21 February, 2007 U. Milan Short Course

Lattice Dynamics

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

– Minimum• Bulk Modulus

– Curvature• Thermal

Expansivity– Beyond harmonic– Molecules– Solids

Potential EnergyDistance

One Dimensional Lattice

€

Vn = V0 + 12

∂ 2Vn,n + p

∂un2 (un + p − un )2 + H.O.T.

p=−N

N

∑

Fn = −∂Vn

∂un

= K p (un + p − un )p=−N

N

∑

K p =∂ 2Vn,n + p

∂un2

m ∂2un

∂t 2 = K p (un + p − un )p=−N

N

∑

un = u0 exp i nka −ωt( )[ ]

ω2 = 4m

K p sin2 pka2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

p=1

N

∑

ω = 2 Km

sin ka2 ⎛ ⎝ ⎜

⎞ ⎠ ⎟

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

1st BrillouinZone

0

Wavevector, k

Frequency,

2(K/m)1/2

One Dimensional Latticek=/a=2/; =2a

k0;

Acoustic Velocities

– k0– 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– kipi transverse (S) waves (2)

€

=2 Km

sin ka2 ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Frequency

1.00.80.60.40.20.0

Wavevector (/a)

Polyatomic Lattice

• Unit cell doubled• Brillouin Zone Halved• Acoustic Branches

folded• New, finite frequency

mode at k=0• Optic Branch

Frequency

1.00.80.60.40.20.0

Wavevector (/a)

a

New BrillouinZone

General Lattice• 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

Fumagalli et al. (2001) EPSL

MgSiO3 Perovskite Movie

Internal Energy

• Sum over all vibrational modes

• Energy of each mode depends on– Frequency– Population

• Frequency• Temperature

Energy

Displacement

n=0

n=1

n=2

n=3

€

En = hω 12

+ n ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Heat Capacity

• or• CV=3R per mol of

atoms (Dulong-Petit)

€

Uvib = 12

hωi + n ihωii=1

3N

∑i=1

3N

∑

n i = exp hωi

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟−1

⎡ ⎣ ⎢

⎤ ⎦ ⎥

−1

Uvib = 12

hωi + hωi

exp hωi

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟−1i=1

3N

∑i=1

3N

∑

High Temperature

Uvib ≈ 12

hωi + kTi=1

3N

∑i=1

3N

∑

CV = ∂U∂T ⎛ ⎝ ⎜

⎞ ⎠ ⎟V

= 3Nk

Thermal Pressure 1

• Compression Increases– Vibrational

frequencies– Vibrational energy

Thermal pressure

€

FTH = 12

hωii=1

3N

∑ + kT ln 1− exp − hωi

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥

i=1

3N

∑

PTH = − ∂FTH

∂V ⎛ ⎝ ⎜

⎞ ⎠ ⎟T

PTH = γV

UTH

γ = γ iωii=1

3N

∑ ωii=1

3N

∑

γ i = −∂ lnωi

∂ lnV

Energy

Displacement

Thermal Pressure 2Thermal Pressure

Bulk Modulus

Volume dependence of

€

PTH ≈ γρ 3RT

KTH ≈ γρ γ +1− q( )

q = − ∂ lnγ∂ lnρ

Thermal pressure

Interatomic Forces• Ambient Structure

– Minimum• Bulk Modulus

– Curvature• Thermal

Expansivity– Beyond harmonic– Molecules– Solids

Potential EnergyDistance

Fundamental Thermodynamic Relation

€

F(V ,T)

Helmholtz free energy as a function of volume and temperatureComplete information of equilibrium states/properties

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

140

120

100

80

60

40

20

0

Pressure (GPa)

1.000.950.900.850.800.750.70

Volume, V/V0

MgSiO3Perovskite

300 K

F=af 2

Cold part• Start from fundamental relation• Helmholtz free energy

– F=F(V,T,Ni)

• Isotherm, fixed composition– F=F(V)

• Taylor series expansion• Expansion variable must be V or

function of V– F = af2 + bf3 + …

• f = f(V) Eulerian finite strain• a = 9K0V0

Thermal part

€

FTH = 12

hωii=1

3N

∑ + kT ln 1− exp − hωi

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥

i=1

3N

∑

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

Comparison to experiment

70

60

50

40

30

20

10

02000150010005000

Temperature (K)

Anorthite

S

CP

(H-H0)/T

-10

70

60

50

40

30

20

10

02000150010005000

Temperature (K)

Forsterite

S

CP

(H-H0)/T

-10

70

60

50

40

30

20

10

02000150010005000

Temperature (K)

CP

(H-H0)/T

Corundum

S

-10

Characteristic (Debye) frequencies

Anorthite 522 cm-1

Forsterite 562 cm-1

Corundum 647 cm-1