White Dwarf stabilized against collapse by degeneracy pressure of - - PowerPoint PPT Presentation

White Dwarf

stabilized against collapse by degeneracy pressure of electrons radius R, e mass me, nucleon mass Mp, e’s per nucleon q assume constant density: V = 4

3πR3,

ρ = NMp

V

E =

2 10π2me

(3π2Nq)

5/3

V 2/3

=

2 10π2me

(3π2Nq)

5/3

(π/3)2/3R2 = 22 15πme

( 9

4 πNq) 5/3

R2

dEgrav = − GM(r)

r

dM = −

G(ρ 4

3 πr3)

r

ρ 4πr2dr = − 16

3 π3Gρ2 r4dr

Egrav = − 16

3 π3Gρ2 R 0 r4dr = − 16 15π3Gρ2 R5 = − 3 5π3 GN 2M 2

p

R

Etot = E + Egrav =

A R2 − B R dEtot dR

= −2 A

R3 + B R2 = 0

2A = BR

White Dwarf

R =

2A B = 42 15πme

9

4πNq

5/3

5 3π3GN 2m2

= 9π

4

2/3

2 GM 2

pme

q5/3 N1/3 = 7.6×1025m N1/3

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

2 2me

• 3π2 Nq

V

2/3 =

2 2meR2

9π

4 Nq

2/3 = 1.9 × 105eV Erest = mec2 = 5.11 × 105eV ↑ N ⇒ R ↓ EF ↑, more relativistic

UltraRelativistic

E =

• p2c2 + m2

ec4 − mec2 ≈ pc

dE = Ek V

π2 k2dk = ck V π2 k2dk

E =

cV π2

kF k3dk = c V

4π2 k4 F = cV 4π2

• 3π2 Nq

V

4/3 =

c 4π2 (3π2Nq)

4/3

V 1/3

=

c 3πR

9π

4 Nq

4/3 Etot = E + Egrav = C

R − B R

C > B expand, C < B contract

Chandrasekhar Limit

C = B

c 3π

9π

4 Ncq

4/3 =

3 5π3GN 2 c M 2 p

Nc = 15

16

√ 5π c

G

3/2 q2

M 2

p ≈ 2 × 1057

1.7 solar masses

Chandrasekhar Limit

non-relativistic relativistic

Subramanyan Chandrasekhar

1983 Nobel Prize

White Dwarf

Neutron Star

from core collapse supernovae

p+ + e− → n + ν me → mn , q = 1 N ∼ 1057, R ∼ 12 km EF = 2 2 mn R2 9π 4 2 = 56 MeV Erest = mn c2 = 940 MeV

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 2k sin(ka) ψ(0+) = ψ(0−) ψ′(0+) − ψ′(0−) = 2m 2 0+

0−

V (x)ψ(x)dx ψ′(0+) − ψ′(0−) = 2m 2 α B K = nπ

Band Structure

1 2 3 4 5

• 3
• 2
• 1

1 2 3

cos(Ka) = cos(ka) + mα 2k sin(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 2k sin(ka)

Band Structure

1 2 3 4 5

• 3
• 2
• 1

1 2 3

insulator

cos(Ka) = cos(ka) + mα 2k sin(ka)

Band Structure

1 2 3 4 5

• 3
• 2
• 1

1 2 3

insulator conductor

cos(Ka) = cos(ka) + mα 2k sin(ka)