White dwarf cooling - University of Cambridge

White dwarf cooling: electron-phonon coupling and the metallization of solid helium Bartomeu Monserrat University of Cambridge

QMC in the Apuan Alps VIII 31 July 2013

Outline

White dwarf stars overview Theoretical background Anharmonic energy Phonon expectation values Results Conclusions

B. Monserrat – QMC Apuan Alps VIII – July 2013

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Outline



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Star formation

g

I

Virial theorem: K = −1/2 Vg .

I

Energy expressions:

T

K ∝ N kB T and Vg ∝ − I

GM 2 R

Temperature increases as the star gravitationally collapses.


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Main sequence star

g

H→He

I

Thermonuclear reactions: hydrogen burning.

I

Gravitation balanced by nuclear reactions.

I

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


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White dwarf formation

g

I

Burning material exhausted.

I

Gravitational contraction resumes.

I

High density leads to degenerate electron gas (DEG).

I

White dwarf star balanced by DEG.

I

Complications: mass loss (red giant), further burning cycles, . . .

DEG


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White dwarf structure

I

Degenerate core: He or C/O.

I

Atmosphere: H, He and traces of other elements.

I

Atmosphere represents 10−4 – 10−2 of the total mass.

I

Atmosphere stratification due to strong gravity.

I

Weak energy sources: crystallization, . . .

I

Energy transport: conduction, radiation and convection.

g

DEG


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White dwarf cooling 10

3

Teff (10 K)

8 6 4 2 0 0

2

4

6

8 10 12 14 16 18 20 9 Time (10 years)


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White dwarf cooling Age of the Universe

10

3

Teff (10 K)

8 6 4

Black dwarf

2 0 0

2

4

6

8 10 12 14 16 18 20 9 Time (10 years)


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White dwarf cooling: metallization of solid helium (I)

Helium

Degenerate electron gas Degenerate electron gas: isothermal Core to surface: temperature gradient


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Helium phase diagram 2

10

Metallization

He++

1

Pressure (TPa)

10

100

Solid 4He

10-1

10-2 2 10

He+

He0

103

104 Temperature (K)


Fluid 4He

105

106 10 / 43

White dwarf cooling: metallization of solid helium (II) Insulating Helium Metallic Helium

γ

e− Degenerate electron gas

γ

e−

γ

e−

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



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Harmonic approximation I

Vibrational Hamiltonian in {rα } (or {uα }): X 1 X 1 ˆ vib = − 1 H ∇2pα + upα Φpα;p0 β up0 β 2 mα 2 Rp ,α

I

Rp ,α;Rp0 ,β

Normal mode analysis: {upα } −→ {qks } upα;i = qks =

X 1 qks eik·Rp wks;iα N0 mα k,s 1 X √ √ mα upα;i e−ik·Rp w−ks;iα N0 R ,α,i √

p

I

Vibrational Hamiltonian in {qks }: X 1 ∂2 1 2 2 ˆ H vib = − 2 + 2 ωks qks 2 ∂qks k,s


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Principal axes approximation to the BO energy surface

V ({qks }) = V (0)+

X k,s

Vks (qks )+

1 X X0 Vks;k0 s0 (qks , qk0 s0 )+· · · 2 0 0 k,s k ,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 Schr¨odinger equation:   X 1 ∂2   − 2 + V ({qks }) Φ({qks }) = EΦ({qks }) 2 ∂qks k,s

Q

I

Ground state ansatz: Φ({qks }) =

I

Self-consistent equations: 1 ∂2 − 2 + V ks (qks ) φks (qks ) = λks φks (qks ) 2 ∂qks * + Y0 Y0 V ks (qks ) = φk0 s0 (qk0 s0 ) V ({qk00 s00 }) φk0 s0 (qk0 s0 ) k0 ,s0 k0 ,s0

k,s φks (qks )


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Vibrational self-consistent field equations (II)

I

Approximate vibrational excited states: Y S |ΦS (Q)i = |φksks (qks )i k,s

where S is a vector with elements Sks . I

Anharmonic free energy: F =−

1 X −βES ln e β S


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Diamond independent phonon term (I)

V ({qks }) = V (0)+

X

Vks (qks )+

k,s


1.0

BO energy (eV)

0.8

Harmonic Anharmonic

0.6 0.4 0.2 0 -40

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

40


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Mode wave function (arb. units)

Diamond independent phonon term (II) Harmonic Anharmonic

-40

-20 0 20 Mode amplitude (a.u.) B. Monserrat – QMC Apuan Alps VIII – July 2013

40

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Diamond coupled phonons term V ({qks }) = V (0)+

X

Vks (qks )+

k,s


3

3

40

40

30

2.5

30

20

2.5

20

1.5

10 q2 (a.u.)

q2 (a.u.)

0

-10

2 Energy (eV)

2 10

0

1.5

-10

1 -20

1 -20

-30

0.5

-40

-30

0.5

-40 -40

-30

-20

-10

0 10 q1 (a.u.)

20

30

40

0

-40

-30


-20

-10

0 10 q1 (a.u.)

20

30

40

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0

LiH independent phonon term (I)

V ({qks }) = V (0)+

X

Vks (qks )+

k,s


0.6

BO energy (eV)

Harmonic Anharmonic 0.4

0.2

0

-60

-40

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

60


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Mode wave function (arb. units)

LiH independent phonon term (II) Harmonic Anharmonic

-60

-40

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


60

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LiH coupled phonons term V ({qks }) = V (0)+

X

Vks (qks )+

k,s

k,s k ,s

3.5

60

q2 (a.u.)

1.5

-20

2.5

20 Energy (eV)

2 0

3

40

2.5

20

3.5

60

3

40

q2 (a.u.)

1 X X0 Vks;k0 s0 (qks , qk0 s0 )+· · · 2 0 0

2 0 1.5

-20 1

-40

1 -40

0.5 -60 -60

-40

-20

0 q1 (a.u.)

20

40

60

0

0.5 -60 -60

-40


-20

0 q1 (a.u.)

20

40

60

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0

Anharmonic correction (meV/atom)

Anharmonic ZPE correction

7.5

Diamond 1

7

2

7

H Li

5.0

H Li Graphene Graphane

2.5

0.0 0

200 400 600 Number of normal modes B. Monserrat – QMC Apuan Alps VIII – July 2013

800

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General phonon expectation value I

Phonon expectation value at inverse temperature β: ˆ hO(Q)i Φ,β =

1 X S S ˆ hΦ (Q)|O(Q)|Φ (Q)ie−βES Z S

I

Evaluation: I

Standard theories (Allen-Heine, Gr¨ uneisen): X 2 ˆ ˆ O(Q) = O(0) + aks qks k,s

I

Principal axes expansion: ˆ ˆ O(Q) = O(0)+

X k,s

I

ˆ ks (qks )+ 1 O 2

X X0 k,s

ˆ ks;k0 s0 (qks , qk0 s0 )+· · · O

k0 ,s0

Monte Carlo sampling B. Monserrat – QMC Apuan Alps VIII – July 2013

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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 approximate xc-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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εnk (arb. units)

Diamond thermal band gap (I)

W

Γ


X

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Diamond thermal band gap (II) 6.0 5.9 5.8 Eg (eV)

5.7 5.6

0.462 eV

Static lattice Including el-ph coupling

5.5 5.4 5.3 5.2 5.1 0

200

600 400 Temperature (K)


800

1000

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Diamond thermal band gap (III)

Eg (eV)

5.4

5.3

5.2

Theory Experiment

5.1 0

200


800

1000

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

X

ext σij dij

i,j I

Vibrational stress: dFvib = −Ω

X

vib σij dij

i,j I

Effective stress: dG = dFel − Ω

X

eff σij dij

i,j eff = σ ext + σ vib . where σij ij ij


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Thermal expansion (II) I

Potential part of vibrational stress tensor: vib,V el σij = hΦ(Q)|σij |Φ(Q)i

I

Kinetic part of vibrational stress tensor: + * X 1 vib,T σij =− Φ mα u˙ pα;i u˙ pα;j Φ Ω Rp ,α

I

Total vibrational stress tensor: vib,V vib,T vib σij = σij + σij


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LiH and LiD thermal expansion coefficient 7

H Li 7

7

D Li

6

H Li (exp.)

5

D Li (exp.)

7 7

Lattice parameter (a.u.)

-5

-1

α (10 K )

8

4 3 2 1 0 0

200

8.1 8.0 7.9 7.8 7.7 0

200 400 600 Temperature (K)

400 Temperature (K)

600

800

800

Experimental data from J. Phys. C 15, 6321 (1982)


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Outline



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Enthalpy difference (eV/atom)

Solid helium structural phase diagram 0.5

fcc

0.4 bcc

0.3 0.2

dhcp

0.1

hcp-dhcp continuum

0.0 -0.1

hcp 5

10 15 20 Pressure (TPa) B. Monserrat – QMC Apuan Alps VIII – July 2013

25

30

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Electron-phonon correction (eV)

Solid helium electron-phonon gap correction (I) 1.0 0.5 0.0 T=0K T = 2500 K T = 5000 K T = 7500 K T = 10000 K

-0.5 -1.0 0

5

10 15 Pressure (TPa)


20

25

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Solid helium electron-phonon gap correction (II)

10 TPa to 25 TPa


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Solid helium equilibrium density 1.00 0.98

ρ/ρstatic

0.96 0.94 0.92

T=0K T = 2500 K T = 7500 K T = 5000 K T = 10000 K

0.90 0.88 0.86 0

5

10 15 Pressure (TPa)


20

25

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Solid helium metallization pressure DMC and GW el-ph (T=0K) el-ph (T=5000K) kBT (T=5000K)

2

Eg (eV)

1 0 -1 -2 24

25

26

27 28 29 Pressure (TPa)

30

31

DMC and GW from PRL 101, 106407 (2008)


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Helium phase diagram revisited 31 Metallic solid

30

Pressure (TPa)

29 28 Fluid 27 Insulating solid 26 25 24

0

2000


8000


10000 39 / 43

White dwarf cooling revisited: metallization of solid helium

γ-transport e− -transport DEG Higher metallization pressure

Thinner electron-transport layer


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Outline



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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 I I I I I

I

Prof. Richard J. Needs Dr Neil D. Drummond Dr Gareth J. Conduit Prof. Chris J. Pickard TCM group EPSRC

References: I

I

B. Monserrat, N.D. Drummond, R.J. Needs Physical Review B 87, 144302 (2013) B. Monserrat, N.D. Drummond, C.J. Pickard, R.J. Needs Helium paper, in preparation (2013)


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