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Lattice Dynamics, Thermal Properties, and Density Functional Perturbation Theory
Ronald CohenGeophysical Laboratory
Carnegie Institution of [email protected]
2007 Summer School on Computational Materials ScienceQuantum Monte Carlo: From Minerals and Materials to MoleculesJuly 9 –19, 2007 • University of Illinois at Urbana–Champaignhttp://www.mcc.uiuc.edu/summerschool/2007/qmc/
Lattice Dynamics, Thermal Properties, and Density Functional Perturbation Theory
• Why you need estimates of thermal corrections• Thermodynamics of collection of oscillators• Mathematical description of crystals (review)• Atomic displacement waves (phonons)• Dynamical matrix• Secular equation (equations of motion)• Densities of states• Kieffer models• Examples• Ionic systems—non-analyticity• Born effective charges• Linear response• Density functional perturbation theory• Process for DFPT computations
Kieffer models1. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals: 1. Mineral heat capcities and their relationships to simple lattice vibrational models. Reviews of Geophysics and Space Physics, 1979. 17: p. 1-19.2. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals: 2. Vibrational characteristics of silicates. Reviews of Geophysics and Space Physics, 1979. 17: p. 20-34.3. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals:3. Lattice dynamics and an approximation for minerals with application to simple substances and framework silicates. Reviews of Geophysics and Space Physics, 1979. 17: p. 35-59.4. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals:4. Application to chain and sheet silicates and orthosilicates. Reviews of Geophysics and Space Physics, 1980. 18: p. 862-886.5. Kieffer, S.W., Thermodynamics and lattice vibrations of minerals:5. Applications to phase equilibria, itostopic fractionation, and high-pressure thermodynamic properties. Reviews of Geophysics and Space Physics, 1982. 20: p. 827-849.
Entro
py
Temperature%
err
or
Matas, J., et al., Thermodynamic properties of carbonates at high pressures from vibrational modelling. Eur J Mineral, 2000. 12(4): p. 703-720.
2
Calculated phonon frequencies of bcc, fcc and hcp Fe all show excellent agreements with experiment
0
5
1 0
1 5
2 0
2 5
3 0
3 5L
0 .00 .0 0 .50 .5 1 .0
ω (m
eV)
[1 1 1 ]
Λ
0 .2 5
X
T A
L A
Γ
T A 2
T A 1
L A
T A
Σ
[1 1 0 ]
Γ
L A
0 .5 [0 0 1 ]
Δ
0
50
100
150
200
250
300
0.00.0 0.50.5 1.0
ω (c
m-1
)
[110]
Ν
0.25
TA
LA
Γ
LM T O -G G A Experim ent
P P-G G A P P -LD A
TA2
TA1
LA
TA
Η
[100]
Γ
LA
0.5 [111]
Ρ
. fcc Fe
bcc Fe
0 5 1 0 1 5 2 0 2 5
V = 4 0 a . u .
V = 6 0 a . u . E x p : 5 0 G P a
E (T H z )
V = 7 0 a . u .hcp Fe
sound velocities along the sound velocities along the HugoniotHugoniot
Experiments: Nguyen and Holmes, Nature 2004 Theory: Sha and Cohen
50 100 150 200 250 300 3507.0
7.5
8.0
8.5
9.0
9.5
10.0
Vp
Sou
nd V
eloc
ity (k
m s
-1)
Pressure (GPa)
VB
Temperature in EarthTemperature in Earth’’s inner Cores inner Core
0 2000 4000 60000
250
500
750
1000
1250
1500
Steinle-Neumann et al. Nature 2001
Elas
tic M
odul
i (G
Pa)
Temperature (K)
Ks
μSha and Cohen
Cohen: November, 2006Cohen: November, 2006MRSMRS 1212
Thermoelasticity of bcc FeThermoelasticity of bcc Fe65 au
70 au
80 au75 au
85 au
0 GPa10 GPa20 GPa30 GPa40 GPa
Change in elastic constants withtemperature at constant V (beyond normal quasiharmonic approximation)
Exp: Dever (1972) J. Appl. Phys. 43: 3293; Isaak and Masuda (1995) JGR 100(B9): 17689.
LiNbO3 Thermal Expansivity from Linear Response (LDA) vs. Experiment
Born Effective Charges
P3(1), z(1)
displacements=0.3%
P3(2), z(2)
)1()2()1()2( 333*
zzPP
ezP
eZ
−−Ω
=∂∂Ω
=
Pb
Ti O1
O3
(1/2 1/2 0.530) (1/2 0 0.610)
(1/2 1/2 0.105)(0 0 0)
-2.61⊥ 0 0
0 -5.18⎪⎪ 0
0 0 -2.16
-2.15 0 0
0 -2.15 0
0 0 -4.38
Born effective charge tensors for tetragonal PbTiO3
3.74 0 0
0 3.74 0
0 0 3.52
Effective charges are tensors
6.2 0 0
0 6.2 0
0 0 5.2
PRB, 58, 6224, 1998
Effective chargesEffective charges can be greatly enhanced compared to nominal charges.
