Lattice Dynamics
21 February, 2007
U. Milan Short Course
Thermal Expansion
• As temperature changes,
density changes
• Thermodynamics
– Relates this change to changes
in other properties
– Cannot tell the magnitude or
even the sign!
•
•
•
•
Why positive alpha?
What value vs. P,T,X?
Macroscopic to Microscopic
Thermodynamics to
Statistical Mechanics
Interatomic Forces
• Ambient Structure
• Bulk Modulus
– Curvature
• Thermal Expansivity
– Beyond harmonic
– Molecules
– Solids
Potential Energy
– Minimum
Distance
One Dimensional Lattice
N
 2Vn,n  p
1
Vn  V0  
(un  p  un ) 2  H.O.T.
2
2 pN un
N
Vn
Fn  
  K (u  u )
un pN p n  p n
 2Vn,n  p
Kp 
un2
N
 2 un
m 2   K p (un  p  un )
t
pN
un  u0 exp inka  t 
pka
4 N
   K p sin 2  
 2 
m p1
2
 2
ka
K
sin  
m  2 
One Dimensional Lattice
• Periodicity reflects
that of the lattice
• Brillouin zone
center, k=0: =0
• Brillouin zone edge,
k=/a: =maximum
• All information in
first Brillouin zone
Frequency, 
1st Brillouin
Zone
2(K/m)1/2
0
Wavevector, k
One Dimensional Lattice
k=/a=2/; =2a
k0; 
Acoustic Velocities
K ka
sin  
m  2 
– k0
– w=2(K/m)1/2ka/2
– w/k=dw/dk=a(K/m)1/2
• Acoustic Velocity
– v=a(K/m)1/2
• Three dimensions
– Wavevector, ki
– Polarization vector, pi
– For each ki, 3 acoustic
branches
– ki pi longitudinal (P) wave
– kipi transverse (S) waves (2)
Frequency
 2
0.0
0.2
0.4
0.6
Wavevector (a)
0.8
1.0
Polyatomic Lattice
a
New Brillouin
Zone
Frequency
• Unit cell doubled
• Brillouin Zone Halved
• Acoustic Branches
folded
• New, finite frequency
mode at k=0
• Optic Branch
0.0
0.2
0.4
0.6
Wavevector (a)
0.8
1.0
General Lattice
Fumagalli et al. (2001) EPSL
• Number of modes = 3N
organized into 3Z
branches
– Z= number of atoms in unit
cell
• 3 Acoustic branches
• 3Z-3 optic branches
• Experimental Probes
– Optic zone center
• Raman
• Infrared
– Acoustic near zone center
• Brillouin
– Full phonon spectrum
• Inelastic neutron scattering
Quartz
MgSiO3 Perovskite Movie
Internal Energy
Energy
• Sum over all
vibrational modes
• Energy of each
mode depends on
n=3
– Frequency
– Population
• Frequency
• Temperature

n=2
n=1
1

E n    n
2

n=0
Displacement

Heat Capacity
U vib
3N
1 3N
   i   ni  i
2 i1
i1
1
    
n i  exp i 1
  kT  
3N
U vib
3N
1
  i  
2 i1
i1
i
  i 
exp 1
 kT 
High Temperature
3N
U vib
3N
1
   i   kT
2 i1
i1
U 
CV     3Nk
T V
• or
• CV=3R per mol of atoms
(Dulong-Petit)
Thermal Pressure 1
3N

  i 
1 3N
FTH    i  kT ln 1 exp

 kT 
2 i1

i1
FTH 
PTH  

 V T

PTH 
V
UTH
3N
    i i
i1
i  
Energy
3N

i
i1
 ln  i
 ln V
• Compression Increases
– Vibrational frequencies
– Vibrational energy
•  Thermal pressure
Displacement
Thermal Pressure 2
Thermal Pressure
PTH   3RT
Bulk Modulus
KTH    1 q
Volume dependence of 
 ln 
q
 ln 
Thermal pressure
Interatomic Forces
• Ambient Structure
• Bulk Modulus
– Curvature
• Thermal Expansivity
– Beyond harmonic
– Molecules
– Solids
Potential Energy
– Minimum
Distance
Fundamental Thermodynamic
Relation
Helmholtz free energy as a function of volume and temperature
Complete information of equilibrium states/properties
F(V,T)
Divide into purely volume dependent “cold” part and a thermal part
F(V,T)  F0  F(V,T0 )  FTH (V,T)
Recall we already have an expression for the “cold” part
Cold part
•
•
Start from fundamental relation
Helmholtz free energy
140
F=af
120
•
Isotherm, fixed composition
– F=F(V)
•
•
Taylor series expansion
Expansion variable must be V or function
of V
– F = af2 + bf3 + …
•
•
Pressure (GPa)
– F=F(V,T,Ni)
f = f(V) Eulerian finite strain
a = 9K0V0
2
MgSiO3
Perovskite
300 K
100
80
60
40
20
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00
Volume, V/V 0
Thermal part
FTH
3N

  i 
1 3N
   i  kT ln 1 exp



2 i1
kT

i1
Not easily evaluated, need to know all vibrational frequencies at all pressures
This information not available for ANY mantle mineral!
What to do?
Frequencies only appear in sums
Thermodynamics insensitive to details of distribution of frequencies
Assume i=E, all i, where E is a characteristic frequency of the material
-1
-1
-1
-1
-1
Anorthite 522 cm-1
Forsterite 562 cm-1
Corundum 647 cm-1
-1
-1
Characteristic (Debye) frequencies
-1
-1
CP , (H-H0)/T, S+10 (J mol atom K ) CP , (H-H0)/T, S+10 (J mol atom K ) CP , (H-H0)/T, S+10 (J mol atom K )
Comparison to
experiment
70
Anorthite
60
50
S
40
CP
30
20
(H-H0)/T
10
0
0
70
-10
500
1000
1500
2000
Temperature (K)
Forsterite
60
50
S
40
30
CP
20
(H-H0)/T
10
0
0
70
500
1000
1500
2000
-10
Temperature (K)
Corundum
60
50
S
40
CP
30
20
(H-H0)/T
10
0
-10
0
500
1000
1500
Temperature (K)
2000