White dwarf cooling: electron-phonon coupling and …mdt26/tti_talks/qmcitaa_13/... · White dwarf...
Transcript of White dwarf cooling: electron-phonon coupling and …mdt26/tti_talks/qmcitaa_13/... · White dwarf...
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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
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Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 1 / 43
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Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
B. Monserrat – QMC Apuan Alps VIII – July 2013 2 / 43
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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.
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Main sequence star
g
H→He
I Thermonuclear reactions: hydrogenburning.
I Gravitation balanced by nuclearreactions.
I Main sequence star (e.g. the Sun).
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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, . . .
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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.
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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
)
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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
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White dwarf cooling: metallization of solid helium (I)
Degenerateelectron gas
Helium
Degenerate electron gas: isothermal
Core to surface: temperature gradient
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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
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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
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Metallization pressure
I DFT: 17 TPa at zero temperature.
I DMC and GW : 25.7 TPa at zero temperature.
I Electron-phonon coupling: ?
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Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
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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
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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
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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′)
⟩
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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
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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
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Diamond independent phonon term (II)
-40 -20 0 20 40Mode amplitude (a.u.)
Mod
e w
ave
func
tion
(arb
. uni
ts)
HarmonicAnharmonic
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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
)
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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
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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
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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
)
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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
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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
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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.
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Diamond thermal band gap (I)
W Γ X
ε nk (
arb.
uni
ts)
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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
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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)
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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 .
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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
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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)
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Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
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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
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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
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Solid helium electron-phonon gap correction (II)
10 TPa to 25 TPa
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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
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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)
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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
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White dwarf cooling revisited: metallization of solid helium
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Outline
White dwarf stars overview
Theoretical backgroundAnharmonic energyPhonon expectation values
Results
Conclusions
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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.
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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)
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