The Quasiharmonic Approximation

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The Quasiharmonic Approximation
R. Wentzcovitch U. of Minnesota VLab Tutorial
A simple approximate treatment of
thermodynamical behavior
Born and Huang

• It treats vibrations as if they did not interact
• System is equivalent to a collection of
  independent harmonic oscillators
• These establish the quantum mechanical energy
  levels of the system
• The levels are used to compute the partition
  function, Z, and the Helmoltz
• free energy, F(T,V). From the latter, all
  thermodynamic functions can
• be derived.

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Helmholtz free energy
internal energy
entropy
partition function
sum of Boltzman factors of all energy levels
eigenvalues of energy operator
For a single oscillator with freq. ?i , the
energy levels are
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Therefore
For a lattice of normal modes of vibration with
frequencies ?i
is the vibrational free energy
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Complete free energy
As a solid compresses, deforms, etc,U and ?is
change
From F(T,V,e1,e2,) all the thermodynamical
properties can be dirived.
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Notice

• This (or a more complete) quantum treatment is
  required at low T
• The QHA is not appropriate at high T because
  of phonon-phonon
• interactions
• Tlow lt ?Debyelt Thigh
  lt Tmelt

• Phonon frequencies must be accurate (first
  principles)
• Phonon sampling must be thorough

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Summation (integration) over the Brillouin Zone
Ex square BZ
is the multiplicity of a point determined by
symmetry

• In general
• Compute and diagonalize the dynamical matrix at
  few s
• Extract force constants
• Recompute dynamical matrices at several
  points using those force constants

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MgSiO3 Perovskite
----- Most abundant constituent in the Earths
lower mantle ----- Orthorhombic distorted
perovskite structure (Pbnm, Z4) ----- Its
stability is important for understanding deep
mantle (D layer)
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Mineral sequence II
Lower Mantle



(Mgx,Fe(1-x))O
(Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3
CaSiO3
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Mineral sequence II
Lower Mantle


(Mgx,Fe(1-x))O
(Mgx,Fe(1-x))SiO3
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Phonon dispersion of MgSiO3 perovskite
Calc Exp
Calc Exp
-
0 GPa
Calc Karki, Wentzcovitch, de
Gironcoli, Baroni PRB 62, 14750,
2000 Exp Raman Durben and Wolf 1992
Infrared Lu et al. 1994
-
100 GPa
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MgSiO3-perovskite and MgO
4.8
(256)
Exp. Ross Hazen, 1989 Mao et al., 1991 Wang
et al., 1994 Funamori et al., 1996
Chopelas, 1996 Gillet et al., 2000 Fiquet et
al., 2000
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(No Transcript)
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Umemoto, 2005
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Thermal expansivity and the QHA
? provides an a posteriori criterion for the
validity of the QHA
? (10-5 K-1)
?
?
?
MgSiO3
Karki et al, GRL (2001)
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Validity of the QHA
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Mineral sequence II
Lower Mantle
QHA not-valid for this mineral !!



(Mgx,Fe(1-x))O
(Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3
CaSiO3
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Crystal Structures at High PT
Crystal Structures at High PT
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Quasiharmonic Approximation (QHA)
VDoS and F(T,V) within the QHA
N-th (N3,4,5) order isothermal (eulerian or
logarithm) finite strain EoS

IMPORTANT crystal structure and phonon
frequencies are uniquely
related with volume !!.
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How to get V(P,T)

• Static calculations give Vst(Pst). Vst is the
  volume of the optimized structure at Pst.
• The free energy is obtained after phonon
  frequencies are calculated at each equilibrium
  structure, ?(Vst).
• The relationship between structure and Vst or
  ?(Vst) is not altered by temperature after free
  energy calculations, only the pressure P(V,T)
  Pst Pth. Pth is the contribution of the 2nd and
  3rd terms in the rhs in the free energy formula.

• Therefore
• if
  V(P,T) Vst then
• 
  Structure(P,T) Structure(Vst) and
• 
  ?(P,T) ?(Vst)
• These are Structure(P,T) and ?(P,T) given by the
  statically constrained QHA .
• See Carrier et
  al., PRB 76, 064116 (2007)

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High P,T Experiments x Static LDA
Lattice parameters of MgSiO3 perovskite
Carrier et al. PRB, 2008
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Stress and Strain
Stress and Strain
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Uniform deformations (macroscopic strains)
e dL/L
L
LdL
More formally
2
2
Xi xi uijxj
X2


x2
eij ½ (uij uji)
x1
1
1
X1
Lagrangian strains
Stress
sij (P,T) 1 dF
V deij
T,P
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Thermoelastic constant tensor CijS(T,P)
?kl
equilibrium structure
re-optimize
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cij
300 K 1000K 2000K 3000 K 4000 K
Cij(P,T)
(Oganov et al,2001)
MgSiO3-pv
(Wentzcovitch, Karki, Cococciono, de Gironcoli,
Phys. Rev. Lett. 2004)
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Deviatoric Thermal Stresses
Carrier et al. PRB, 2008
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Deviatoric thermal stresses cause T dependent
X-tal structure and phonon frequencies
QHA should be self-consistent!
Carrier et al. PRB, 2008
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Anharmonicity
Anharmonicity
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410 km discontinuity
QHA
Anharmonic
2.5 MPa/K
3.5 MPa/K
Mg2SiO4? Mg2SiO4
Wu and Wentzcovitch, PRB 2009
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Summary

• QHA is simple useful theory even for lower mantle
  
• QHA LDA gives excellent structural and
  thermodynamic properties
• It is possible to obtain crystal structures at
  high pressures and temperatures using the QHA
• Beware thermal pressure is not isotropic
• Phase boundaries appear to be affected by
  anharmonicity