(A) van der Waals-London Interaction
- present between all atoms
- short range, weak
- responsible for condensation (gas-liquid) and freezing
(liquid-solid) if no stronger forces present
No interaction if charge distribution were rigid.
Ar
Ar
Charge on real atoms fluctuates, induces dipole moments in
each other, and causes an attractive interaction.
+ Ar - + Ar -
A simple argument for van der Waals-London Interaction
<p>=0
p2=E1 ~ p1/R3
p1
At time t
average
+ -
R
atom 1
+ -
is the polarizability of the atom
Electric Field: E1~ p1/R3
atom 2
p2
 p  p 
I t
Interactio
ti n energy   p2 E1 ~  31  13  ~  16
R
 R  R 
The time-averaged interaction energy of the two atoms:
U  c
 p12 
A
 6
6
R
R
 p12   0, U  0
Van der Waals force:
- Always attractive.
- A quantum mechanical effect.
This is called van der Waals interaction, also called the London interaction
or induced dipole-dipole interaction.
1
(B) The Repulsive Interaction
When two atoms are brought sufficiently closer, the interaction energy
becomes repulsive due to Pauli exclusion principle.
R
atom1
atom2
electron charge distribution
Charge distribution overlap
Two electrons cannot occupy the
same quantum state.
For inert gases, the repulsive energy can be fitted well to the form: U= B/R12.
Therefore , the total potential energy of two atoms is:
U(R)
V (R)
4
  12   6 
6 =A
B
A
U ( R)  12  6  4       where 4

2
R
R
 R   R   and 4 = B .
6
2
R
 
 
R
1/
12
 
U rep ( R4)  4    e  R / 
This is the Lennard-Jones potential.
Sometimes, the repulsive energy is written as
R
(C) Equilibrium Lattice Constants
j
In an inert gas crystal, the interaction energy
between atom i and atom j is:
 
U ij  4 
 Rij

12


  
R

 ij





6
Rij= pijR
12
6


 
 
  4        
 pij R   pij R  




R
i
where R is the nearest-neighbor distance.
With N atoms, the total cohesive energy is:
U total
6
   12

 
1
'
'  



 N (4 ) 



2
 j  pij R 
j  pij R  


i is the reference atom, ( j ≠ i).
pij is a pure number determined by the lattice type.

For an FCC lattice:
1
 p 
j 1
12
ij
 12.13188

1
 p 
j 1
6
 14.45392
ij
2
For an FCC lattice:
U total 

1
N ( 4 )  
2
 j

'
  


 p R
 ij 
12
6
   
 
  '
 p R 
j
 ij  
12
6

 
  
 2 N  12 . 13    14 . 45   
R
 R  

Rij= pijR
R
i
When a crystal is in equilibrium, the energy is at a minimum:

dUtotal
 12
6 
 0  2N (12)(12.13) 13  (6)(14.45) 7 
dR
R
R 

Therefore the equilibrium nearest neighbor atomic spacing is
(R0/) = 1.09
For all inert gas elements with an FCC structure at
temperature = 0 k and pressure = 0.
Measured R0/ values:
Ne: R0/=1.14; Ar: (40): R0/=1.11; Kr: R0/=1.10; Xe: R0/=1.09.
The slight differences are due to zero point quantum effects.
(D) Cohesive Energy
From the equilibrium inter atomic spacing, we can calculate the
cohesive energy of an inert gas crystal:
12
6

  
 
U total  2 N  12 . 13    14 . 45   
R
 R  

Using:
We get:
(R0/) = 1.09
12
6

 
  
1
U total ( R0 )  N (4 ) (12.13)   (14.45)    (2.15)(4 N )
2

 R0 
 R0  
same for all inert gases, (at 0 K and 0 pressure)
Quantum mechanical corrections (KE contributions) to the cohesive energy of
inert gases:
Ne (AW=4): -28%;
Ar (40): -10%; Kr (84): -6%; Xe (131): -4%.
3
2. Ionic Crystals - electrostatic interaction
Ionic crystals => made up of positive and negative ions.
Ionic bonds => electrostatic interaction between the ions.
Complication – long range interaction U ~1/r
NaCl
Simplicity – Coulomb interaction dominates.
Ionic crystals form between low
ionization energy atoms (e.g. alkalis:
Li, Na) and high electron affinity
atoms (e.g. halogens: F, Cl)
+ 5.14 eV
+electron
+
electron
+
Requires energy (ionization energy)
+ 3.61 eV
Releases energy (electron affinity)
+ 7.9 eV
Releases energy (cohesive energy)
Net energy released per NaCl unit = 7.9 eV + 3.6 eV - 5.1 eV = 6.4 eV
4
The binding energy of ionic crystals is predominantly electrostatic, and is
called the Madelung energy, which is defined relative to the energy of
the ions at infinite separation.
(A) The Madelung Energy
If Uij is interaction energy between ion i and j, then the total
energy involving ion i is:
rij= pij R
j
U i   'U ij
j i
For each pair: U ij  e
Repulsive
(short range)
For each ion pair:
i
 rij / 
q2

rij
R
(in CGS units)
Repulsive or attractive
(long range)
U ij  e
 rij / 
rij= pij R
q2

rij
j
i
If we measure distances in R (nearest neighbor distance),
and consider the repulsive interaction only for the nearest
neighbors, then

q2
R / 
 ,
  e
R

U ij  
2
q
1
 
,

pij R
R




(otherwise , where rij  pij R )

(nearestt neighbors
i hb )
For a crystal
y
of N molecules ((2N ions),
), the total lattice energy
gy is:


q2 
U total  NU i  N  U ij  N    e  R /    
R
j
 j  ( n.n.) 
2
 1 q 2 


   N  z e  R /    q 




pij R 
R 
j  other 


where z = number of nearest neighbors of an ion, and
 is called the madelung constant.
 '
j
 
pij
The sum includes
nearest-neighbor
contributions.
5
The madelung constants of several lattices
 '
j
NaCl:
CsCl:
z=6
z=8
z=4
=1.747
=1.762
=1.638
 
pij
ZnS:
2
The total lattice energy U total  NU i  N  ze  R /    q 

R 

At the equilibrium inter-ionic separation,
We have: R02 e  R
0
At equilibrium:
/

q 2
z
dU total
0
dR
where R0 is the equilibrium nearestneighbor distance.

q 2 
N q 2
  
U total  N  ze  R0 /  
R0 
R0


 
1  
 R0 
The attractive component, -Naq2/R0, is called the Madelung energy.
The repulsive interaction is short- ranged.
Calculation of the Madelung constant    

pij
2
For a crystal to be stable, the total energy must U  N  ze  R /   q 
total

R 
contain both attractive and repulsive terms

j i
Thus  must be positive !
rij= pij R
Assume that ion i is negative, then in the definition of 
j
rj
i


q2 
U total  NU i  N    e  R /    
R
 j ( n.n.) 
R
2
 1 q 2 



   N  z e  R /    q 




p
R
R
j  other 


ij


+ is for a positive ion (j) and – for a negative ion (j).
we can write

R

j i
  rj => distance of ion j to the reference ion i.
R => nearest neighbor distance.
rj
Example: find the Madelung constant for a 1-d ionic crystal.
+
-
+
-
+
-
+
-
+
-
R
+
-
+
Reference ion

R

j i
 
rj
1
1
1
 2 ln 2
1
 2 


    
R

 R 2 R 3R 4 R
-
+
-
For 3-d lattices,  can
be calculated from the
structure in a similar
fashion.
6
3. Covalent Crystals
Energy
The covalent bond between two atoms is usually formed by two electrons,
one from each atom. The electrons forming the bond tend to be partly
localized in the region between the two atoms.
1S
1S
H atom
H atom
H2
molecule
Features of covalent crystals:
- Quantum mechanics is needed to calculate the binding energy.
- Covalent bonds are highly directional (low APF, low density).
- Only few crystals are covalently bound (diamond, Si, Ge, SiC)
- Covalent bonds are strong (hardest materials, high melting
points, insoluble).
- All covalent crystals have tetrahedral (diamond) structure.
- There is a continuous range of crystals between the ionic and
covalent limits.
APF = 0.34
7