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Chapter 3. Crystal Binding & Elastic Constants
Solid State Physics by Heesang Kim at SSU
Chapter 3. Crystal Binding
Solid State Physics by Heesang Kim at SSU
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.
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
Van der Waals-London interaction (attractive interaction)
Simple argument
Solid State Physics by Heesang Kim at SSU
When there is no interaction between the atoms,
)(2
1000 wwE Ground state energy
Solid State Physics by Heesang Kim at SSU
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)
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
Pauli exclusion principle (repulsive interaction)
Two electrons can not have their quantum numbers equal.
12/~ RBEmpirical form of the interaction
Solid State Physics by Heesang Kim at SSU
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)
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
)4)(15.2()( 0 NRUtotal
Calculate Xe case as an example,……
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU
Hermite polynomial
Quantum harmonic oscillator problem
*** Helium, He ***
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
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU
Electron density distribution By x-ray study
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
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
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
Solid State Physics by Heesang Kim at SSU
Covalent Crystals
312222 221221 PSSPSS
exchange interaction ←spin dependent Coulomb energy
Si, Ge, C Share electrons=> fill their shells Strong bond Directionality Tetrahedral bond
Solid State Physics by Heesang Kim at SSU
exchange interaction ←spin dependent Coulomb energy
a way to avoid Pauli repulsion
Solid State Physics by Heesang Kim at SSU
If the bonding is not symmetric, it looks ionic as well.
Solid State Physics by Heesang Kim at SSU
Metals
High electrical conductivity Valence e get delocalized into conduction e (e’s delocalization reduces K.E.
Think of uncertainty principle)
Sea of e
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
Elastic properties of Crystal
Assume a homogeneous, continuous medium; Valid for elastic waves ; Consider small strain so that Hooke’s law may apply.
Solid State Physics by Heesang Kim at SSU
Displacement of the vector
Solid State Physics by Heesang Kim at SSU
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.
Solid State Physics by Heesang Kim at SSU
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 ; ;
Solid State Physics by Heesang Kim at SSU
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.
Solid State Physics by Heesang Kim at SSU
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.
Solid State Physics by Heesang Kim at SSU
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
Solid State Physics by Heesang Kim at SSU
Elastic energy density
eeCU2
1We might expect elastic waves, which we call phonon, i.e., quantum lattice vibration.
Solid State Physics by Heesang Kim at SSU
Solid State Physics by Heesang Kim at SSU