white dwarf cooling: electron-phonon coupling and the … · 2013. 8. 14. · white dwarf structure...

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



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 Atmosphere stratification due tostrong gravity.

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

)


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 (I)

Degenerateelectron gas

Helium

Degenerate electron gas: isothermal

Core to surface: temperature gradient


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


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


Metallization pressure

I DFT: 17 TPa at zero temperature.

I DMC and GW : 25.7 TPa at zero temperature.

I Electron-phonon coupling: ?


Outline



Results

Conclusions


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


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′)

⟩


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


Diamond independent phonon term (I)

V (qks) = V (0)+∑k,s

Vks(qks)+1

2

∑k,s

∑k′,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


Diamond independent phonon term (II)

-40 -20 0 20 40Mode amplitude (a.u.)

Mod

e w

ave

func

tion

(arb

. uni

ts)

HarmonicAnharmonic


Diamond coupled phonons term

V (qks) = V (0)+∑k,s

Vks(qks)+1

2

∑k,s

∑k′,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

)


LiH independent phonon term (I)

V (qks) = V (0)+∑k,s

Vks(qks)+1

2

∑k,s

∑k′,s′


-60 -40 -20 0 20 40 60Mode amplitude (a.u.)

0

0.2

0.4

0.6

BO

ene

rgy

(eV

)

HarmonicAnharmonic


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


LiH coupled phonons term

V (qks) = V (0)+∑k,s

Vks(qks)+1

2

∑k,s

∑k′,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

)


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


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


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.


Diamond thermal band gap (I)

W Γ X

ε nk (

arb.

uni

ts)


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


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)


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 .


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


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)


Outline



Results

Conclusions


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


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


Solid helium electron-phonon gap correction (II)

10 TPa to 25 TPa


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


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 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 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)


white dwarf cooling: electron-phonon coupling and the … · 2013. 8. 14. · white dwarf structure...

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