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.
Helmholtz free energy:
E  internal energy
S  entropy
F   k B T ln Z
Z = partition function
Z  sum of Boltzman factors of all energy levels
Z 
e
i

kBT
 i  eigenvalues of energy operator
i
For a single oscillator with freq. ωi , the energy levels are:
1
2
i ,
3
2
i ,
5
2
 i , ...
Therefore:
Zi  e
i

2 kBT

e
s0

s i
k BT
i


2 kBT
e
1 e
i


1
k T
Fi 
 i  k B T ln  1  e B

2


i
kBT




For a lattice of normal modes of vibration with frequencies ωi:
F 
F
i
i
is the vibrational free energy
Complete free energy:
F U 
1
2
i
 i  k BT 
i
i


k T
ln  1  e B






As a solid compresses, deforms, etc…,U and ωi’s change
From F(T,V,ε1,ε2,…) all the thermodynamical properties
can be dirived.
 F 
S  


T

V
E  F  TS
 F 
P  


V

T
H  E  PV
G  F  PV
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 < θDebye< Thigh < Tmelt
-Phonon frequencies must be accurate (first principles)
-Phonon sampling must be thorough
Summation (integration) over the Brillouin Zone
F U 
1
2
 i  k BT 
i
i
i


k T
ln  1  e B






i  q, n
M
Ex: square BZ

f ( i ) 
i

3 
d
q
f
(

)
q ,n 
  n

d
3

X

i
f ( i ) 
w
j
f ( j )
j  i
q
w i is the “multiplicity” of a point
determined by symmetry
In general:
1) Compute and diagonalize the dynamical matrix at few q ’s
2) Extract “force constants”
3) Recompute dynamical matrices at several q points using those force constants
MgSiO3 Perovskite
----- Most abundant constituent in the Earth’s lower mantle
----- Orthorhombic distorted perovskite structure (Pbnm, Z=4)
----- Its stability is important for understanding deep mantle (D” layer)
Mineral sequence II
Lower Mantle
+
+
(Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3
(Mgx,Fe(1-x))O
CaSiO3
Mineral sequence II
Lower Mantle
+
(Mgx,Fe(1-x))SiO3
(Mgx,Fe(1-x))O
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
MgSiO3-perovskite and MgO

(gr/cm-3)
V
(A3)
KT
(GPa)
d KT/dP
3.580
18.80
159
4.30
3.601
18.69
160
4.15
4.210
164.1
247
4.0
4.8
162.3
246
|
266
3.7
|
4.0
4.247
d KT2/dP2
(GPa-1)
d KT/dT
(Gpa K-1)
10-5 K-1
-0.030
-0.014
3.12
Calc.
pc
-0.0145
3.13
Exp.
pc
-0.016
-0.031
2.1
Calc.
Pv
~
-0.02
|
-0.07
1.7
|
2.2
Exp.
Pv
~

(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]
1  V 
  

V  T  P
Umemoto, 2005
Thermal expansivity and the QHA
 (10-5 K-1)
 provides an a posteriori criterion for the validity of the QHA



MgSiO3
Karki et al, GRL (2001)
Validity of the QHA
Mineral sequence II
Lower Mantle
QHA not-valid
for this mineral !!
+
+
(Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3
(Mgx,Fe(1-x))O
CaSiO3
Crystal Structures at High PT
Quasiharmonic Approximation (QHA)
• VDoS and F(T,V) within the QHA
F (V , T )  U (V ) 

qj
  qj (V )
2
 k BT 
qj

   qj (V )  


ln 1  exp  


k B T  


N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS
P  
 F 
 V 

T
S  
 F 
 T 

V
G  F  TS  PV
IMPORTANT: crystal structure and phonon frequencies
are uniquely related with volume !!….
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, ω(Vst).
•
The relationship between structure and Vst or ω(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.
F (Vst , T )  U (V ) 

qj
•
  qj (V )
2

   qj (V )  


 k B T  ln 1  exp  


k B T  
qj


Therefore:
if
V(P,T) = Vst
then
Structure(P,T) = Structure(Vst)
and
ω(P,T) = ω(Vst)
These are Structure(P,T) and ω(P,T) given by the statically constrained QHA .
See Carrier et al., PRB 76, 064116 (2007)
High P,T Experiments x Static LDA
Lattice parameters of
MgSiO3 perovskite
Carrier et al. PRB, 2008
Stress and
and Strain
Strain
Stress
Uniform deformations (macroscopic strains)
ε = dL/L
L
L+dL
More formally
2
x2
2
X2
•
Xi – xi = uijxj
•
εij = ½ (uij + uji)
x1
Stress
1
X1
1
σij (P,T)= 1 δF
V δεij
“Lagrangian” strains
T,P
Thermoelastic constant tensor CijS(T,P)
2


G
T
c ij ( T , P )  
   i  
kl
S
T
c ij ( T , P )  c ij ( T , P ) 
equilibrium
structure
re-optimize
i 
S
 i
T


j 
P
 i  jVT
CV
c
i
j
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)
Deviatoric Thermal Stresses
i  
1 F
V  i
 i   i  P
Carrier et al. PRB, 2008
i 

1
C ij  
j
j
a i  a (1   i )
0
i
Deviatoric thermal stresses
cause T dependent X-tal
structure and phonon
frequencies
QHA should be self-consistent!
Carrier et al. PRB, 2008
Anharmonicity
410 km discontinuity
QHA
Anharmonic
2.5 MPa/K
3.5 MPa/K
Mg2SiO4→ Mg2SiO4
Wu and Wentzcovitch, PRB 2009
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