white dwarf cooling: electron-phonon coupling and the ...mdt26/esdg_slides/tomeu_jun13.pdf · white...

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


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.


Main sequence star

g

H→He

I Thermonuclear reactions: hydrogenburning.

I Gravitation balanced by nuclearreactions.

I Main sequence star (e.g. the Sun).


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, . . .


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.


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


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


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)


Outline



Results

Conclusions


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 .


Outline



Results

Conclusions


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


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


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)


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


White dwarf cooling revisited: metallization of solid helium


Outline



Results

Conclusions


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.


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)


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