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White dwarf cooling: electron-phonon couplingand the metallization of solid helium
Bartomeu Monserrat
University of Cambridge
QMC in the Apuan Alps VIII31 July 2013
Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 1 / 43
Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 2 / 43
Star formation
g
T
I Virial theorem: K = −1/2Vg.
I Energy expressions:
K ∝ NkBT and Vg ∝ −GM2
R
I Temperature increases as the stargravitationally collapses.
B. Monserrat – QMC Apuan Alps VIII – July 2013 3 / 43
Main sequence star
g
H→He
I Thermonuclear reactions: hydrogenburning.
I Gravitation balanced by nuclearreactions.
I Main sequence star (e.g. the Sun).
B. Monserrat – QMC Apuan Alps VIII – July 2013 4 / 43
White dwarf formation
g
DEG
I Burning material exhausted.
I Gravitational contraction resumes.
I High density leads to degenerateelectron gas (DEG).
I White dwarf star balanced by DEG.
I Complications: mass loss (redgiant), further burning cycles, . . .
B. Monserrat – QMC Apuan Alps VIII – July 2013 5 / 43
White dwarf structure
g
DEG
I Degenerate core: He or C/O.
I Atmosphere: H, He and traces ofother elements.
I Atmosphere represents10−4 – 10−2 of the total mass.
I Atmosphere stratification due tostrong gravity.
I Weak energy sources:crystallization, . . .
I Energy transport: conduction,radiation and convection.
B. Monserrat – QMC Apuan Alps VIII – July 2013 6 / 43
White dwarf cooling
0 2 4 6 8 10 12 14 16 18 20Time (10
9 years)
0
2
4
6
8
10T
eff (
103 K
)
B. Monserrat – QMC Apuan Alps VIII – July 2013 7 / 43
White dwarf cooling
0 2 4 6 8 10 12 14 16 18 20Time (10
9 years)
0
2
4
6
8
10T
eff (
103 K
)
Age
of t
he U
nive
rse
Black dwarf
B. Monserrat – QMC Apuan Alps VIII – July 2013 8 / 43
White dwarf cooling: metallization of solid helium (I)
Degenerateelectron gas
Helium
Degenerate electron gas: isothermal
Core to surface: temperature gradient
B. Monserrat – QMC Apuan Alps VIII – July 2013 9 / 43
Helium phase diagramP
ress
ure
(T
Pa)
Temperature (K)
10-2
10-1
100
101
102
102
103
104
105
106
Solid 4He
Fluid 4HeHe
0
He+
He++Metallization
B. Monserrat – QMC Apuan Alps VIII – July 2013 10 / 43
White dwarf cooling: metallization of solid helium (II)
e−e− e−
γ γ γ
Degenerateelectron gas
Insulating Helium
Metallic Helium
Degenerate electron gas: isothermal
Core to surface: temperature gradient
B. Monserrat – QMC Apuan Alps VIII – July 2013 11 / 43
Metallization pressure
I DFT: 17 TPa at zero temperature.
I DMC and GW : 25.7 TPa at zero temperature.
I Electron-phonon coupling: ?
B. Monserrat – QMC Apuan Alps VIII – July 2013 12 / 43
Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 13 / 43
Harmonic approximation
I Vibrational Hamiltonian in rα (or uα):
Hvib = −1
2
∑Rp,α
1
mα∇2pα +
1
2
∑Rp,α;Rp′ ,β
upαΦpα;p′βup′β
I Normal mode analysis: upα −→ qks
upα;i =1√
N0mα
∑k,s
qkseik·Rpwks;iα
qks =1√N0
∑Rp,α,i
√mα upα;ie
−ik·Rpw−ks;iα
I Vibrational Hamiltonian in qks:
Hvib =∑k,s
(−1
2
∂2
∂q2ks
+1
2ω2ksq
2ks
)B. Monserrat – QMC Apuan Alps VIII – July 2013 14 / 43
Principal axes approximation to the BO energy surface
V (qks) = V (0)+∑k,s
Vks(qks)+1
2
∑k,s
∑k′,s′
′Vks;k′s′(qks, qk′s′)+· · ·
I Static lattice DFT total energy
I DFT total energy along frozen independent phonon
I DFT total energy along frozen coupled phonons
B. Monserrat – QMC Apuan Alps VIII – July 2013 15 / 43
Vibrational self-consistent field equations
I Phonon Schrodinger equation:∑k,s
−1
2
∂2
∂q2ks
+ V (qks)
Φ(qks) = EΦ(qks)
I Ground state ansatz: Φ(qks) =∏
k,s φks(qks)
I Self-consistent equations:(−1
2
∂2
∂q2ks
+ V ks(qks)
)φks(qks) = λksφks(qks)
V ks(qks) =
⟨∏k′,s′
′φk′s′(qk′s′)
∣∣∣∣∣∣V (qk′′s′′)
∣∣∣∣∣∣∏k′,s′
′φk′s′(qk′s′)
⟩
B. Monserrat – QMC Apuan Alps VIII – July 2013 16 / 43
Vibrational self-consistent field equations (II)
I Approximate vibrational excited states:
|ΦS(Q)〉 =∏k,s
|φSksks (qks)〉
where S is a vector with elements Sks.
I Anharmonic free energy:
F = − 1
βln∑S
e−βES
B. Monserrat – QMC Apuan Alps VIII – July 2013 17 / 43
Diamond independent phonon term (I)
V (qks) = V (0)+∑k,s
Vks(qks)+1
2
∑k,s
∑k′,s′
′Vks;k′s′(qks, qk′s′)+· · ·
-40 -20 0 20 40Mode amplitude (a.u.)
0
0.2
0.4
0.6
0.8
1.0
BO
ene
rgy
(eV
)
HarmonicAnharmonic
B. Monserrat – QMC Apuan Alps VIII – July 2013 18 / 43
Diamond independent phonon term (II)
-40 -20 0 20 40Mode amplitude (a.u.)
Mod
e w
ave
func
tion
(arb
. uni
ts)
HarmonicAnharmonic
B. Monserrat – QMC Apuan Alps VIII – July 2013 19 / 43
Diamond coupled phonons term
V (qks) = V (0)+∑k,s
Vks(qks)+1
2
∑k,s
∑k′,s′
′Vks;k′s′(qks, qk′s′)+· · ·
-40
-30
-20
-10
0
10
20
30
40
-40 -30 -20 -10 0 10 20 30 40
q2 (
a.u
.)
q1 (a.u.)
0
0.5
1
1.5
2
2.5
3
Ener
gy
(eV
)
-40
-30
-20
-10
0
10
20
30
40
-40 -30 -20 -10 0 10 20 30 40
q2 (
a.u
.)
q1 (a.u.)
0
0.5
1
1.5
2
2.5
3
Ener
gy
(eV
)
B. Monserrat – QMC Apuan Alps VIII – July 2013 20 / 43
LiH independent phonon term (I)
V (qks) = V (0)+∑k,s
Vks(qks)+1
2
∑k,s
∑k′,s′
′Vks;k′s′(qks, qk′s′)+· · ·
-60 -40 -20 0 20 40 60Mode amplitude (a.u.)
0
0.2
0.4
0.6
BO
ene
rgy
(eV
)
HarmonicAnharmonic
B. Monserrat – QMC Apuan Alps VIII – July 2013 21 / 43
LiH independent phonon term (II)
-60 -40 -20 0 20 40 60Mode amplitude (a.u.)
Mod
e w
ave
func
tion
(arb
. uni
ts)
HarmonicAnharmonic
B. Monserrat – QMC Apuan Alps VIII – July 2013 22 / 43
LiH coupled phonons term
V (qks) = V (0)+∑k,s
Vks(qks)+1
2
∑k,s
∑k′,s′
′Vks;k′s′(qks, qk′s′)+· · ·
-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60
q2 (
a.u
.)
q1 (a.u.)
0
0.5
1
1.5
2
2.5
3
3.5
Ener
gy
(eV
)
-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60
q2 (
a.u
.)
q1 (a.u.)
0
0.5
1
1.5
2
2.5
3
3.5
Ener
gy
(eV
)
B. Monserrat – QMC Apuan Alps VIII – July 2013 23 / 43
Anharmonic ZPE correction
0 200 400 600 800Number of normal modes
0.0
2.5
5.0
7.5
Anh
arm
onic
cor
rect
ion
(meV
/ato
m)
Diamond1H
7Li
2H
7Li
GrapheneGraphane
B. Monserrat – QMC Apuan Alps VIII – July 2013 24 / 43
General phonon expectation value
I Phonon expectation value at inverse temperature β:
〈O(Q)〉Φ,β =1
Z∑S
〈ΦS(Q)|O(Q)|ΦS(Q)〉e−βES
I Evaluation:I Standard theories (Allen-Heine, Gruneisen):
O(Q) = O(0) +∑k,s
aksq2ks
I Principal axes expansion:
O(Q) = O(0)+∑k,s
Oks(qks)+1
2
∑k,s
∑k′,s′
′Oks;k′s′(qks, qk′s′)+· · ·
I Monte Carlo sampling
B. Monserrat – QMC Apuan Alps VIII – July 2013 25 / 43
Band gap renormalization
I Band gap problem (LDA, PBE, . . . ): underestimation of gaps.
I Caused by the lack of a discontinuity in approximatexc-functionals with respect to particle number: correction ∆xc
to band gap.
I Approximate systematic shift in all displaced configurations.
I Error disappears in change in band gap.
B. Monserrat – QMC Apuan Alps VIII – July 2013 26 / 43
Diamond thermal band gap (I)
W Γ X
ε nk (
arb.
uni
ts)
B. Monserrat – QMC Apuan Alps VIII – July 2013 27 / 43
Diamond thermal band gap (II)
0 200 400 600 800 1000Temperature (K)
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0E
g (eV
)
Static latticeIncluding el-ph coupling0.462 eV
B. Monserrat – QMC Apuan Alps VIII – July 2013 28 / 43
Diamond thermal band gap (III)
0 200 400 600 800 1000Temperature (K)
5.1
5.2
5.3
5.4
Eg (
eV)
Theory Experiment
Experimental data from Proc. R. Soc. London, Ser. A 277, 312 (1964)
B. Monserrat – QMC Apuan Alps VIII – July 2013 29 / 43
Thermal expansion (I)
I Gibbs free energy:
dG = dFel + dFvib − Ω∑i,j
σextij dεij
I Vibrational stress:
dFvib = −Ω∑i,j
σvibij dεij
I Effective stress:
dG = dFel − Ω∑i,j
σeffij dεij
where σeffij = σext
ij + σvibij .
B. Monserrat – QMC Apuan Alps VIII – July 2013 30 / 43
Thermal expansion (II)
I Potential part of vibrational stress tensor:
σvib,Vij = 〈Φ(Q)|σel
ij |Φ(Q)〉
I Kinetic part of vibrational stress tensor:
σvib,Tij = − 1
Ω
⟨Φ
∣∣∣∣∣∣∑Rp,α
mαupα;iupα;j
∣∣∣∣∣∣Φ⟩
I Total vibrational stress tensor:
σvibij = σvib,V
ij + σvib,Tij
B. Monserrat – QMC Apuan Alps VIII – July 2013 31 / 43
LiH and LiD thermal expansion coefficient
0 200 400 600 800Temperature (K)
0
1
2
3
4
5
6
7
8α
(10-5
K-1
)
H7Li
D7Li
H7Li (exp.)
D7Li (exp.)
0 200 400 600 800Temperature (K)
7.7
7.8
7.9
8.0
8.1
Latti
ce p
aram
eter
(a.
u.)
Experimental data from J. Phys. C 15, 6321 (1982)
B. Monserrat – QMC Apuan Alps VIII – July 2013 32 / 43
Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 33 / 43
Solid helium structural phase diagram
5 10 15 20 25 30Pressure (TPa)
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Ent
halp
y di
ffere
nce
(eV
/ato
m)
hcp
dhcp
bcc
fcc
hcp-dhcp continuum
B. Monserrat – QMC Apuan Alps VIII – July 2013 34 / 43
Solid helium electron-phonon gap correction (I)
0 5 10 15 20 25 Pressure (TPa)
-1.0
-0.5
0.0
0.5
1.0
Ele
ctro
n-ph
onon
cor
rect
ion
(eV
)
T = 0 KT = 2500 KT = 5000 KT = 7500 KT = 10000 K
B. Monserrat – QMC Apuan Alps VIII – July 2013 35 / 43
Solid helium electron-phonon gap correction (II)
10 TPa to 25 TPa
B. Monserrat – QMC Apuan Alps VIII – July 2013 36 / 43
Solid helium equilibrium density
0 5 10 15 20 25 Pressure (TPa)
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00ρ/
ρ stat
ic
T = 0 KT = 2500 KT = 7500 KT = 5000 KT = 10000 K
B. Monserrat – QMC Apuan Alps VIII – July 2013 37 / 43
Solid helium metallization pressure
24 25 26 27 28 29 30 31Pressure (TPa)
-2
-1
0
1
2
Eg (
eV)
DMC and GWel-ph (T=0K)
el-ph (T=5000K)
kBT (T=5000K)
DMC and GW from PRL 101, 106407 (2008)
B. Monserrat – QMC Apuan Alps VIII – July 2013 38 / 43
Helium phase diagram revisited
Pre
ssur
e (T
Pa)
Temperature (K)
24
25
26
27
28
29
30
31
0 2000 4000 6000 8000 10000
Insulating solid
Metallic solid
Fluid
B. Monserrat – QMC Apuan Alps VIII – July 2013 39 / 43
White dwarf cooling revisited: metallization of solid helium
B. Monserrat – QMC Apuan Alps VIII – July 2013 40 / 43
Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 41 / 43
Conclusions
I Theory for anharmonic vibrational energy of solids.
I General framework for phonon-dependent expectation values.
I Metallization of solid helium.
I White dwarf energy transport and cooling.
B. Monserrat – QMC Apuan Alps VIII – July 2013 42 / 43
I Acknowledgements:I Prof. Richard J. NeedsI Dr Neil D. DrummondI Dr Gareth J. ConduitI Prof. Chris J. PickardI TCM groupI EPSRC
I References:I B. Monserrat, N.D. Drummond, R.J. Needs
Physical Review B 87, 144302 (2013)I B. Monserrat, N.D. Drummond, C.J. Pickard, R.J. Needs
Helium paper, in preparation (2013)
B. Monserrat – QMC Apuan Alps VIII – July 2013 43 / 43
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