)   i j   ji
X y  Yx
Elastic Compliance & Stiffness Constants
ei j  Si j k l  k l
S = elastic compliance tensor
Contracted indices
i
j 
1 1
2 2
1
2
3
2
3
1
3
4
5
6
3
3
1
2
1  11  X x
 4   23   32  Yz  Z y
e  S   
  C  e
C = elastic stiffness tensor
Elastic Energy Density
Let
then

U
1
C  e e
2
 
1
U
 C  e  C  e
2
 e
C  


1
C   C 
2





1
C   C  e  C  e
2
 C  C 
Landau’s notations:
1
U  Ci j k l ui j uk l
2

1
C  u u
2
1
 C  e e
2
i j

 eii
  
for
1

ei j

i j
2
 1, 2,3
for   
u  e
4,5,
6

1  u u j
ui j   i 
2   x j  xi
 uii
u  
ui j  u ji
Elastic Stiffness Constants for Cubic Crystals
Invariance under reflections xi → –xi  C with odd numbers of like indices vanishes
Invariance under C3 , i.e.,
x yzx
x  z  y  x
x  z  y  x
x  y  z  x
 All C i j k l = 0 except for (summation notation suspended):
Ciiii  C1111
Ciik k  C1122
C  C11
C   C12
Ci k ik  C1212
 ,   1, 2,3
C  C44
1
1
U  C11  e12  e22  e32   C12  e1e2  e2 e3  e3e1   C44  e42  e52  e62 
2
2
  1   C11
  
  2   C12
  3   C12
 
 4   0
5   0
  
 6   0
C12
C12
0
0
C11
C12
0
0
C12
0
C11
0
0
C44
0
0
0
0
0
C44
0
0
0
0
0   e1 
 
0   e2 
0   e3 
 
0   e4 
0   e5 
 
C44   e6 
  4,5, 6
 C11 C12 C12

 C12 C11 C12
 C12 C12 C11

0
0
 0
 0
0
0

0
0
 0
where

S11 
0
0
0
C44
0
0
0
0
0
0
C44
0
1
0 
 S11


0 
 S12
 S12
0 



0 
 0
 0
0 


C44 
 0
C11  C12
 C11  2C12  C11  C12 
 S11  S12 
1
 C11  C12
S12
S11
S12
0
0
0
S12
S12
S11
0
0
0
S12  
0
0
0
S44
0
0
0
0
0
0
S44
0
0 

0 
0 

0 
0 

S44 
C12
 C11  2C12  C11  C12 
 S11  2S12 
1
 C11  2C12
S44 
1
C44
Bulk Modulus & Compressibility
1
1
U  C11  e12  e22  e32   C12  e1e2  e2 e3  e3e1   C44  e42  e52  e62 
2
2
Uniform dilation:
e1  e2  e3 

e4  e5  e6  0
3
δ = Tr eik = fractional volume change
U
B
1
 C11  2C12   2
6

1
B2
2
1
 C11  2C12  = 1/κ
3
B = Bulk modulus
κ = compressibility
U
p

See table 3 for values of B & κ .

1 V
V p
 2 U    p  V  p
B

V
 2
Elastic Waves in Cubic Crystals
Newton’s
2nd
 2 ui  i k
 2 
t
 xk
law:
 ik  Cik jl u jl
→
don’t confuse ui with uα
u jl
  2 ul
 2 ui
2 u j
1
 2  Ci k jl
 Ci k jl 

  x x  x x
t
 xk
2
k
l
 k j

 2 ul
  Ci k jl
 xk x j

 2 u3
 2 u3
 2 u1
 2 u1
 2 u2
 2 u2
 2 u1
 2 u1
 2  C1111 2  C1122
 C1133
 C1212
 C1221 2  C1313
 C1331 2
t
 x1
 x1x2
 x1x3
 x2x1
 x2
 x3x1
 x3
  2 u2
  2 u2  2 u1  2 u3  2 u1 
 2 u3 
 2 u1
 C1111 2  C1122 

 2 
 2 
  C1212 
 x1
  x1x2  x1x3 
  x2x1  x2  x3x1  x3 

Similarly
  2 u2
  2 u1  2 u1 
 2 u3 
 2 u1
 2 u1
 2  C11 2   C12  C44  

  C44  2  2 
t
 x1

x

x

x

x
1
3 
 1 2
  x2  x3 
  2 u3
  2 u2  2 u2 
 2 u2
 2 u2
 2 u1 
 2  C11 2   C12  C44  

  C44  2 
2 
t
 x2

x

x

x

x

x

x
2
1 
3 
 2 3
 1
  2 u2
  2 u3  2 u3 
 2 u3
 2 u3
 2 u1 
 2  C11 2   C12  C44  

  C44  2  2 
t
 x3

x

x

x

x
 x1 
3 1 
  x2
 3 2
Dispersion Equation
 2 ui
 2 ul
 2  Cik jl
t
 xk x j
ui  u0 i ei  k r   t 
→
  
2
il
 Ci k j l kk k j
2  u0i  Cik jl kk k j u0l
u
 2   i l  Ci k j l kk k j
 2  I  C k   0
0l
0
0
dispersion equation
Ci j k   Cimn j km kn
Waves in the [100] direction
 2  I  C k   0
k  k 1,0,0
→
Ci j k   Cimn j km kn
Ci j k   Ci11 j k 2
 C1111 C1112

C  k   k 2  C2111 C2112
C
 3111 C3112
C1113 
0
 C1111


C2113   k 2  0
C2112
 0
C3113 
0


L 
C11
k
T 
C44
k


u0  1,0,0
u0   0,1,0
u0   0,0,1
 C11 C16

 k 2  C61 C66
C
 51 C56
0 
 C11 0


0   k 2  0 C44
 0
C3113 
0

C15 

C65 
C55 

Longitudinal
Transverse, degenerate
0 

0 
C44 
Waves in the [110] direction
 2  I  C k   0
k
k
1,1, 0 
2
→
Ci j k   Cimn j km kn
Ci j  k    Ci11 j
 C1111  C1221 C1122  C1212
k 
C  k    C2121  C2211 C2112  C2222
2 
0
0

2
L 
T 1 
1
C11  C12  2C44  k
2
1
 C11  C12  k
2
T 2 
C44

k
k2
 Ci12 j  Ci 21 j  Ci 22 j 
2

 C11  C44
2
k 

0

C12  C44


2 
C3113  C3223 
0

0
u0  1,1,0
u0  1, 1,0
u0   0,0,1
C12  C44
C11  C44
0
Lonitudinal
Transverse
Transverse
0 

0 
2C44 
Prob 3.10