this differs from 03._crystalbindingandelasticconstants.ppt only in the section “analysis of...
TRANSCRIPT
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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.
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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
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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.
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Kcal/mol = 0.0434 eV/molecule KJ/mol = 0.0104 eV/molecule
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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).
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Van der Waals – London Interaction
Van der Waals forces = induced dipole – dipole interaction between neutral atoms/molecules.
Ref: A.Haug, “Theoretical Solid State Physics”, §30, Vol I, Pergamon Press (1972).
2 2 2 2
2 1 1 2
Q Q Q QV
R
R x x R x R x
Atom i charge +Q at Ri and charge –Q at Ri + xi.( center of charge distributions )
2 22
2 1 2 1 2 1ˆ ˆ ˆ ˆ ˆ2 R x x R x R x R x R x
2 1 R R R
1/22 212R a
R a
R a 2
2
2 2
ˆ3ˆ11
2 2
a
R R R R
R aR a ˆRR R
2 2 22 1 2 1 1 22 x x x x x x
2
2 1 1 23ˆ ˆ3
QV
R R x R x x x
2
1 2 1 2 1 232
Qx x y y z z
R ˆRR z
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0H H V H0 = sum of atomic hamiltonians
2
0
0 0 00 0
j
j j
VE E V
E E
0 = antisymmetrized product of ground state atomic functions
1st order term vanishes if overlap of atomic functions negligible.2nd order term is negative & R6 (van der Waals binding).
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Repulsive Interaction
Pauli exclusion principle (non-electrostatic) effective repulsion
Lennard-Jones potential: 12 6
4VR R
, determined from gas phase data
/Re
Alternative repulsive term:
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Equilibrium Lattice Constants
Neglecting K.E. 12 6
14
2toti j i j i j
E U Np R p R
For a fcc lattice:
12
12
112.13188
i j i jp
6
6
114.45392
i j i jp
For a hcp lattice:
12 12.13229 6 14.45489
R n.n. dist
At equilibrium:
0dE
dR
12 6
12 613 7
14 12 6
2N
R R
1/6
0 12
6
2R
1.09 for fcc lattices
Experiment (Table 4):
Error due to zero point motion
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Cohesive Energy
12 6
12 6
14
2totU R NR R
26
012
1
2totU R N
26
12
48
N
4 2.15N for fcc lattices
For low T, K.E. zero point motion.
For a particle bounded within length , p
2 2
2. .
2 2
pK E
m m
1
2
quantum correction is inversely proportional to the atomic mass:~ 28, 10, 6, & 4% for Ne, Ar, Kr, Xe.
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Ionic Crystals
ions: closed outermost shells ~ spherical charge distribution
Cohesive/Binding energy = 7.9+3.615.14 = 6.4 eV
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Electrostatic (Madelung) Energy
Interactions involving ith ion: i i jj i
U U
2/
2
. .R
i j
i j
qn ne
RU
qotherwise
p R
tot iU NUFor N pairs of ions:2
/R qN z e
R
z number of n.n.﹦ρ ~ .1 R0
j i i jp
﹦Madelung constant
At equilibrium: 0totdU
dR
2/
2Rz q
N eR
→ 0
2/2
0R q
R ez
2
00 0
1tot
N qU
R R
2
0
N qMadelung Energy
R
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Evaluation of Madelung Constant
App. B: Ewald’s method j i i jp
1 1 12 1
2 3 4
2ln 2
KCl
i fixed
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Kcal/mol = 0.0434 eV/molecule Prob 3.6
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Covalent Crystals
• 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
H2
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Ar : Filled outermost shell → van der Waal interaction (3.76A)Cl2 : Unfilled outermost shell → covalent bond (2A)
s2 p2 → s p3 → tetrahedral bonds
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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.
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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.
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Atomic Radii
Na+ = 0.97AF = 1.36ANaF = 2.33Aobs = 2.32A
Standard ionic radii ~ cubic (N=6)
Bond lengths:F2 = 1.417ANa –Na = 3.716A NaF = 2.57A
Tetrahedral:C = 0.77ASi = 1.17ASiC = 1.94AObs: 1.89A
Ref: CRC Handbook of Chemistry & Physics
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Ionic Crystal Radii
E.g. BaTiO3 : a = 4.004ABa++ – O– – : D12 = 1.35 + 1.40 + 0.19 = 2.94A → a = 4.16ATi++++ – O – – : D6 = 0.68 + 1.40 = 2.08A → a = 4.16ABonding has some covalent character.
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Analysis of Elastic Strains
ˆ ixLet be the Cartesian axes of the unstrained state
ix be the the axes of the stained state
Using Einstein’s summation notation, we have
ˆi j i j j xˆ ˆi i i j j x x x
1 1 11 1 11 2 k k x x 2 2 211 11 12 131 2
ˆi irr x
i ir r x
R r r ˆi i j jr x
Position of atom in unstrained lattice:
Its position in the strained lattice is defined as
Displacement due to deformation:
i j j iu r
iii ii
i
ue
x
jii j i j j i
j i
uue
x x
ˆi i ir x x ˆi iu x
Define ( Einstein notation suspended ):
i j
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Dilation
1 1 2 2 3 3ˆ ˆ ˆi i i j j j k k ka b c x x x
1 1 2 2 3 3i i j j k k i j kV
1 2 3ˆ ˆ ˆV a b c abc x x x
1 2 3V x x x
211 22 33
V VO
V
211 22 331V O
1 even permutation of 123
1 for odd permutation of 123
0 otherwisei j k ijk
2123 1 23 2 1 3 3i i j j k i jkV V O
2Tr O
where
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Stress Components
Xy = fx on plane normal to y-axis = σ12 .
(Static equilibrium → Torqueless) i j j i
y xX Y
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Elastic Compliance & Stiffness Constants
i j i j k l k le S
1 11 xX 1 1 1
2 2 2
3 3 3
2 3 4
3 1 5
1 2 6
i j
e S
4 23 32 z yY Z
C e
S = elastic compliance tensor
Contracted indices
C = elastic stiffness tensor
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Elastic Energy Density1
2U C e e
1
2C e C e U
e
C e 1
2C C e
C 1
2C C C
Let
then
1
2 i j k l i j k lU C u u
1
2C u u
Landau’s notations:
1
2ji
i jj i
uuu
x x
1
2
ii
i j
i jefor
ei j
1,2,3for
4,5,6ii
i j j i
uu
u u
u e
C
1
2C e e
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Elastic Stiffness Constants for Cubic Crystals
Invariance under reflections xi → –xi C with odd numbers of like indices vanishes
Invariance under C3 , i.e.,
1111iiiiC C
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):
1122ii k kC C 1212i k ikC C
2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6
1 1
2 2U C e e e C e e e e e e C e e e
1 111 12 12
2 212 11 12
3 312 12 11
4 444
5 544
6 644
0 0 0
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
eC C C
eC C C
eC C C
eC
eC
eC
11C C 12C C 44C C , 1, 2,3 4,5,6
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1
11 12 12 11 12 12
12 11 12 12 11 12
12 12 11 12 12 11
44 44
44 44
44 44
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
C C C S S S
C C C S S S
C C C S S S
C S
C S
C S
where 11 12
1111 12 11 122
C CS
C C C C
12
1211 12 11 122
CS
C C C C
4444
1S
C
1
11 12 11 12S S C C 1
11 12 11 122 2S S C C
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Bulk Modulus & Compressibility
2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6
1 1
2 2U C e e e C e e e e e e C e e e
Uniform dilation:
211 12
12
6U C C
1 2 3 3e e e
4 5 6 0e e e
δ = Tr eik = fractional volume change
21
2B B = Bulk modulus
1 V
V p
11 12
12
3B C C = 1/κ κ = compressibility
See table 3 for values of B & κ .
Up
2
2
UB
p
pV
V
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Elastic Waves in Cubic Crystals
Newton’s 2nd law:2
2
i ki
k
u
t x
don’t confuse ui with uα
i k ik j l j lC u →2
2
jlii k j l
k
uuC
t x
221
2jl
i k j lk j k l
uuC
x x x x
2l
i k j lk j
uC
x x
2 22 2 2 2 2 23 31 1 2 2 1 1
1111 1122 1133 1212 1221 1313 13312 2 2 21 1 2 1 3 2 1 2 3 1 3
u uu u u u u uC C C C C C C
t x x x x x x x x x x x
2 22 2 2 2 23 31 2 2 1 1
1111 1122 12122 2 21 1 2 1 3 2 1 2 3 1 3
u uu u u u uC C C
x x x x x x x x x x x
22 2 2 2 2
31 1 2 1 111 12 44 442 2 2 2
1 1 2 1 3 2 3
uu u u u uC C C C
t x x x x x x x
Similarly 22 2 2 2 2
32 2 1 2 211 12 44 442 2 2 2
2 2 3 2 1 1 3
uu u u u uC C C C
t x x x x x x x
2 2 2 22 2
3 3 3 32 111 12 44 442 2 2 2
3 3 2 3 1 2 1
u u u uu uC C C C
t x x x x x x x
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Dispersion Equation2 2
2i l
i k j lk j
u uC
t x x
0
i ti iu u e k r
→2
0 0i i k j l k j lu C k k u
20 0il i k j l k j lC k k u
2 0i l i k j l k jC k k dispersion equation
2 0I kC i j imn j m nC k kkC
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Waves in the [100] direction
2 0I kC i j imn j m nC k kkC
1,0,0kk → 211i j i jC kkC
1111 1112 1113
22111 2112 2113
3111 3112 3113
C C C
k C C C
C C C
kC11
244
44
0 0
0 0
0 0
C
k C
C
11L
Ck
0 1,0,0u Longitudinal
44T
Ck
0 0,1,0uTransverse, degenerate 0 0,0,1u
11112
2112
3113
0 0
0 0
0 0
C
k C
C
11 16 152
61 66 65
51 56 55
C C C
k C C C
C C C
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Waves in the [110] direction
2 0I kC i j imn j m nC k kkC
1,1,02
kk →
2
11 12 21 22 2i j i j i j i j i j
kC C C C kC
1111 1221 1122 12122
2121 2211 2112 2222
3113 3223
0
02
0 0
C C C Ck
C C C C
C C
kC11 44 12 442
12 44 11 44
44
0
02
0 0 2
C C C Ck
C C C C
C
11 12 44
12
2L C C C k
0 1,1,0u Lonitudinal
442T
Ck
0 0,0,1u
Transverse 1 11 12
1
2T C C k
0 1, 1,0 u
Transverse
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Prob 3.10
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