Мартыненко Т.П. и др. Практический курс физики. Квантовая физика. Элементы физики твёрдого тела и ядерной физики
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101
V = L
3
,
.
mV
2
)2(n
E
3/2
222
n
π
=
(4.10)
, ,
( n
x
, n
y
, n
z
),
m
s
(m
s
= ±1/2). ,
.
g
n
= 2 (
).
.
,
– . ,
, , ,
= 0 ,
–
F
(0).
n
0
. ,
,
A0
N·
M
n
ρ
=
(4.11)
– , - , N
A
- .
n
x
, n
y
, n
z
, ( . 4.5),
(g
n
= 2).
,
n, n
F
,
n
3
F
gn
3
4
π
.
,
n
0
L
3
, ,
3
0n
3
F
Lngn
3
4
=π
. H
,L
g4
n3
n
3/1
n
0
F
π
=
(4.9)
, g
n
=2,
(
)
.n3
m2
)0(E
3/2
0
2
2
F
π=
(4.12)
g(E).
n
F
n
Z
n
Y
n
X
. 4.5
102
g( )
, .
d g(E)dE,
n
n
E
0
g(E)dE
. ,
, 4/3πn
3
g
n
,
.gn4/3g(E)dE
n
E
0
n
3
π=
n
n
( (4.10)),
π=
n
E
0
2/3
2
n
n
;V
mE2
g4/3g(E)dE
,
n
g
n
= 2,
.E
m2
2
V
)E(g
2/1
2/3
22
n
⋅
π
=
(4.13)
n
, V.
n
- , n .
.E
m2
2
1
)E(g
2/3
22
1
⋅
π
= (4.14)
.
-
, d ,
dn(E) = g(E)⋅f(E,T)dE, (4.15)
0
⋅=
E
0
dE)T,E(f)E(gN
(4.16)
g( ) – (4.13).
f(E,T) – - ,
. ,
,
103
,
1
e
1
)T,E(f
kT
EE
F
+
=
−
(4.17)
, ,
F
–
, k – , – .
( , , ,
,H
1
1
+
,He
3
2
+
He
4
2
) .
( , ,
, ,
2
4
)
- ,
,
1
e
1
)T,E(f
kT
EE
F
−
=
−
(4.18)
,
, = 0
.
-
.
1). T = 0 K.
,
, = 0
.
.
= 0
. 4. 6. , = 0 =
max
.
= 0 >
F
(4.17) ,
∞→
kT
-
F
f(E,T) = 0. ,
, ,
.
<
F
,
= 0
0
kT
EE
exp
F
→
−
f(E,T) = 1.
,
, .
= 0
. 4.7.
2). > 0 .
. 4.6
E
F
f(E,T)
E
F
E
4.7
1
104
=
F
,
(1.17),
,1
kT
EE
exp
F
=
−
f( , ) = 1/2. ,
50 %. > 0
,
. ,
,
.
. 4.8.
,
f( , ) = 0 ( . 4.7)
±kT .
, -
, .
.
<<
F
,
F
E
F
(0) = kT
F
(4.19)
(
F
> 10
4
)
,
.
,
–
F
>> kT.
(4.17)
−
kT
EE
exp
F
>> 1,
.econsteee)T,E(f
kT
E
kT
E
kT
E
kT
EE
FF
−−
−
−
⋅=⋅=≅ (4.20)
−⋅=
kT
E
expA)T,E(f
- .
, -
- . ,
- , .
, -
1.
, , .
E
F
-kT E
F
E
F
+kT E
1
f(E,T)
. 4.8
105
, d ,
(4.15), (4.17)
- (4.13), . .
.dE
1kT/)EEexp(
E
V
m2
2
1
)T,E(dN
F
2/1
2/3
22
⋅
+−
⋅
π
=
(4.21)
0 E
.
1kT/)EEexp(
dEE
V
m2
2
1
N
E
0
F
2/1
2/3
22
+−
⋅
⋅
π
=∆
(4.22)
= 0 0
F
(4.21)
.dEEV
m2
2
1
)E(dN
2/1
2/3
22
⋅⋅⋅
π
=
(4.23)
∆N
1
,
2
⋅⋅
π
=∆
2
1
E
E
2/1
2/3
22
EdEV
m2
2
1
N
(4.24)
,
.
2
mv
E
2
=
,
v
2
m
E
2/1
⋅=
( ),
dvmv
2
mv
ddE
2
⋅==
( ).
( ) ( ) (4.21)
, :
.dv
1kT/)E2/mvexp(
1
v
m1
)v(dN
F
2
2
2
3
2
+−
⋅⋅
π
= (4.25)
, ,
:
.dp
1kT/)Em2/pexp(
1p1
)p(dN
F
23
2
2
+−
⋅⋅
π
=
(4.26)
2. , , .
.
dE)T,E(f)E(g
dE)T,E(f)E(gA
A
0
0
∞
∞
⋅
⋅⋅
=
(4.27)
106
,
.
N
dE)T,E(f)E(gE
E
0
∞
⋅⋅
=
(4.28)
N – .
= 0 <
F
f( , ) = 1 f( , ) = 0 E >
F
,
(4.28) :
.
dE)E(g
dE)E(gE
E
F
F
E
0
E
0
⋅
=
(4.29)
(4.27), , = v:
.
dv)T,v(f)v(g
dv)T,v(f)v(gv
v
0
0
∞
∞
⋅
⋅⋅
=
(4.30)
g(v) – .
= 0
,
dv)v(g
dv)v(gv
v
max
max
v
0
v
0
⋅
=
(4.31)
v
max
– :
).0(E
2
mv
F
2
max
=
(4.32)
.
.
– ,
( = 1/ ).
/ ( / ).
, . 1
107
, n
.
1.
=300
( ⋅ )
(
-3
)
∆E
.
( )
10
-8
10
22
– 10
23
0
10
-4
– 10
7
10
14
– 10
15
0,3 – 3
10
12
– 10
20
10
9
– 10
11
3
10
,
, – ,
.
,
.
1 2 .
1- ( , , , , )
.
, ,
N –
, . ,
, N –
,
, – N.
, N/2 –
. 2-
( , , ,
, )
.
, ,
N – (N –
)
. , 2-
( ,
1- )
. 4.9
E
F
108
.
2-
, .
. 4.9.
( ,
).
,
( ).
, ,
.
.
, ,
,
.
,
,
( . 1).
. 4.10.
∆
F
= ∆ /2.
, ,
. ,
,
∆ .
(∆
~
4÷6 ),
( ~ 10
7
÷10
8
/ ).
( ).
, ,
2
E
E
F
∆
=
E
. 4.10
109
, " "
.
.
(∆ 3 )
.
, , III V
(G s, InSb) II VI ( dSe).
(∆ = 1÷2 ),
,
10
14
÷10
15 -3
.
–
. ,
, ,
" ". " "
, " " . .
" " –
.
,
" "
. -
n,p
= e⋅n
n,p
(
n
+
p
), (4.33)
n
n,p
– (
),
n
,
p
– .
,
( ),
= 1 / , = v/ .
,
T
p,n
vm2
e λ
=µ
(4.34)
< > – , < V
T
> –
, m –
.
110
. ,
,
, , ,
, ,
. . .
.
" " m*.
. ,
. m* ≠ m ,
. m*
2.
2.
Be Li Na Cu K Ca Al Zn
m*/m
e
1,6 1,2 1,2 1,0 1,1 1,4 0,57 0,85
(m* = m
= 9,1⋅10
-31
).
.
,
j
,
B
,
( . 4.11).
]B,v[qF ⋅= (v – ).
,
,
= U/d (U –
, d – ). ,
q⋅v⋅B = q⋅E. ,
j = q⋅n⋅v (n – ),
U = U :
,dB
qn
j
U ⋅⋅=
(4.35)
d
j
B
. 4.11