Кашурников В.А., Красавин А.В. Современные проблемы физики твердого тела. Часть 2 - Высокотемпературная сверхпроводимость
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ɑɚɫɬɶ 2. ȼɕɋɈɄɈɌȿɆɉȿɊȺɌɍɊɇȺə ɋȼȿɊɏɉɊɈȼɈȾɂɆɈɋɌɖ
120
q
ɡɚɪɹɞ ɱɟɪɟɡ ɩɨɥɧɵɣ, ɢ ɩɨɥɭɱɚɟɦ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɢ
(ɩɪɨɞɨɥɶɧɨɣ):
).,(/),(1),( ZUZU ZH
oo
qq
totind
ɍɦɧɨɠɢɜ ɮɭɪɶɟ-ɭɪɚɜɧɟɧɢɹ ɫɤɚɥɹɪɧɨ ɧɚ ɢɦɩɭɥɶɫ
o
q , ɜɵɪɚɠɚɟɦ ɢɧɞɭɰɢɪɨɜɚɧɧɵɣ
o
1)( ZH
.
2
2
Z
Z
pl
Ɂɞɟɫɶ ɜɟɥɢɱɢɧɚ
Z
pl
-
ɩɥɚɡɦɟɧɧɚɹ ɱɚɫɬɨɬɚ
:
*4
22
mne
pl
S Z
.
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
7. Ɇɨɞɟɥɶ ɏɚɛɛɚɪɞɚ. Ɉɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ
ȿɳɟ ɜ 1987 ɝɨɞɭ Ⱥɧɞɟɪɫɨɧ ɩɪɟɞɥɨɠɢɥ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ
ɨɛɴɹɫɧɟɧɢɹ ɫɜɨɣɫɬɜ ȼɌɋɉ ɦɨɞɟɥɶ ɏɚɛɛɚɪɞɚ. Ȼɨɥɟɟ ɬɨɝɨ, ɨɧ ɧɟɨɞɧɨ-
ɤɪɚɬɧɨ ɩɨɞɱɟɪɤɢɜɚɥ, ɱɬɨ ɜɫɟ ɢɧɵɟ ɦɨɞɟɥɢ (ɨɫɨɛɟɧɧɨ ɬɪɚɞɢɰɢɨɧɧɚɹ
ɷɥɟɤɬɪɨɧ-ɮɨɧɨɧɧɚɹ) ɤ ȼɌɋɉ ɧɢɤɚɤɨɝɨ ɨɬɧɨɲɟɧɢɹ ɧɟ ɢɦɟɸɬ. Ʉɚɤ ɦɵ
ɨɛɫɭɞɢɦ ɞɚɥɟɟ, ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɨɧɚ ɧɚɢɛɨɥɟɟ ɤɨɪɪɟɤɬɧɨ ɨɩɢɫɵɜɚɟɬ
ɩɪɢɪɨɞɭ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ȼɌɋɉ, ɚ ɡɧɚɱɢɬ, ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɡɧɚ-
ɧɚ ɛɚɡɢɫɧɨɣ, ɨɬɤɭɞɚ ɢ ɫɥɟɞɭɟɬ ɫɬɚɪɬɨɜɚɬɶ ɥɸɛɨɦɭ ɬɟɨɪɟɬɢɱɟɫɤɨɦɭ
ɩɨɞɯɨɞɭ. Ɇɵ ɭɠɟ ɜɢɞɟɥɢ, ɱɬɨ ɩɨɱɬɢ ɜɫɟ ɚɜɬɨɪɵ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜ
ɧɚɲɟɦ ɨɛɡɨɪɟ ɦɨɞɟɥɟɣ ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɦɟɪɟ ɨɩɢɪɚɥɢɫɶ ɧɚ ɩɪɢɛɥɢ-
ɠɟɧɢɟ ɫɢɥɶɧɨɣ ɫɜɹɡɢ, ɫɢɥɶɧɨɟ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ, ɱɬɨ ɢ
ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ, ɢɥɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɟɟ ɢɫɩɨɥɶɡɨ-
ɜɚɥɢ ɧɚ ɩɨɫɥɟɞɧɟɣ ɫɬɚɞɢɢ ɤɨɧɤɪɟɬɢɡɚɰɢɢ ɫɜɨɟɝɨ ɩɨɞɯɨɞɚ ɞɥɹ ɫɨɩɨɫ-
ɬɚɜɥɟɧɢɹ ɫ ɷɤɫɩɟɪɢɦɟɧɬɨɦ.
ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ ɦɵ ɩɨɞɪɨɛɧɟɟ ɪɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ ɏɚɛɛɚɪɞɚ
ɢ ɟɟ ɨɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ, ɬɚɤ ɤɚɤ ɷɬɨ ɧɚ ɫɟɝɨɞɧɹɲɧɢɣ ɞɟɧɶ ɫɚɦɚɹ ɩɨ-
ɩɭɥɹɪɧɚɹ ɦɨɞɟɥɶ ɫɢɥɶɧɨɣ ɫɜɹɡɢ, ɢɫɩɨɥɶɡɭɟɦɚɹ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ
ɦɟɯɚɧɢɡɦɨɜ ɫɩɚɪɢɜɚɧɢɹ ɧɨɫɢɬɟɥɟɣ ɡɚɪɹɞɚ ɜ ȼɌɋɉ.
Ⱦɥɹ ɨɛɴɹɫɧɟɧɢɹ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɦɟɬɚɥɥ–ɢɡɨɥɹɬɨɪ ɜ ɩɟ-
ɪɟɯɨɞɧɵɯ
ɦɟɬɚɥɥɚɯ ɫ ɭɡɤɢɦɢ ɡɨɧɚɦɢ ɏɚɛɛɚɪɞ [84] ɩɪɟɞɥɨɠɢɥ ɦɨ-
ɞɟɥɶ, ɤɨɬɨɪɚɹ ɜ ɪɟɠɢɦɟ ɫɢɥɶɧɨɣ ɫɜɹɡɢ, ɭɱɢɬɵɜɚɹ ɩɟɪɟɫɤɨɤɢ ɷɥɟɤ-
ɬɪɨɧɨɜ ɧɚ ɫɨɫɟɞɧɢɟ ɚɬɨɦɵ ɢ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ ɧɚ ɭɡɥɟ, ɩɨ-
ɡɜɨɥɢɥɚ ɨɩɢɫɚɬɶ ɩɟɪɟɯɨɞ ɢɡ ɩɪɨɜɨɞɹɳɟɝɨ ɜ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɫɬɨɹ-
ɧɢɟ. Ⱦɥɹ ɜɵɜɨɞɚ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞ ɢɫɯɨɞɢɥ ɢɡ ɫɬɚɧɞɚɪɬɧɨɝɨ ɝɚɦɢɥɶ-
ɬɨɧɢɚɧɚ ɮɟɪɦɢ-ɝɚɡɚ ɫ ɤɭɥɨɧɨɜɫɤɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ:
V
c
V
cc
V
c
V
V
c
V
cc
cc
V
V
V
¦¦
H
1221
2121
2121
2
1
ˆ
pppp
pppp
pppp
p
ppp
aaaaVaaH
. (7.1)
Ʉɭɥɨɧɨɜɫɤɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɜ (7.1) ɜɵɩɢɫɚɧɨ ɜ ɧɚɢɛɨɥɟɟ ɨɛɳɟɦ
ɜɢɞɟ. Ɂɞɟɫɶ ɭɱɬɟɧɨ ɢ ɩɪɹɦɨɟ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ, ɢ ɨɛɦɟɧɧɵɟ
ɩɪɨɰɟɫɫɵ. ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫɢɫɬɟɦɵ ɫ ɭɡɤɢɦɢ ɡɨɧɚɦɢ (~0.1–
1 ɷȼ), ɬ.ɟ. ɷɥɟɤɬɪɨɧɵ ɢɦɟɸɬ ɛɨɥɶɲɭɸ ɷɮɮɟɤɬɢɜɧɭɸ ɦɚɫɫɭ ɢ ɩɨɱɬɢ
ɥɨɤɚɥɢɡɨɜɚɧɵ, ɬɨ ɯɨɪɨɲɢɦ ɤɜɚɧɬɨɜɵɦ ɱɢɫɥɨɦ ɹɜɥɹɟɬɫɹ ɧɨɦɟɪ ɭɡɥɚ.
ɉɟɪɟɣɞɟɦ ɨɬ ɢɦɩɭɥɶɫɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ (7.1) ɤ ɭɡɟɥɶɧɵɦ ɨɩɟɪɚɬɨ-
ɪɚɦ:
121
ɑɚɫɬɶ 2. ȼɕɋɈɄɈɌȿɆɉȿɊȺɌɍɊɇȺə ɋȼȿɊɏɉɊɈȼɈȾɂɆɈɋɌɖ
¦
oo
VV
i
iip
rpia
N
a }exp{
1
.
Ɂɞɟɫɶ ɩɨɞ “”ɦɵ ɩɨɧɢɦɚɟɦ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɩɥɨɫɤɢɟ ɜɨɥɧɵ,
ɚ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ȼɚɧɧɶɟ ɜ ɤɪɢɫɬɚɥɥɟ, ɫɨɜɩɚɞɚɸɳɢɟ ɫ ɭɡɟɥɶɧɵɦɢ
ɮɭɧɤɰɢɹɦɢ ɬɨɱɧɨ ɧɚ ɭɡɥɟ ɢ ɢɦɟɸɳɢɟ ɚɫɢɦɩɬɨɬɚɦɢ ɩɥɨɫɤɢɟ ɜɨɥɧɵ
ɜɞɚɥɢ ɨɬ ɚɬɨɦɚ. Ɉɧɢ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɵ ɢ ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɫɢɫɬɟ-
ɦɭ, ɤɚɤ ɢ ɩɥɨɫɤɢɟ ɜɨɥɧɵ. Ɍɨɝɞɚ ɤɢɧɟɬɢɱɟɫɤɚɹ ɱɚɫɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧɚ
ɩɨɫɥɟ ɩɪɨɫɬɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɢɦɟɟɬ ɜɢɞ:
}exp{
oo
i
rpi
./exp~][exp
1
,
ˆ
,
Bji
p
jipij
ji
jiijkin
arrrrpi
N
t
aatH
&&&&&
H
¦
¦
V
Vz
V
V
(7.2)
Ⱥɦɩɥɢɬɭɞɚ ɩɟɪɟɫɤɨɤɚ, ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ ɩɟɪɟɤɪɵɬɢɹ
ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ ɷɥɟɤɬɪɨɧɚ ɧɚ ɭɡɥɚɯ i ɢ j, ɛɵɫɬɪɨ ɡɚɬɭɯɚɟɬ ɜ ɭɡɤɨ-
ɡɨɧɧɨɦ ɜɟɳɟɫɬɜɟ ɫ ɫɢɥɶɧɨ ɥɨɤɚɥɢɡɨɜɚɧɧɵɦɢ ɧɨɫɢɬɟɥɹɦɢ, ɢ ɨɛɵɱɧɨ
ɞɨɫɬɚɬɨɱɧɨ ɭɱɟɫɬɶ ɩɟɪɟɫɤɨɤɢ ɷɥɟɤɬɪɨɧɨɜ ɬɨɥɶɤɨ ɧɚ ɛɥɢɠɚɣɲɢɟ ɫɨ-
ɫɟɞɢ.
ɉɨɞɫɬɚɜɢɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɨɩɟɪɚɬɨɪɨɜ ɤ ɭɡɟɥɶɧɨɦɭ ɜɢɞɭ ɜ
ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɱɚɫɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧɚ, ɩɨɥɭɱɢɦ, ɱɬɨ ɜ ɭɡɟɥɶɧɨɦ
ɩɪɟɞɫɬɚɜɥɟɧɢɢ
ɨɧɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
}].{exp[
1
,
2
1
ˆ
'
1
'
22
''
1''
2
'
''int
2121
2121
o
o
o
o
oooo
VV
V
VV
V
¦
¦
klji
pppp
ppppijkl
ijkl
kljiijkl
rprprprpiV
N
V
aaaaVH
ɉɨɥɚɝɚɹ, ɱɬɨ ɝɥɚɜɧɵɣ ɜɤɥɚɞ ɞɚɟɬ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɧɚ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ
ɭɡɥɚɯ ɫ i=j=k=l (V
ijkl
{V
iiii
=V(|r|=0)=U), ɚ ɧɚ ɛɨɥɶɲɢɯ ɪɚɫɫɬɨɹɧɢɹɯ V
ɦɧɨɝɨ ɦɟɧɶɲɟ, ɩɨɥɭɱɚɟɦ ɨɤɨɧɱɚɬɟɥɶɧɨ ɝɚɦɢɥɶɬɨɧɢɚɧ ɦɨɞɟɥɢ ɏɚɛ-
ɛɚɪɞɚ:
¦¦
V
V
VV
V
Vz
V
V
i
iiii
ji
jiij
aaaUaaatH
2
1
ˆ
,
.
122
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
Ɂɞɟɫɶ ɩɟɪɜɵɣ ɱɥɟɧ ɝɚɦɢɥɶɬɨɧɢɚɧɚ (ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ) ɨɩɢɫɵɜɚ-
ɟɬ ɩɟɪɟɫɤɨɤɢ ɧɚ ɫɨɫɟɞɧɢɟ ɭɡɥɵ ɫ ɚɦɩɥɢɬɭɞɨɣ t
ij
. ȼɬɨɪɨɣ ɱɥɟɧ ɨɩɢɫɵ-
ɜɚɟɬ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ ɧɚ ɭɡɥɟ ɫ ɩɨɬɟɧɰɢɚɥɨɦ U ɢɭɱɢɬɵɜɚ-
ɟɬ, ɱɬɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɚ ɭɡɥɟ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɱɚɫɬɢɰɵ ɬɨɥɶɤɨ ɫ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɫɩɢɧɨɦ. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɡɚɩɢɫɵɜɚɸɬ ɜ ɛɨɥɟɟ
ɭɞɨɛɧɨɣ ɮɨɪɦɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɪɢɛɥɢɠɟɧɢɹ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ
Z (t
ij
=–t, i,j=Z), ɢ ɷɬɨɬ ɜɢɞ ɦɨɞɟɥɢ ɧɚɢɛɨɥɟɟ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɦ:
.
,
ˆ
,
V
VV
pn
Vz
V
V
¦¦
iii
i
ii
ji
ji
aan
nUnaatH
(7.3)
ȼɨ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ ɱɚɫɬɢ (7.3) ɤɨɷɮɮɢɰɢɟɧɬ 1/2 ɫɤɨɦɩɟɧɫɢɪɨ-
ɜɚɥɫɹ ɡɚ ɫɱɟɬ ɫɭɦɦɢɪɨɜɚɧɢɹ ɩɨ ɫɩɢɧɚɦ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɦɨɞɟɥɢ
ɜɫɟɝɨ ɞɜɚ ɩɚɪɚɦɟɬɪɚ: ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɩɟɪɟɫɤɨɤɚ ɧɚ ɫɨɫɟɞɧɢɣ
ɭɡɟɥ t ɢ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ ɧɚ ɭɡɥɟ U. Ȼɨɥɟɟ ɬɨɝɨ, ɟɫɥɢ ɨɬ-
ɫɱɢɬɵɜɚɬɶ ɜɫɟ ɷɧɟɪɝɢɢ ɜ ɟɞɢɧɢɰɚɯ t, ɬɨ ɨɫɬɚɟɬɫɹ ɟɞɢɧɫɬɜɟɧɧɵɣ ɩɚ-
ɪɚɦɟɬɪ
t/U.
ɑɬɨ ɤɚɫɚɟɬɫɹ ȼɌɋɉ, ɬɨ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɪɢ ɞɨɩɢɪɨɜɚ-
ɧɢɢ ɫɜɨɛɨɞɧɵɟ ɞɵɪɤɢ ɛɭɞɭɬ ɜ ɨɫɧɨɜɧɨɦ ɨɬɬɚɥɤɢɜɚɬɶɫɹ ɧɚ ɦɟɞɧɵɯ
ɭɡɥɚɯ, ɢ ɫɨɜɟɪɲɚɬɶ ɷɮɮɟɤɬɢɜɧɵɟ ɩɪɵɠɤɢ ɫ ɦɟɞɢ ɧɚ ɦɟɞɶ ɱɟɪɟɡ ɤɢ-
ɫɥɨɪɨɞ, ɪɟɚɥɢɡɭɹ ɩɪɨɫɬɭɸ ɞɜɭɦɟɪɧɭɸ ɷɮɮɟɤɬɢɜɧɭɸ ɦɨɞɟɥɶ ɏɚɛ-
ɛɚɪɞɚ ɧɚ ɤɜɚɞɪɚɬɧɨɣ ɪɟɲɟɬɤɟ. Ʉɨɧɟɱɧɨ, ɷɬɚ ɭɩɪɨɳɟɧɧɚɹ ɫɯɟɦɚ ɧɟ
ɨɱɟɧɶ ɤɨɪɪɟɤɬɧɚ, ɬɚɤ ɤɚɤ
ɢɡɜɟɫɬɧɨ, ɱɬɨ ɞɨɩɢɪɨɜɚɧɧɵɟ ɞɵɪɤɢ ɫɚɞɹɬɫɹ
ɧɚ ɤɢɫɥɨɪɨɞ. ɉɨɡɠɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɛɨɥɟɟ ɪɟɚɥɢɫɬɢɱɧɭɸ ɦɨɞɟɥɶ
ɗɦɟɪɢ, ɭɱɢɬɵɜɚɸɳɭɸ ɨɫɨɛɟɧɧɨɫɬɢ ɩɥɨɫɤɨɫɬɢ CuO
2
ɛɨɥɟɟ ɤɨɪɪɟɤɬ-
ɧɨ.
Ɂɧɚɤ ɩɟɪɟɫɤɨɤɚ (ɦɢɧɭɫ) ɜɵɛɢɪɚɟɬɫɹ ɢɡ ɭɞɨɛɫɬɜɚ ɨɬɫɱɟɬɚ ɩɨ-
ɥɭɱɚɸɳɢɯɫɹ ɡɨɧ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬ ɰɟɧɬɪɚ ɡɨɧɵ Ȼɪɢɥɥɸɷɧɚ. Ɍɚɤɚɹ
ɜɨɥɶɧɨɫɬɶ ɜ ɜɵɛɨɪɟ ɡɧɚɤɚ ɫɜɹɡɚɧɚ ɫɨ ɫɥɟɞɭɸɳɢɦ ɫɜɨɣɫɬɜɨɦ ɝɚɦɢɥɶ-
ɬɨɧɢɚɧɚ: ɡɚɦɟɧɚ to
–t ɧɟ ɦɟɧɹɟɬ ɫɩɟɤɬɪ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ, ɟɫɥɢ ɩɟ-
ɪɟɫɤɨɤ ɩɪɨɢɫɯɨɞɢɬ ɬɨɥɶɤɨ ɦɟɠɞɭ ɛɥɢɠɚɣɲɢɦɢ ɫɨɫɟɞɹɦɢ (ɫɦ. ɡɚɞɚɱɢ
ɤ ɩɚɪɚɝɪɚɮɭ).
Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ ɬɚɤɠɟ, ɱɬɨ ɝɚɦɢɥɶɬɨɧɢɚɧ ɤɨɦɦɭɬɢɪɭɟɬ ɫ
ɨɩɟɪɚɬɨɪɨɦ ɩɨɥɧɨɝɨ ɱɢɫɥɚ ɱɚɫɬɢɰ N ɢ ɩɨɥɧɨɣ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɧɚ
ɨɫɶ z – S
Z
:
123
ɑɚɫɬɶ 2. ȼɕɋɈɄɈɌȿɆɉȿɊȺɌɍɊɇȺə ɋȼȿɊɏɉɊɈȼɈȾɂɆɈɋɌɖ
¦¦¦
pnp
p
n
n
V
V
V
i
ii
i
i
i
i
i
Z
i
ii
nnaaaaSaaN )(
2
1
)(
2
1
,
, (7.4)
ɬ.ɟ. ɡɚɞɚɱɭ ɦɨɠɧɨ ɪɟɲɚɬɶ ɩɪɢ ɤɨɧɤɪɟɬɧɨɦ ɡɚɩɨɥɧɟɧɢɢ ɢ ɩɪɨɟɤɰɢɢ
ɫɩɢɧɚ ɧɟɡɚɜɢɫɢɦɨ, ɬɚɤ ɤɚɤ ɷɬɢ ɜɟɥɢɱɢɧɵ ɫɨɯɪɚɧɹɸɬɫɹ ɜ ɦɨɞɟɥɢ ɏɚɛ-
ɛɚɪɞɚ.
ɉɭɫɬɶ ɱɢɫɥɨ ɭɡɥɨɜ ɪɚɜɧɨ N
a
. ɉɪɢ ɩɨɥɨɜɢɧɧɨɦ ɡɚɩɨɥɧɟɧɢɢ,
ɬ.ɟ. ɤɨɝɞɚ N=N
a
, ɢ ɛɨɥɶɲɨɦ ɤɭɥɨɧɨɜɫɤɨɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ U/t>>1
ɜɫɟ ɷɥɟɤɬɪɨɧɵ ɩɪɚɤɬɢɱɟɫɤɢ “ɡɚɩɟɪɬɵ” ɧɚ ɫɜɨɢɯ ɭɡɥɚɯ, ɨɛɪɚɡɭɹ ɚɧ-
ɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɟ ɭɩɨɪɹɞɨɱɟɧɢɟ:
npnpnpnpnpn
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɝɚɦɢɥɶɬɨɧɢɚɧ ɏɚɛɛɚɪɞɚ ɷɤɜɢɜɚɥɟɧɬɟɧ ɝɟɣɡɟɧɛɟɪɝɨɜ-
ɫɤɨɦɭ ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɦɭ ɝɚɦɢɥɶɬɨɧɢɚɧɭ (ɫɦ. ɞɚɥɟɟ ɜɵɜɨɞ
t-J-ɦɨɞɟɥɢ ɢɡ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɏɚɛɛɚɪɞɚ).
Ɂɚɦɟɬɢɦ, ɱɬɨ ɦɨɞɟɥɶ ɏɚɛɛɚɪɞɚ ɫɢɦɦɟɬɪɢɱɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɷɥɟɤɬɪɨɧ-ɞɵɪɤɚ (a
+
ĺb, aĺb
+
). ɉɨɷɬɨɦɭ, ɩɨɥɶɡɭɹɫɶ
ɫɜɨɣɫɬɜɨɦ (tĺ–t) ɢ ɷɬɢɦɢ ɩɪɟɨɛɪaɡɨɜɚɧɢɹɦɢ, ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ ɫɥɟ-
ɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ.
ɉɭɫɬɶ Ɇ – ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɜɟɪɯ;
M
c
– ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɧɢɡ,
M+
M
c
=N. Ɍɨɝɞɚ ɢɦɟɸɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ (ɜɩɟɪɜɵɟ
ɩɨɥɭɱɟɧɧɵɟ Ʌɢɛɨɦ ɢ ȼɭ ɜ 1968 ɝ. [85]):
E(M,
M
c
, U)=–(N
a
–M–
M
c
)U+E(N
a
–M, N
a
–
M
c
, U),
E(M,
M
c
, U)=MU+E(M, N
a
–
M
c
, –U), (7.5)
E(M,
M
c
, U)=
M
c
U+E(N
a
–M,
M
c
, –U).
ɉɨɥɭɱɢɦ, ɧɚɩɪɢɦɟɪ, ɩɟɪɜɨɟ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ (7.5). ɉɨɞɫɬɚɜɢɜ ɜ (7.3)
ɞɵɪɨɱɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ a
+
ĺb, aĺb
+
, ɢɦɟɟɦ:
.
,
¦¦¦
¦¦
p
pn
n
p
pn
n
Vz
V
V
ii
iiii
i
i
iiii
ji
ji
bUbbUbU
bbbUbbbtH
(7.6)
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɱɢɫɥɨ ɞɵɪɨɤ ɫɨ ɫɩɢɧɨɦ ɜɜɟɪɯ
,MNbb
a
i
ii
¦
n
n
ɚ ɱɢɫɥɨ ɞɵɪɨɤ ɫɨ ɫɩɢɧɨɦ ɜɧɢɡ
,MNbb
a
i
ii
c
¦
p
p
124
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
ɩɨɥɭɱɚɟɦ ɩɟɪɜɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (7.5) (E=<H>).
ɉɨɥɭɱɢɦ ɫɩɟɤɬɪ ɷɥɟɤɬɪɨɧɨɜ ɜ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ ɜ ɩɪɨɫɬɟɣ-
ɲɟɦ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ, “ɪɚɡɪɟɡɚɜ” ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫɥɟ-
ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
¦¦
V
VV
Vz
V
V
!
i
ii
ji
ji
nUnaatH
,
ˆ
.
ɉɟɪɟɣɞɹ ɜ ɮɭɪɶɟ-ɩɪɟɞɫɬɚɜɥɟɧɢɟ, ɧɚɯɨɞɢɦ:
! HH
VV
V
V
VV
¦
nUakaktaaH
yxk
k
kkk
)cos(cos2;.
Ɇɵ ɜɵɩɢɫɚɥɢ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ
ɢ ɞɥɹ ɞɜɭɦɟɪɧɨɣ ɪɟɲɟɬɤɢ. ɋɪɚɡɭ ɨɬɦɟɬɢɦ ɨɫɨɛɟɧɧɨɫɬɶ ɫɩɟɤɬɪɚ – ɨɧ
ɧɚɩɪɹɦɭɸ ɡɚɜɢɫɢɬ ɨɬ ɫɩɢɧɚ ɱɚɫɬɢɰɵ ɢ ɡɚɩɨɥɧɟɧɢɹ ɩɪɨɬɢɜɨɩɨɥɨɠ-
ɧɨɣ ɩɨ ɫɩɢɧɭ ɩɨɞɫɢɫɬɟɦɵ, ɬ.ɟ. ɞɥɹ ɨɤɨɧɱɚɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ɧɟɨɛɯɨ-
ɞɢɦɨ ɫɚɦɨɫɨɝɥɚɫɨɜɚɬɶ ɷɥɟɤɬɪɨɧɵ ɫ ɪɚɡɧɵɦɢ ɫɩɢɧɚɦɢ. Ɍɚɤɚɹ ɤɨɪɪɟ-
ɥɹɰɢɹ ɩɪɢɫɭɳɚ ɦɨɞɟɥɹɦ ɫ ɫɢɥɶɧɨɣ
ɫɜɹɡɶɸ. ȼɢɞɧɨ ɬɚɤɠɟ, ɱɬɨ ɫɩɟɤɬɪ
– ɬɢɩɢɱɧɵɣ ɞɥɹ ɦɨɞɟɥɢ ɫɢɥɶɧɨɣ ɫɜɹɡɢ, ɨɛɥɚɞɚɸɳɢɣ ɤɚɤ ɹɜɥɟɧɢɟɦ
ɧɟɫɬɢɧɝɚ, ɬɚɤ ɢ ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɦɢ ɨɫɨɛɟɧɧɨɫɬɹɦɢ ɩɥɨɬɧɨɫɬɢ ɫɨ-
ɫɬɨɹɧɢɣ ɜɛɥɢɡɢ ɩɨɥɨɜɢɧɧɨɝɨ ɡɚɩɨɥɧɟɧɢɹ ɡɨɧɵ.
Ɉɛɫɭɞɢɦ, ɤɚɤ ɨɛɴɹɫɧɹɟɬɫɹ ɜ ɪɚɦɤɚɯ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ ɮɚɡɨ-
ɜɵɣ ɩɟɪɟɯɨɞ “ɦɟɬɚɥɥ-ɞɢɷɥɟɤɬɪɢɤ” ɜ ɩɪɨɫɬɟɣɲɟɦ ɩɪɢɛɥɢɠɟɧɢɢ. Ɋɚɫ-
ɫɦɨɬɪɢɦ ɫɧɚɱɚɥɚ ɩɪɟɞɟɥ ɫɢɥɶɧɨɝɨ ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
U>>t. ɉɭɫɬɶ ɢɦɟɟɬɫɹ
ɩɨɥɨɜɢɧɧɨɟ ɡɚɩɨɥɧɟɧɢɟ N=N
a
. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɷɥɟɤɬɪɨɧɵ ɡɚɩɟɪɬɵ ɫɢɥɶɧɵɦ ɤɭɥɨɧɨɜɫɤɢɦ ɨɬɬɚɥɤɢɜɚɧɢɟɦ ɩɨ ɨɞɧɨ-
ɦɭ ɧɚ ɭɡɥɟ, ɬɚɤ ɤɚɤ ɥɸɛɨɟ ɩɟɪɟɞɜɢɠɟɧɢɟ ɩɪɢɜɨɞɢɬ ɤ ɩɨɜɵɲɟɧɢɸ
ɷɧɟɪɝɢɢ ɧɚ U. Ɍɨɝɞɚ ɜ ɝɚɦɢɥɶɬɨɧɢɚɧɟ ɏɚɛɛɚɪɞɚ (7.3) ɦɨɠɧɨ ɩɪɟɧɟɛ-
ɪɟɱɶ ɩɟɪɟɫɤɨɤɨɜɵɦ ɱɥɟɧɨɦ. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɞɢɚɝɨɧɚɥɟɧ, ɢ ɷɧɟɪɝɢɹ
ɪɚɜɧɚ:
)2(})({
02010
UENENnUnnnEE
i
iiii
¦
pnpn
, (7.7)
ɝɞɟ N
1
– ɱɢɫɥɨ ɭɡɥɨɜ ɫ ɨɞɧɢɦ ɷɥɟɤɬɪɨɧɨɦ; N
2
– c ɞɜɭɦɹ. ȼ (7.7) ɦɵ
ɧɟɹɜɧɨ ɞɨɛɚɜɢɥɢ ɨɬɫɱɟɬ ɷɧɟɪɝɢɢ ɨɬ E
0
(ɱɬɨ ɷɤɜɢɜɚɥɟɧɬɧɨ ɞɨɛɚɜɤɟ ɤ
ɝɚɦɢɥɶɬɨɧɢɚɧɭ (7.3) ɱɥɟɧɚ
¦
V
V
V
i
ii
aaE
0
,
125
ɑɚɫɬɶ 2. ȼɕɋɈɄɈɌȿɆɉȿɊȺɌɍɊɇȺə ɋȼȿɊɏɉɊɈȼɈȾɂɆɈɋɌɖ
ɬ.ɟ. E
0
– ɷɧɟɪɝɢɹ, ɧɟɨɛɯɨɞɢɦɚɹ, ɱɬɨɛɵ ɫɜɹɡɚɬɶ ɷɥɟɤɬɪɨɧ ɜ ɢɡɨɥɢɪɨ-
ɜɚɧɧɨɦ ɚɬɨɦɟ; U+E
0
– ɷɧɟɪɝɢɹ, ɧɟɨɛɯɨɞɢɦɚɹ ɞɥɹ ɩɪɢɫɨɟɞɢɧɟɧɢɹ
ɜɬɨɪɨɝɨ ɷɥɟɤɬɪɨɧɚ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜɥɟɧɧɵɦ ɫɩɢɧɨɦ. Ɂɨɧ-
ɧɚɹ ɤɚɪɬɢɧɚ ɬɚɤɨɣ ɫɢɫɬɟɦɵ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɡɨɧ (ɜɵɪɨɠɞɟɧɧɵɯ ɜ
ɥɨɤɚɥɢɡɨɜɚɧɧɵɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɭɪɨɜɧɢ) ɫ ɷɧɟɪɝɢɹɦɢ E
0
ɢ U+E
0
. ȼ
ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ N ɢɦɟɸɳɢɯɫɹ ɷɥɟɤɬɪɨɧɨɜ ɨɛɥɚɞɚɸɬ ɷɧɟɪɝɢɟɣ
E
0
, ɢ ɜ ɤɚɠɞɨɦ ɚɬɨɦɟ ɪɚɡɦɟɳɟɧɨ ɩɨ ɨɞɧɨɦɭ ɷɥɟɤɬɪɨɧɭ (N
1
=N, N
2
=0).
ȼ ɷɬɨɦ ɩɪɟɞɟɥɶɧɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɦɟɫɬɨ ɫɬɪɨɝɚɹ ɥɨɤɚɥɢɡɚɰɢɹ ɷɥɟɤ-
ɬɪɨɧɨɜ.
Ɋɢɫ. 7.1.
ɉɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ ɜ ɦɨɞɟɥɢ
ɏɚɛɛɚɪɞɚ
E
N(E)
ȿɫɥɢ ɦɵ ɧɚɱɢɧɚɟɦ “ɜɤɥɸɱɚɬɶ” ɩɟɪɟɫɤɨɤ t, ɬɨ ɷɬɢ ɥɨɤɚɥɢɡɨ-
ɜɚɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɧɚɱɢɧɚɸɬ ɪɚɫɳɟɩɥɹɬɶɫɹ ɜ ɡɨɧɵ ɲɢɪɢɧɨɣ ɩɨɪɹɞɤɚ
2Zt, ɝɞɟ Z – ɱɢɫɥɨ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ (ɷɬɨ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɧɢɠɧɹɹ
ɢ ɜɟɪɯɧɹɹ ɏɚɛɛɚɪɞɨɜɫɤɢɟ ɡɨɧɵ, ɪɢɫ. 7.1). ɉɨɤɚ ɡɨɧɵ ɧɟ ɫɥɢɥɢɫɶ,
ɜɟɳɟɫɬɜɨ ɨɫɬɚɟɬɫɹ ɞɢɷɥɟɤɬɪɢɤɨɦ. ɉɪɢ ɤɪɢɬɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɟ ɪɚɫ-
ɳɟɩɥɟɧɢɹ (ɤɪɢɬɢɱɟɫɤɨɦ ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɚ t/U~Z), ɨɩɪɟɞɟɥɹɟɦɨɝɨ
ɩɨɫɬɨɹɧɧɨɣ ɪɟɲɟɬɤɢ (ɱɟɪɟɡ t) ɢ ɤɭɥɨɧɨɜɫɤɢɦ ɭɡɟɥɶɧɵɦ ɜɡɚɢɦɨɞɟɣ-
ɫɬɜɢɟɦ, ɳɟɥɶ ɢɫɱɟɡɚɟɬ. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɦɟɫɬɨ ɩɟɪɟɯɨɞ ɨɬ ɥɨ-
ɤɚɥɢɡɨɜɚɧɧɨɝɨ ɨɩɢɫɚɧɢɹ ɤ ɡɨɧɧɨɣ ɦɨɞɟɥɢ, ɬ.ɟ. ɨɬ ɞɢɷɥɟɤɬɪɢɤɚ ɤ ɦɟ-
ɬɚɥɥɭ. ɋɚɦ ɏɚɛɛɚɪɞ ɜ 1963 ɢ 1964 ɝɝ. ɩɪɨɜɟɥ ɚɧɚɥɢɡ ɡɨɧɧɨɣ ɤɚɪɬɢɧɵ
ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ (ɷɬɢ ɩɨɞɯɨɞɵ ɬɚɤ ɢ ɧɚɡɵɜɚɸɬ: ɩɪɢ-
ɛɥɢɠɟɧɢɟ
“ɏɚɛɛɚɪɞ-1”, “ɏɚɛɛɚɪɞ-3” ɢɬ.ɞ.), ɢ ɩɨɤɚɡɚɥ, ɱɬɨ ɫ ɢɡɦɟ-
ɧɟɧɢɟɦ ɤɨɧɰɟɧɬɪɚɰɢɢ ɧɨɫɢɬɟɥɟɣ ɫɚɦ ɜɢɞ ɡɨɧɧɨɣ ɤɚɪɬɢɧɵ ɧɟɫɤɨɥɶɤɨ
ɢɫɤɚɠɚɟɬɫɹ, ɬ.ɟ. ɜ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ, ɫɬɪɨɝɨ ɝɨɜɨɪɹ, ɧɟ ɩɪɢɦɟɧɢɦɨ
ɩɪɢɛɥɢɠɟɧɢɟ “ɠɟɫɬɤɢɯ ɡɨɧ”.
126
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ ɜɨɡɦɨɠɟɧ ɮɚɡɨɜɵɣ ɩɟɪɟ-
ɯɨɞ ɦɟɬɚɥɥ – ɞɢɷɥɟɤɬɪɢɤ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɚɦɩɥɢɬɭɞɵ
ɩɟɪɟɫɤɨɤɚ t ɢ ɤɭɥɨɧɨɜɫɤɨɝɨ ɨɬɬɚɥɤɢɜɚɧɢɹ U. Ⱥɦɩɥɢɬɭɞɚ ɩɟɪɟɫɤɨɤɚ
ɱɭɜɫɬɜɢɬɟɥɶɧɚ ɤ ɩɨɫɬɨɹɧɧɨɣ ɪɟɲɟɬɤɢ a, ɬɚɤ ɤɚɤ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ
(7.2) ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ
B
aat /exp~
, ɩɨɷɬɨɦɭ
ɩɟɪɟɯɨɞ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ, ɧɚɩɪɢɦɟɪ, ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɜɧɟɲɧɟɝɨ
ɩɪɢɥɨɠɟɧɧɨɝɨ ɞɚɜɥɟɧɢɹ, ɢɥɢ ɢɡ-ɡɚ ɫɬɪɭɤɬɭɪɧɨɣ ɩɟɪɟɫɬɪɨɣɤɢ ɫ ɢɡ-
ɦɟɧɟɧɢɟɦ ɩɨɫɬɨɹɧɧɨɣ ɪɟɲɟɬɤɢ, ɢɥɢ ɢɡ-ɡɚ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɪɚɫɲɢɪɟ-
ɧɢɹ.
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɜ
ɨɞɧɨɦɟɪɧɨɣ
ɫɢɬɭɚɰɢɢ ɩɨɥɭɱɟɧɨ
ɬɨɱɧɨɟ
ɪɟɲɟɧɢɟ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ [85]. Ɉɫɧɨɜɧɨɣ ɜɵɜɨɞ – ɨɬɫɭɬɫɬ-
ɜɢɟ ɩɟɪɟɯɨɞɚ ɦɟɬɚɥɥ–ɞɢɷɥɟɤɬɪɢɤ ɩɪɢ ɥɸɛɨɦ ɩɚɪɚɦɟɬɪɟ U/t>0. ɉɨ
-
ɡɧɚɤɨɦɢɦɫɹ ɫ ɷɬɢɦ ɪɟɲɟɧɢɟɦ.
ȼ 1968ɝ. Ʌɢɛ ɢ ȼɭ (Lieb, Wu) ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɥɢ
ɬɨɱɧɨɟ
ɪɟɲɟɧɢɟ ɨɞɧɨɦɟɪɧɨɣ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ. Ɉɧɢ ɢɫɫɥɟɞɨɜɚɥɢ ɦɨɞɟɥɶ
ɦɟɬɨɞɨɦ ɚɧɡɚɰ-Ȼɟɬɟ, ɨɫɧɨɜɵɜɚɹɫɶ ɧɚ ɪɚɧɟɟ ɪɟɲɟɧɧɨɣ əɧɝɨɦ ɩɪɨ-
ɛɥɟɦɟ ɨɞɧɨɦɟɪɧɨɣ ɫɢɫɬɟɦɵ ɦɧɨɝɢɯ ɱɚɫɬɢɰ ɫ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɨɧɧɵɦ
ɨɬɬɚɥɤɢɜɚɧɢɟɦ ɢ ɧɚ ɤɜɚɧɬɨɜɨɦ ɦɟɬɨɞɟ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ ɪɚɫɫɟɹɧɢɹ
(ɄɆɈɁ). Ȼɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɨɞɧɨɦɟɪɧɚɹ ɫɢɫɬɟɦɚ,
ɨɩɢɫɵɜɚɟɦɚɹ
ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɏɚɛɛɚɪɞɚ, ɧɟ ɩɟɪɟɯɨɞɢɬ ɜ ɦɟɬɚɥɥɢɱɟɫɤɨɟ ɫɨɫɬɨɹɧɢɟ
ɧɢ ɩɪɢ ɤɚɤɨɦ ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɚ t/U, ɬ.ɟ. ɜɫɟɝɞɚ ɹɜɥɹɟɬɫɹ
ɞɢɷɥɟɤɬɪɢɤɨɦ.
Ƚɚɦɢɥɶɬɨɧɢɚɧ ɨɞɧɨɦɟɪɧɨɣ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɏɚɛɛɚɪɞɨɜɫɤɨɣ
ɰɟɩɨɱɤɢ ɫ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɬɨɥɶɤɨ ɫ ɛɥɢɠɚɣɲɢɦɢ ɫɨɫɟɞɹɦɢ ɜɵɩɢ-
ɲɟɦ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
¦¦
pn
V
V
VV
V
i
ii
i
iiii
nUnaaaatH
,
11
)(
ˆ
. (7.8)
ɍɱɢɬɵɜɚɹ ɫɨɨɬɧɨɲɟɧɢɹ (7.5), ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɦɟɧɟɟ
ɱɟɦ ɧɚɩɨɥɨɜɢɧɭ ɡɚɩɨɥɧɟɧɧɭɸ ɡɨɧɭ. Ɇɵ ɧɟ ɛɭɞɟɦ ɩɨɞɪɨɛɧɨ ɨɩɢɫɵ-
ɜɚɬɶ ɝɪɨɦɨɡɞɤɢɟ ɜɵɱɢɫɥɟɧɢɹ ɢ ɢɫɫɥɟɞɭɟɦ ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɪɟɡɭɥɶɬɚɬ
(ɞɨɫɬɚɬɨɱɧɨ ɩɨɞɪɨɛɧɨ ɪɟɲɟɧɢɟ Ʌɢɛɚ ɢ ȼɭ ɢɡɥɨɠɟɧɨ ɜ ɤɧɢɝɟ [3]).
Ɉɤɨɧɱɚɬɟɥɶɧɵɣ ɨɬɜɟɬ ɜ ɢɧɬɟɝɪɚɥɶɧɨɦ ɜɢɞɟ ɢɦɟɬ ɜɢɞ:
127
ɑɚɫɬɶ 2. ȼɕɋɈɄɈɌȿɆɉȿɊȺɌɍɊɇȺə ɋȼȿɊɏɉɊɈȼɈȾɂɆɈɋɌɖ
³
³
³
O
c
O
O
c
VO
c
OSV
O
U
O
OOV SU
B
B
Q
Q
B
B
U
U
d
pU
U
pdp
pU
U
dp
.
)(4
4
)(
)(2
)sin(16
8
)(2
;
)sin(16
8
)(1)(2
22
22
22
(7.9)
(U=U/t). ȼɯɨɞɹɳɢɟ ɜ ɩɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɟɥɢɱɢɧɵ Q ɢ B ɨɩɪɟ-
ɞɟɥɹɸɬɫɹ ɧɨɪɦɢɪɨɜɤɨɣ ɧɚ ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɜɟɪɯ M ɢ ɩɨɥɧɨɟ ɱɢɫɥɨ
ɷɥɟɤɬɪɨɧɨɜ N:
a
Q
Q
a
B
B
NNpdpNMd /)(,/)( U OOV
³³
. (7.10)
ȼ ɬɚɤɨɦ ɤɨɧɬɢɧɭɚɥɶɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɷɧɟɪɝɢɹ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹ-
ɧɢɹ ɫɢɫɬɟɦɵ
³
U
Q
Q
a
ppdpNE cos)(2
0
. (7.11)
ȿɫɬɟɫɬɜɟɧɧɨ, ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɨɬ M ɢ N ɱɟɪɟɡ ɩɚɪɚ-
ɦɟɬɪ Q ɫɭɱɟɬɨɦ (7.10), ɬɚɤ ɱɬɨ
E
0
=E
0
(M,
M
c
, U),
M
c
=N–M.
Ʌɢɛ ɢ Ɇɚɬɬɢɫ ɞɨɤɚɡɚɥɢ, ɱɬɨ ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ ɮɟɪɦɢɨɧ-
ɧɨɣ ɫɢɫɬɟɦɵ – ɫɢɧɝɥɟɬɧɨɟ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɫɟ ɱɢɫɥɚ O
ɡɚɩɨɥɧɹɸɬ
ɜɟɫɶ ɢɧɬɟɪɜɚɥ (–f,+f, ɬ.ɟ. ɜɫɸɞɭ ɜ ɢɧɬɟɝɪɚɥɚɯ ɧɚɞɨ ɩɨɥɨɠɢɬɶ
B=f, ɚ ɢɫɯɨɞɹ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ (7.10) ɞɥɹ O ɢ p ɫɥɟɞɭɟɬ ɬɚɤɠɟ, ɱɬɨ
Q=S. Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɹ (7.9) ɦɨɠɧɨ ɪɟɲɢɬɶ ɦɟɬɨɞɨɦ Ɏɭɪɶɟ ɢ ɩɨɥɭ-
ɱɢɬɶ
.
)2exp(1
)()sincos(
cos
1
2
1
)(
,
)4exp()4exp(
)()cos(
1
)(
0
0
0
0
³
³
f
f
Z
Z
ZZ
S
S
U
Z
ZZ
ZZO
S
OV
d
U
Jp
pp
d
UU
J
(7.12)
128
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
Ⱦɥɹ ɫɢɧɝɥɟɬɧɨɝɨ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ (M=
M
c
=N
a
/2) ɷɧɟɪɝɢɸ ɦɨɠ-
ɧɨ ɪɚɫɫɱɢɬɚɬɶ ɢɡ (7.11):
³
f
Z
ZZ
ZZ
0
10
0
)]2exp(1[
)()(
4),
2
1
,
2
1
( d
U
JJ
NUNNE
aaa
. (7.13)
ȼ ɮɨɪɦɭɥɚɯ (7.12)–(7.13) J
0
ɢ J
1
– ɮɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ ɧɭɥɟɜɨɝɨ ɢ ɩɟɪ-
ɜɨɝɨ ɩɨɪɹɞɤɚ.
ɂɫɫɥɟɞɭɟɦ ɬɟɩɟɪɶ ɫ ɩɨɦɨɳɶɸ ɩɨɥɭɱɟɧɧɵɯ ɬɨɱɧɵɯ ɪɟɡɭɥɶ-
ɬɚɬɨɜ, ɹɜɥɹɟɬɫɹ ɥɢ ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ ɩɪɨɜɨɞɹɳɢɦ ɢɥɢ ɞɢɷɥɟɤɬɪɢ-
ɱɟɫɤɢɦ. Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ ɫɥɟɞɭɸɳɢɟ ɯɢɦɢɱɟɫɤɢɟ
ɩɨɬɟɧɰɢɚɥɵ (M=N
a
/2):
).,,1(),,(
),,,(),,1(
00
00
UMMEUMME
UMMEUMME
P
P
(7.14)
ȼ ɨɞɧɨɱɚɫɬɢɱɧɨɦ ɫɩɟɤɬɪɟ ɜɟɥɢɱɢɧɚ P
+
–P
–
ɹɜɥɹɟɬɫɹ ɳɟɥɶɸ ɦɟɠɞɭ
ɜɟɪɯɧɟɣ ɢ ɧɢɠɧɟɣ ɯɚɛɛɚɪɞɨɜɫɤɢɦɢ ɩɨɞɡɨɧɚɦɢ (ɫɦ. ɪɢɫ. 7.1). ɉɨɷɬɨ-
ɦɭ ɩɪɢ ɩɨɥɨɜɢɧɧɨɦ ɡɚɩɨɥɧɟɧɢɢ (ɡɚɩɨɥɧɟɧɚ ɧɢɠɧɹɹ ɩɨɞɡɨɧɚ), ɟɫɥɢ
P
+
>P
–
, ɬɨ ɫɨɫɬɨɹɧɢɟ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɟ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɦɟɬɚɥ-
ɥɢɱɟɫɤɨɟ. ɂɡ ɪɟɡɭɥɶɬɚɬɨɜ (7.12)–(7.13) ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ:
.]
2
1
)
4
1
1[()1(4
)]2exp(1[
)(
42
,
1
2/122
0
1
¦
³
f
f
Z
ZZ
Z
P
P P
n
n
nUUn
d
U
J
U
(7.15)
ɂɡ (7.15) ɜ ɩɪɟɞɟɥɶɧɵɯ ɫɥɭɱɚɹɯ ɩɨɥɭɱɚɟɦ:
°
°
¯
°
°
®
oo
S
S
!!fo
PP
¦
f
.0,0
)12(
]}12[/4exp{
2
4
;1,2ln
16
4
0
3
U
n
nUU
U
U
U
n
(7.16)
129
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, P
+
>P
–
, ɬ.ɟ. ɨɞɧɨɦɟɪɧɚɹ ɦɨɞɟɥɶ ɏɚɛɛɚɪɞɚ
ɜɫɟɝɞɚ ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɢɡɨɥɹɬɨɪɚ. ɗɬɨ ɫɨɫɬɨɹɧɢɟ, ɤɨɝɞɚ ɢɦɟ-
ɸɬɫɹ ɞɜɟ ɩɨɞɡɨɧɵ, ɪɚɫɳɟɩɥɟɧɧɵɟ ɤɭɥɨɧɨɜɫɤɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ U
ɢ ɫ ɡɚɩɨɥɧɟɧɧɨɣ ɧɢɠɧɟɣ ɩɨɞɡɨɧɨɣ, ɧɚɡɵɜɚɸɬ “ɯɚɛɛɚɪɞɨɜɫɤɢɣ ɞɢ-
ɷɥɟɤɬɪɢɤ”.