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White dwarf cooling: electron-phonon couplingand the metallization of solid helium
Bartomeu Monserrat
University of Cambridge
Electronic Structure Discussion Group5 June 2013
B. Monserrat – ESDG – June 2013
Outline
White dwarfs overview
Theoretical background
Results
Conclusions
B. Monserrat – ESDG – June 2013
Outline
White dwarfs overview
Theoretical background
Results
Conclusions
B. Monserrat – ESDG – June 2013
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 – ESDG – June 2013
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 – ESDG – June 2013
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 – ESDG – June 2013
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 Weak energy sources:crystallization, . . .
I Energy transport: conduction,radiation and convection.
B. Monserrat – ESDG – June 2013
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 – ESDG – June 2013
White dwarf cooling: metallization of solid helium
e−e− e−
γ γ γ
Degenerateelectron gas
Insulating Helium
Metallic Helium
Degenerate electron gas: isothermal
Core to surface: temperature gradient
B. Monserrat – ESDG – June 2013
Metallization pressure
I DFT: 17 TPa at all temperatures.
I DMC and GW : 25.7 TPa at all temperatures.
I Electron-phonon coupling: ?
DMC and GW from PRL 101, 106407 (2008)
B. Monserrat – ESDG – June 2013
Outline
White dwarfs overview
Theoretical background
Results
Conclusions
B. Monserrat – ESDG – June 2013
Theoretical overview
I Principal axes approximation to the BO energy surface:
V (Q) = V (0)+∑k,s
Vks(qks)+1
2
∑k,s
∑k′,s′
′Vks;k′s′(qks, qk′s′)+· · ·
I Vibrational self-consistent field method.
I Phonon expectation values:
〈O(Q)〉Φ,β =1
Z∑S
〈ΦS(Q)|O(Q)|ΦS(Q)〉e−βES .
B. Monserrat – ESDG – June 2013
Outline
White dwarfs overview
Theoretical background
Results
Conclusions
B. Monserrat – ESDG – June 2013
Solid helium stable phase
0 5 10 15 20 25Pressure (TPa)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6E
ntha
lpy
diffe
renc
e (e
V/a
tom
)
hcpfccdhcp
B. Monserrat – ESDG – June 2013
Solid helium electron-phonon gap correction
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 – ESDG – June 2013
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 – ESDG – June 2013
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 – ESDG – June 2013
White dwarf cooling revisited: metallization of solid helium
B. Monserrat – ESDG – June 2013
Outline
White dwarfs overview
Theoretical background
Results
Conclusions
B. Monserrat – ESDG – June 2013
Conclusions
I Summary: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.
I Outlook:I Low pressure solid helium.
B. Monserrat – ESDG – June 2013
I Acknowledgements:I Prof Richard J. NeedsI Dr Neil D. DrummondI TCM groupI EPSRC
I References:I B. Monserrat, N.D. Drummond, R.J. Needs
Physical Review B 87, 144302 (2013)
B. Monserrat – ESDG – June 2013