Alprazolam is a white crystalline powder, which is soluble ...
White Dwarfparticle.physics.ucdavis.edu/teaching/115B/slides/lec11.pdf · 2010. 10. 18. · White...
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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: V = 4 3 π R 3 , ρ = NM p V E = 2 10π 2 m e ( 3π 2 Nq ) 5/3 V 2/3 = 2 10π 2 m e ( 3π 2 Nq ) 5/3 (π/3) 2/3 R 2 = 2 2 15πm e ( 9 4 πNq ) 5/3 R 2 dE grav = - GM (r) r dM = - G ( ρ 4 3 πr 3 ) r ρ 4π r 2 dr = - 16 3 π 3 Gρ 2 r 4 dr E grav = - 16 3 π 3 Gρ 2 R 0 r 4 dr = - 16 15 π 3 Gρ 2 R 5 = - 3 5 π 3 GN 2 M 2 p R E tot = E + E grav = A R 2 - B R dE tot dR = -2 A R 3 + B R 2 =0 2A = BR
Transcript of White Dwarfparticle.physics.ucdavis.edu/teaching/115B/slides/lec11.pdf · 2010. 10. 18. · White...
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)
