Electrons libres, gaz de Fermi

1s22s22p63s1
N N
N+1s22s2
2p6 + 0.098
0.185 +15%
N
1
108109
m L
V(x) = 0,06x6L
∞, x < 0x>L
−~2
2m
d2Ψ
dx2=EΨ

Ψ(0) = Ψ(L) = 0
Ψ(x) = r2
Lsin nπx L, n
E=~2π2n2
2mL2=h2 2mn 2L2
Ψ
n+1/2
−1/2
N L N N
n= 1 n=nF
2nF=N
EF
EF=h2
2mnF 2L2
EF=h2
2mN 4L2
N/4L
N/L ≈4EF≈1
4 4 ·109
1
E0
E0= 2
N/2
X
n=1 h2
2m 1 2L2
n2
s
X
1
n2=1
6s(2s2+ 3s+ 1) ≈1 3s3s1
E0≈2
3h2 2m 1 2L2N 23
=1
3NEF
D(E)
E n
E=h2
2m1 4L2n2

dE =h2
m
n
2L
dn
2L
dn
dE =4L2 h2
1
n
dn/dE
D(E)=2dn
dE =8L2m h2
1
n
D(E) = 4L
hm 2E1/2
E−1/2
0E
D(E)
← D(E) ∼ E−1/2
EF
D(E)EF

−~2
2m∂2 ∂x2+∂2 ∂y2+∂2 ∂z2Ψ = EΨ
Ψ=0 L
Ψ( ) = r8
L3sin nxπx Lsin nyπy Lsin nzπz L
nxnynz
E
E=h2
2mnx 2L2
+ny
2L2
+nz
2L2
L
Ψ(x+L, y, z) = Ψ(x, y, z)
y z
Ψ
Ψk( ) = 1
V1/2
exp{i· }, V =L3
kx= 0 ±2π
L±4π L±6π L. . . kykz
Ψ(x+L, y, z) = 1
V1/2
exp{ikx(x+L)}exp{i(kyy+kzz)}
=1
V1/2
exp{i(kxx+kyy+kzz)}exp{ikxL}
exp{ikxL}= exp in2π
LL= 1

Ψ(x+L, y, z) = Ψ(x, y, z)
= (kx, ky, kz)
E=~2
2mk2
=~=m
= (kx, ky, kz)
(2π/L)3F
2×=V
2
4
3πk3
F
(2π/L)3=V 3π2k3
F=N
kF=3π2N
V1/3
EF=~2
2m3π2N V2/3
vF~kF=mvF
vF=~kF
m=~ m3π2N V1/3
~/m = 1.16 2−1
kB= = 1.381 ·10−23