White Dwarfstabilized against collapse by degeneracy pressure of electronsradius R, e mass me, nucleon mass Mp, e’s per nucleon q

assume constant density:

V = 43πR3, ρ = NMp

V

E = !2

10π2me

(3π2Nq)5/3

V 2/3 = !2

10π2me

(3π2Nq)5/3

(π/3)2/3R2 = 2!2

15πme

( 94 πNq)5/3

R2

dEgrav = −GM(r)r dM = −G(ρ 4

3 πr3)r ρ 4πr2dr = − 16

3 π3Gρ2 r4dr

Egrav = − 163 π3Gρ2

∫ R0 r4dr = − 16

15π3Gρ2 R5 = − 35π3 GN2M2

p

R

Etot = E + Egrav = AR2 − B

R

dEtotdR = −2 A

R3 + BR2 = 0

2A = BR

White Dwarf

R = 2AB = 4!2

15πme

(94πNq

)5/3 53π3GN2m2

=(

9π4

)2/3 !2

GM2pme

q5/3

N1/3 = 7.6×1025mN1/3

for a solar mass N ≈ 1.2× 1057, R ≈ 7× 106 m

EF = !2

2me

(3π2 Nq

V

)2/3= !2

2meR2

(9π4 Nq

)2/3 = 1.9× 105eV

Erest = mec2 = 5.11× 105eV

↑ N ⇒ R ↓ EF ↑, more relativistic

UltraRelativistic

E =√

p2c2 + m2ec

4 −mec2 ≈ pc

dE = EkVπ2 k2dk = !ck V

π2 k2dk

E = !cVπ2

∫ kF

0 k3dk = ! c V4π2 k4

F = !cV4π2

(3π2 Nq

V

)4/3

= ! c4π2

(3π2Nq)4/3

V 1/3 = ! c3πR

(9π4 Nq

)4/3

Etot = E + Egrav = CR −

BR

C > B expand, C < B contract

Chandrasekhar LimitC = B

! c3π

(9π4 Ncq

)4/3 = 35π3GN2

c M2p

Nc = 1516

√5π

(! cG

)3/2 q2

M2p≈ 2× 1057

1.7 solar masses

Chandrasekhar Limit

non-relativistic

relativistic

SubramanyanChandrasekhar

1983 Nobel Prize

White Dwarf

Neutron Starfrom core collapse supernovae

p+ + e− → n + ν

me → mn , q = 1

N ∼ 1057, R ∼ 12 km

EF =!2

2 mn R2

(9π

4

)2

= 56MeV

Erest = mn c2 = 940MeV

non-relativistic

Band Structure

ψ(x + a) = eiKaψ(x)

V (x + a) = V (x)

Bloch’s Theorem

K =2πj

Na

Band Structure

0 < x < a

ψ(x) = A sin(kx) + B cos(kx)

ψ(x) = e−iKa [A sin k(x + a) + B cos k(x + a)]

−a < x < 0

Band Structure

cos(Ka) = cos(ka) +mα

!2ksin(ka)

ψ(0+) = ψ(0−)

ψ′(0+)− ψ′(0−) =2m

!2

∫ 0+

0−

V (x)ψ(x)dx

ψ′(0+)− ψ′(0−) =2m

!2α B

K = nπ

Band Structure

1 2 3 4 5

-3

-2

-1

1

2

3

cos(Ka) = cos(ka) +mα

!2ksin(ka)

band edge: K = nπ

-0.2 0.2 0.4 0.6 0.8 1 1.2

-1.5

-1

-0.5

0.5

1

1.5

-0.2 0.2 0.4 0.6 0.8 1 1.2

-1.5

-1

-0.5

0.5

1

1.5

-0.2 0.2 0.4 0.6 0.8 1 1.2

-1.5

-1

-0.5

0.5

1

1.5

Bottom of Band Structure

1st band 2nd band

3rd band

K = nπ

Band Structure

1 2 3 4 5

-3

-2

-1

1

2

3

cos(Ka) = cos(ka) +mα

!2ksin(ka)

Band Structure

1 2 3 4 5

-3

-2

-1

1

2

3

insulator

cos(Ka) = cos(ka) +mα

!2ksin(ka)

Band Structure

1 2 3 4 5

-3

-2

-1

1

2

3

insulator conductor

cos(Ka) = cos(ka) +mα

!2ksin(ka)