Download Lecture 4

PHYS 342/555
Introduction to solid state physics
Instructor: Dr. Pengcheng Dai
Associate Professor of Physics
The University of Tennessee
(Room 407A, Nielsen, 974-1509)
(Office hours: TR 1:10AM-2:00 PM)
Lecture 4, room 512 Nielsen
Chapter 3: Crystal Binding and Elastic Constants
Lecture in pdf format will be available at:
http://www.phys.utk.edu
1
FCC considered as simple cubic with a basis
Primitive vectors axˆ , ayˆ , and azˆ,
with four atoms per unit cell at
(000),(0
11 1 1 11
), ( 0 ), (
0)
22 2 2 22
4
SK = 1 + ∑ exp[iK idi ]
i =1
2π
K=
(n1 xˆ + n 2 yˆ + n3 zˆ )
a
SK = 1 + (−1)( n1+ n 2 ) + (−1)( n 2+ n 3) + (−1)( n 3+ n1)
⎧4, n1, n 2, n3 are even or odd ⎫
=⎨
⎬
1,
2,
3 mixed even/odd
0,
n
n
n
⎩
⎭
Dai/PHYS 342/555 Spring 2006
Chapter 3-2
Atomic form factor for x-rays
n
SK = ∑ fj ( K )eiK ⋅dj
j =1
where fj ( K ) is the atomic form factor, and is
determined by the internal structure of the ion that
occupies position dj in the basis. The atomic form
factor at K is taken to be proportional to the Fourier
transform of the electron charge distribution of the
corresponding ion:
fj ( K ) = ∫ dVnj (r ) exp(−iK ir )
Here nj (r ) is electron density concentration.
Dai/PHYS 342/555 Spring 2006
Chapter 3-3
Crystal binding:
The cohesive energy of a
crystal is defined as the
energy that must be added
to the crystal to separate its
components into neutral free
atoms at rest, at infinite
separation, with the same
electronic configuration.
Lattice energy is the energy
required to separate its
component ions into free
ions.
Dai/PHYS 342/555 Spring 2006
Chapter 3-4
Van de Waals-London interaction
Consider two atoms (1 and 2) separated by a distance r.
If the instantaneous dipole moment of atom 1 is p1, then
there will be an electric field proportional to p1 / r 3 at a
distance r from the atom. This will induce a dipole
moment in atom 2 proportional to the field:
p2 = α E ∼
α p1
r
3
, where α is the polarizability of the atom.
Dai/PHYS 342/555 Spring 2006
Chapter 3-5
Since two dipoles have an energy of interaction proportional
to the product of their moments divided by the cubic of
the distance between them, there will be a lowering of
energy of order
p1 p 2 α p12
∼ 6 associated with the induced moment.
3
r
r
Since the energy drop depends on p12 , its time average
does not vanish, even though the average value of p1 is zero.
Dai/PHYS 342/555 Spring 2006
Chapter 3-6
Assuming the force constant is C and p1 , p2 denotes the momenta, the
hamiltonian of the unperturbed system is
H 0 = p12 / 2m + Cx12 / 2 + p22 / 2m + Cx22 / 2;
The coulomb interaction energy of the two oscillators is:
e2
e2
e2
e2
−
−
H1 = +
R R + x1 − x2 R + x1 R − x2
in the limit | x1 |,| x2 |
R, we have (using Taylor expansion
1
= 1 + x + x 2 + x 3 + for small x.
1-x
2e 2 x1 x2
H1 = −
. The total Hamiltonian of the system is then:
3
R
H = H 0 + H1
Dai/PHYS 342/555 Spring 2006
Chapter 3-7
Define symmetric and antisymmetric modes of motion as:
xs ≡ ( x1 + x2 ) / 2; xa ≡ ( x1 − x2 ) / 2.
The momenta associated with the two modes:
p1 ≡ ( ps + pa ) / 2; p2 ≡ ( ps − pa ) / 2.
The total Hamiltonian of the system is then:
⎡ 2
1
2e 2 2 ⎤ ⎡ 2
1
2e 2 2 ⎤
H = H 0 + H1 = ⎢ ps /2m + (C − 3 ) xs ⎥ + ⎢ pa /2m + (C − 3 ) xa ⎥
2
R
2
R
⎣
⎦ ⎣
⎦
C ' = mω02
1/ 2
⎡
⎤
2e
ω = ⎢ (C ± 3 ) / m ⎥
R
⎣
⎦
2
⎡ 1 ⎛ 2e 2 ⎞ 1 ⎛ 2e 2 ⎞ 2
≅ ω0 ⎢1 ± ⎜
− ⎜
+
3 ⎟
3 ⎟
⎢⎣ 2 ⎝ CR ⎠ 8 ⎝ CR ⎠
Dai/PHYS 342/555 Spring 2006
⎤
⎥
⎥⎦
Chapter 3-8
Repulsive interaction
The Pauli exclusion principle prevents multiple occupancy,
electron distributions of atoms with closed shells can overlap
only if accompanied by the partial promotion of electrons
to unoccupied high energy states of the atoms.
Experimentally, one finds Lennard-Jones potential,
⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤
U ( R ) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥ , where ε and σ are the new
⎝ R ⎠ ⎥⎦
⎢⎣⎝ R ⎠
parameters with 4εσ 6 ≡ A and 4εσ 12 ≡ B.
Dai/PHYS 342/555 Spring 2006
Chapter 3-9
Equilibrium lattice constants
If there are N atoms in the crystal, the total potential energy is:
12
6
⎡
⎤
⎛
⎞
⎛
⎞
1
σ
σ
'
'
U total = N (4ε ) ⎢ ∑ ⎜
⎟⎟ − ∑ ⎜⎜
⎟⎟ ⎥ ,
⎜
2
pij R ⎠ ⎥
⎢ j ⎝ pij R ⎠
j
⎝
⎣
⎦
where pij R is the distance between reference atom i and
other atom j in terms of the nearest neighbor distance R.
For FCC structure, ∑ ' pij −12 = 12.13188,
j
∑
'
pij −6 = 14.45392
j
If there are N atoms in the crystal, the total potential energy is:
⎡
dU total
σ 12
σ6⎤
= 0 = −2 N ε ⎢ (12)(12.13) 13 − (6)(14.45) 7 ⎥ ,
dR
R
R ⎦
⎣
therefore R0 / σ = 1.09.
Dai/PHYS 342/555 Spring 2006
Chapter 3-10
Cohesive Energy
If there are N atoms in the crystal, the total potential energy is:
12
6
⎡
σ
σ
⎛ ⎞
⎛ ⎞ ⎤
U total ( R) = 2 N ε ⎢(12.13) ⎜ ⎟ − (14.45) ⎜ ⎟ ⎥ ,
⎝R⎠
⎝ R ⎠ ⎥⎦
⎢⎣
and, at R = R0 , U total ( R0 ) = −(2.15)(4 N ε ).
Dai/PHYS 342/555 Spring 2006
Chapter 3-11
Ionic crystals
Ionic crystals are made up of
positive and negative ions. The
ionic bond results from the
electrostatic interaction of
oppositely charged ions.
Cohesive energy is given by the potential
energy of classical particles localized at
the equilibrium positions. The total cohesive
energy per ion pair is:
U (r ) = U core (r ) + U coul (r ), where r is the
nearest-neighbor distance.
U core (r ) = λ exp( − rij / ρ ), where λ and ρ are
empirical parameters.
Dai/PHYS 342/555 Spring 2006
Chapter 3-12
U coul (r ) = ± q 2 / rij , In NaCl structure, we have N molecules
or 2 N ions. The total lattice energy is defined as the energy
required to separate the crystal into individual ions.
If we define rij ≡ pij R, where R is the nearest-neighbor
separation.
⎧λ exp(− R / ρ ) − q 2 / R (nearest neighbor) ⎫
⎪
⎪
2
U ij = ⎨ q
⎬
(otherwise)
⎪± p R
⎪
ij
⎩
⎭
U tot = N ( zλ exp(− R / ρ ) − α q 2 / R ),
where z is # of nearest neighbors, and α is Madelung
constant. α ≡ ∑
j
'
(±)
.
pij
Dai/PHYS 342/555 Spring 2006
Chapter 3-13
dU tot
= 0, or R02 exp(− R0 / ρ ) = ρα q 2 / zλ.
dR
Nα q 2 ⎛
ρ ⎞
=−
⎜1 − ⎟
R0 ⎝ R0 ⎠
At equilibrium,
U tot
Evaluation of the Madelung constant
α
R
≡∑'
j
(±)
. For one-dimensional NdCl, one have
Rj
α
1
1
⎡1 1
⎤
= 2⎢ −
+
−
+ ⎥,
R
⎣ R 2 R 3R 4 R
⎦
x 2 x3 x 4
ln(1 + x) = x − + − + . and α =2ln2.
2 3 4
Dai/PHYS 342/555 Spring 2006
Chapter 3-14
Structure
NaCl
CsCl
α
1.747565
1.762675
Dai/PHYS 342/555 Spring 2006
Chapter 3-15
Covalent crystals
The binding of
molecular hydrogen is
a covalent bond. The
strongest binding
occurs when the spins
of the two electrons are
antiparallel because of
the Pauli contribution
to the repulsion is
reduced in antiparallel
spins. The spindependent coulomb
energy is called the
exchange interaction.
Dai/PHYS 342/555 Spring 2006
Chapter 3-16