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ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

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Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ

< <

. ɋ ɩɨɦɨɳɶɸ

ɝɪɚɞɢɟɧɬɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɢɞɚ

]exp[;

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

ɭɛɢɪɚɟɦ ɜɟɤɬɨɪɧɵɣ ɩɨɬɟɧɰɢɚɥ ɢɡ ɭɪɚɜɧɟɧɢɹ, ɩɪɢ ɷɬɨɦ ɫɜɹɡɶ ɜɨɥɧɨɜɨɣ

ɮɭɧɤɰɢɢ ɛɟɡ ɬɪɭɛɨɤ ɦɚɝɧɢɬɧɨɝɨ ɩɨɬɨɤɚ ɢ ɫ ɧɢɦɢ ɢɦɟɟɬ ɜɢɞ:

< <



]exp[

,, jiji

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɹɜɥɹɟɬɫɹ ɮɚɡɨɜɵɣ ɦɧɨɠɢɬɟɥɶ, ɤɨɬɨɪɵɣ ɢ ɫɢɦɜɨɥɢɡɢ-

ɪɭɟɬ ɞɪɨɛɧɭɸ (ɷɧɢɨɧɧɭɸ) ɫɬɚɬɢɫɬɢɤɭ: ɩɪɢ ɥɸɛɨɣ ɩɟɪɟɫɬɚɧɨɜɤɟ ɞɜɭɯ

ɱɚɫɬɢɰ ɨɧ ɜɧɨɫɢɬ ɫɜɨɣ ɜɤɥɚɞ. ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɜɜɟɞɟɧɢɹ ɨɞɧɨɣ ɬɪɭɛɤɢ

ɦɚɝɧɢɬɧɨɝɨ ɩɨɬɨɤɚ (ɤɚɤ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɜɨɡɛɭɠɞɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ

Ʌɚɮɥɢɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ) ɦɵ ɧɟɦɟɞɥɟɧɧɨ ɩɨɥɭɱɚɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ

(3.8).

4. ȼɢɝɧɟɪɨɜɫɤɚɹ ɤɪɢɫɬɚɥɥɢɡɚɰɢɹ. Ɏɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ

“ɤɪɢɫɬɚɥɥ ȼɢɝɧɟɪɚ - ɠɢɞɤɨɫɬɶ Ʌɚɮɥɢɧɚ”. ɑɢɫɥɟɧɧɨɟ

ɦɨɞɟɥɢɪɨɜɚɧɢɟ

Ɉɫɬɪɢɟ ɫɨɜɪɟɦɟɧɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɞɪɨɛɧɨɝɨ ɤɜɚɧɬɨɜɨɝɨ

ɷɮɮɟɤɬɚ ɏɨɥɥɚ ɧɚɩɪɚɜɥɟɧɨ ɟɳɟ ɧɚ ɨɞɧɨ ɢɧɬɟɪɟɫɧɨɟ ɹɜɥɟɧɢɟ

 ɪɟɚɥɢɡɚɰɢɸ ɪɟɝɭɥɹɪɧɨɣ ɪɟɲɟɬɤɢ ɢɡ ɧɨɫɢɬɟɥɟɣ ɡɚɪɹɞɚ (ɜ

ɞɜɭɦɟɪɢɢ  ɩɥɨɫɤɨɣ ɬɪɟɭɝɨɥɶɧɨɣ, ɝɟɤɫɚɝɨɧɚɥɶɧɨɣ ɪɟɲɟɬɤɢ).

ȼ ɧɚɱɚɥɟ 90-ɯ ɝɨɞɨɜ ɜɫɟ ɛɨɥɟɟ ɞɨɤɚɡɚɬɟɥɶɧɨ ɫɬɚɥɢ ɡɜɭɱɚɬɶ

ɫɥɨɜɚ ɨ ɪɟɚɥɢɡɚɰɢɢ ɜɛɥɢɡɢ FQHE ɷɬɨɝɨ ɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ

ɞɜɭɦɟɪɧɨɝɨ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ  ɬɚɤ ɧɚɡɵɜɚɟɦɨɝɨ ɤɪɢɫɬɚɥɥɚ

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ȼɢɝɧɟɪɚ. ȼɩɟɪɜɵɟ ɜɨɡɦɨɠɧɨɫɬɶ ɬɚɤɨɣ ɜɢɝɧɟɪɨɜɫɤɨɣ ɤɪɢ-

ɫɬɚɥɥɢɡɚɰɢɢ ɜ ɞɜɭɦɟɪɧɨɣ ɷɥɟɤɬɪɨɧɧɨɣ ɫɢɫɬɟɦɟ ɜ ɫɢɥɶɧɨɦ

ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɟɳɟ ɜ 1975 ɝɨɞɭ ɪɚɫɫɦɨɬɪɟɥɢ ȿ.ɘ. Ʌɨɡɨɜɢɤ

ɢȼ.ɂ. ɘɞɫɨɧ (ɢ ɷɬɭ ɫɬɚɬɶɸ, ɨɩɭɛɥɢɤɨɜɚɧɧɭɸ ɜ ɠɭɪɧɚɥɟ

“ɉɢɫɶɦɚ ɜ ɀɗɌɎ”, ɨɛɹɡɚɬɟɥɶɧɨ ɰɢɬɢɪɭɸɬ ɫɟɣɱɚɫ, ɤɚɤ ɬɨɥɶ-

ɤɨ ɞɟɥɨ ɤɚɫɚɟɬɫɹ ɨɛɫɭɠɞɟɧɢɹ ɜɨɩɪɨɫɨɜ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ

ȼɢɝɧɟɪɚ).

Ⱦɟɥɨ ɜ ɬɨɦ

, ɱɬɨ ɤɚɤ ɛɵɥɨ ɭɩɨɦɹɧɭɬɨ ɜ ɧɚɱɚɥɟ ɩɪɟɞɵɞɭ-

ɳɟɝɨ ɪɚɡɞɟɥɚ, ɜɛɥɢɡɢ ɮɚɤɬɨɪɨɜ ɡɚɩɨɥɧɟɧɢɹ

=1/3, 1/5 (ɧɨ

ɜɧɟ ɪɟɚɥɢɡɚɰɢɢ FQHE) ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɧɚɛɥɸɞɚɸɬ ɪɟɡ-

ɤɨɟ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɟ ɩɨɜɵɲɟɧɢɟ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫ

ɜɨɡɪɚɫɬɚɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ (ɚɤɬɢɜɚɰɢɨɧɧɵɣ ɡɚɤɨɧ). Ɍɚɤ,

ɀɢɚɧɝ ɢ ɞɪ. (Jiang, Willett, Stormer, Tsui, 1990) ɜɛɥɢɡɢ ɮɚɤ-

ɬɨɪɚ ɡɚɩɨɥɧɟɧɢɹ

=1/5 (ɚ ɬɨɱɧɟɟ, ɩɪɢ

=0.19 ɢ 0.21) ɧɚɛɥɸ-

ɞɚɥɢ ɚɤɬɢɜɚɰɢɨɧɧɵɣ ɡɚɤɨɧ ɜɢɞɚ

~ exp(E

/T), ɩɪɢɱɟɦ

ɷɧɟɪɝɢɹ ɚɤɬɢɜɚɰɢɢ E

~ 1K. Ƚɨɥɞɦɚɧ ɫ ɫɨɬɪɭɞɧɢɤɚɦɢ

(Goldman, Santos, Shauegau, Gunningha, 1990) ɩɨɥɭɱɢɥɢ ɬɚ-

ɤɨɟ ɩɨɜɟɞɟɧɢɟ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɪɢ 2/9 >

> 1/5 ɢ ɩɪɢ

<~1/5. Ⱦɨɪɨɠɤɢɧ ɢ ɞɪ. (Ⱦɨɪɨɠɤɢɧ, ɒɚɲɤɢɧ ɢ ɞɪ., 1992)

ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɞɨɤɚɡɵɜɚɸɬ ɞɢɷɥɟɤɬɪɢɡɚɰɢɸ ɩɪɢ

<0.28.

ɋɚɞɠɨɬɨ ɫ ɫɨɬɪ. (Sajoto, Li et.al., 1993) ɧɚɛɥɸɞɚɸɬ ɞɢɷɥɟɤ-

ɬɪɢɡɚɰɢɸ ɨɤɨɥɨ ɫɬɭɩɟɧɟɤ FQHE ɩɪɢ

=1/5 ɢ 2/9.

Ɏɚɤɬɢɱɟɫɤɢ ɤ ɧɚɫɬɨɹɳɟɦɭ ɦɨɦɟɧɬɭ ɧɚɞɟɠɧɨ ɷɤɫɩɟɪɢ-

ɦɟɧɬɚɥɶɧɨ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɨɤɨɥɨ ɮɚɤɬɨɪɨɜ ɡɚɩɨɥɧɟɧɢɹ

=1/3 ɢ 1/5 ɢɦɟɟɬ ɦɟɫɬɨ ɚɤɬɢɜɚɰɢɨɧɧɵɣ ɡɚɤɨɧ ɦɚɝɧɢɬɨɫɨ-

ɩɪɨɬɢɜɥɟɧɢɹ

. ɗɬɨ ɤɨɫɜɟɧɧɨ ɫɜɢɞɟɬɟɥɶɫɬɜɭɟɬ ɨ ɞɢɷɥɟɤ-

ɬɪɢɡɚɰɢɢ ɫɢɫɬɟɦɵ, ɨ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤɬɪɨɧɨɜ ɢ ɪɟɚɥɢɡɚɰɢɢ

Ɋɢɫ. 4.1

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

-8 -4 0 4 8

x /a

-8

-4

y /a

QUANTUM

Ʉɥɚɫɫɢɱɟɫɤɚɹ ɬɪɟɭɝɨɥɶɧɚɹ

ɪɟɲɟɬɤɚ  ɜɢɝɧɟɪɨɜɫɤɢɣ

ɤɪɢɫɬɚɥɥ

ɪɟɝɭɥɹɪɧɨɣ ɪɟɲɟɬɤɢ ɢɡ ɧɨɫɢɬɟɥɟɣ ɡɚɪɹɞɚ. ȼ ɩɪɢɧɰɢɩɟ, ɫɨ-

ɨɛɪɚɠɟɧɢɹ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɜɨɡɦɨɠɧɨɫɬɢ ɥɨɤɚɥɢɡɚɰɢɢ (ɜɩɟɪ-

ɜɵɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ Ʌɨɡɨɜɢɤɨɦ ɢ ɘɞɫɨɧɨɦ) ɞɨɫɬɚɬɨɱɧɨ

ɨɱɟɜɢɞɧɵ: ɩɪɢ ɭɫɢɥɟɧɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɦɚɝɧɢɬɧɚɹ ɞɥɢɧɚ

- ɦɚɫɲɬɚɛ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɥɨɤɚɥɢɡɚɰɢɢ ɜɨɥɧɨɜɨɣ

ɮɭɧɤɰɢɢ ɷɥɟɤɬɪɨɧɚ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɭɦɟɧɶ-

ɲɚɟɬɫɹ. Ɍ.ɟ. ɩɪɢ ɫɜɨɟɦ ɞɜɢɠɟɧɢɢ ɜɨɤɪɭɝ ɥɢɧɢɣ ɦɚɝɧɢɬɧɨɝɨ

ɩɨɬɨɤɚ ɧɨɫɢɬɟɥɢ ɡɚɪɹɞɚ ɜɫɟ ɛɨɥɟɟ ɫɠɢɦɚɸɬ ɪɚɞɢɭɫ «ɜɢɬɤɨɜ»

(ɟɫɥɢ ɩɨɥɶɡɨɜɚɬɶɫɹ ɤɥɚɫɫɢɱɟɫɤɨɣ ɚɧɚɥɨɝɢɟɣ ɞɜɢɠɟɧɢɹ), ɢ

ɮɚɤɬɢɱɟɫɤɢ ɢɦɟɸɬ ɨɩɪɟɞɟɥɟɧɧɵɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɤɨɨɪɞɢɧɚ-

ɬɵ. . ɉɪɢ ɧɚɥɢɱɢɢ ɫɢɥɶɧɨɝɨ ɤɭɥɨɧɨɜɫɤɨɝɨ ɨɬɬɚɥɤɢɜɚɧɢɹ ɦɵ

ɢɦɟɟɦ ɝɚɡ ɩɪɚɤɬɢɱɟɫɤɢ

ɤɥɚɫɫɢɱɟɫɤɢɯ ɱɚɫɬɢɰ ɜ ɞɜɭɦɟɪɧɨɦ

ɩɪɨɫɬɪɚɧɫɬɜɟ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɜɵɫɬɪɨɢɬɶɫɹ ɜ ɪɟɝɭɥɹɪɧɭɸ

ɩɥɨɫɤɭɸ ɬɪɟɭɝɨɥɶɧɭɸ ɫɬɪɭɤɬɭɪɭ ɞɥɹ ɦɢɧɢɦɢɡɚɰɢɢ ɷɧɟɪɝɢɢ,

“ɡɚɩɢɪɚɹ” ɫɟɛɹ ɜ ɬɨɱɤɚɯ ɪɚɜɧɨɜɟɫɢɹ (ɫɦ., ɧɚɩɪɢɦɟɪ, ɪɢɫ.4.1).

Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɤɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɞɜɭɦɟɪɧɚɹ ɤɨɧ-

ɰɟɧɬɪɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ n=

. ɋ

ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɜ ɬɪɟɭɝɨɥɶɧɨɣ ɩɥɨɫɤɨɣ ɪɟɲɟɬɤɟ ɫ ɩɟɪɢɨɞɨɦ

a ɢɦɟɟɦ ɢɡ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɫɨɨɛɪɚɠɟɧɢɣ n=

3/2 a

. Ɉɬɫɸ-

ɞɚ ɩɨɥɭɱɚɟɦ ɫɨɨɬɧɨɲɟɧɢɟ

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

PS 3/4)/(

. (4.1)

ɍɫɢɥɢɜɚɹ ɩɨɥɟ, ɦɵ ɭɦɟɧɶɲɚɟɦ

, ɢ ɷɮɮɟɤɬɢɜɧɨ ɭɜɟɥɢɱɢɜɚ-

ɟɦ ɩɟɪɢɨɞ a. Ɏɢɡɢɱɟɫɤɢɟ ɪɚɡɦɟɪɵ ɷɥɟɤɬɪɨɧɚ (ɜɟɪɧɟɟ, ɷɥɟɤ-

ɬɪɨɧɧɨɝɨ “ɛɥɢɧɚ”), ɨɩɪɟɞɟɥɹɟɦɵɟ ɜɟɥɢɱɢɧɨɣ ɦɚɝɧɢɬɧɨɣ

ɞɥɢɧɵ l

, ɭɦɟɧɶɲɚɸɬɫɹ (ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɯ ɪɚɫɫɬɨɹɧɢɹ ɞɪɭɝ

ɨɬ ɞɪɭɝɚ a), ɢ ɨɧɢ ɜɫɟ ɛɨɥɟɟ ɥɨɤɚɥɢɡɭɸɬɫɹ. Ɍɚɤɚɹ ɫɬɪɭɤɬɭɪɚ

- ɞɢɷɥɟɤɬɪɢɤ, ɫ ɯɚɪɚɤɬɟɪɧɨɣ ɷɧɟɪɝɢɟɣ ɚɤɬɢɜɚɰɢɢ Eg ~ 1-5Ʉ,

ɤɚɤ ɢ ɧɚɛɥɸɞɚɟɬɫɹ ɧɚ ɷɤɫɩɟɪɢɦɟɧɬɟ. Ɂɚɦɟɬɢɦ, ɱɬɨ Eg ɨɤɚɡɵ-

ɜɚɟɬɫɹ ɛɥɢɡɤɨɣ ɩɨ ɩɨɪɹɞɤɭ ɜɟɥɢɱɢɧɵ ɯɚɪɚɤɬɟɪɧɵɦ ɡɧɚɱɟɧɢ-

ɹɦ ɷɧɟɪɝɢɢ ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜ ɧɚɲɟɣ ɞɜɭɦɟɪ-

ɧɨɣ ɡɚɞɚɱɟ ~ e

), ɝɞɟ

 ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɨɫɬɨɹɧɧɚɹ.

ȿɫɥɢ ɭɱɟɫɬɶ, ɱɬɨ ɜ GaAs

~ 13, ɢɦɟɟɦ ɦɚɫɲɬɚɛ <~ 10

-3

ɷȼ (ɜ

ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ~ 1-5 Ɍɥ).

ɋɥɟɞɭɟɬ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɬɨɱɧɨɟ ɬɟɨɪɟɬɢɱɟɫɤɨɟ ɨɛɨɫɧɨ-

ɜɚɧɢɟ ɜɢɝɧɟɪɨɜɫɤɨɣ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ  ɨɱɟɧɶ ɧɟɩɪɨɫɬɚɹ ɡɚ-

ɞɚɱɚ. Ʉɚɤɢɟ-ɥɢɛɨ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɨɰɟɧɤɢ ɜ ɫɢɥɶɧɨ-

ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɦ ɷɥɟɤɬɪɨɧɧɨɦ ɝɚɡɟ ɩɪɨɛɥɟɦɚɬɢɱɧɵ ɢɡ-ɡɚ

ɨɬɫɭɬɫɬɜɢɹ ɩɚɪɚɦɟɬɪɨɜ ɦɚɥɨɫɬɢ ɢ ɧɟɜɨɡɦɨɠɧɨɫɬɢ ɢɫɩɨɥɶɡɨ-

ɜɚɧɢɹ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ. ȼ ɩɨɫɥɟɞɧɟɟ ɜɪɟɦɹ ɪɚɡɜɢɬɢɟ

ɤɨɦɩɶɸɬɟɪɧɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɩɪɢɜɟɥɨ ɤ ɜɨɡɦɨɠɧɨɫɬɢ

ɱɢɫɥɟɧɧɨɝɨ

ɚɧɚɥɢɡɚ ɨɝɪɚɧɢɱɟɧɧɵɯ ɧɢɡɤɨɪɚɡɦɟɪɧɵɯ ɫɢɫɬɟɦ.

Ɋɚɫɱɟɬɵ ɦɟɬɨɞɨɦ Ɇɨɧɬɟ-Ʉɚɪɥɨ ɢ ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ

ɝɚɦɢɥɶɬɨɧɨɜɨɣ ɦɚɬɪɢɰɵ ɜ ɞɜɭɦɟɪɧɨɣ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ

ɷɥɟɤɬɪɨɧɧɨɣ ɫɢɫɬɟɦɟ ɩɪɨɜɨɞɹɬɫɹ ɧɚ ɤɨɧɟɱɧɵɯ (10 - 30 ɷɥɟɤ-

ɬɪɨɧɨɜ) ɤɥɚɫɬɟɪɚɯ ɜ ɪɚɡɥɢɱɧɨɣ ɝɟɨɦɟɬɪɢɢ. Ɋɚɫɫɱɢɬɵɜɚɸɬ

ɷɧɟɪɝɢɸ, ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɢ ɪɚɡɥɢɱɧɵɟ ɤɨɪɪɟɥɹɰɢɨɧɧɵɟ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ. Ɂɚɞɚɱɭ ɞɥɹ ɬɚɤɢɯ ɦɚɥɵɯ ɫɢɫɬɟɦ ɦɨɠɧɨ ɪɟ-

ɲɢɬɶ

ɬɨɱɧɨ

, ɛɟɡ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɤɚɤɢɯ-ɥɢɛɨ ɩɪɢɛɥɢɠɟɧɢɣ.

Ʉɪɚɬɤɨ ɩɪɢɜɟɞɟɦ ɪɟɡɭɥɶɬɚɬɵ ɧɟɤɨɬɨɪɵɯ ɧɚɢɛɨɥɟɟ ɯɚɪɚɤ-

ɬɟɪɧɵɯ ɱɢɫɥɟɧɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɩɨ ɜɢɝɧɟɪɨɜɫɤɨɣ ɤɪɢɫɬɚɥ-

ɥɢɡɚɰɢɢ. Ɍɚɤ, Ʌɚɦ ɢ Ƚɢɪɜɢɧ (Lam, Girvin, 1984) ɜ ɩɪɢɛɥɢ-

ɠɟɧɢɢ ɏɚɪɬɪɢ-Ɏɨɤɚ ɪɚɫɫɱɢɬɚɥɢ ɦɟɬɨɞɨɦ Ɇɨɧɬɟ-Ʉɚɪɥɨ

ɜɛɥɢɡɢ

=1/3, 1/5 ɷɧɟɪɝɢɢ ɜɢɝɧɟɪɨɜɫɤɨɝɨ ɤɪɢɫɬɚɥɥɚ (WC) ɢ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɥɚɮɥɢɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ (LL). ɋɨɩɨɫɬɚɜɥɟɧɢɟ ɷɬɢɯ ɷɧɟɪɝɢɣ

ɩɪɢɜɨɞɢɬ ɤ ɜɵɜɨɞɭ ɨ ɜɨɡɦɨɠɧɨɫɬɢ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ ɩɪɢ

<1/5 (ɷɧɟɪɝɢɹ ɧɢɠɟ). ɉɪɚɣɫ, ɉɥɚɰɦɚɧ ɢ ɏɟ (Price, Platzman,

He, 1993) ɩɪɨɜɟɥɢ ɜɚɪɢɚɰɢɨɧɧɵɟ ɪɚɫɱɟɬɵ Ɇɨɧɬɟ-Ʉɚɪɥɨ ɢ

ɩɨɤɚɡɚɥɢ ɩɟɪɟɯɨɞ ɜ ɤɪɢɫɬɚɥɥ ɨɤɨɥɨ

=1/3, 1/5, 1/7. Ɍɚɤɠɟ

ɉɥɚɰɦɚɧ ɢ ɉɪɚɣɫ (Platzman, Price, 1993) ɜ ɩɪɢɛɥɢɠɟɧɧɨɦ

ɩɨɞɯɨɞɟ ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɥɢ ɩɟɪɟɯɨɞ ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ ɢɡ LL ɜ

WC, ɬ.ɟ. ɨɬɨɝɪɟɜ ɢ ɡɚɦɨɪɚɠɢɜɚɧɢɟ ɫɢɫɬɟɦɵ ɩɪɢ

=1/3 ɢ 1/5.

ȼ ɷɬɨɣ ɪɚɛɨɬɟ ɛɵɥɚ ɩɨɫɬɪɨɟɧɚ ɮɚɡɨɜɚɹ ɞɢɚɝɪɚɦɦɚ ɧɚ ɩɥɨɫ-

ɤɨɫɬɢ T

 ɩɚɪɚɦɟɬɪ r

(ɧɚɩɪɹɦɭɸ ɫɜɹɡɚɧɧɵɣ ɫ ɩɥɨɬɧɨɫɬɶɸ

ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ ɱɟɪɟɡ ɫɨɨɬɧɨɲɟɧɢɹ n=1/2

P

), ɩɨɤɚɡɵɜɚɸɳɚɹ ɨɛɥɚɫɬɢ ɪɟɚɥɢɡɚɰɢɢ ɠɢɞɤɨɫɬɢ

ɢ ɤɪɢɫɬɚɥɥɚ. Ɂɭ ɢ Ʌɨɣɟ (Zhu, Louie, 1984) ɜɚɪɢɚɰɢɨɧɧɵɦ

ɦɟɬɨɞɨɦ Ɇɨɧɬɟ-Ʉɚɪɥɨ ɫɪɚɜɧɢɥɢ ɷɧɟɪɝɢɸ ɫɨɫɬɨɹɧɢɣ WC ɢ

LL. ɉɪɢ ɩɟɪɟɫɟɱɟɧɢɢ ɩɨ ɜɟɥɢɱɢɧɟ ɩɨɥɹ ɮɚɤɬɨɪɚ ɡɚɩɨɥɧɟɧɢɹ

=1/5 ɨɧɢ ɧɚɛɥɸɞɚɥɢ ɩɟɪɟɯɨɞ WC-LL-WC.

ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɩɪɢɛɥɢɠɟɧɧɵɟ ɜɚɪɢɚɰɢɨɧɧɵɟ

ɪɚɫɱɟɬɵ (ɧɚɩɪɢɦɟɪ, Ɂɭ ɢ Ʌɨɣɟ, ɢɬ.ɩ.) ɧɟ ɩɨɡɜɨɥɹɸɬ ɞɨɫɬɨ-

ɜɟɪɧɨ ɨɬɥɢɱɢɬɶ ɷɬɢ ɫɨɫɬɨɹɧɢɹ, ɬɚɤ ɤɚɤ ɠɢɞɤɨɫɬɶ Ʌɚɮɥɢɧɚ

ɢɦɟɟɬ ɩɥɨɬɧɨɫɬɶ, ɛɥɢɡɤɭɸ ɤ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɮɚɡɟ, ɢ ɷɧɟɪ-

ɝɢɢ ɫɨɫɬɨɹɧɢɣ WC ɢ LL ɫɪɚɜɧɢɦɵ. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɧɚɢ-

ɛɨɥɟɟ ɭɫɩɟɲɧɵɣ ɢɡɜɟɫɬɧɵɣ

ɬɨɱɧɵɣ

ɱɢɫɥɟɧɧɵɣ ɪɚɫɱɟɬ (Gros

and MacDonald, 1990  ɦɟɬɨɞ ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ) ɛɵɥ

ɩɪɨɜɟɞɟɧ ɧɟ ɧɚ ɩɥɨɫɤɨɫɬɢ, ɚ ɧɚ ɫɮɟɪɟ

ɫ N=12 ɱɚɫɬɢɰɚɦɢ ɢ ɧɚ

m=25 ɨɪɛɢɬɚɥɹɯ. Ɉɞɧɚɤɨ ɛɟɡ ɫɨɦɧɟɧɢɹ ɨɩɬɢɦɚɥɶɧɚɹ ɷɥɟɤ-

ɬɪɨɧɧɚɹ ɤɨɧɮɢɝɭɪɚɰɢɹ ɜ ɜɢɝɧɟɪɨɜɫɤɨɦ ɤɪɢɫɬɚɥɥɟ ɞɨɥɠɧɚ

ɢɦɟɬɶ ɝɟɤɫɚɝɨɧɚɥɶɧɭɸ (ɩɥɨɫɤɭɸ

) ɷɥɟɦɟɧɬɚɪɧɭɸ ɹɱɟɣɤɭ.

Ɉɬɫɸɞɚ ɱɢɫɥɨ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟ ɞɨɥɠɧɨ ɛɵɬɶ ɛɥɢɡɤɨ ɤ 7, 13

ɢɬ.ɞ. ɞɥɹ ɚɞɟɤɜɚɬɧɨɝɨ ɨɩɢɫɚɧɢɹ ɫɨɫɬɨɹɧɢɹ WC (ɫɦ., ɧɚɩɪɢ-

ɦɟɪ, Ʉɚɲɭɪɧɢɤɨɜ, ɉɪɨɤɨɮɶɟɜ, ɋɜɢɫɬɭɧɨɜ ɢ Ɍɭɩɢɰɢɧ ,1996).

ɗɬɢ ɡɧɚɱɟɧɢɹ

ɩɪɢɧɰɢɩɢɚɥɶɧɨ

ɧɟɥɶɡɹ ɪɟɚɥɢɡɨɜɚɬɶ ɧɚ ɫɮɟɪɟ.

Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɱɢɬɚɬɟɥɸ ɛɵɥɨ ɛɨɥɟɟ ɩɨɧɹɬɧɵ ɩɪɨɛɥɟ-

ɦɵ ɫɨɜɪɟɦɟɧɧɨɝɨ ɤɨɦɩɶɸɬɟɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɤɪɚɬɤɨ

ɩɪɨɞɟɦɨɧɫɬɪɢɪɭɟɦ ɨɫɨɛɟɧɧɨɫɬɢ ɢ ɫɥɨɠɧɨɫɬɶ ɱɢɫɥɟɧɧɨɝɨ

“ɷɤɫɩɟɪɢɦɟɧɬɚ” ɩɪɢ ɢɫɫɥɟɞɨɜɚɧɢɢ ɩɪɨɛɥɟɦɵ ɜɢɝɧɟɪɨɜɫɤɨɣ

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ, ɚ ɬɚɤɠɟ ɮɢɡɢɱɟɫɤɢɟ ɤɪɢɬɟɪɢɢ, ɧɟɨɛɯɨɞɢ-

ɦɵɟ ɩɪɢ ɢɡɭɱɟɧɢɢ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ “ɠɢɞɤɨɫɬɶ-ɤɪɢɫɬɚɥɥ”.

Ɋɚɫɫɦɨɬɪɢɦ ɞɜɭɦɟɪɧɵɣ Ɏɟɪɦɢ-ɝɚɡ ɫ ɤɭɥɨɧɨɜɫɤɢɦ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɟɦ, ɬ.ɟ. ɦɨɞɟɥɶ (3.4), ɤɚɤ ɩɪɟɞɥɚɝɚɥ Ʌɚɮɥɢɧ. ɑɢɫ-

ɥɟɧɧɨ ɪɟɲɚɬɶ ɬɚɤɭɸ ɫɢɫɬɟɦɭ ɦɨɠɧɨ, ɩɟɪɟɣɞɹ ɨɬ ɤɨɨɪɞɢɧɚɬ-

ɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɜ ɤɚɤɨɣ ɧɢɛɭɞɶ ɞɢɫɤɪɟɬɧɵɣ ɛɚɡɢɫ. ɉɨ-

ɷɬɨɦɭ ɫɥɟɞɭɟɬ ɩɟɪɟɩɢɫɚɬɶ Ƚɚɦɢɥɶɬɨɧɢɚɧ (3.4) ɜ ɩɪɟɞɫɬɚɜɥɟ-

ɧɢɢ ɜɬɨɪɢɱɧɨɝɨ ɤɜɚɧɬɨɜɚɧɢɹ ɩɨ ɧɚɢɛɨɥɟɟ ɚɞɟɤɜɚɬɧɨɦɭ ɡɚɞɚ-

ɱɟ ɩɨɥɧɨɦɭ ɛɚɡɢɫɭ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ. Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ,

ɜ ɫɢɥɶɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ (ɢɦɟɟɬ ɫɦɵɫɥ ɜɵɛɪɚɬɶ ɬɨɥɶɤɨ

ɨɞɧɭ ɨɪɢɟɧɬɚɰɢɸ ɫɩɢɧɚ, ɬ.ɟ. ɛɟɫɫɩɢɧɨɜɭɸ ɫɢɬɭɚɰɢɸ  ɜɫɟ

ɫɩɢɧɵ ɱɚɫɬɢɰ ɧɚɩɪɚɜɥɟɧɵ ɩɨ ɩɨɥɸ) ɢɦɟɟɬɫɹ ɯɨɪɨɲɟɟ ɤɜɚɧ-

ɬɨɜɨɟ ɱɢɫɥɨ  ɭɝɥɨɜɨɣ ɦɨɦɟɧɬ m ɜ ɩɟɪɜɨɣ ɡɨɧɟ Ʌɚɧɞɚɭ. ɉɨ-

ɷɬɨɦɭ ɭɞɨɛɧɨ ɩɨɥɧɵɣ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɵɣ ɛɚɡɢɫ ɜɵɛɪɚɬɶ

ɩɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɨɞɧɨɱɚɫɬɢɱɧɵɦ ɫɨɫɬɨɹɧɢɹɦ ɷɥɟɤɬɪɨɧɚ

ɜ ɷɬɨɣ ɡɨɧɟ |m> (3.3).

ɍɱɢɬɵɜɚɹ ɬɨɥɶɤɨ ɤɭɥɨɧɨɜɫɤɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɦɟɠɞɭ

ɱɚɫɬɢɰɚɦɢ, ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧ ɮɟɪɦɢ-ɝɚɡɚ

ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɜ ɷɬɨɦ ɛɚɡɢɫɟ:



mmm

mmmm

mmm

aaaaV

,,,

4321

432

(4.2)

ɝɞɟ a

- ɨɩɟɪɚɬɨɪ ɪɨɠɞɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɜ ɫɨɫɬɨɹɧɢɢ (ɧɚ ɨɪ-

ɛɢɬɚɥɢ) |m>, Vm

 ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɤɭɥɨɧɨɜ-

ɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜ ɞɜɭɦɟɪɧɨɦ ɫɥɭɱɚɟ (ɨ ɪɚɫɱɟɬɟ ɦɚɬ-

ɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ ɢ ɟɝɨ ɹɜɧɵɣ ɜɢɞ ɫɦ. ɡɚɞɚɱɭ). Ⱦɥɹ ɝɚɦɢɥɶ-

ɬɨɧɢɚɧɚ (4.2) ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɭɝɥɨɜɨɝɨ ɦɨ-

ɦɟɧɬɚ m

+ m

= m

+ m

. Ɍɚɤ ɤɚɤ ɦɵ ɢɧɬɟɪɟɫɭɟɦɫɹ ɩɟɪɟɯɨ-

ɞɨɦ “ɤɪɢɫɬɚɥɥ-ɠɢɞɤɨɫɬɶ”, ɬɨ ɭɩɪɨɫɬɢɦ ɡɚɞɚɱɭ ɢ ɪɚɫɩɨɥɨ-

ɠɢɦ ɢɫɯɨɞɧɨ ɱɚɫɬɢɰɵ ɜ ɤɥɚɫɫɢɱɟɫɤɭɸ ɬɪɟɭɝɨɥɶɧɭɸ ɪɟɲɟɬ-

ɤɭ (ɪɢɫ. 4.1) ɫɭɡɥɚɦɢ, ɩɨɦɟɳɟɧɧɵɦɢ ɧɚ ɧɚɢɧɢɡɲɭɸ ɧɭɥɟ-

ɜɭɸ ɨɪɛɢɬɚɥɶ (m=0) ɜ ɫɨɫɬɨɹɧɢɹ

( Z

) =

m=0



( z - Z

) ɜ

ɩɨɡɢɰɢɢ Z

, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɡɥɚɦ ɪɟɲɟɬɤɢ – ɬ.ɟ. ɤɚɤ ɛɵ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

«ɡɚɦɨɪɚɠɢɜɚɟɦ» ɱɚɫɬɢɰɵ. ɉɟɪɢɨɞ ɬɚɤɨɝɨ ɜɢɝɧɟɪɨɜɫɤɨɝɨ

ɤɪɢɫɬɚɥɥɚ a ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɷɥɟɤɬɪɨɧɧɭɸ ɩɥɨɬɧɨɫɬɶ n

ɮɚɤɬɨɪ ɡɚɩɨɥɧɟɧɢɹ ɨɪɛɢɬɚɥɟɣ

= 2

S

ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ

ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɫɨɨɬɧɨɲɟɧɢɟɦ (4.1). Ⱦɚɥɟɟ ɜɫɟ ɪɚɫɫɬɨɹɧɢɹ

ɛɭɞɟɦ ɢɫɱɢɫɥɹɬɶ ɜ ɦɚɝɧɢɬɧɵɯ ɞɥɢɧɚɯ, ɚ ɜɫɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ

ɜɟɥɢɱɢɧɵ  ɜ ɟɞɢɧɢɰɚɯ e

H

- ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɨɫɬɨ-

ɹɧɧɚɹ. ȼɚɠɧɨ, ɱɬɨ ɦɟɧɹɹ ɩɟɪɢɨɞ, ɦɵ ɦɨɠɟɦ ɜ ɞɚɧɧɨɣ ɫɯɟɦɟ

ɦɟɧɹɬɶ ɮɚɤɬɨɪ ɡɚɩɨɥɧɟɧɢɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɥɨɠɟɧɧɵɦ

ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ.

Ⱦɥɹ ɬɨɱɧɨɝɨ ɤɜɚɧɬɨɜɨɦɟɯɚɧɢɱɟɫɤɨɝɨ ɪɚɫɱɟɬɚ ɦɵ ɞɨɥɠɧɵ

“ɪɚɡɦɨɪɨɡɢɬɶ” ɰɟɧɬɪɚɥɶɧɭɸ ɱɚɫɬɢɰɭ, ɡɚɬɟɦ ɩɟɪɜɭɸ ɤɨɨɪɞɢ-

ɧɚɰɢɨɧɧɭɸ ɫɮɟɪɭ (ɟɳɟ 6 ɱɚɫɬɢɰ), ɢ ɟɫɥɢ ɩɨɡɜɨɥɢɥɢ ɛɵ ɜɨɡ-

ɦɨɠɧɨɫɬɢ ɤɨɦɩɶɸɬɟɪɚ, ɬɨ ɢ ɫɥɟɞɭɸɳɭɸ (ɟɳɟ 6 ) ɢ

ɬ.ɞ., ɢ

ɞɥɹ ɧɢɯ ɬɨɱɧɨ ɪɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ. Ɉɞɧɚɤɨ ɱɢɫ-

ɥɨ ɨɪɛɢɬ ɜ ɜɵɪɚɠɟɧɢɢ (4.2) ɞɨɥɠɧɨ ɛɵɬɶ ɨɝɪɚɧɢɱɟɧɨ ɢɡ ɮɢ-

ɡɢɱɟɫɤɢɯ ɫɨɨɛɪɚɠɟɧɢɣ ɢ ɢɡ ɨɝɪɚɧɢɱɟɧɢɣ ɤɨɦɩɶɸɬɟɪɧɨɣ

ɩɚɦɹɬɢ ɢ ɛɵɫɬɪɨɞɟɣɫɬɜɢɹ. Ɇɵ ɞɨɥɠɧɵ ɞɨɛɢɬɶɫɹ ɬɚɤɨɣ ɫɢ-

ɬɭɚɰɢɢ, ɱɬɨɛɵ ɩɨɫɥɟɞɭɸɳɟɟ ɭɜɟɥɢɱɟɧɢɟ ɦɚɤɫɢɦɚɥɶɧɨɝɨ

ɡɧɚɱɟɧɢɹ ɨɪɛɢɬɵ ɧɟ ɦɟɧɹɥɨ ɫɭɳɟɫɬɜɟɧɧɨ ɢɯ ɡɚɩɨɥɧɟɧɢɹ –

ɷɬɨ, ɤ ɫɨɠɚɥɟɧɢɸ, ɦɨɠɧɨ ɩɪɨɜɟɪɢɬɶ

ɬɨɥɶɤɨ ɧɟɩɨɫɪɟɞɫɬɜɟɧ-

ɧɨ ɜ ɪɚɫɱɟɬɟ. Ɍɚɤɢɟ ɨɰɟɧɤɢ ɞɚɸɬ, ɱɬɨ ɭɠɟ ɩɪɢ ɭɱɟɬɟ ɞɜɭɯ

ɤɨɨɪɞɢɧɚɰɢɨɧɧɵɯ ɫɮɟɪ (N=13) ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ 65 ɨɪɛɢɬ

ɧɟɨɛɯɨɞɢɦɵ ɞɥɹ ɪɚɫɱɟɬɚ ɫ ɮɚɤɬɨɪɨɦ ɡɚɩɨɥɧɟɧɢɹ 1/5 (ɢɥɢ 40

ɞɥɹ

=1/3), ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɧɟɞɨɫɬɢɠɢɦɨɦɭ ɞɥɹ ɢɫɩɨɥɶɡɭɟ-

ɦɨɝɨ ɡɞɟɫɶ ɦɟɬɨɞɚ ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ ɪɚɡɦɟɪɭ ɝɢɥɶɛɟɪ-

ɬɨɜɚ ɩɪɨɫɬɪɚɧɫɬɜɚ (ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɧɟɨɛɯɨɞɢɦɨɣ ɞɥɹ

ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ ɝɚɦɢɥɶɬɨɧɨɜɨɣ ɦɚɬɪɢɰɵ).

ɉɨɹɫɧɢɦ ɡɞɟɫɶ ɤɪɚɬɤɨ ɫɭɬɶ ɦɟɬɨɞɚ ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚ-

ɰɢɢ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɧɚɲɟɣ ɡɚɞɚɱɟ. ɇɚɦ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ

ɷɧɟɪɝɢɸ ɢ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ Ƚɚɦɢɥɶɬɨɧɢɚɧɚ (4.2), ɬ.ɟ. ɪɟ-

ɲɢɬɶ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ, ɩɪɢɱɟɦ ɬɨɱɧɨ,

ɧɨ ɞɥɹ ɨɝɪɚɧɢ-

ɱɟɧɧɨɣ ɫɢɫɬɟɦɵ. Ɂɞɟɫɶ ɨɝɪɚɧɢɱɟɧɧɨɫɬɶ ɞɨɫɬɢɝɚɟɬɫɹ ɤɨɧɟɱ-

ɧɵɦ ɱɢɫɥɨɦ ɨɪɛɢɬɚɥɟɣ m ɢ ɱɢɫɥɚ ɷɥɟɤɬɪɨɧɨɜ N, ɡɚɩɨɥɧɹɸ-

ɳɢɯ ɷɬɢ ɨɪɛɢɬɚɥɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɩɨɥɧɨɝɨ ɩɟɪɟɛɨɪɚ

ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɫɨɫɬɨɹɧɢɣ ɫɢɫɬɟɦɵ ɦɵ ɞɨɥɠɧɵ ɪɚɫɩɨɥɨ-

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɠɢɬɶ N ɷɥɟɤɬɪɨɧɨɜ ɧɚ m ɨɪɛɢɬɚɥɹɯ ɜɫɟɦɢ ɜɨɡɦɨɠɧɵɦɢ ɫɩɨ-

ɫɨɛɚɦɢ ɫ ɭɱɟɬɨɦ ɬɨɠɞɟɫɬɜɟɧɧɨɫɬɢ. Ʌɟɝɤɨ ɩɨɧɹɬɶ, ɱɬɨ ɬɚɤɢɯ

ɜɚɪɢɚɧɬɨɜ C

. ɗɬɨ ɢ ɟɫɬɶ ɪɚɡɦɟɪ ɝɢɥɶɛɟɪɬɨɜɚ ɩɪɨɫɬɪɚɧɫɬɜɚ.

Ⱦɚɥɟɟ, ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɜɫɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɩɟɪɟɯɨɞɚ

ɢɡ ɨɞɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ ɞɪɭɝɢɟ ɡɚ ɫɱɟɬ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚ

(4.2), ɢ ɫɬɪɨɢɬɫɹ Ƚɚɦɢɥɶɬɨɧɨɜɚ ɦɚɬɪɢɰɚ. Ɂɚɞɚɱɚ ɫɜɨɞɢɬɫɹ ɤ

ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ ɷɬɨɣ ɦɚɬɪɢɰɵ (ɬ.ɟ. ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɟɟ ɜ ɞɢɚ-

ɝɨɧɚɥɶɧɨɦ ɜɢɞɟ, ɱɬɨ ɟɫɬɶ ɩɟɪɟɯɨɞ ɤ ɫɨɛɫɬɜɟɧɧɨ-

ɷɧɟɪɝɟɬɢɱɟɫɤɨɦɭ ɛɚɡɢɫɭ). ɉɨɥɭɱɚɸɳɢɟɫɹ ɩɪɢ ɷɬɨɦ ɫɨɛɫɬ-

ɜɟɧɧɵɟ ɱɢɫɥɚ ɢ ɜɟɤɬɨɪɚ ɟɫɬɶ ɢɫɤɨɦɵɣ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ

ɫɩɟɤɬɪ ɢ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ (ɪɚɡɥɨɠɟɧɧɵɟ ɩɨ ɢɫɯɨɞɧɨɦɭ

ɛɚɡɢɫɭ |m>). Ɉɞɧɚɤɨ ɪɚɡɦɟɪ Ƚɚɦɢɥɶɬɨɧɨɜɨɣ ɦɚɬɪɢɰɵ ɫɢɥɶɧɨ

ɪɚɫɬɟɬ ɫ ɭɜɟɥɢɱɟɧɢɟɦ m ɢ N. Ɍɚɤ, ɩɪɢ ɱɢɫɥɟ ɨɪɛɢɬɚɥɟɣ

m=25 ɢ ɱɢɫɥɟ ɷɥɟɤɬɪɨɧɨɜ N=7 ɪɚɡɦɟɪ ɦɚɬɪɢɰɵ ɪɚɜɟɧ ɭɠɟ

480700. ɉɨɷɬɨɦɭ ɩɪɢɯɨɞɢɬɫɹ ɨɝɪɚɧɢɱɢɬɶ ɪɚɫɱɟɬ ɩɟɪɜɨɣ

ɹɱɟɣɤɨɣ ɜɢɝɧɟɪɨɜɫɤɨɝɨ ɤɪɢɫɬɚɥɥɚ ɢ 7-ɸ ɱɚɫɬɢɰɚɦɢ, ɧɨ ɞɥɹ

ɧɢɯ ɤɜɚɧɬɨɜɭɸ ɡɚɞɚɱɭ ɪɟɲɢɬɶ ɬɨɱɧɨ

. ɂɬɚɤ, ɜɵɞɟɥɹɟɦ 7

ɷɥɟɤɬɪɨɧɨɜ ɫ Z

= 0 ɢ Z

= a ɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɞɥɹ ɧɢɯ ɩɨɥ-

ɧɭɸ ɤɜɚɧɬɨɜɭɸ ɡɚɞɚɱɭ (4.2), ɭɞɟɪɠɢɜɚɹ ɞɪɭɝɢɟ ɷɥɟɤɬɪɨɧɵ ɜ

ɫɨɫɬɨɹɧɢɢ c m=0 ɢ ɨɫɬɚɜɥɹɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɢɯ ɫ ɩɟɪɜɵɦɢ

ɫɟɦɶɸ ɱɚɫɬɢɰɚɦɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (4.2) (ɪɢɫ.4.1). “Ɂɚɦɨɪɨ-

ɠɟɧɧɚɹ” ɬɪɟɭɝɨɥɶɧɚɹ ɪɟɲɟɬɤɚ ɫɬɚɛɢɥɢɡɢɪɭɟɬ ɡɚɞɚɱɭ, ɹɜɥɹ-

ɟɬɫɹ ɤɚɤ ɛɵ “ɫɪɟɞɧɢɦ ɩɨɥɟɦ” ɞɥɹ ɤɜɚɧɬɨɜɨɣ ɡɚɞɚɱɢ, ɢɫɫɥɟ-

ɞɭɟɦɨɣ ɜ ɩɪɟɞɟɥɚɯ ɩɟɪɜɨɣ ɤɨɨɪɞɢɧɚɰɢɨɧɧɨɣ ɫɮɟɪɵ ɢ ɞɨ

ɩɭɫɤɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɜɢɝɧɟɪɨɜɫɤɨɣ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ. ȼ

ɩɪɢɧɰɢɩɟ, ɭɠɟ ɩɟɪɜɨɣ ɤɨɨɪɞɢɧɚɰɢɨɧɧɨɣ ɫɮɟɪɵ ɞɨɫɬɚɬɨɱɧɨ,

ɱɬɨɛɵ ɮɢɤɫɢɪɨɜɚɬɶ ɩɟɪɟɯɨɞ ɨɬ ɤɨɪɪɟɥɹɰɢɣ ɜ ɤɪɢɫɬɚɥɥɟ, ɥɨ-

ɤɚɥɢɡɨɜɚɧɧɵɯ ɧɚ ɦɚɫɲɬɚɛɟ ɩɟɪɢɨɞɚ ɪɟɲɟɬɤɢ, ɤ ɛɨɥɟɟ ɞɥɢɧ-

ɧɨɜɨɥɧɨɜɵɦ ɤɨɪɪɟɥɹɰɢɹɦ ɜ ɠɢɞɤɨɫɬɢ Ʌɚɮɥɢɧɚ. Ɉɬɦɟɬɢɦ,

ɱɬɨ ɨɛɵɱɧɨ ɪɚɞɢɭɫ ɤɪɭɝɚ, ɜɵɪɟɡɚɧɧɨɝɨ ɢɡ ɛɟɫɤɨɧɟɱɧɨɣ

ɩɥɨɫɤɨɫɬɢ ɞɥɹ “ɫɪɟɞɧɟɝɨ ɩɨɥɹ”, ɜɵɛɢɪɚɥɫɹ ɪɚɡɦɟɪɨɦ ɨɤɨɥɚ

100 ɩɟɪɢɨɞɨɜ ɪɟɲɟɬɤɢ, ɬɚɤ

ɱɬɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɭɠɟ ɧɟ

ɜɥɢɹɥɢ ɧɚ ɪɟɡɭɥɶɬɚɬɵ.

Ⱦɥɹ 7-ɦɢ ɷɥɟɤɬɪɨɧɨɜ ɜɵɛɢɪɚɥɚɫɶ 31-ɚ ɨɪɛɢɬɚɥɶ (m=0-

30), ɢɯ ɱɢɫɥɨ ɩɨɞɛɢɪɚɥɨɫɶ ɢɡ ɭɫɥɨɜɢɹ, ɱɬɨ ɱɢɫɥɚ ɡɚɩɨɥɧɟɧɢɹ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

>, ɨɩɪɟɞɟɥɹɸɳɢɟ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɯɨɠɞɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɜ

ɫɨɫɬɨɹɧɢɢ ɫ ɨɪɛɢɬɚɥɶɧɵɦ ɦɨɦɟɧɬɨɦ m, ɧɟ ɢɡɦɟɧɹɸɬɫɹ (ɫ

ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɬɨɱɧɨɫɬɶɸ 0.001) ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ ɨɪ-

ɛɢɬ, ɢ ɫɥɟɞɭɸɳɢɟ ɫɨɫɬɨɹɧɢɹ (ɫ m >30) ɧɟ ɡɚɩɨɥɧɟɧɵ ɫ ɷɬɨɣ

ɠɟ ɬɨɱɧɨɫɬɶɸ. Ⱦɚɠɟ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɪɚɡɦɟɪ ɝɚɦɢɥɶɬɨɧɨɜɨɣ

ɦɚɬɪɢɰɵ (ɬ.ɟ. ɝɢɥɶɛɟɪɬɨɜɚ ɩɪɨɫɬɪɚɧɫɬɜɚ) ɪɚɜɟɧ

2629575 x

2629575 ! Ʉɪɨɦɟ ɬɨɝɨ, ɢɡ-ɡɚ ɤɭɥɨɧɨɜɫɤɨɝɨ ɞɚɥɶɧɨɞɟɣɫɬɜɢɹ

(ɜɫɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɟɬ ɫɨ ɜɫɟɦ) ɝɚɦɢɥɶɬɨɧɨɜɚ ɦɚɬɪɢɰɚ ɹɜɥɹ-

ɟɬɫɹ ɛɨɥɟɟ ɩɥɨɬɧɨɣ, ɱɟɦ ɜ ɫɬɚɧɞɚɪɬɧɵɯ ɭɡɟɥɶɧɵɯ ɡɚɞɚɱɚɯ

(ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ, t-J, Ȼɨɡɟ-ɏɚɛɛɚɪɞ, ɫɩɢɧɨɜɵɟ ɢ ɬ.ɞ), ɱɬɨ

ɭɫɥɨɠɧɹɟɬ ɟɟ ɯɪɚɧɟɧɢɟ ɜ ɩɚɦɹɬɢ ɤɨɦɩɶɸɬɟɪɚ ɢ ɨɛɪɚɛɨɬɤɭ.

ɂɬɚɤ, ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ ɚɧɚɥɢɡɚ ɞɢɚɝɨɧɚɥɢɡɨɜɚɥɚɫɶ ɦɚɬɪɢ-

ɰɚ ɫɥɟɞɭɸɳɟɝɨ ɝɚɦɢɥɶɬɨɧɢɚɧɚ, ɢɦɟɸɳɚɹ

ɥɢɧɟɣɧɵɣ ɪɚɡɦɟɪ

ɛɨɥɟɟ 2.5 ɦɢɥɥɢɨɧɨɜ:







mmmm

mmmmmmmm

aaV

aaaaV

)(

,,,

4321

43214321

(4.3)

ɝɞɟ V

(MF)

 ɫɬɚɛɢɥɢɡɢɪɭɸɳɟɟ “ɫɪɟɞɧɟɟ ɩɨɥɟ” ɜɧɟɲɧɢɯ “

ɡɚɦɨɪɨɠɟɧɧɵɯ” ɷɥɟɤɬɪɨɧɨɜ (ɟɝɨ ɪɚɫɱɟɬ  ɫɚɦɨɫɬɨɹɬɟɥɶɧɚɹ

ɡɚɞɚɱɚ). ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɨɧɨ ɧɟ ɫɢɦɦɟɬɪɢɱɧɨ, ɢ ɜɵɡɵɜɚɟɬ

ɩɟɪɟɯɨɞɵ ɫ ɢɡɦɟɧɟɧɢɟɦ ɦɨɦɟɧɬɚ ɭ ɜɧɭɬɪɟɧɧɢɯ ɷɥɟɤɬɪɨɧɨɜ

ɧɚ

nm 6r '

, ɝɞɟ n  ɰɟɥɨɟ (ɷɬɨ ɩɪɨɢɫɯɨɞɢɬ ɢɡ-ɡɚ ɫɢɦɦɟɬ-

ɪɢɢ ɬɪɟɭɝɨɥɶɧɨɣ ɪɟɲɟɬɤɢ). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɷɬɚ ɤɨɪɪɟɥɹɰɢɹ

ɜɵɝɨɞɧɚ ɞɥɹ “ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɝɨ” ɫɨɫɬɨɹɧɢɹ ɰɟɧɬɪɚɥɶɧɨɝɨ

ɤɥɚɫɬɟɪɚ. ɑɬɨɛɵ ɨɛɨɣɬɢ ɷɬɭ ɩɪɨɛɥɟɦɭ, ɢ ɧɟ ɨɛɥɟɝɱɚɬɶ ɫɢɫ-

ɬɟɦɟ ɤɪɢɫɬɚɥɥɢɡɚɰɢɸ, ɦɵ ɢɝɧɨɪɢɪɭɟɦ ɜɫɟ ɩɟɪɟɯɨɞɵ, ɤɪɨɦɟ

'm = 0. Ɍɚɤɚɹ ɩɪɨɰɟɞɭɪɚ ɷɤɜɢɜɚɥɟɧɬɧɚ ɭɫɪɟɞɧɟɧɢɸ ɩɨ ɭɝ-

ɥɚɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɪɚɳɟɧɢɹ ɫɬɚɛɢɥɢɡɢɪɭɸɳɟɝɨ ɩɨɬɟɧɰɢɚɥɚ

ɜɨɤɪɭɝ

ɰɟɧɬɪɚɥɶɧɨɝɨ ɤɥɚɫɬɟɪɚ, ɜ ɢɬɨɝɟ ɫɢɫɬɟɦɚ ɫɨɯɪɚɧɹɟɬ

ɩɨɥɧɵɣ ɦɨɦɟɧɬ, ɢ ɦɨɠɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ ɷɬɢɦ ɢɧɜɚɪɢɚɧɬɨɦ

ɩɪɢ ɪɚɫɱɟɬɟ.

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɑɬɨɛɵ ɧɟ ɭɬɨɦɥɹɬɶ ɱɢɬɚɬɟɥɹ ɰɟɥɵɦ ɧɚɛɨɪɨɦ ɩɪɨɛɥɟɦ,

ɫɜɹɡɚɧɧɵɯ ɫ ɨɫɨɛɟɧɧɨɫɬɹɦɢ ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ ɞɚɧɧɨɣ

ɨɱɟɧɶ ɫɥɨɠɧɨɣ ɡɚɞɚɱɢ (ɫɬɨɢɬ ɥɢɲɶ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɬɨɥɶɤɨ

“ɩɨɞɝɨɬɨɜɢɬɟɥɶɧɚɹ” ɡɚɞɚɱɚ ɤɨɪɪɟɤɬɧɨɝɨ ɫɭɦɦɢɪɨɜɚɧɢɹ ɦɚɬ-

ɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ (ɹɜɧɵɣ ɜɢɞ ɤɨɬɨɪɵɯ ɩɨɤɚɡɚɧ ɜ ɡɚɞɚɱɚɯ

ɧɢɠɟ) ɹɜɥɹɟɬɫɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɧɟɬɪɢɜɢɚɥɶɧɨɣ ɩɪɨɛɥɟɦɨɣ,

ɩɨ ɫɥɨɠɧɨɫɬɢ, ɫɨɩɨɫɬɚɜɢɦɨɣ ɫ ɫɚɦɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɟɣ),

ɫɪɚɡɭ ɩɟɪɟɣɞɟɦ ɤ ɧɚɢɛɨɥɟɟ ɡɧɚɱɢɦɨɦɭ

ɪɟɡɭɥɶɬɚɬɭ ɪɚɫɱɟɬɚ.

ɇɚɢɛɨɥɟɟ ɚɞɟɤɜɚɬɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɩɨɡɜɨɥɹɸɳɟɣ ɜɵɹɜɢɬɶ

ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ “ɬɜɟɪɞɵɦ” ɢ “ɠɢɞɤɢɦ” ɫɨɫɬɨɹɧɢɹɦɢ ɜ ɪɚɫ-

ɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɟ, ɹɜɥɹɟɬɫɹ ɩɚɪɧɚɹ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ

ɮɭɧɤɰɢɹ, ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɯɨɠɞɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɜ ɬɨɱɤɟ z

ɩɪɢ ɭɫɥɨɜɢɢ ɧɚɯɨɠɞɟɧɢɹ ɜɬɨɪɨɝɨ ɷɥɟɤɬɪɨɧɚ ɜ ɬɨɱɤɟ z

. ɂɡ

ɫɨɨɛɪɚɠɟɧɢɣ ɫɢɦɦɟɬɪɢɢ ɦɨɠɧɨ ɨɠɢɞɚɬɶ (ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ

ɞɥɹ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɝɨ ɮɚɤɬɨɪɚ ɡɚɩɨɥɧɟɧɢɹ), ɱɬɨ ɨɞɧɚ ɱɚɫ-

ɬɢɰɚ ɛɭɞɟɬ ɪɚɫɩɨɥɨɠɟɧɚ ɜ ɰɟɧɬɪɟ ɤɥɚɫɬɟɪɚ, ɚ ɨɫɬɚɥɶɧɵɟ 6

ɱɚɫɬɢɰ ɛɭɞɭɬ ɢɦɟɬɶ ɦɚɤɫɢɦɭɦ ɩɥɨɬɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚ

ɪɚɫɫɬɨɹɧɢɢ ɩɪɢɦɟɪɧɨ a(

), ɬ.ɟ. ɧɚ ɩɟɪɜɨɣ ɤɨɨɪɞɢɧɚɰɢɨɧɧɨɣ

ɫɮɟɪɟ. ɋɨɫɬɨɹɧɢɟ ɷɬɢɯ ɷɥɟɤɬɪɨɧɨɜ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɢɬ ɨɬ

ɭɝɥɚ

ɦɟɠɞɭ ɤɨɨɪɞɢɧɚɬɚɦɢ ɱɚɫɬɢɰ ɧɚ ɩɟɪɜɨɣ ɤɨɨɪɞɢɧɚɰɢ-

ɨɧɧɨɣ ɫɮɟɪɟ. Ɍɨɝɞɚ ɚɞɟɤɜɚɬɧɭɸ ɩɚɪɧɭɸ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ

ɮɭɧɤɰɢɸ ɨɩɪɟɞɟɥɢɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

(

) = <

)<(z

>, |z

|= |z

| = a , (4.4)

ɝɞɟ ɜɚɪɶɢɪɭɟɬɫɹ ɜ ɢɧɬɟɪɜɚɥɟ (0,

), |

> -

ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɩɨɥɭɱɟɧɧɨɟ ɢɡ ɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢɡɚɰɢɢ.

ȼ ɬɜɟɪɞɨɦ ɫɨɫɬɨɹɧɢɢ, ɨɛɥɚɞɚɸɳɢɦ ɞɚɥɶɧɢɦ ɩɨɪɹɞɤɨɦ, ɫɥɟ-

ɞɭɟɬ ɨɠɢɞɚɬɶ ɬɪɢ ɯɨɪɨɲɨ ɨɩɪɟɞɟɥɟɧɧɵɟ ɨɫɰɢɥɥɹɰɢɢ g(

) , ɜ

ɬɨ ɜɪɟɦɹ ɤɚɤ ɜ ɠɢɞɤɨɣ ɮɚɡɟ, ɢɦɟɸɳɟɣ ɬɨɥɶɤɨ ɛɥɢɠɧɢɣ ɩɨ-

ɪɹɞɨɤ, ɷɬɢ ɤɨɥɟɛɚɧɢɹ ɞɨɥɠɧɵ ɛɵɬɶ ɫɢɥɶɧɨ ɩɨɞɚɜɥɟɧɵ. ȼ

ɥɸɛɨɦ ɫɥɭɱɚɟ ɪɟɡɤɨɟ (ɫɤɚɱɤɨɨɛɪɚɡɧɨɟ) ɢɡɦɟɧɟɧɢɟ ɤɨɪɪɟɥɹ-

ɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ g

)arg()arg(

zz  T

(

) ɜ ɡɚɜɢɫɢɦɨɫɬɢ

Кашурников В.А., Маймистов А.И. Современные проблемы физики твердого тела. Часть1 - Целый и дробный квантовые эффекты Холла

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