White Dwarf stabilized against collapse by degeneracy pressure of electrons radius R , e mass m e , nucleon mass M p , e’s per nucleon q assume constant density: ρ = NM p V = 4 3 π R 3 , V 5 / 3 5 / 3 5 / 3 ( 3 π 2 Nq ) ( 3 π 2 Nq ) ( 9 4 π Nq ) � 2 � 2 2 � 2 = = ( π / 3) 2 / 3 R 2 = E 10 π 2 m e V 2 / 3 10 π 2 m e 15 π m e R 2 G ( ρ 4 3 π r 3 ) 3 π 3 G ρ 2 r 4 dr dE grav = − GM ( r ) ρ 4 π r 2 dr = − 16 dM = − r r 3 π 3 G ρ 2 � R 5 π 3 GN 2 M 2 15 π 3 G ρ 2 R 5 = − 3 E grav = − 16 0 r 4 dr = − 16 p R R 2 − B A E tot = E + E grav = R dE tot = − 2 A R 3 + B R 2 = 0 dR 2 A = BR
White Dwarf � 9 � 5 / 3 4 � 2 2 A 5 = B = R 4 π N q 3 π 3 GN 2 m 2 15 π m e � 9 π � 2 / 3 q 5 / 3 � 2 N 1 / 3 = 7 . 6 × 10 25 m = GM 2 p m e N 1 / 3 4 for a solar mass N ≈ 1 . 2 × 10 57 , R ≈ 7 × 10 6 m � 2 / 3 � 2 / 3 = 1 . 9 × 10 5 eV � 9 π � � 2 � 2 3 π 2 Nq E F = = 4 Nq 2 m e V 2 m e R 2 E rest = m e c 2 = 5 . 11 × 10 5 eV ↑ N ⇒ R ↓ E F ↑ , more relativistic
UltraRelativistic p 2 c 2 + m 2 � e c 4 − m e c 2 ≈ pc E = dE = E k V π 2 k 2 dk = � ck V π 2 k 2 dk � 4 / 3 � k F � 3 π 2 Nq � cV k 3 dk = � c V F = � cV 4 π 2 k 4 = E π 2 4 π 2 V 0 4 / 3 4 π 2 ( 3 π 2 Nq ) � 9 π � 4 / 3 � c � c = = 4 Nq V 1 / 3 3 π R E tot = E + E grav = C R − B R C > B expand, C < B contract
Chandrasekhar Limit C B = � 9 π � 4 / 3 � c 3 5 π 3 GN 2 c M 2 4 N c q = p 3 π √ � � c � 3 / 2 q 2 N c = 15 p ≈ 2 × 10 57 5 π G M 2 16 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 m e → m n , q = 1 N ∼ 10 57 , R ∼ 12 km � 2 � 2 � 9 π E F = = 56 MeV 2 m n R 2 4 E rest = m n c 2 = 940 MeV non-relativistic
Band Structure V ( x + a ) = V ( x ) Bloch’ s Theorem ψ ( x + a ) = e iKa ψ ( x ) K = 2 π j Na
Band Structure 0 < x < a ψ ( x ) = A sin( kx ) + B cos( kx ) − a < x < 0 ψ ( x ) = e − iKa [ A sin k ( x + a ) + B cos k ( x + a )]
Band Structure ψ (0 + ) = ψ (0 − ) � 0 + ψ ′ (0 + ) − ψ ′ (0 − ) = 2 m V ( x ) ψ ( x ) dx � 2 0 − ψ ′ (0 + ) − ψ ′ (0 − ) = 2 m � 2 α B cos( Ka ) = cos( ka ) + m α � 2 k sin( ka ) K = n π
Band Structure cos( Ka ) = cos( ka ) + m α � 2 k sin( ka ) 3 2 1 1 2 3 4 5 -1 -2 -3 band edge: K = n π
Bottom of Band K = n π Structure 1.5 1.5 1 1 0.5 0.5 -0.2 0.2 0.4 0.6 0.8 1 1.2 -0.2 0.2 0.4 0.6 0.8 1 1.2 -0.5 1st band -0.5 2nd band -1 -1 -1.5 -1.5 1.5 1 0.5 3rd band -0.2 0.2 0.4 0.6 0.8 1 1.2 -0.5 -1 -1.5
Band Structure 3 2 1 1 2 3 4 5 -1 -2 -3 cos( Ka ) = cos( ka ) + m α � 2 k sin( ka )
Band Structure 3 2 1 1 2 3 4 5 -1 -2 -3 cos( Ka ) = cos( ka ) + m α � 2 k sin( ka ) insulator
Band Structure 3 2 1 1 2 3 4 5 -1 -2 -3 cos( Ka ) = cos( ka ) + m α � 2 k sin( ka ) insulator conductor
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