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ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
ɞɵɜɚɸɬ
U
XY
ɨɬ H ɢ ɜɢɞɹɬ ɫɬɭɩɟɧɶɤɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɟɪɟɯɨɞɭ ɨɬ ɨɞ-
ɧɨɝɨ ɩɨɞɭɪɨɜɧɹ Ʌɚɧɞɚɭ ɤ ɞɪɭɝɨɦɭ.
3. Ⱦɪɨɛɧɵɣ ɤɜɚɧɬɨɜɵɣ ɷɮɮɟɤɬ ɏɨɥɥɚ. ɂɫɬɨɪɢɹ ɨɬɤɪɵ-
ɬɢɹ ɢ ɫɨɜɪɟɦɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ. ɋɢɫɬɟɦɚ
ɭɪɨɜɧɟɣ ɜ ɩɟɪɜɨɣ ɡɨɧɟ Ʌɚɧɞɚɭ. Ʌɚɮɥɢɧɨɜɫɤɚɹ ɠɢɞ-
ɤɨɫɬɶ ɤɚɤ ɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɞɜɭɦɟɪɧɨɝɨ ɷɥɟɤɬɪɨɧɧɨɝɨ
ɝɚɡɚ. ȼɨɡɛɭɠɞɟɧɢɹ ɫ ɞɪɨɛɧɵɦ ɡɚɪɹɞɨɦ.
Ⱦɚɥɶɧɟɣɲɟɟ ɪɚɡɜɢɬɢɟ ɬɟɯɧɨɥɨɝɢɢ ɩɨɡɜɨɥɢɥɨ ɫɞɟɥɚɬɶ ɝɟɬɟ-
ɪɨɫɬɪɭɤɬɭɪɵ ɧɚ ɨɫɧɨɜɟ ɚɪɫɟɧɢɞɚ ɝɚɥɥɢɹ ɫ ɩɟɪɢɨɞɨɦ (ɩɨɩɟ-
ɪɟɱɧɵɦ ɪɚɡɦɟɪɨɦ, ɪɚɜɧɵɦ ɪɚɡɦɟɪɭ ɨɞɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
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ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ
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ɹɦɵ) d < 100 ɚɧɝɫɬɪɟɦ, ɱɬɨ ɩɨɡɜɨɥɢɥɨ ɢɫɫɥɟɞɨɜɚɬɶ ɞɜɭɦɟɪ-
ɧɵɣ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ ɩɪɢ ɦɚɝɧɢɬɧɵɯ ɩɨɥɹɯ H > 10 Ɍɥ. ɗɬɨ,
ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɩɨɡɜɨɥɹɟɬ ɢɫɫɥɟɞɨɜɚɬɶ ɤɨɧɰɟɧɬɪɚɰɢɸ n <
1/2
S
l
2
H
, ɬ.ɟ. ɤɨɝɞɚ ɧɟ ɡɚɥɢɬɚ ɞɚɠɟ ɩɟɪɜɚɹ ɡɨɧɚ Ʌɚɧɞɚɭ ( n ~
10
11
ɫɦ
-2
).
ȼ 1982 ɝɨɞɭ Ⱦ.ɋ.Ɍɫɭɢ, ɏ.Ʌ.ɒɬɟɪɦɟɪ ɢ Ⱥ.ɋ.Ƚɨɫɫɚɪɞ ɨɛ-
ɧɚɪɭɠɢɥɢ, ɱɬɨ ɩɪɢ ɡɚɩɨɥɧɟɧɢɢ
P
= 2
S
l
H
2
n = 1/3, ɬ.ɟ. “ɡɚɥɢɜ-
ɤɟ” ɪɨɜɧɨ 1/3 ɩɟɪɜɨɣ ɡɨɧɵ Ʌɚɧɞɚɭ ɧɚ ɡɚɜɢɫɢɦɨɫɬɢ
U
XY
(H)
ɢɦɟɟɬɫɹ ɩɥɚɬɨ, ɩɪɢɱɟɦ ɜɟɥɢɱɢɧɚ
U
XY
= ɩɪɢ
P
=1/3 !
ɇɚɛɥɸɞɚɥɨɫɶ ɷɬɨ ɧɚ ɫɬɪɭɤɬɭɪɟ GaAs - Al
PS
2
/2 e!
0.3
Ga
0.7
As ɩɪɢ ɩɨ-
ɥɹɯ H ~ 10 Ɍɥ, ɩɪɢ ɝɟɥɢɟɜɵɯ ɬɟɦɩɟɪɚɬɭɪɚɯ (T=5K), ɩɪɢɱɟɦ
ɞɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɷɥɟɤɬɪɨɧɨɜ n > 10
11
ɫɦ
-2
. ɗɬɨ ɛɵɥɨ
ɫɨɜɟɪɲɟɧɧɨ ɧɟɩɨɧɹɬɧɨ, ɬ.ɤ. ɧɢɤɚɤɨɝɨ ɤɜɚɧɬɨɜɚɧɢɹ ɜɧɭɬɪɢ
ɩɟɪɜɨɣ ɡɨɧɵ Ʌɚɧɞɚɭ ɜɪɨɞɟ ɛɵɬɶ ɧɟ ɞɨɥɠɧɨ. Ɉɞɧɚɤɨ ɜ ɞɚɥɶ-
ɧɟɣɲɟɦ ɷɬɢ ɪɟɡɭɥɶɬɚɬɵ ɩɨɞɬɜɟɪɞɢɥɢɫɶ (D.Yochioka and
P.A.Lee, 1983). ɉɨɫɥɟ ɩɪɟɞɫɤɚɡɚɧɢɹ Ʌɚɮɥɢɧɵɦ (ɫɦ. ɧɢɠɟ)
ɧɚɛɥɸɞɟɧɢɹ ɷɬɨɬ ɷɮɮɟɤɬ (ɞɪɨɛɧɵɣ ɤɜɚɧɬɨɜɵɣ ɷɮɮɟɤɬ ɏɨɥ-
ɥɚ FQHE) ɢ ɩɪɢ ɞɪɭɝɢɯ ɮɚɤɬɨɪɚɯ ɡɚɩɨɥɧɟɧɢɹ
P
= 1/5, 1/7,
ɨɧ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɧɚɛɥɸɞɚɥɫɹ ɩɪɢ
P
=1/5, ɚ ɩɨɡɞɧɟɟ, ɩɨɫɥɟ
ɪɚɫɲɢɪɟɧɢɹ ɏɨɥɞɟɣɧɨɦ (Haldane, 1983) ɢɟɪɚɪɯɢɢ ɱɢɫɟɥ
ɡɚɩɨɥɧɟɧɢɹ, ɢ ɩɪɢ
P
=1/7, 2/7, 1/3, 2/3, 4/3, 5/3, 7/3, 2/9, 2/11...
(ɪɢɫ. 3.1). ( ɧɚɩɪɢɦɟɪ, H.V.Jiang, R.L.Willett, H.L.Stormer,
D.C.Tsui (1990) FQHE ɩɪɢ
P
=1/5, 1/7, 2/7, 1/3, 2/3, 2/9;
V.J.Goldman, J.K. Wang, Bo Su, M.Hajegai (1993) FQHE
ɩɪɢ
P
=2/5, 1/3, 2/7, 3/11, 2/9, 1/5). Ɉɞɧɨɜɪɟɦɟɧɧɨ ɛɵɥɨ ɨɬ-
ɦɟɱɟɧɨ, ɱɬɨ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɟ
U
XX
ɨɱɟɧɶ ɦɚɥɨ ɩɪɢ ɪɟɚ-
ɥɢɡɚɰɢɢ ɫɬɭɩɟɧɟɤ FQHE (ɪɢɫ.3.1), ɩɪɢ ɷɬɨɦ ɬɟɦɩɟɪɚɬɭɪɧɭɸ
ɡɚɜɢɫɢɦɨɫɬɶ ɦɨɠɧɨ ɤɚɱɟɫɬɜɟɧɧɨ ɨɩɢɫɚɬɶ ɜ ɜɢɞɟ
U
XX
~ exp (-
'
/T), ɝɞɟ
'
~5K (ɬɚɤ, H.V.Jiang, R.L.Willett, H.L.Stormer,
D.C.Tsui (1990) ɧɚɛɥɸɞɚɥɢ ɷɬɨ ɩɪɢ
P
=1/5). Ʉɪɨɦɟ ɬɨɝɨ, ɜ
ɩɨɫɥɟɞɧɟɟ ɜɪɟɦɹ ɷɤɫɩɟɪɢɦɟɧɬɚɬɨɪɵ ɜɢɞɹɬ ɪɨɫɬ ɫ ɬɟɦɩɟɪɚɬɭ-
ɪɨɣ ɷɬɨɣ ɜɟɥɢɱɢɧɵ ɜɛɥɢɡɢ
P
=1/3, 1/5 (ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɚɤɬɢ-
ɜɚɰɢɨɧɧɵɣ ɡɚɤɨɧ
U
XX
~ exp (Eg/T), ɝɞɟ Eg ~ 1Ʉ - ɷɧɟɪɝɢɹ
ɚɤɬɢɜɚɰɢɢ).
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ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
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Ɋɢɫ. 3.1.
ɉɨɩɟɪɟɱɧɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ
U
XY
ɢ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟ-
ɧɢɟ
U
XX
ɜ ɞɪɨɛɧɨɦ ɤɜɚɧɬɨ-
ɜɨɦ ɷɮɮɟɤɬɟ ɏɨɥɥɚ
Ɍɚɤɨɟ ɧɟɨɛɵɱɧɨɟ ɩɨɜɟɞɟɧɢɟ ɞɜ ɟɪɧɨɣ ɷɥɟɤɬɪɨɧɧɨɣ
ɡ
ɬɦ
ɭɦ
ɫɢɫɬɟɦɵ ɜ ɫɢɥɶɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɧɟ ɨɫɬɚɥɨɫɶ ɛɟɡ ɜɧɢɦɚ-
ɧɢɹ ɬɟɨɪɟɬɢɤɨɜ. ȼ 1983 ɝɨɞɭ Ɋ.ȼ. Ʌɚɮɥɢɧ ɩɪɟɞɩɨɥɨɠɢɥ, ɱɬɨ
ɨɩɢɫɚɬɶ ɷɥɟɤɬɪɨɧɵ ɜ ɞɜɭɦɟɪɧɨɦ ɫɥɨɟ ɜ ɭɫɥɨɜɢɹɯ ɧɟɩɨɥɧɨɣ
ɡɚɥɢɜɤɢ ɭɪɨɜɧɹ Ʌɚɧɞɚɭ ɧɟɥɶɡɹ ɛɟɡ ɭɱɟɬɚ ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢ-
ɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɧɢɦɢ. Ɉɧ ɩɨɤɚɡɚɥ, ɱɬɨ ɜɫɥɟɞɫɬɜɢɟ
ɫɢɥɶɧɨɝɨ ɤɭɥɨɧɨɜɫɤɨɝɨ ɨɬɬɚɥɤɢɜɚɧɢɹ ɨɧɢ ɨɛɪɚɡɭɸɬ ɧɟɫɠɢ-
ɦɚɟɦɭɸ ɤɜɚɧɬɨɜɭɸ ɠɢɞɤɨɫɬɶ ɠɢɞɤɨɫɬɶ Ʌɚɮɥɢɧɚ. Ɉɧ ɩɨ-
ɫɬɪɨɢɥ ɜɚɪɢɚɰɢɨɧɧɭɸ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ ɷɥɟɤɬɪɨɧɨɜ ɢ
ɩɨɤɚɡɚɥ ɮɨɪɦɚɥɶɧɭɸ ɚɧɚɥɨɝɢɸ ɦɟɠɞɭ ɷɬɨɣ ɫɢɫɬɟɦɨɣ ɢ ɨɞ-
ɧɨɤɨɦɩɨɧɟɧɬɧɨɣ ɞɜɭɦɟɪɧɨɣ ɩɥɚɡɦɨɣ, ɜ ɤɨɬɨɪɨɣ ɜɡɚɢɦɨɞɟɣ-
ɫɬɜɢɟ ɦɟɠɞɭ ɡɚɪɹɠɟɧɧɵɦɢ ɱɚɫɬɢɰɚɦɢ ɨɩɢɫɵɜɚɟɬɫɹ
ɩɨɬɟɧɰɢɚɥɨɦ, ɥɨɝɚɪɢɮɦɢɱɟɫɤɢ ɚɜɢɫɹɳɢɦ ɨɬ ɪɚɫɫɬɨɹɧɢɹ
ɦɟɠɞɭ ɱɚɫɬɢɰɚɦɢ. Ɍɚɤ ɤɚɤ ɤ ɷɬɨɦɭ ɜɪɟɦɟɧɢ ɩɥɚɡɦɟɧɧɵɟ
ɦɨɞɟɥɢ ɛɵɥɢ ɩɨɞɪɨɛɧɨ ɢɡɭɱɟɧɵ, Ʌɚɮɥɢɧ ɜɨɫɩɨɥɶɡɨɜɚɥɫɹ
ɢɡɜɟɫɬɧɵɦɢ ɪɟɡɭɥɶɬɚɬɚɦɢ, ɢ ɫɮɨɪɦɭɥɢɪɨɜɚɥ ɭɫɥɨɜɢɟ
ɦɢɧɢɦɭɦɚ ɷɧɟɪɝɢɢ ɬɚɤɨɣ ɫɢɫ ɟɦɵ. Ɉɤɚɡɚɥɨɫɶ, ɱɬɨ ɢɧɢɦɭɦ
33
ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
ɷɧɟɪɝɢɢ ɬɚɤɨɣ ɫɢɫɬɟɦɵ. Ɉɤɚɡɚɥɨɫɶ, ɱɬɨ ɦɢɧɢɦɭɦ ɞɨɫɬɢɝɚɟɬ-
ɫɹ ɩɪɢ ɮɚɤɬɨɪɟ ɡɚɩɨɥɧɟɧɢɹ
P
=1/m, ɝɞɟ m ɧɟɱɟɬɧɨɟ (ɧɟɱɟɬ-
ɧɨɫɬɶ ɢɡ-ɡɚ ɚɧɬɢɫɢɦɦɟɬɪɢɢ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɷɥɟɤɬɪɨ-
ɧɨɜ). ɉɨɡɠɟ ɏɨɥɞɟɣɧ (F.D.M.Haldane) ɪɚɫɲɢɪɢɥ ɢɟɪɚɪɯɢɸ
ɱɢɫɟɥ ɡɚɩɨɥɧɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɧɚɛɥɸɞɚɟɬɫɹ FQHE, ɬɚɤ ɱɬɨ
ɷɬɨɬ ɷɮɮɟɤɬ ɦɨɠɟɬ ɪɟɚɥɢɡɨɜɚɬɶɫɹ ɩɪɢ
mp /
P , p ɰɟɥɨɟ,
m ɧɟɱɟɬɧɨɟ (ɚ ɬɚɤɠɟ ɩɪɢ
p1 m/ P ɢɡ-ɡ
ɜɢɞɟɬɶ, ɱɬɨ
ɬɚɤɨɟ ɫɜɨɟɨɛ-
ɚ ɷɥɟɤɬɪɨɧ-
ɞɵɪɨɱɧɨɣ ɫɢɦɦɟɬɪɢɢ). Ʌɟɝɤɨ ɜɫɟ ɭɩɨɦɹɧɭɬɵɟ
ɜɵɲɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɧɚɛɥɸɞɚɟɦɵɟ ɮɚɤɬɨɪɵ ɡɚɩɨɥɧɟɧɢɹ
FQHE ɧɟ ɩɪɨɬɢɜɨɪɟɱɚɬ ɷɬɢɦ ɫɨɨɬɧɨɲɟɧɢɹɦ.
Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɩɨɧɹɬɶ, ɤɚɤ ɩɪɨɢɫɯɨɞɢɬ
ɪɚɡɧɨɟ ɤɜɚɧɬɨɜɚɧɢɟ ɜɧɭɬɪɢ ɩɟɪɜɨɣ ɡɨɧɵ Ʌɚɧɞɚɭ, ɪɚɫɫɦɨɬ-
ɪɢɦ ɜ ɞɜɭɦɟɪɧɨɦ ɫɥɭɱɚɟ ɷɥɟɤɬɪɨɧ ɜ ɫɢɥɶɧɨɦ ɩɨɩɟɪɟɱɧɨɦ
ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɛɨɥɟɟ ɭɞɨɛɧɨ ɜɵɛɪɚɬɶ ɧɟ
ɬɨɪɨɢɞɚɥɶɧɭɸ ɝɟɨɦɟɬɪɢɸ (ɩɟɪɢɨɞɢɱɟɫɤɢɣ “ɹɳɢɤ”
ZYX
LLL uu
), ɚ ɰɢɥɢɧɞɪɢɱɟɫɤɭɸ ɝɟɨɦɟɬɪɢɸ. ȼɟɤɬɨɪɧɵɣ
ɜɵɛɪɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ (ɫɢɦɦɟɬɪɢɱ-
ɧɚɹ ɤɚɥɢɛɪɨɜɤɚ ɞɥɹ ɩɨɥɹ
],0,0[ HH
ɦɨɠɧɨɩɨɬɟɧɰɢɚɥ
&
):
2/),(2/;0
U
MU
HxyHAAA
Z
. (3.1)
ɪɟɞɫɬɚɜɥɹɹ ɨɩɟɪɚɬɨɪ Ƚɚɦɢɥɶɬɨɧɚɉ
2
])/(/)/*][(2/1[
ˆ
AcerimH
&
&
!
ww
ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ, ɤɨɬɨɪɵɟ ɧɚɢɥɭɱɲɢɦ ɨɛɪɚ-ɜ
ɡɨɦ ɨɬɜɟɱɚɸɬ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɝɟɨɦɟɬɪɢɢ ɫɢɫɬɟɦɵ (ɩɨɞ-
ɪɨɛɧɨɫɬɢ ɩɪɢɜɟɞɟɧɵ ɜ ɡɚɞɚɱɟ ɜ ɤɨɧɰɟ ɩɚɪɚɝɪɚɮɚ), ɢɳɟɦ ɪɟ-
ɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɜ ɜɢɞɟ:
exp[)(),,(]MU MU\ zikRzim
Z
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ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
Ɍɚɤ ɠɟ, ɤɚɤ ɢ ɩɪɢ ɜɵɜɨɞɟ ɭɪɨɜɧɟɣ Ʌɚɧɞɚɭ, ɫɨɯɪɚɧɹɟɬɫɹ ɩɨ-
ɩɟɪɟɱɧɵɣ ɢɦɩɭɥɶɫ ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɭɝɥɨɜɨɣ ɦɨɦɟɧɬ m ɤɜɚɧ-
ɬɨɜɨɟ ɱɢɫɥɨ, ɫɢɦɜɨɥɢɡɢɪɭɸɳɟɟ ɫɜɨɣɫɬɜɚ ɫɢɦɦɟɬɪɢɢ ɜɨɥɧɨ-
ɜɨɣ ɮɭɧɤɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɪɚɳɟɧɢɹ ɜɨɤɪɭɝ ɨɫɢ Z.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (ɫɦ. ɜ ɤɨɧɰɟ ɩɚɪɚɝɪɚɮɚ)
ɩɨɥɭɱɚɟɦ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ, ɤɨɬɨɪɚɹ ɨɤɚɡɵɜɚɟɬɫɹ ɜɵɪɨɠ-
ɞɟɧɧɨɣ ɝɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ, ɚ ɭɫɥɨɜɢɹ ɨɝɪɚɧɢ-
ɱɟɧɧɨɫɬɢ ɪɟɲɟɧɢɹ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɩɪɢɜɨɞɹɬ ɤ ɜɵɪɚɠɟ-
ɧɢɸ, ɨɩɪɟɞɟɥɹɸɳɟɦɭ ɫɩɟɤɬɪ ɷɥɟɤɬɪɨɧɚ:
*2/)2/12/2/||()(
22
mkmmnkE
ZCZnm
!!
Z
(3.2)
Ɂɞɟɫɶ n ɧɨɦɟɪ ɡɨɧɵ Ʌɚɧɞɚɭ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɪɨɦɟ ɨɛɵɱ-
ɧɨɝɨ ɤɜɚɧɬɨɜɚɧɢɹ ɩɨ ɡɨɧɚɦ Ʌɚɧɞɚɭ ɢɦɟɟɬ ɦɟɫɬɨ ɤɜɚɧɬɨɜɚɧɢɟ
ɩɨ ɭɝɥɨɜɨɦɭ ɦɨɦɟɧɬɭ m. Ȼɨɥɟɟ ɬɨɝɨ, ɫɨɝɥɚɫɧɨ (3.2), ɷɧɟɪɝɢɹ
ɜɵɪɨɠɞɟɧɚ ɩɪɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɯ m=0, 1, 2, …(ɩɪɢ m<0 ɡɧɚ-
ɱɟɧɢɹ ɷɧɟɪɝɢɢ ɩɪɨɫɬɨ ɫɨɜɩɚɞɚɸɬ ɫɨ ɡɧɚɱɟɧɢɹɦɢ ɷɧɟɪɝɢɣ
ɭɪɨɜɧɟɣ Ʌɚɧɞɚɭ, ɬɨɥɶɤɨ ɫ ɤɜɚɧɬɨɜɵɦ ɱɢɫɥɨɦ n+|m|). Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ, ɜ ɤɚɠɞɨɣ ɡɨɧɟ Ʌɚɧɞɚɭ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ n ɢɦɟɟɬɫɹ
ɛɟɫɤɨɧɟɱɧɵɣ ɧɚɛɨɪ ɜɵɪɨɠɞɟɧɧɵɯ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ
m ɫ ɷɧɟɪɝɢɟɣ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɨɛɵɱɧɨɣ ɷɧɟɪɝɢɟɣ ɭɪɨɜɧɹ
Ʌɚɧɞɚɭ:.ɍɱɢɬɵɜɚɹ ɬɨɱ-
ɧɨɟ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ ɝɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ, ɞɥɹ
ɩɟɪɜɨɣ ɡɨɧɵ Ʌɚɧɞɚɭ (n=0) ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ (ɫɦ. ɡɚɞɚɱɭ) ɹɜ-
ɧɵɣ ɜɢɞ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ:
*22
2/)2/1()( mknkE
ZCZnm
!!
Z
>
4/||exp
!22
)(
2
z
m
z
zm
m
m
m
S
\
@
, (3.3)
ɝɞɟ ɢ ɨɬɜɟɱɚɸɬ ɤɨɨɪɞɢɧɚɬɚɦ ɷɥɟɤɬɪɨɧɚ
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɜɨɞɢɬɫɹ ɛɟɡɪɚɡɦɟɪɧɚɹ ɤɨɦɩɥɟɤɫɧɚɹ ɤɨɨɪ-
ɞɢɧɚɬɚ z, ɨɬɫɱɢɬɵɜɚɟɦɚɹ ɜ ɟɞɢɧɢɰɚɯ ɦɚɝɧɢɬɧɨɣ ɞɥɢɧɵ l
H
liyxz /)(
yx,
H
).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɜɢɞɢɦ, ɱɬɨ ɷɥɟɤɬɪɨɧ ɥɨɤɚɥɢɡɨɜɚɧ ɜ ɩɪɨ-
ɫɬɪɚɧɫɬɜɟ, ɢɦɟɸɳɟɦ ɥɢɧɟɣɧɵɣ ɪɚɡɦɟɪ ɦɚɝɧɢɬɧɨɣ ɞɥɢɧɵ
35
ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
(ɬɨɱɧɟɟ, ɤɜɚɞɪɚɬ ɪɚɞɢɭɫɚ ɨɪɛɢɬɵ <
U
2
>=<x
2
+y
2
> ɪɚɜɟɧ
2(m+1)l
H
2
(ɫɦ. ɡɚɞɚɱɭ 3.3).)
ȼɵɲɟ ɛɵɥɚ ɪɚɫɫɦɨɬɪɟɧɚ ɨɞɧɨɷɥɟɤɬɪɨɧɧɚɹ ɡɚɞɚɱɚ. ɉɟɪɟ-
ɯɨɞɹ ɤ ɛɨɥɟɟ ɪɟɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ, Ʌɚɮɥɢɧ ɪɚɫɫɦɨɬɪɟɥ ɞɜɭ-
ɦɟɪɧɵɣ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ ɜ (x,y) ɩɥɨɫɤɨɫɬɢ, ɧɚɯɨɞɹɳɢɣɫɹ ɜ
ɩɨɩɟɪɟɱɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ, ɢ ɫ ɜɟɤɬɨɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ
(3.1). ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɣ Ƚɚɦɢɥɶɬɨɧɢɚɧ ɬɚɤɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ
ɜɢɞ:
,
||
)(
*2
1
ˆ
2
2
¦¦
!
°
¿
°
¾
½
°
¯
°
®
kj
kj
j
jjj
rr
e
rVA
c
e
im
H
&&
&
&
!
(3.4)
ɝɞɟ ɢɧɞɟɤɫɵ j,k ɩɪɨɛɟɝɚɸɬ ɩɨ ɜɫɟɦ N ɷɥɟɤɬɪɨɧɚɦ, ɚ V ɩɨ-
ɬɟɧɰɢɚɥ ɨɞɧɨɪɨɞɧɨɝɨ ɧɟɣɬɪɚɥɢɡɭɸɳɟɝɨ ɮɨɧɚ. Ɍɨɱɧɨ ɪɟ-
ɲɢɬɶ ɬɚɤɭɸ ɡɚɞɚɱɭ ɧɟɜɨɡɦɨɠɧɨ. ɉɨɷɬɨɦɭ Ʌɚɮɥɢɧ ɩɪɟɞɥɨ-
ɠɢɥ ɢɡɭɱɚɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɫɢɫɬɟɦɭ ɜɚɪɢɚɰɢɨɧɧɵɦ ɦɟ-
ɬɨɞɨɦ. ȼɢɞ ɜɚɪɢɚɰɢɨɧɧɨɣ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɨɧ ɩɪɟɞɥɨɠɢɥ
ɜ ɬɚɤ ɧɚɡɵɜɚɟɦɨɦ ɞɠɚɫɬɪɨɜɫɤɨɦ (Jastrow) ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɜ
ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɩɨ ɩɚɪɚɦ ɱɚɫɬɢɰ:
»
¼
º
«
¬
ª
¿
¾
½
¯
®
¦
N
i
i
kj
kj
zzzf
1
2
||
4
1
exp)(
\
(3.5)
ɝɞɟ , ɢ ɷɤɫɩɨɧɟɧɬɚ ɜ (3.5) ɨɬɪɚɠɚɟɬ ɚɫɢɦɩɬɨ-
ɬɢɱɟɫɤɨɟ ɩɨɜɟɞɟɧɢɟ ɬɨɱɧɨɣ ɨɞɧɨ-ɱɚɫɬɢɱɧɨɣ ɮɭɧɤɰɢɢ (3.3) ɜ
ɩɟɪɜɨɣ ɡɨɧɟ Ʌɚɧɞɚɭ. əɜɧɵɣ ɜɢɞ ɩɨɤɚ ɧɟ ɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ
f(z) ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɫɥɟɞɭɸɳɢɯ ɫɨɨɛɪɚɠɟɧɢɣ: ɜɨ-
ɩɟɪɜɵɯ, ɢɡ ɜɢɞɚ ɨɞɧɨ-ɱɚɫɬɢɱɧɨɣ ɮɭɧɤɰɢɢ (3.3) ɮɭɧɤɰɢɹ f(z)
ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɧɢɱɟɦ ɢɧɵɦ, ɤɚɤ
H
liyxz /)(
ɤɨɧɟɱɧɵɦ ɩɨɥɢɧɨɦɨɦ ɩɨ z.
Ɂɚɬɟɦ, ɢɡ ɭɫɥɨɜɢɹ ɚɧɬɢɫɢɦɦɟɬɪɢɢ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɡɚ-
ɤɥɸɱɚɟɦ, ɱɬɨ f(z) ɞɨɥɠɧɚ ɛɵɬɶ ɧɟɱɟɬɧɚ
. Ⱦɚɥɟɟ, ɭɫɥɨɜɢɹ ɫɨ-
ɯɪɚɧɟɧɢɹ ɩɨɥɧɨɝɨ ɭɝɥɨɜɨɝɨ ɦɨɦɟɧɬɚ M ɬɪɟɛɭɸɬ, ɱɬɨɛɵ ɜɟɫɶ
ɩɪɟɞɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɣ ɦɧɨɠɢɬɟɥɶ ɜ (3.5) ɛɵɥ ɨɞɧɨɪɨɞɧɵɦ
36
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
ɩɨɥɢɧɨɦɨɦ ɫɬɟɩɟɧɢ M. ȿɫɥɢ ɫɨɛɫɬɜɟɧɧɵɣ ɭɝɥɨɜɨɣ ɦɨɦɟɧɬ
ɷɥɟɤɬɪɨɧɚ ɧɚ ɧɢɠɧɟɦ ɭɪɨɜɧɟ Ʌɚɧɞɚɭ m, ɬɨ ɩɨɥɧɵɣ ɭɝɥɨ-
ɜɨɣ ɦɨɦɟɧɬ N ɷɥɟɤɬɪɨɧɨɜ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɤɚɤ
M=(1/2)mN(N-1). ɂɧɚɱɟ, ɷɬɨ ɦɨɠɧɨ ɩɨɹɫɧɢɬɶ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ. ȿɫɥɢ ɭ ɧɚɫ ɟɫɬɶ N ɷɥɟɤɬɪɨɧɨɜ ɜ ɩɟɪɜɨɣ ɡɨɧɟ Ʌɚɧ-
ɞɚɭ, ɬɨ ɜ ɨɞɧɨɪɨɞɧɨɦ (ɤɚɤ ɭɜɢɞɢɦ ɞɚɥɟɟ ɠɢɞɤɨɦ) ɫɨɫɬɨɹ-
ɧɢɢ ɨɧɢ ɡɚɩɨɥɧɹɸɬ ɪɚɜɧɨɦɟɪɧɨ ɫɧɢɡɭ ɡɨɧɭ ɢ ɡɚɧɢɦɚɸɬ ɜ
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɉɚɭɥɢ ɩɟɪɜɵɟ N-ɨɪɛɢɬɚɥɟɣ (ɨɬ
m=0 ɞɨ m=N-1). ȿɫɥɢ ɠɟ ɫɪɟɞɧɹɹ ɩɥɨɬɧɨɫɬɶ 1/m, ɬɨ ɩɨɥɧɵɣ
ɦɨɦɟɧɬ ɜ m ɪɚɡ ɛɨɥɶɲɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɧɵɣ ɦɨɦɟɧɬ
. ȼ ɢɬɨɝɟ ɟɞɢɧɫɬɜɟɧɧɨɟ ɩɪɟ-
ɫɬɚɜɥɟɧɢɟ ɮɭɧɤɰɢɢ f(z), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɩɟɪɟɱɢɫɥɟɧɧɵɦ
ɜɵɲɟ ɫɜɨɣɫɬɜɚɦ, ɦɨɠɟɬ ɛɵɬɶ ɬɨɥɶɤɨ f(z)=z
2/)1(
1
0
NNmnimM
N
i
i
!
¦
m
. Ɉɤɨɧɱɚɬɟɥɶɧɨ
ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ:
»
¼
º
«
¬
ª
¿
¾
½
¯
®
¦
N
i
i
kj
m
kjNm
zzzzz
1
2
1
||
4
1
exp)(),...(
\
, (3.6)
ɩɪɢ ɷɬɨɦ m ɧɟɱɟɬɧɨɟ.
Ɏɚɤɬɢɱɟɫɤɢ, ɩɨɥɭɱɟɧɚ ɜɚɪɢɚɰɢɨɧɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ
ɦɧɨɝɨɷɥɟɤɬɪɨɧɧɨɣ ɞɜɭɦɟɪɧɨɣ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɫɬɟ-
ɦɵ, ɫ ɟɞɢɧɫɬɜɟɧɧɵɦ ɩɚɪɚɦɟɬɪɨɦ m, ɤɨɬɨɪɵɣ ɫɥɟɞɭɟɬ ɩɨɥɭ-
ɱɢɬɶ ɢɡ ɦɢɧɢɦɭɦɚ ɷɧɟɪɝɢɢ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ, ɤɚɤɨɟ ɠɟ ɡɧɚ-
ɱɟɧɢɟ m ɦɢɧɢɦɢɡɢɪɭɟɬ ɷɧɟɪɝɢɸ, ɩɨɥɨɠɢɦ, ɱɬɨ ɚɧɫɚɦɛɥɶ
ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ (3.6) ɤɚɤ ɮɭɧɤɰɢɣ ɤɨɨɪɞɢɧɚɬ ɱɚɫɬɢɰ -
ɩɟɪɟɦɟɧɧɵɯ z
j
, ɪɚɫɩɪɟɞɟɥɟɧɵ ɩɨ Ƚɢɛɛɫɭ ɫ ɧɟɤɨɬɨɪɨɣ ɮɢɤ-
ɬɢɜɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ, ɬ.ɟ. ɫɤɨɧɫɬɪɭɢɪɭɟɦ ɚɧɚɥɨɝ ɤɥɚɫɫɢɱɟ-
ɫɤɨɝɨ ɞɟɣɫɬɜɢɹ ɩɨ ɫɥɟɞɭɸɳɟɦɭ ɫɨɨɬɧɨɲɟɧɢɸ, ɡɚɩɢɫɚɜ
ɤɜɚɞɪɚɬ ɦɨɞɭɥɹ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɜ ɜɢɞɟ (ɫɦ.ɧɢɠɟ ɡɚɞɚɱɭ):
].exp[||
2
)E \
m
Ɂɞɟɫɶ ) - ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɪɚɫɩɪɟɞɟɥɟ-
ɧɢɸ ɱɚɫɬɢɰ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ ɜ ɫɨɫɬɨɹɧɢɢ <
m
. ɉɨɥɚɝɚɹ
37
ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
m/1 E ɜ ɪɨɥɢ ɨɛɪɚɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ, ɧɚɯɨɞɢɦ ɜɵɪɚɠɟɧɢɟ
ɞɥɹ :
)
¦¦
)
!
j
j
kj
kj
zmzzm .||
2
1
||ln2
22
(3.7)
ȼɵɪɚɠɟɧɢɟ (3.7) ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ N ɬɨɠɞɟɫɬɜɟɧɧɵɯ ɱɚɫ-
ɬɢɰ ɧɚ ɩɥɨɫɤɨɫɬɢ, ɢɦɟɸɳɢɯ ɡɚɪɹɞ e* = m ɢ ɜɡɚɢɦɨɞɟɣɫɬ-
ɜɭɸɳɢɯ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɤɭɥɨɧɨɜɫɤɢɦ ɨɬɬɚɥɤɢɜɚɸɳɢɦ ɩɨɬɟɧ-
ɰɢɚɥɨɦ:
||ln2)(
2
kjkj
zzezzv
.
(ɷɬɨ ɫɩɪɚɜɟɞɥɢɜɨ ɜ ɩɥɨɫɤɨɣ ɝɟɨɦɟɬɪɢɢ ɫɦ. ɡɚɞɚɱɭ). Ʉɪɨɦɟ
ɬɨɝɨ, ɩɨɫɥɟɞɧɢɣ ɱɥɟɧ ɜ (3.7) ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɷɬɢɯ ɱɚɫɬɢɰ ɫ
ɨɞɧɨɪɨɞɧɵɦ ɧɟɣɬɪɚɥɢɡɭɸɳɢɦ ɮɨɧɨɦ ɫ ɩɨɜɟɪɯɧɨɫɬɧɨɣ
ɩɥɨɬɧɨɫɬɶɸ
V
=1/2
S
l
H
2
(ɫɦ. ɬɚɤɠɟ ɡɚɞɚɱɭ ɜ ɤɨɧɰɟ ɩɚɪɚɝɪɚɮɚ).
Ʌɚɮɥɢɧ ɨɬɦɟɱɚɟɬ, ɱɬɨ ɜ ɤɚɱɟɫɬɜɟ ɜɵɪɚɠɟɧɢɹ (3.7) ɦɵ
ɩɨɥɭɱɢɥɢ ɷɧɟɪɝɢɸ ɤɥɚɫɫɢɱɟɫɤɨɣ ɨɞɧɨɤɨɦɩɨɧɟɧɬɧɨɣ ɩɥɚɡɦɵ
ɜ ɞɜɭɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɝɞɟ ɤɭɥɨɧɨɜɫɤɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ
ɨɩɢɫɵɜɚɟɬɫɹ ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɦ ɩɨɬɟɧɰɢɚɥɨɦ. Ʉ ɦɨɦɟɧɬɭ
ɧɚɩɢɫɚɧɢɹ ɫɬɚɬɶɢ (1983) ɷɬɚ ɫɢɫɬɟɦɚ ɛɵɥɚ ɯɨɪɨɲɨ ɢɡɭɱɟɧɚ ɜ
ɫɜɹɡɢ ɫ ɢɧɬɟɧɫɢɜɧɵɦ ɪɚɡɜɢɬɢɟɦ ɪɚɛɨɬ ɩɨ ɭɩɪɚɜɥɹɟɦɨɦɭ
ɬɟɪɦɨɹɞɟɪɧɨɦɭ ɫɢɧɬɟɡɭ. ɑɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ (ɦɟɬɨɞɨɦ Ɇɨɧ-
ɬɟ-Ʉɚɪɥɨ) ɧɚɞɟɠɧɨ ɭɫɬɚɧɨɜɢɥɢ, ɱɬɨ ɬɚɤɚɹ ɫɢɫɬɟɦɚ ɹɜɥɹɟɬɫɹ
ɠɢɞɤɨɫɬɶɸ ɩɪɢ m < 70 ɢ ɝɟɤɫɚɝɨɧɚɥɶɧɵɦ ɤɪɢɫɬɚɥɥɨɦ ɜ ɩɪɨ-
ɬɢɜɧɨɦ ɫɥɭɱɚɟ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ m < 70 ɩɥɨɬɧɨɫɬɶ ɷɥɟɤɬɪɨɧɨɜ ɨɞɧɨ-
ɪɨɞɧɚ ɢ ɞɨɥɠɧɚ ɛɵɬɶ ɪɚɜɧɚ ɡɚɪɹɞɨɜɨɣ ɩɥɨɬɧɨɫɬɢ ɧɟɣɬɪɚɥɢ-
ɡɭɸɳɟɝɨ ɮɨɧɚ, ɞɟɥɟɧɧɨɣ ɧɚ ɜɟɥɢɱɢɧɭ ɮɢɤɬɢɜɧɨɝɨ ɡɚɪɹɞɚ
ɩɥɚɡɦɟɧɧɵɯ ɱɚɫɬɢɰ: n=
V
/e*=1/[2
S
ml
H
2
]. ɉɨɫɥɟɞɧɟɟ ɨɡɧɚɱɚɟɬ,
ɱɬɨ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɫɨɨɬɧɨɲɟɧɢɟɦ (2.3), ɬɚɤɨɟ ɠɢɞɤɨɟ
ɷɥɟɤɬɪɨɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɪɟɚɥɢɡɭɟɬɫɹ ɩɪɢ ɮɚɤɬɨɪɟ ɡɚɩɨɥɧɟɧɢɹ
P
=1/m, m=3,5,7,...
38
ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
Ʌɚɮɥɢɧ ɩɪɨɜɟɪɢɥ ɱɢɫɥɟɧɧɨ ɧɚ ɫɢɫɬɟɦɟ ɢɡ 3-ɯɢ 4-ɯ ɱɚɫ-
ɬɢɰ, ɧɚɫɤɨɥɶɤɨ ɜɚɪɢɚɰɢɨɧɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ (3.6) ɛɥɢɡɤɚ
ɤ ɬɨɱɧɨɣ ɢ ɧɚɲɟɥ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɧɟ ɛɨɥɟɟ
0.05 ɞɥɹ m=3, 5. Ʉɪɨɦɟ ɬɨɝɨ, ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɨɫɧɨɜɧɨɟ
ɫɨɫɬɨɹɧɢɟ (3.6) ɢɦɟɟɬ ɛɨɥɟɟ ɧɢɡɤɭɸ ɷɧɟɪɝɢɸ, ɱɟɦ ɷɧɟɪɝɢɹ
ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɝɨ ɫɨɫɬɨɹɧɢɹ ɞɢɷɥɟɤɬɪɢɤɚ ɜ ɞɚɧɧɨɦ ɫɥɭ-
ɱɚɟ. ɇɟɡɚɜɢɫɢɦɵɟ ɩɨɫɥɟɞɭɸɳɢɟ ɱɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ (ɦɟɬɨɞ
ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ ɧɚ ɫɮɟɪɟ ɞɥɹ 7-12 ɱɚɫɬɢɰ) ɩɨɞɬɜɟɪ-
ɞɢɥɢ ɢɟɪɚɪɯɢɸ ɱɢɫɟɥ ɡɚɩɨɥɧɟɧɢɹ Ʌɚɮɥɢɧɚ ɢ ɏɨɥɞɟɣɧɚ. Ɍɚɤ,
Ƚɪɨɫ ɢ Ɇɚɤɞɨɧɚɥɶɞ (Gros and MacDonald, 1990) ɩɨɥɭɱɢɥɢ
ɮɚɤɬɨɪ ɡɚɩɨɥɧɟɧɢɹ
P
=n/2n+1, n=0,1,2... ɞɥɹ 12-ɬɢ ɱɚɫɬɢɰ ɧɚ
ɫɮɟɪɟ.
ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ 1998 ɝɨɞɭ ɩɪɨɢɡɨɲɥɨ ɞɚɜɧɨ
ɨɠɢɞɚɟɦɨɟ ɫɨɛɵɬɢɟ, ɥɨɝɢɱɟɫɤɢ ɜɟɧɱɚɸɳɟɟ ɷɬɨ ɧɨɜɨɟ ɧɚ-
ɩɪɚɜɥɟɧɢɟ ɜ ɫɨɜɪɟɦɟɧɧɨɣ ɮɢɡɢɤɟ ɤɨɧɞɟɧɫɢɪɨɜɚɧɧɨɝɨ ɜɟɳɟ-
ɫɬɜɚ ɇɨɛɟɥɟɜɫɤɚɹ ɩɪɟɦɢɹ ɩɨ ɮɢɡɢɤɟ ɩɪɢɫɭɠɞɟɧɚ ɩɪɨɮɟɫ-
ɫɨɪɭ ɋɬɷɧɮɨɪɞɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ Ɋɨɛɟɪɬɭ Ʌɚɮɥɢɧɭ (Robert
B.Laughlin), ɩɪɨɮɟɫɫɨɪɭ Ʉɨɥɭɦɛɢɣɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ ɏɨɪ-
ɫɬɭ ɒɬɟɪɦɟɪɭ (Horst L. Stɨrmer) ɢ ɩɪɨɮɟɫɫɨɪɭ ɉɪɢɧɫɬɨɧ-
ɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ Ⱦɷɧɢɷɥɭ Ɍɫɭɢ (Daniel C. Tsui) ɡɚ ɨɬɤɪɵ-
ɬɢɟ ɢ ɨɛɴɹɫɧɟɧɢɟ ɞɪɨɛɧɨɝɨ ɤɜɚɧɬɨɜɨɝɨ ɷɮɮɟɤɬɚ ɏɨɥɥɚ. Ɍɨɱ-
ɧɟɟ, ɫɨɝɥɚɫɧɨ ɨɮɢɰɢɚɥɶɧɨɣ ɮɨɪɦɭɥɢɪɨɜɤɟ, ɡɚ ”ɨɬɤɪɵɬɢɟ
ɧɨɜɨɝɨ ɬɢɩɚ ɤɜɚɧɬɨɜɨɣ ɠɢɞɤɨɫɬɢ ɫ ɞɪɨɛɧɵɦ ɡɚɪɹɞɨɦ ɷɥɟ-
ɦɟɧɬɚɪɧɵɯ ɜɨɡɛɭɠɞɟɧɢɣ”.
ɀɢɞɤɨɫɬɶ Ʌɚɮɥɢɧɚ ɨɤɚɡɚɥɚɫɶ ɧɨɜɵɦ ɤɜɚɧɬɨɜɵɦ ɫɨɫɬɨɹ-
ɧɢɟɦ ɞɜɭɦɟɪɧɨɣ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɫɬɟɦɵ. ɋɩɟɰɢɮɢɱɟ-
ɫɤɚɹ ɨɫɨɛɟɧɧɨɫɬɶ ɫɨɫɬɨɹɧɢɹ Ʌɚɮɥɢɧɚ ɷɥɟɦɟɧɬɚɪɧɵɟ ɜɨɡ-
ɛɭɠɞɟɧɢɹ ɜ ɧɟɣ ɦɨɝɭɬ ɢɦɟɬɶ ɞɪɨɛɧɵɣ ɡɚɪɹɞ. Ɍɚɤ, Ʌɚɮɥɢɧ
ɩɪɟɞɥɨɠɢɥ ɪɚɫɫɦɨɬɪɟɬɶ ɫɥɟɞɭɸɳɭɸ ɫɢɬɭɚɰɢɸ ɞɥɹ ɩɨɫɬɪɨɟ-
ɧɢɹ ɜɨɡɛɭɠɞɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ: ɜ ɤɚɤɨɣ-ɥɢɛɨ ɬɨɱɤɟ ɩɥɨɫɤɨ-
ɫɬɢ, z
0
ɜɧɨɫɢɬɫɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɟɣ ɛɟɫɤɨɧɟɱɧɵɣ ɬɨɧɤɢɣ
ɫɨɥɟɧɨɢɞ ɢ ɱɟɪɟɡ ɧɟɝɨ ɚɞɢɚɛɚɬɢɱɟɫɤɢ ɦɟɞɥɟɧɧɨ ɩɪɨɩɭɫɤɚɟɬ-
ɫɹ ɟɞɢɧɢɱɧɵɣ ɤɜɚɧɬ ɦɚɝɧɢɬɧɨɝɨ ɩɨɬɨɤɚ . ɋɨ-
ɨɬɜɟɬɫɬɜɭɸɳɭɸ ɞɨɛɚɜɤɭ ɜ ɜɟɤɬɨɪɧɵɣ ɩɨɬɟɧɰɢɚɥ ɦɨɠɧɨ
ec /2
0
!
S
)
39
ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ
_________________________________________________________________________________________________________________________________________________________________________________________________
ɭɱɟɫɬɶ ɱɢɫɬɨ ɢɡ ɤɚɥɢɛɪɨɜɨɱɧɨɝɨ ɫɞɜɢɝɚ, ɱɬɨ ɩɪɨɫɬɨ ɩɪɢɜɟ-
ɞɟɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɧɚ ɟɞɢɧɢɰɭ ɭɝɥɨɜɨɝɨ ɦɨɦɟɧɬɚ ɜ ɨɞɧɨ-
ɱɚɫɬɢɱɧɨɣ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ:
]||4/1exp[)(]||4/1exp[)(
21
0
2
0
zzzzzz
mm
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɚɪɢɚɰɢɨɧɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ (3.6)
ɞɨɥɠɧɚ ɛɵɬɶ ɩɟɪɟɩɢɫɚɧɚ ɜ ɜɢɞɟ:
»
¼
º
«
¬
ª
¿
¾
½
¯
®
¦
N
i
i
kj
m
kj
i
imZ
z
m
zzzzzA
1
2
0
||
4
1
exp)()(
0
0
\\
.
(3.8)
ɗɬɚ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɷɥɟɦɟɧɬɚɪɧɨɦɭ ɜɨɡɛɭ-
ɠɞɟɧɢɸ, ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɸɬ “ɤɜɚɡɢɷɥɟɤɬɪɨɧ”. Ⱥɧɚɥɨɝɢɱɧɨ
ɦɨɠɧɨ ɜɜɟɫɬɢ ɷɥɟɦɟɧɬɚɪɧɨɟ ɜɨɡɛɭɠɞɟɧɢɟ “ɤɜɚɡɢɞɵɪɤɚ”, ɫ
ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɟɣ . Ʌɚɮɥɢɧ ɩɪɨɜɟɪɢɥ ɷɬɢ
ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ, ɫɩɪɨɟɤɬɢɪɨɜɚɜ ɢɯ ɧɚ ɬɨɱɧɵɟ ɜɨɥɧɨɜɵɟ
ɮɭɧɤɰɢɢ ɩɟɪɜɵɯ ɜɨɡɛɭɠɞɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɢɡ ɱɢɫɥɟɧɧɨɝɨ
ɚɧɚɥɢɡɚ ɱɟɬɵɪɟɯ ɱɚɫɬɢɰ ɢ ɩɨɥɭɱɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɨɬɤɥɨɧɟ-
ɧɢɟ ɧɟ ɛɨɥɟɟ 0.005! ɉɨɡɞɧɟɟ, Ɏɚɧɨ ɢ ɞɪ. (Fano, Ortolani,
Colombo, 1986) ɩɪɨɜɟɥɢ ɱɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ ɦɟɬɨɞɨɦ ɬɨɱ-
ɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ ɧɚ ɫɮɟɪɟ ɢ ɞɢɫɤɟ ɢ ɩɨɥɧɨɫɬɶɸ ɩɨɞɬɜɟɪ-
ɞɢɥɢ ɜɢɞ ɜɨɡɛɭɠɞɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ (3.8) ɩɪɢ ɮɚɤɬɨɪɚɯ ɡɚ-
ɩɨɥɧɟɧɢɹ
P
=1/3, 1/5.
mZ
z
m
A
\\
*
0
0
ɉɨɥɭɱɟɧɧɵɟ ɷɥɟɦɟɧɬɚɪɧɵɟ ɜɨɡɛɭɠɞɟɧɢɹ ɹɜɥɹɸɬɫɹ ɱɚɫ-
ɬɢɰɚɦɢ ɫ ɡɚɪɹɞɨɦ e*=e/m. ɗɬɨ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɫɤɨɧɫɬɪɭɢ-
ɪɨɜɚɜ ɢɡ ɢɯ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɤɥɚɫɫɢɱɟɫɤɨɟ ɞɟɣɫɬɜɢɟ ɩɨ
ɮɨɪɦɭɥɟ ,
E
=1/m. Ɍɨɝɞɚ, ɩɨɞɫɬɚɜɢɜ
(3.8), ɧɚɯɨɞɢɦ:
]exp[||
2
0
)
c
E\
z
m
¦
) )
c
i
zzm ||ln2
0
, (3.9)
40