What you have to learn from this chapter

1
Chapter 3. Crystal Binding
What you have to learn from this chapter? ?
Classification of Solid ? Energies of
Crystals ? Van der Waals crystals ? Repulsive
Interaction between Atoms ? Ionic Crystals ?
Debye Frequency ? Differences between Covalent
and Metallic Bonding
2
? References C. Kittel, Introduction to Solid
State Physics 6th ed. (or newer), Chapter
3., John Wiley Sons, Inc., Singapore,
1986.
Classification of crystalline solid (a) ionic,
(b) van der Waals, (c) covalent, and (d)
metallic ? Internal energy of a crystal includes
(1) lattice energy (cohesive energy) and (2)
thermal energy (lattice vibration). Consider
lattice energy only for simplicity! Crystal at
0 K! Neglecting zero-point energy in most
cases! ? The cohesive energy of a crystal is
defines as the energy that must be supplied to
the crystal to separate its components into
neutral free atoms (at rest, at infinite
separation, with the same electronic
configuration).
3
? The structure parameter of a system is set by
the lowest energy configuration.
? Van der Waals crystals Van der
Waals-London Interaction
Simple model ? two oscillator with ? charge ?
the force constant for the spring is C
x1
x2
R

-

-
The Hamiltonian of the system (classical
mechanics) kinetic energy (p2/2m) potential
energy (Cx2/2)
When this two oscillators are uncoupled, the
natural frequency is ?0 for each oscillators
?0(C/m)1/2.
4
F (-Cx) ma (md2x/dt2) gt md2x/dt2Cx0 Assume
xAcos ?0t gt d2x/dt2 -?02Acos ?0t gt-m?02Acos
?0t CAcos ?0t 0 gt ?0(C/m)1/2 meaning the up
down motion of a spring is similar to a
circular motion projected in one axis! ?0
is independent of initial condition of the
spring, but amplitude is!
When this two oscillators are uncoupled, the
Hamiltonian of the system is H0
When this two oscillators are coupled, the
Hamiltonian of the coupling term is H1
5
Assume x1, x2 ltlt R
Define symmetric and antisymmetric mode of
motion (xs, xa) and momentua (ps, pa)
gt Two oscillators with different spring
constant the resonant frequencies gt
?s(C2e2/R3)/m1/2 ?p (C-2e2/R3)/m1/2
6
When this two oscillators are uncoupled, the zero
point energy of the system is (??0/2 ??0/2)
??0 after interactions gt (??s/2 ??p/2)
?0
The zero point energy change between these
two systems
The lowering of the system energy by van der
Waals interaction is proportional to R-6.
7
a
Potential at the point p
e

-
T
l2r-acosT/2
-e
r
p
l1racosT/2
For r gtgt a ,
The electric field in the radial and transverse
direction
µea dipole moment.
The force which acts on a charge e at point p is
8

• ? Inert gas is the prototype for the van der
  Waals crystal
• closed shell no average dipole moment only
  have
• instantaneous dipole moment!
• Induced dipoles the presence of an external
  could
• cause the center of gravity of the electron
  cloud to be
• different from the nucleus gt dipole

r
The total force on the induced dipole due to the
field of the other dipole
-e
e
µ
a
µI aE
a polarizability
9
The energy of a pair of inert-gas atoms due to
dipole interaction is now
Inverse sixth power of their distance
The same conclusion as before!
? Dipole-Quadrupole and Quadrupole-Quadrupole
terms Synchronization of the motion of the
electrons on the various atoms of a solid gt
van der Waals attractive energy (lower
energy). More complex charge distribution
exists in real atoms. Calculations show
Dipole-Quadrupole interaction ? -r-8
Quadrupole-Quadrupole interaction ? -r-10
Modern QM gt van der Waals attractive energy of
an ion ?(r)-(c1/r6c2/r8 c3/r10)
83, 16, lt1.3
10
? Molecular Crystal van der Waals forces are the
major attractive forces for typical covalent
nonpolar molecules (like N2, H2, CH4, etc no
permanent dipole moments) polar molecules
(like H2O) possess permanent dipoles gt much
stronger binding (van der Waals binding) in
their respective crystals gthigher melting
temperature and boiling points. ? Repulsive
Interaction bring two atoms together gt
overlapping of the charge distribution gt
changing the electrostatic energy of the
system At very distance gt Pauli exclusion
principle (no two electrons could be at the
same quantum state) gt charge distribution
adjusting is required gt electrons will
occupy higher energy state gt repulsive force!
11
Experimental data on the inert gases shows
the repulsive potenetial to be the form of
B/R12.
The total potential of two atoms (inert gas
atoms) at separation R


U
R
Lennard-Jones Potential
Equilibrium Lattice Constants of inert gas
crystal neglect kinetic energy of the atoms,
assuming there are N atoms in the crystal, the
total potential of the system is
12
Pij the distance betwwen atom i and j in terms
of the nearest neighbor distance R
for fcc structure
Minimize total energy with respect to R
Ne Ar Kr Xe R0/s
1.14 1.11 1.10 1.09
Cohesive energy U(R0) -2.15(4Ne)
13
? Ionic crystal Combination of a highly
electropositive metallic element with a
highly electronegative element.
NaCl CN6
CsCl CN8
Zincblend CN4
There are still others types of ionic crystals!
? Attraction force Coulomb force ? Repulsive
force ion resists overlap with the electron
distributions of neighboring ions ? Empirical
potential for repulsive term
14
nearest neighbor
otherwise
The total energy of an ionic crystal with N
molecules (assume z 1)
C number of nearest Neighbors a Madelung
constant
Madelung constant in 1-D

-

-

-
R
a 2ln21.386
15
For NaCl structure, 6 nearest neighbors with
distance r 12 second nearest neighbors with
distance 21/2r 8 third nearest neighbors with
distance 31/2r ...
a 1.762675 for CsCl a 1.6381 for Zincblend
The parameter used in repulsive term must be
evaluated Two experimental quantities could be
used (1) equilibrium interionic separation at
0K, (2) The compressibility of the solid at 0K.
U? CBgtB aA
16
? Refinement ionic crystals also have some
portions of van der Waals energy. See Table
3.2 as an example.
? The Debye Frequency ? The zero point
energy of a crystal is its thermal
(vibrating) energy at 0K for an oscillator the
zero point energy is ??0/2. In
a crystal solid (3-D), three vibrational degrees
of freedom per atom gt N atoms gt 3N
oscillators! ? Interaction between
neighboring pair of atoms (assume to be
a linear spring) gt Debye proposed that
the entire lattice could be considered to be a
3-D array of masses
interconnected by spring!
3D
1D
17
? These interconnecting atoms gt collective
behavior! gt phonon ? The vibrations of this
atoms has constrains there is no real
physical translation of the objects! gt Boundary
condition! gt Standing waves
Shortest wavelength (maximum frequency) allowed
?min 2a gt ?m v/?min v sound velocity v
5x103 m/s assume a 0.25 nm gt ?m 1013 Hz
attempt frequency used in diffusion order of
magnitude correct only.
a
In 1-D, e.g. number of mode allowed in a 4
atoms system is 4 (1-4 half wave) gt N atoms gt N
modes of vibration gt 3-D ?3 gt 3N modes
(Polarization)
18
? In 1-D, the density of vibrational modes
(density of states) is the same in any
frequency in 3-D, the density of vibrational
mode in the range of ? to ? d?.
Area 3N
f(?)
?
?m
Phonon quantized elastic wave fill the modes
from low energy portions
This quantities should be added to
experimentally determined cohesive energies in
comparing with computed static lattice energies.
19
? Covalent and Metallic Bonding ? Quantum
mechanics is an essential when it comes to
studying the bonding in covalent and metallic
crystals. In both crystals, the bonding
is associated with the valence electrons
that are not considered to be
permanently bound to specific atoms in the
solids (i.e. valence electrons are
shared between atoms). ? e.g. diamond C
1S2, 2S2, 2P2 2S2, 2P2 gt 4 hybrid
bonds , to share electron with the 4 nearest
neighbor gt achieve the electron
configuration of 1S2, 2S2, 2P6. These
valence electrons belong to the crystal as
a whole! ? Covalent bond is very strong
and directional in nature.
20
? The simplest covalent bond system is Hydrogen
molecule. (see. Fig.3.12) (1) Two hydrogen
atoms with the same spin are brought
together gt one electron has to be move to a
higher energy state gt increase energy
(2) Two hydrogen atoms with the opposite spin
are brought together gt each nucleus can
accommodate two electrons gt ,- charge
pair (exists for limited periods of
time) gt 5 of the total binding energy
The electron pair could move from one nuclei to
another gt resonance effect at a very
fast rate gt 80 of the total binding
energy (3) From QM gt electrons spend more of
their time is the region between the
two protons.
21
(time-average position gt phase space
position and momenta average over a
long time is the same as average over
all positions.
? Valence electrons are also shared between atoms
in metallic crystal. The shared electrons are
near free, while electrons can move from one
bond to another in covalent bond gt metallic
crystal tends to be close- packed in which the
directionality of the bonds between atoms is
of secondary importance.
? Metallic crystal lattice of positively charged
nuclei in a near free electron sea. The
nuclei and electron hold each other together!
22
Supplement vibration mode and density of state
a
us
s 0
s 10
Fixed
L
Fixed
Standing wave
Condition
s 0
s 10
K
..
0
Link together
Another way to express the condition is periodic
boundary condition
Wave
23
.
.
0
Interval between successive value of K
Number of mode per unit length of K
In 2D ? Kx, Ky. The allowed mode is a 2D K
space. Each mode occupied an K space area of
K
K?K
24
In 3D ? Kx, Ky, Kz.
The allowed mode is a 3D K space. Each mode
occupied an K space volume of There are
modes per unit volume of K space.
The total number of modes with vector less than K
is
for each polarization
Density of states
25
Polarization
Transverse wave (optical mode) x2
Longitudinal wave sound wave (acoustic mode) x1