Chapter 3. Crystal Binding & Elastic...
Transcript of Chapter 3. Crystal Binding & Elastic...
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Chapter 3. Crystal Binding & Elastic Constants
Solid State Physics by Heesang Kim at SSU
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Chapter 3. Crystal Binding
Solid State Physics by Heesang Kim at SSU
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Ionization energy ? Energy required in taking an electron away from an atom.
Cohesive energy? Energy required in taking a compound away from a crystal. Inert gas crystal < alkali metal < C, Si, Ge, … < transition metal
Melting temp. & bulk modulus vary roughly as cohesive energies.
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Crystals of inert gases
Simplest crystal, transparent insulator, weakly bound, low melting point, closed packed (fcc, hcp)
Inert gas atoms : completely filled shell, stable, very high ionization energy.
What holds them together? Van der Waals-London interaction
Induced dipole dipole interaction
atom1 atom2
+ +
weak bonding between neutral atoms, between molecules
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Van der Waals-London interaction (attractive interaction)
Simple argument
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When there is no interaction between the atoms,
)(2
1000 wwE Ground state energy
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Now, turning on the coulomb interaction between them.
Assuming that
3
21
22
2
2
1
2
2
2
110
2
2
1
2
1
2
1
2
1
R
xxeCxCxP
mP
mHHH
This is nothing but a coupled oscillator problem. (mech. Chap. 12)
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3
21
22
2
2
1
2
2
2
110
2
2
1
2
1
2
1
2
1
R
xxeCxCxP
mP
mHHH
One of the ways to solve it: normal mode transformation
)(2
11 as xxx )(
2
12 as xxx )(
2
11 as ppp )(
2
12 as ppp
2
3
222
3
22
)2
(2
1
2
1)
2(
2
1
2
1aass x
R
eCp
mx
R
eCp
mH
2
3
2
3
2
0
2
1
3
2
)2
(8
1)
2(
2
11/)
2(
CR
e
CR
ewm
R
eCw
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6
2
3
2
00 )2
(8
1
R
A
CR
ewEEU
Therefore, Van der Waals interaction (London int., induced dipole dipole int.) lowers the ground state energy : quantum effects( ).
2
3
2
00
2
3
2
0 )2
(8
1))
2(
8
11()(
2
1
CR
ewE
CR
ewwwE
Ground state energy
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Pauli exclusion principle (repulsive interaction)
Two electrons can not have their quantum numbers equal.
12/~ RBEmpirical form of the interaction
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All together gives the Lennard-Jones potential :
612
4)(RR
RU
6
'
12
')4(
2
1
RPRPNU
ijj
ijj
total
N=# of atoms ; R=nn distance ; Pij R= distance btw I & j atoms
; 13188.1212'
jij
P 45392.146'
jij
PFor fcc
; 13229.1212'
jij
P 45489.146'
jij
PFor hcp
Cohesive energy (total)
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0)45.14)(6()13.12)(12(20
0
0 7
6
13
12
RR
RRtot
RRN
dR
dU 09.10
R
6
'
12
')45.14()13.12(2
RRNU
jjtotal
Exp. values from gas phase
Ne Ar Kr Xe
R0/σ 1.14 1.11 1.10 1.09
)4)(15.2(09.1
1)45.14(
09.1
1)13.12(2)(
6
'
12
'
0 NNRUjj
total
for all inert gases
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)4)(15.2()( 0 NRUtotal
Calculate Xe case as an example,……
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Hermite polynomial
Quantum harmonic oscillator problem
*** Helium, He ***
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Ionic Crystals : ionic bond
1622 3221 : spssNa
52622 33221 : pspssCl
Energy 손익계산서: Na + Cl = NaCl + 6.4 eV
-5.14 + 3.61 + 7.9 = 6.4
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Electron density distribution By x-ray study
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Interaction btw/ i-th & j-th ions (cgs unit)
Attractive interaction : Coulomb interaction Repulsive interaction : Pauli exclusion principle
jij
i UU'
ij
ijij r
qrU
2
exp
It is found that exponential form works better in ionic crystal case.
RqR
2
exp
ijU
R
q
Pij
21
nearest neighbors
otherwise
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R
qezNNUU
R
itotal
2
Cohesive energy (total)
0)('
ijj P
Madelung constant
01
00
2
2
RR
R
RR
i
R
qez
dR
dU
R0
Notice that there are 2N ions.
0
2
0
2
0 1)(0
RR
Nq
R
qezNRU
o
R
total
Madelung (electrostatic) energy
Solid State Physics by Heesang Kim at SSU
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How to evaluate Madelong constant
Let us consider a 1-dim. Ionic crystal as in the figure.
4
1
3
1
2
11
2
4
1
3
1
2
112
RRRRRR
432
)1ln(432 xxx
xx
2ln2
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Covalent Crystals
312222 221221 PSSPSS
exchange interaction ←spin dependent Coulomb energy
Si, Ge, C Share electrons=> fill their shells Strong bond Directionality Tetrahedral bond
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exchange interaction ←spin dependent Coulomb energy
a way to avoid Pauli repulsion
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If the bonding is not symmetric, it looks ionic as well.
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Metals
High electrical conductivity Valence e get delocalized into conduction e (e’s delocalization reduces K.E.
Think of uncertainty principle)
Sea of e
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Hydrogen Bonds
A type of bond formed when the partially positive hydrogen atom of a polar covalent bond in one molecule is attracted to the partially negative atom of a polar covalent bond in another.
Responsible for the strange behavior of water, and for the DNA double helix structure & reproduction
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Elastic properties of Crystal
Assume a homogeneous, continuous medium; Valid for elastic waves ; Consider small strain so that Hooke’s law may apply.
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Displacement of the vector
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zzzyzx
yzyyyx
xzxyxx
eee
eee
eee
Strain
ze
ye
x
ue zzzzyyyyxxxx
;;
xz
uxze
yzzye
xy
uyxe
xzzxzx
yzzyyz
xyyxxy
Dilation : fractional increase of volume
zyxV zzyyxx eee 1
zzyyxx eeeV
VV
Dilation
Symmetric matrix : only 6 components.
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Stress : force acting on a unit area (a bit different from pressure)
zyx
zyx
zyx
ZZZ
YYY
XXX Capital : direction of force Sub: direction of plane
Symmetric matrix w.r.t. the diagonal: Thus, there are only 6 components.
yzxzxy ZYZXYX ; ;
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xy
zx
yz
zz
yy
xx
y
x
z
z
y
x
e
e
e
e
e
e
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
X
Z
Y
Z
Y
X
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
Hooke’s law gives
C : Elastic stiffness constants, moduli of elasticity (somewhat like spring constant)
(e) = (S)(X…) = (C)^{-1}(X…)
S : Elastic compliance constants, elastic constant
Symmetric matrix : only 21 components.
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xy
zx
yz
zz
yy
xx
y
x
z
z
y
x
e
e
e
e
e
e
C
C
C
CCC
CCC
CCC
X
Z
Y
Z
Y
X
44
44
44
113212
121112
121211
00000
00000
00000
000
000
000
Cubic crystal case : symmetry consideration gives
There are only 4 Cs.
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Elastic energy density
eeCU2
1
Bulk modulus, B
Compressibility, K
2
2
1BU
BK
1
dV
dpVB or equivalently
변형에 대해 얼마나 rigid 한가를 나타내는 수치
얼마나 쉽게 압축되는가를 나타내는 수치
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Elastic energy density
eeCU2
1We might expect elastic waves, which we call phonon, i.e., quantum lattice vibration.
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