Елесин В.Ф., Кашурников В.А. Физика фазовых переходов
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7.1.1.1. ɂɫɯɨɞɹ ɢɡ ɭɪɚɜɧɟɧɢɹ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ, ɪɚɫɫɱɢ-
ɬɚɬɶ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ T
c
, ɤɪɢɬɢɱɟɫɤɢɣ ɨɛɴɟɦ
X
c
, ɢ
ɤɪɢɬɢɱɟɫɤɨɟ ɞɚɜɥɟɧɢɟ P
c
, ɨɩɪɟɞɟɥɹɸɳɢɟ ɬɨɱɤɭ ɧɚ ɮɚɡɨɜɨɣ
ɞɢɚɝɪɚɦɦɟ, ɡɚ ɤɨɬɨɪɨɣ ɢɦɟɟɬ ɦɟɫɬɨ ɬɨɥɶɤɨ ɝɚɡɨɨɛɪɚɡɧɨɟ ɫɨ-
ɫɬɨɹɧɢɟ.
7.1.2. Ɇɨɞɟɥɶ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ
ɉɨɤɚɠɟɦ ɤɚɤ, ɢɫɩɨɥɶɡɭɹ ɦɨɞɟɥɶ ɂɡɢɧɝɚ, ɦɨɠɧɨ ɨɩɢɫɚɬɶ
ɤɨɧɞɟɧɫɚɰɢɸ ɢ ɩɨɥɭɱɢɬɶ ɭɪɚɜɧɟɧɢɟ ɫɨɫɬɨɹɧɢɹ, ɚɧɚɥɨɝɢɱɧɨɟ
ɭɪɚɜɧɟɧɢɸ ȼɚɧ ɞɟɪ ȼɚɚɥɶɫɚ (Ʌɢ, əɧɝ, 1952ɝ.). ɗɬɨɬ ɩɨɞɯɨɞ
ɧɚɡɵɜɚɟɬɫɹ ɦɨɞɟɥɶɸ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ.
Ɋɚɡɨɛɴɟɦ ɜɟɫɶ ɨɛɴɟɦ ɧɚ ɹɱɟɣɤɢ (ɭɡɥɵ), ɨɛɪɚɡɭɸɳɢɟ
ɩɪɚɜɢɥɶɧɭɸ ɤɪɢɫɬɚɥɥɢɱɟɫɤɭɸ ɪɟɲɟɬɤɭ, ɡɚɩɨɥɧɹɸɳɭɸ ɜɫɟ
ɩɪɨɫɬɪɚɧɫɬɜɨ. ɉɭɫɬɶ ɱɢɫɥɨ ɹɱɟɟɤ ɪɚɜɧɨ N
0
. ɑɚɫɬɢɰɵ ɩɟɪɟɞɜɢ-
ɝɚɸɬɫɹ ɩɨ ɷɬɨɦɭ ɩɪɨɫɬɪɚɧɫɬɜɭ, ɡɚɧɢɦɚɹ ɢ ɩɨɤɢɞɚɹ ɭɡɥɵ (ɹɱɟɣ-
ɤɢ) ɜ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ. ɋɨɩɨɫɬɚɜɢɦ ɧɚɩɪɚɜɥɟɧɧɨɦɭ
ɜɜɟɪɯ ɫɩɢɧɭ ɡɚɧɹɬɵɣ ɱɚɫɬɢɰɟɣ ɭɡɟɥ, ɚ ɧɚɩɪɚɜɥɟɧɧɨɦɭ ɜɧɢɡ -
ɩɭɫɬɨɣ ɭɡɟɥ ɢ ɜɜɟɞɟɦ ɜɟɥɢɱɢɧɭ
H
i
, ɪɚɜɧɭɸ 1, ɟɫɥɢ ɭɡɟɥ i ɡɚɧɹɬ,
ɢ ɪɚɜɧɭɸ ɧɭɥɸ, ɟɫɥɢ ɨɧ ɩɭɫɬ:
H
P
P
P
i
i
i
i
®
¯
1
2
11
00
,;
,.
(7.14)
Ɍɨɝɞɚ ɝɚɦɢɥɶɬɨɧɢɚɧ ɂɡɢɧɝɚ ɫɨ ɫɩɢɧɚɦɢ
P
i
ɩɨ ɬɚɤɨɣ ɪɟɲɟɬɤɟ
ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ
H
i
:
HV H VV
VH
ij i j
ij
iijij
ij
ij i j
ij
ij
ij
i
ij
¦¦ ¦¦
¦¦
12 12 4
12 2 1
//
/().
PP P HH H H
H
()
(7.15)
120
ɉɟɪɟɨɛɨɡɧɚɱɢɦ ɩɚɪɚɦɟɬɪ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɢ ɜɜɟɞɟɦ ɟɝɨ ɧɭɥɟ-
ɜɭɸ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɭ
121
V4VVVq Virqu
ij
ij
ij
i
i q ij ij
() ( ) exp( )| ,00
0
¦¦
oo
. (7.16)
ɂɫɩɨɥɶɡɭɹ (7.16), ɢɡ (7.15) ɩɨɥɭɱɚɟɦ:
Hu HVCCNVN
H
ij i j
ij
i
i
¦¦
12 2 0 0 2
00
/ { ()} , ()/
HH H
.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɧɟɫɭɳɟɫɬɜɟɧɧɨɣ
ɤɨɧɫɬɚɧɬɵ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɨɛɵɱɧɵɦ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɂɡɢɧɝɚ,
ɧɨ ɬɨɥɶɤɨ ɩɨ ɛɨɥɶɲɨɦɭ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɚɧɫɚɦɛɥɸ
(
ɱɢɫɥɨ ɧɟ
-
ɧɭɥɟɜɵɯ
“
ɫɩɢɧɨɜ
”
H
i
ɧɟ ɮɢɤɫɢɪɨɜɚɧɨ) ɫ ɯɢɦɢɱɟɫɤɢɦ ɩɨɬɟɧ-
ɰɢɚɥɨɦ
P
=2{H-V(0)}:
HuNN
ij i j
ij
i
i
¦¦
12/,
HH P H
. (7.17)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɬɚɬɢɫɬɢɱɟɫɤɚɹ ɫɭɦɦɚ ɞɥɹ ɫɢɫɬɟɦɵ
ɫɩɢɧɨɜ ɩɟɪɟɯɨɞɢɬ ɜ ɛɨɥɶɲɭɸ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ ɞɥɹ ɪɟ-
ɲɟɬɨɱɧɨɝɨ ɝɚɡɚ. Ɍɟɩɟɪɶ ɦɨɠɧɨ ɩɪɨɞɨɥɠɢɬɶ ɚɧɚɥɨɝɢɸ ɞɚɥɶɲɟ ɢ
ɩɨɥɭɱɢɬɶ ɭɪɚɜɧɟɧɢɟ ɫɨɫɬɨɹɧɢɹ. ɇɚɦɚɝɧɢɱɟɧɧɨɫɬɶ
R ɜ ɦɨɞɟɥɢ
ɂɡɢɧɝɚ ɫɜɹɡɚɧɚ ɫ ɩɥɨɬɧɨɫɬɶɸ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ
U
:
UH
H
!
¦¦
N
NN N
R
i
i
i
i
00 0
111
22
1
2
{}
. (7.18)
Ɇɵ ɦɨɠɟɦ ɬɟɩɟɪɶ, ɢɫɩɨɥɶɡɭɹ ɞɥɹ (7.15) ɩɪɢɛɥɢɠɟɧɢɟ ȼɟɣɫɫɚ,
ɡɚɩɢɫɚɬɶ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ
RVR th[ H
E
(() )]0 ɢ ɩɨɥɭɱɢɬɶ ɢɡ ɧɟɝɨ ɭɪɚɜɧɟɧɢɟ ɞɥɹ
ɩɥɨɬɧɨɫɬɢ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ
U
ɜ ɬɟɪɦɢɧɚɯ ɩɟɪɟɧɨɪɦɢɪɨɜɚɧɧɨ-
ɝɨ ɩɨɬɟɧɰɢɚɥɚ u ɢ ɯɢɦɢɱɟɫɤɨɝɨ ɩɨɬɟɧɰɢɚɥɚ
P
:
21
2
0
U
E
U
th[ ( ( ) )]u
P
, (7.19)
ɢɥɢ
EP EU
U
U
u() ln0
1
. (7.20)
ɉɨɥɭɱɢɦ ɞɚɜɥɟɧɢɟ P ɢ ɭɪɚɜɧɟɧɢɟ ɫɨɫɬɨɹɧɢɹ, ɭɱɢɬɵɜɚɹ ɫɨɨɬ-
ɧɨɲɟɧɢɹ
w
P
wU
w
P
w
w
P
w
w
P
w
w
wU
w
P
wU U
w
P
wU
N
NNNP
P
T0
0
11
;()()(
T
)
. (7.21)
ɂɫɩɨɥɶɡɭɹ (7.21), ɩɨɥɭɱɚɟɦ ɨɤɨɧɱɚɬɟɥɶɧɨ ɢɡ (7.20) ɭɪɚɜɧɟɧɢɟ
ɫɨɫɬɨɹɧɢɹ ɜ ɦɨɞɟɥɢ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ
PukT
U
U
2
2
01( ) ln( ) . (7.22)
ɍɪɚɜɧɟɧɢɟ (7.22) ɢɦɟɟɬ ɤɚɱɟɫɬɜɟɧɧɨ ɨɞɢɧɚɤɨɜɵɟ ɢɡɨ-
ɬɟɪɦɵ ɫ ɭɪɚɜɧɟɧɢɟɦ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ (ɫɦ. ɪɢɫ.13). ɋɪɚɜɧɢɦ
ɟɝɨ ɫ ɭɪɚɜɧɟɧɢɟɦ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ (7.11), ɩɟɪɟɩɢɫɚɜ (7.11) ɜ
ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
PukT aub
UU
U
2
2
0
1
02 1() ; ()/ ,
. (7.23)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɚɫɢɦɩɬɨɬɵ ɩɨɫɥɟɞɧɢɯ ɱɥɟɧɨɜ (ɬɟɩɥɨɜɨɟ ɞɚɜɥɟ-
ɧɢɟ) ɜ (7.22) ɢ (7.23) ɩɪɢ
Uo
0 (ɩɪɟɞɟɥ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ) ɨɞɢɧɚ-
ɤɨɜɵ ɢ ɩɪɢ
Uo
(ɫɥɭɱɚɣ ɛɟɫɤɨɧɟɱɧɨɝɨ ɞɚɜɥɟɧɢɹ) ɛɥɢɡɤɢ.
Ɉɩɪɟɞɟɥɢɦ ɤɪɢɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɟɲɟɬɨɱɧɨɣ ɦɨɞɟɥɢ
(7.22). ɍɱɢɬɵɜɚɹ, ɱɬɨ ɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ ɢɡ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɨɩ-
ɪɟɞɟɥɹɟɬɫɹ ɤɚɤ R=0, ɢɦɟɟɦ:
UUUX
1
2
0121
R
R
cc
,//
X
2
1)
(7.24)
(ɢɡ ɭɪɚɜɧɟɧɢɹ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ ). Ɂɚɦɟɬɢɦ, ɱɬɨ
ɡɧɚɱɟɧɢɟ
X
c
b 3(
X
c
2 ɞɥɹ ɪɟɲɟɬɨɱɧɨɣ ɦɨɞɟɥɢ ɦɨɠɧɨ ɬɚɤɠɟ ɩɨɥɭ-
ɱɢɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɡ ɭɫɥɨɜɢɣ (7.12).
Ɉɛɫɭɞɢɦ ɜɚɠɧɵɣ ɜɨɩɪɨɫ ɨ ɪɨɞɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. ɇɟ-
ɫɥɨɠɧɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɢɡ ɪɚɜɟɧɫɬɜɚ ɯɢɦɢɱɟɫɤɢɯ ɩɨɬɟɧɰɢɚɥɨɜ
P
ɢ ɞɚɜɥɟɧɢɣ P ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɮɚɡ (ɠɢɞɤɨɫɬɢ ɢ
122
ɩɚɪɚ) ɫɥɟɞɭɟɬ:
P
uH()02 0.Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɚɧɚɥɨɝ
ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ (ɮɟɪɪɨɦɚɝɧɟɬɢɤ) ɧɚɯɨɞɢɬɫɹ ɜ ɧɭɥɟɜɨɦ ɜɧɟɲ-
ɧɟɦ ɩɨɥɟ. Ⱦɥɹ ɮɟɪɪɨɦɚɝɧɟɬɢɤɚ ɩɪɢ T<T
c
ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ ɦɟ-
ɧɹɟɬɫɹ ɫɤɚɱɤɨɦ ɨɬ +R ɞɨ -R ɩɪɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɢɡɦɟɧɟɧɢɢ
ɩɨɥɹ H ɨɬ +0 ɞɨ -0 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɷɬɨ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɩɟɪ-
ɜɨɝɨ ɪɨɞɚ, ɩɪɢ ɤɨɬɨɪɨɦ ɬɟɪɩɢɬ ɪɚɡɪɵɜ ɩɟɪɜɚɹ ɩɪɨɢɡɜɨɞɧɚɹ
ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ F.
Ɂɚɞɚɱɢ
7.1.2.
1. ɉɨɤɚɡɚɬɶ ɫɨɜɩɚɞɟɧɢɟ (ɢɥɢ ɛɥɢɡɨɫɬɶ) ɚɫɢɦɩɬɨɬ
ɭɪɚɜɧɟɧɢɹ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ ɢ ɭɪɚɜɧɟɧɢɹ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ ɜ
ɩɪɟɞɟɥɚɯ ɩɥɨɬɧɨɫɬɟɣ
Uo
0 (ɢɞɟɚɥɶɧɵɣ ɝɚɡ) ɢ
Uo
(ɛɟɫɤɨɧɟɱ-
ɧɨɟ ɞɚɜɥɟɧɢɟ). ɉɨɥɭɱɢɬɶ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɢ ɤɪɢɬɢɱɟ-
ɫɤɨɟ ɞɚɜɥɟɧɢɟ ɜ ɦɨɞɟɥɢ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ.
7.1.2.
2. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɢɡ ɪɚɜɟɧɫɬɜɚ ɯɢɦɢɱɟɫɤɢɯ ɩɨɬɟɧ-
ɰɢɚɥɨɜ
P
ɢ ɞɚɜɥɟɧɢɣ P ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɮɚɡ (ɠɢɞɤɨ-
ɫɬɢ ɢ ɩɚɪɚ) ɜ ɪɟɲɟɬɨɱɧɨɣ ɦɨɞɟɥɢ ɫɥɟɞɭɟɬ, ɱɬɨ ɯɢɦɢɱɟɫɤɢɣ ɩɨ-
ɬɟɧɰɢɚɥ
P
u()02.
7.1.3. Ɏɥɭɤɬɭɚɰɢɢ ɜ ɦɨɞɟɥɢ ɪɟɲɟɬɨɱɧɨɝɨ ɝɚɡɚ
ɉɨ ɚɧɚɥɨɝɢɢ ɫ ɦɨɞɟɥɶɸ ɂɡɢɧɝɚ ɧɚɣɞɟɦ ɮɥɭɤɬɭɚɰɢɢ ɜ
ɪɟɲɟɬɨɱɧɨɣ ɦɨɞɟɥɢ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ (3.15ɚ) ɞɥɹ
ɮɥɭɤɬɭɚɰɢɣ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɜɛɥɢɡɢ ɬɨɱɤɢ ɮɚɡɨɜɨɝɨ ɩɟ-
ɪɟɯɨɞɚ:
!
o
||
()( )
P
E
q
R
Vq R
2
2
2
1
11
. (7.25)
ɉɟɪɟɩɢɲɟɦ ɟɝɨ ɜ ɬɟɪɦɢɧɚɯ ɪɟɲɟɬɨɱɧɨɣ ɦɨɞɟɥɢ. ɍɱɢɬɵɜɚɹ
ɫɨɨɬɧɨɲɟɧɢɹ (7.14), (7.16) ɢ (7.18), ɩɨɥɭɱɚɟɦ:
123
! !
o
|| ||
()
()( )
HP
U
U
EUU
qq
uq
22
1
4
1
11
. (7.26)
Ɋɨɥɶ ɬɜɟɪɞɵɯ ɹɞɟɪ ɨɬɪɚɠɚɟɬɫɹ ɧɚ ɩɨɹɜɥɟɧɢɢ ɜ ɡɧɚɦɟɧɚ-
ɬɟɥɟ (7.26) ɦɧɨɠɢɬɟɥɹ (1-
U
). ȼɢɞɢɦ, ɱɬɨ ɜ ɠɢɞɤɨɣ ɮɚɡɟ, ɤɨɝɞɚ
U|
1, ɷɬɨɬ ɦɧɨɠɢɬɟɥɶ ɩɨɞɚɜɥɹɟɬ ɮɥɭɤɬɭɚɰɢɢ, ɩɪɢɜɨɞɹ ɤ ɭɫɬɨɣ-
ɱɢɜɨɣ ɠɢɞɤɨɫɬɢ.
7.2. ɉɟɪɟɯɨɞ ɠɢɞɤɨɫɬɶ - ɬɜɟɪɞɨɟ ɬɟɥɨ.
Ʉɪɢɫɬɚɥɥɢɡɚɰɢɹ
ɉɟɪɟɯɨɞ “ɠɢɞɤɨɫɬɶ-ɬɜɟɪɞɨɟ ɬɟɥɨ” ɷɬɨ ɩɟɪɟɯɨɞ ɢɡ ɩɨɥ-
ɧɨɣ ɫɢɦɦɟɬɪɢɢ ɠɢɞɤɨɣ ɮɚɡɵ ɜ ɩɟɪɢɨɞɢɱɟɫɤɭɸ ɫɢɦɦɟɬɪɢɸ
ɬɜɟɪɞɨɝɨ ɬɟɥɚ. ɉɭɫɬɶ ɱɚɫɬɢɰɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɫ ɩɚɪɧɵɦ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ v(|r
i
-r
j
|) ɜ ɜɢɞɟ, ɩɨɤɚɡɚɧɧɨɦ ɧɚ ɪɢɫ. 11. ɉɪɟɞ-
ɩɨɥɨɠɢɦ, ɱɬɨ ɱɢɫɥɨ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ ɜɟɥɢɤɨ, ɬɚɤ ɱɬɨ ɦɨɠ-
ɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢɛɥɢɠɟɧɢɟ ɫɪɟɞɧɟɝɨ ɩɨɥɹ. ɏɚɪɚɤɬɟɪɧɨɣ
ɱɟɪɬɨɣ, ɨɬɥɢɱɚɸɳɟɣ ɬɜɟɪɞɨɟ ɫɨɫɬɨɹɧɢɟ ɨɬ ɠɢɞɤɨɝɨ ɹɜɥɹɟɬɫɹ
ɩɟɪɢɨɞɢɱɧɨɫɬɶ ɩɥɨɬɧɨɫɬɢ
U
r). Ɍɨɝɞɚ ɩɨɬɟɧɰɢɚɥ, ɞɟɣɫɬɜɭɸɳɢɣ
ɧɚ ɱɚɫɬɢɰɭ ɜ ɬɨɱɤɟ r ɨɬ ɩɨɥɹ ɞɪɭɝɢɯ ɱɚɫɬɢɰ, ɬɚɤɠɟ ɛɭɞɟɬ ɩɟ-
ɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ:
Vr . (7.27) dr r v r r() ' (')( ')
oo
³
U
o
ɍɱɬɟɦ, ɱɬɨ ɞɥɹ ɦɨɞɟɥɢ ɬɜɟɪɞɵɯ ɲɚɪɨɜ (ɫɦ. ɪɢɫ.11), ɡɚɥɨɠɟɧ-
ɧɨɣ ɜɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ , ɨɧɨ ɦɟɧɶɲɟ ɧɭɥɹ ɩɪɢ r>C, ɝɞɟ C
- ɪɚɞɢɭɫ ɲɚɪɨɜ (ɫɦ. ɩɚɪɚɝɪɚɮ 7.1.1), ɚ ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɤɨɪ-
ɪɟɥɹɰɢɨɧɧɚɹ ɮɭɧɤɰɢɹ g ɪɚɜɧɚ ɧɭɥɸ (ɫɦ. ɪɢɫ.12). Ɉɬɫɸɞɚ ɬɨɝɞɚ
ɜɦɟɫɬɨ ɩɨɬɟɧɰɢɚɥɚ (7.27) ɩɢɲɟɦ:
vr()
Vr dr r gr r vr r dr r vr r() '(')( ')( ') '(')( ')
ooooo oo
³³
UU
o
, (7.28)
124
ɝɞɟ
vr() -ɩɟɪɟɧɨɪɦɢɪɨɜɚɧɧɨɟ ɫ ɭɱɟɬɨɦ ɡɧɚɤɚ ɩɚɪɧɨɟ ɜɡɚɢɦɨ-
ɞɟɣɫɬɜɢɟ. ȼ ɬɜɟɪɞɨɦ ɬɟɥɟ , ɬ.ɟ. ɩɥɨɬɧɨɫɬɶ ɩɟ-
ɪɢɨɞɢɱɧɚ ɫ ɩɟɪɢɨɞɨɦ ɪɟɲɟɬɤɢ a. ȿɟ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɪɚɡ-
ɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɜɟɤɬɨɪɚɦ ɨɛɪɚɬɧɨɣ ɪɟɲɟɬɤɢ:
UU
() ( )rra
ooo
n
U
)
iG r
o
UU U
() exp[()] exp[](ra iqra iqr r
q
q
q
q
oo ooo oo o
¦¦
. (7.29)
Ɉɬɫɸɞɚ ɢ ɫɥɟɞɭɟɬ, ɱɬɨ - ɜɟɤɬɨɪ ɨɛɪɚɬɧɨɣ
ɪɟɲɟɬɤɢ, ɢ
exp( )iqa q G
oo o o
1
UU
() exp[ ]r
G
G
oo
¦
. (7.30)
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ, ɱɬɨ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɵɣ ɩɨɬɟɧɰɢɚɥ (7.28) ɢɦɟ-
ɟɬ ɬɭ ɠɟ ɩɟɪɢɨɞɢɱɧɨɫɬɶ, ɱɬɨ ɢ
U
, ɫɥɟɞɭɟɬ ɢɡ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
Ɏɭɪɶɟ:
Vq qvq q G() ()() ;
oo
U
0 z
o
, (7.30ɚ)
ɬɚɤ ɤɚɤ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɥɟɝɤɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ V(r)=V(r+a).
Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɪɚɫɫɦɨɬɪɢɦ ɨɞɧɨɦɟɪɧɭɸ ɫɢɬɭɚɰɢɸ. Ɍɨ-
ɝɞɚ ɜɟɤɬɨɪ ɨɛɪɚɬɧɨɣ ɪɟɲɟɬɤɢ G={2
S
/2a}n=(
S
/a)n, ɝɞɟ ɱɟɪɟɡ 2a
ɨɛɨɡɧɚɱɢɦ ɩɟɪɢɨɞ ɪɟɲɟɬɤɢ. Ɉɬɫɸɞɚ
UU
S
() exp[ ]x
n
a
x
n
n
¦
i
, (7.31)
U
S
U
n
a
a
a
dx i
n
a
xx
³
1
2
exp[ ] ( )
. (7.32)
Ɏɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɛɭɞɟɬ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶɫɹ ɫɭɳɟɫɬɜɨɜɚ-
ɧɢɟɦ ɧɟɧɭɥɟɜɵɯ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬ ɩɥɨɬɧɨɫɬɢ
U
n
z
0 ɩɪɢ n
z
0, ɬɚɤ
ɤɚɤ ɟɫɥɢ
U
n
=0, n
z
0, ɬɨ ɨɫɬɚɟɬɫɹ ɬɨɥɶɤɨ ɧɭɥɟɜɚɹ ɤɨɦɩɨɧɟɧɬɚ
U
0
=const, ɬ.ɟ. ɠɢɞɤɚɹ ɮɚɡɚ.
ɋɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ
U
(r) ɩɨɥɭɱɢɦ ɢɡ ɪɚɫ-
ɩɪɟɞɟɥɟɧɢɹ Ȼɨɥɶɰɦɚɧɚ ɱɚɫɬɢɰ ɜ ɩɨɥɟ ɫ ɩɨɬɟɧɰɢɚɥɨɦ V(r):
125
U
E
E
()
exp[ ( )]
exp[ ( )]
x
Vx
Vx dx
³
, (7.33)
ɝɞɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɢɞɟɬ ɩɨ ɷɥɟɦɟɧɬɚɪɧɨɣ ɹɱɟɣɤɟ (-ɚ,ɚ). ɉɟɪɟ-
ɩɢɲɟɦ ɩɨɬɟɧɰɢɚɥ V(x) (7.28):
Vx dxvx x x v n i
a
nx
n
n
() '( ')(') ( )exp[ ]
³
¦
UU
S
. (7.34)
ɩɪɢ ɷɬɨɦ ɧɟɫɥɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɢɡ
vx v x() ( )
ɫɥɟɞɭɟɬ
vn v n() ( )
. Ⱥɧɚɥɨɝɢɱɧɚɹ ɱɟɬɧɨɫɬɶ ɢɦɟɟɬ ɦɟɫɬɨ ɢ ɞɥɹ
U
. Ɍɨ-
ɝɞɚ:
Vx v
a
nx
dx x
a
nx
nn
n
n
() cos( ),
()cos( ).
¦
³
U
S
UU
S
(7.35)
ɉɨɞɫɬɚɜɢɜ (7.33) ɜ (7.35), ɢɦɟɟɦ ɞɥɹ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɵ:
U
S
EU
S
EU
S
n
mm
m
a
a
mm
m
a
a
dx
a
nx v
a
nx
dx v
a
nx
¦
³
¦
³
cos( )exp{ cos( )}
exp{ cos( )}
. (7.36)
ɉɨɥɚɝɚɟɦ, ɱɬɨ ɜɛɥɢɡɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɟɪɟɯɨɞɚ, ɤɨɝɞɚ ɤɪɢɫɬɚɥɥ
ɟɳɟ ɧɟ ɫɬɚɥ ɪɟɝɭɥɹɪɧɨɣ ɪɟɲɟɬɤɨɣ, ɨɫɧɨɜɧɨɣ ɜɤɥɚɞ ɞɚɸɬ ɤɨɦ-
ɩɨɧɟɧɬɵ
U
0
ɢ
U
1
. ɇɚɥɢɱɢɟ ɩɟɪɟɯɨɞɚ ɦɨɠɧɨ ɬɟɫɬɢɪɨɜɚɬɶ ɩɨ ɨɬ-
ɥɢɱɢɸ ɨɬ ɧɭɥɹ ɤɨɦɩɨɧɟɧɬɵ
U
1
, ɤɨɬɨɪɚɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɭɞɨɜɥɟ-
ɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
U
S
EU
S
EU
S
1
11
11
2
2
³
³
dx
a
xv
a
x
dx v
a
x
a
a
a
a
cos( )exp{ cos( )}
exp{ cos( )}
(7.37)
126
(ɜ (7.37) ɦɵ ɨɫɬɚɜɢɥɢ ɬɨɥɶɤɨ
U
0
ɢ
U
1
, ɩɪɢ ɷɬɨɦ ɱɥɟɧ ɫ
U
0
ɫɨɤɪɚ-
ɬɢɥɫɹ, ɚ ɤɨɷɮɮɢɰɢɟɧɬ “2” ɜ ɷɤɫɩɨɧɟɧɬɟ - ɢɡ-ɡɚ ɩɨɥɨɠɢɬɟɥɶɧɨɣ
ɢ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɱɚɫɬɢ ɫɭɦɦɵ ɩɨ m:
U
1
+
U
-1
=2
U
1
). ɗɬɨ ɭɪɚɜɧɟ-
ɧɢɟ ɫɨɜɩɚɞɚɟɬ ɫ ɭɪɚɜɧɟɧɢɟɦ ȼɟɣɫɫɚ ɞɥɹ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ R ɜ
ɦɨɥɟɤɭɥɹɪɧɨɦ ɩɨɥɟ
vR
1
(ɟɫɥɢ R ɫɨɩɨɫɬɚɜɢɬɶ ɫ
U
1
) ɜ ɞɜɭɦɟɪ-
ɧɨɦ ɫɥɭɱɚɟ ɜ ɩɪɟɞɟɥɟ ɧɟɩɪɟɪɵɜɧɨɝɨ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɫɩɢɧɚ S.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ɤɥɚɫɫɢɱɟɫɤɢɣ Ƚɟɣɡɟɧɛɟɪɝɨɜɫɤɢɣ ɝɚɦɢɥɶ-
ɬɨɧɢɚɧ ɢɦɟɟɬ ɜɢɞ
Hv
ij i j
ij
oo
z
¦
12/ SS
, (7.38)
ɬɨ ɦɨɥɟɤɭɥɹɪɧɨɟ ɩɨɥɟ ȼɟɣɫɫɚ ɜ ɬɨɱɤɟ i ɪɚɜɧɨ
vS
ij j
j
!
o
¦
,
ɚ ɷɧɟɪɝɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɩɢɧɚ <S
i
> ɫ ɷɬɢɦ ɩɨɥɟɦ ɪɚɜɧɚ:
VvSSvS
ij i j
j
i
!
oo o
¦
11
UT
||cos
,
ɝɞɟ
4
- ɭɝɨɥ ɦɟɠɞɭ S
i
ɢ ɩɨɥɧɨɣ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶɸ. Ɍɨɝɞɚ
ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ ɞɥɹ
U
1
=|<S
i
>| ɫɨɜɩɚɞɚɟɬ ɫ (7.37).
Ɋɚɫɫɱɢɬɚɟɦ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ. ȿɫɥɢ ɭɫɬɪɟɦɢɬɶ
ɜ (7.36) ɤɧɭɥɸ
U
n
, ɬɨ
UEU
nn
n
c
n
vTv , . (7.39)
ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɤɪɢɬɢɱɟɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɩɢɫɚɧɨ ɜ ɬɨɦ
ɫɦɵɫɥɟ, ɱɬɨ ɟɟ ɩɪɢɪɚɜɧɢɜɚɸɬ ɤ ɦɚɤɫɢɦɚɥɶɧɨɣ ɩɨ ɜɟɥɢɱɢɧɟ
ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɟ ( ɢɫɤɥɸɱɚɹ ɧɭɥɟɜɭɸ) ɩɨɬɟɧɰɢɚɥɚ .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɟɫɥɢ ɛɵ ɨɛɴɟɦ ɫɢɫɬɟɦɵ ɧɟ ɭɦɟɧɶɲɚɥɫɹ
ɩɪɢ ɨɬɜɟɪɞɟɜɚɧɢɢ, ɦɵ ɢɦɟɥɢ ɛɵ ɞɟɥɨ ɫ ɨɛɵɱɧɵɦ ɮɚɡɨɜɵɦ ɩɟ-
ɪɟɯɨɞɨɦ ɜɬɨɪɨɝɨ ɪɨɞɚ. Ⱥ ɬɚɤ ɤɚɤ ɤɪɢɫɬɚɥɥɢɡɚɰɢɹ ɨɛɹɡɚɬɟɥɶɧɨ
ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɭɦɟɧɶɲɟɧɢɟɦ ɨɛɴɟɦɚ, ɬɨ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ
“ɠɢɞɤɨɫɬɶ-ɤɪɢɫɬɚɥɥ” - ɷɬɨ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɩɟɪɜɨɝɨ ɪɨɞɚ.
127
Ɂɚɞɚɱɢ
7.2.
1. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ ɩɟɪɢɨɞɢɱɟɫɤɭɸ (ɫ ɩɟɪɢɨɞɨɦ a)
ɜɟɥɢɱɢɧɭ
U
, ɪɚɡɥɨɠɢɬɶ ɜ ɨɞɧɨɦɟɪɧɵɣ ɪɹɞ Ɏɭɪɶɟ
UU
S
() exp[ ]x
n
a
x
n
n
¦
i
, ɬɨ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
U
S
U
n
a
a
a
dx i
n
a
xx
³
1
2
exp[ ] ( )
, ɢɢɡ
ɱɟɬɧɨɫɬɢ
U
U
() ( )x x
ɫɥɟɞɭɟɬ
U
U
() ( )nn
.
7.2.
2. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ (7.37)
ɧɚ Ɏɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɭ
U
1
ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ
UEU EU
11 110 11
22 IvI()()v , ɝɞɟ I
0
ɢ I
1
- ɮɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ
ɦɧɢɦɨɝɨ ɚɪɝɭɦɟɧɬɚ. ɉɨɥɶɡɭɹɫɶ ɚɫɢɦɩɬɨɬɚɦɢ I
0
(x), I
1
(x) ɩɪɢ
x
o
0 ɢ x
of
, ɩɨɥɭɱɢɬɶ ɬɟɦɩɟɪɚɬɭɪɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ
U
1
ɩɪɢ
T
o
0 ( ~exp(/1
1
K
vT))ɢ T
o
T
c
,
(
U
11
21 (/ ),TT T v
cc
).
7.2.
3
. Ⱦɨɤɚɡɚɬɶ ɢɡ ɭɪɚɜɧɟɧɢɹ (7.36) ɧɚ ɮɭɪɶɟ-
ɤɨɦɩɨɧɟɧɬɵ
U
n
, ɱɬɨ ɬɨɱɤɚ ɩɟɪɟɯɨɞɚ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɨɨɬɧɨɲɟ-
ɧɢɸ
UEU
nn
n
v ɩɪɢ ɥɸɛɨɦ n.
ɍɤɚɡɚɧɢɟ
: ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ (ɢ ɞɨɤɚɡɚɬɶ
ɟɝɨ):
dx nx mx
nm n m
cos cos
,,
SSGG
³
1
1
.
____________
128
129
8. ɎȺɁɈȼɕȿ ɉȿɊȿɏɈȾɕ
ɆȿɌȺɅɅ - ɂɁɈɅəɌɈɊ
Ɋɚɫɫɦɨɬɪɢɦ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɜ ɩɟɪɟ-
ɯɨɞɧɵɯ ɦɟɬɚɥɥɚɯ ɢ ɫɩɥɚɜɚɯ. ɗɬɢ ɜɟɳɟɫɬɜɚ ɢɫɩɵɬɵɜɚɸɬ ɩɟɪɟ-
ɯɨɞ ɜ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɫɬɨɹɧɢɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɪɚɡɥɢɱɧɵɯ
ɩɪɢɱɢɧ: ɩɪɢɦɟɫɟɣ, ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɢ ɬ.ɞ. Ɇɵ
ɢɫɫɥɟɞɭɟɦ ɩɟɪɟɯɨɞ Ɇɨɬɬɚ, ɩɟɪɟɯɨɞ Ʉɟɥɞɵɲɚ-Ʉɨɩɚɟɜɚ, ɦɨɞɟɥɶ
Ⱥɧɞɟɪɫɨɧɚ ɢ ɦɨɞɟɥɶ ɏɚɛɛɚɪɞɚ. Ʉɥɸɱɟɜɵɦ ɜ ɩɨɧɢɦɚɧɢɢ ɩɟɪɟ-
ɯɨɞɚ ɢɡ ɦɟɬɚɥɥɢɱɟɫɤɨɝɨ ɩɪɨɜɨɞɹɳɟɝɨ ɫɨɫɬɨɹɧɢɹ ɜ ɞɢɷɥɟɤɬɪɢ-
ɱɟɫɤɨɟ ɫɨɫɬɨɹɧɢɟ ɹɜɥɹɟɬɫɹ ɬɨ, ɱɬɨ ɩɟɪɟɯɨɞ ɩɪɨɢɫɯɨɞɢɬ ɢɡ-ɡɚ
ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤɬɪɨɧɨɜ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɟɪɟɫɬɪɨɣɤɢ ɢɯ ɫɩɟɤɬɪɚ ɜ
ɫɢɥɭ ɪɚɡɥɢɱɧɵɯ ɩɪɢɱɢɧ.
8.1. ɉɟɪɟɯɨɞ Ɇɨɬɬɚ
ɍ ɪɹɞɚ ɩɟɪɟɯɨɞɧɵɯ ɫɨɟɞɢɧɟɧɢɣ ɧɚɛɥɸɞɚɟɬɫɹ ɫɥɟɞɭɸ-
ɳɟɟ: ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɨɢɫɯɨɞɢɬ ɫɤɚɱɨɤ ɩɪɨɜɨɞɢ-
ɦɨɫɬɢ, ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɞɨɫɬɢɝɚɬɶ ɦɧɨɝɢɯ ɩɨɪɹɞɤɨɜ ɜɟɥɢɱɢɧɵ.
ȼɨɡɦɨɠɧɨɟ ɨɛɴɹɫɧɟɧɢɟ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɬɟɦ-
ɩɟɪɚɬɭɪɵ ɜɟɥɢɱɢɧɚ ɩɨɫɬɨɹɧɧɨɣ ɪɟɲɟɬɤɢ ɩɟɪɟɯɨɞɢɬ ɱɟɪɟɡ ɩɨ-
ɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɥɨɤɚɥɢɡɨɜɚɧɧɵɟ ɷɥɟɤɬɪɨɧɵ
ɫɬɚɧɨɜɹɬɫɹ ɞɟɥɨɤɚɥɢɡɨɜɚɧɧɵɦɢ. ɗɬɨɬ ɚɪɝɭɦɟɧɬ, ɜɵɞɜɢɧɭɬɵɣ
Ɇɨɬɬɨɦ, ɦɨɠɟɬ ɨɛɴɹɫɧɢɬɶ ɪɟɡɤɨɟ ɭɜɟɥɢɱɟɧɢɟ ɩɪɨɜɨɞɢɦɨɫɬɢ.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫ ɩɪɢɥɨɠɟɧɢɟɦ ɞɚɜɥɟɧɢɹ ɬɚɤɠɟ ɦɨɠɧɨ ɢɡɦɟ-
ɧɢɬɶ ɩɨɫɬɨɹɧɧɭɸ ɪɟɲɟɬɤɢ, ɬɚɤ ɱɬɨ ɧɚɛɥɸɞɚɟɦɨɟ ɪɟɡɤɨɟ ɭɜɟɥɢ-
ɱɟɧɢɟ ɩɪɨɜɨɞɢɦɨɫɬɢ ɜ ɷɬɢɯ ɫɨɟɞɢɧɟɧɢɹɯ ɩɪɢ ɞɨɫɬɢɠɟɧɢɢ ɧɟ-
ɤɨɝɨ ɤɪɢɬɢɱɟɫɤɨɝɨ ɡɧɚɱɟɧɢɹ ɜɧɟɲɧɟɝɨ ɞɚɜɥɟɧɢɹ ɦɨɠɧɨ ɬɚɤɠɟ
ɨɛɴɹɫɧɢɬɶ ɦɟɯɚɧɢɡɦɨɦ Ɇɨɬɬɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɨɫɬɨɣ ɦɨɞɟɥɢ ɜɨɡɦɨɠɧɨɫɬɶ ɪɟɚɥɢɡɚ-
ɰɢɢ ɩɟɪɟɯɨɞɚ Ɇɨɬɬɚ, ɧɟ ɢɧɬɟɪɟɫɭɹɫɶ ɤɨɧɤɪɟɬɧɵɦ ɫɩɨɫɨɛɨɦ
ɢɡɦɟɧɟɧɢɹ ɩɨɫɬɨɹɧɧɨɣ ɪɟɲɟɬɤɢ (ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɩɥɨɬɧɨɫɬɢ
ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ).