Елесин В.Ф., Кашурников В.А. Физика фазовых переходов

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8.3. Ɇɨɞɟɥɶ ɏɚɛɛɚɪɞɚ

Ⱦɥɹ ɨɛɴɹɫɧɟɧɢɹ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɦɟɬɚɥɥ-ɢɡɨɥɹɬɨɪ ɜ

ɩɟɪɟɯɨɞɧɵɯ ɦɟɬɚɥɥɚɯ c ɭɡɤɢɦɢ ɡɨɧɚɦɢ ɏɚɛɛɚɪɞ ɜ 1964ɝ. ɩɪɟɞ-

ɥɨɠɢɥ ɦɨɞɟɥɶ, ɤɨɬɨɪɚɹ ɜ ɪɟɠɢɦɟ ɫɢɥɶɧɨɣ ɫɜɹɡɢ, ɭɱɢɬɵɜɚɹ ɩɟ-

ɪɟɫɤɨɤɢ ɷɥɟɤɬɪɨɧɨɜ ɧɚ ɫɨɫɟɞɧɢɟ ɚɬɨɦɵ ɢ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥ-

ɤɢɜɚɧɢɟ ɧɚ ɭɡɥɟ, ɩɨɡɜɨɥɢɥɚ ɨɩɢɫɚɬɶ ɩɟɪɟɯɨɞ ɢɡ ɩɪɨɜɨɞɹɳɟɝɨ ɜ

ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɫɬɨɹɧɢɟ. ɗɬɚ ɦɨɞɟɥɶ (ɢ ɟɟ ɪɚɫɲɢɪɟɧɧɵɟ

ɚɧɚɥɨɝɢ) ɜ

ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɫɬɚɥɚ ɩɨɩɭɥɹɪɧɨɣ ɜ ɫɜɹɡɢ ɫ ɢɫɫɥɟ-

ɞɨɜɚɧɢɟɦ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɫɜɟɪɯɩɪɨɜɨɞɧɢɤɨɜ, ɬɚɤ ɤɚɤ ɜ

ɧɢɯ ɢɦɟɸɬ ɦɟɫɬɨ ɫɢɥɶɧɵɟ ɤɭɥɨɧɨɜɫɤɢɟ ɤɨɪɪɟɥɹɰɢɢ.

8.3.1. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ.

Ɉɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ

Ⱦɥɹ ɜɵɜɨɞɚ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞ ɢɫɯɨɞɢɥ ɢɡ ɫɬɚɧɞɚɪɬɧɨɝɨ

ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɮɟɪɦɢ-ɝɚɡɚ ɫ ɤɭɥɨɧɨɜɫɤɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ:

Haa

Vaaaa

pp p

pp p p p p p p

pp p p





VV VV

'' , ,' '' '

'' '

/.12

121 2 1 2 2 1

121 2

(8.33)

Ʉɭɥɨɧɨɜɫɤɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɜ (8.33) ɜɵɩɢɫɚɧɨ ɜ ɫɭɳɟɫɬɜɟɧɧɨ

ɛɨɥɟɟ ɨɛɳɟɦ ɜɢɞɟ, ɱɟɦ, ɧɚɩɪɢɦɟɪ, ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɥɢ ɩɪɢ ɢɫ-

ɫɥɟɞɨɜɚɧɢɢ ɦɨɞɟɥɢ ɋɬɨɧɟɪɚ ɜ ɝɥɚɜɟ 6 (ɫɦ. (6.12)). Ɂɞɟɫɶ ɭɱɬɟɧɨ

ɢ ɩɪɹɦɨɟ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ (ɤɚɤ ɜ ɦɨɞɟɥɢ ɋɬɨɧɟɪɚ), ɢ

ɨɛɦɟɧɧɵɟ ɩɪɨɰɟɫɫɵ; ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫɢɫɬɟɦɵ ɫ ɭɡɤɢɦɢ ɡɨ-

ɧɚɦɢ (~ 0.1-1 ɷȼ), ɬ.ɟ. ɷɥɟɤɬɪɨɧɵ ɢɦɟɸɬ ɛɨɥɶɲɭɸ ɷɮɮɟɤɬɢɜ-

ɧɭɸ ɦɚɫɫɭ ɢ

ɩɨɱɬɢ ɥɨɤɚɥɢɡɨɜɚɧɵ. ɉɨɷɬɨɦɭ ɯɨɪɨɲɢɦ ɤɜɚɧɬɨ-

ɜɵɦ ɱɢɫɥɨɦ ɹɜɥɹɟɬɫɹ ɧɨɦɟɪ ɭɡɥɚ. ɉɟɪɟɣɞɟɦ ɨɬ ɢɦɩɭɥɶɫɧɨɝɨ

ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɤ ɭɡɟɥɶɧɵɦ ɨɩɟɪɚɬɨɪɚɦ:

aip

exp{ }r

140

Ɂɞɟɫɶ ɩɨɞ “” - ɦɵ ɩɨɧɢɦɚɟɦ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɩɥɨɫɤɢɟ

ɜɨɥɧɵ, ɚ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ȼɚɧɧɶɟ ɜ ɤɪɢɫɬɚɥɥɟ, ɫɨɜɩɚɞɚɸɳɢɟ

ɫ ɭɡɟɥɶɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɬɨɱɧɨ ɧɚ ɭɡɥɟ ɢ ɢɦɟɸɳɢɟ ɚɫɢɦɬɨɬɚɦɢ

ɩɥɨɫɤɢɟ ɜɨɥɧɵ ɜɞɚɥɢ ɨɬ ɚɬɨɦɚ. Ɉɧɢ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɵ ɢ ɨɛɪɚ-

ɡɭɸɬ ɩɨɥɧɭɸ ɫɢɫɬɟɦɭ, ɤɚɤ ɢ ɩɥɨɫɤɢɟ ɜɨɥɧɵ. Ɍɨɝɞɚ ɤɢɧɟɬɢɱɟ-

ɫɤɚɹ ɱɚɫɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɢɦɟɟɬ ɜɢɞ:

exp{ }ipr

Htaa

iprr rra

kin

ij i j

ij p

ij ijB





oo o o o

exp( [ ]) ~ exp( | | ).

(8.34)

Ⱥɦɩɥɢɬɭɞɚ ɩɟɪɟɫɤɨɤɚ ɛɵɫɬɪɨ ɡɚɬɭɯɚɟɬ ɜ ɭɡɤɨɡɨɧɧɨɦ ɜɟɳɟɫɬɜɟ

ɫ ɫɢɥɶɧɨ ɥɨɤɚɥɢɡɨɜɚɧɧɵɦɢ ɧɨɫɢɬɟɥɹɦɢ, ɢ ɨɛɵɱɧɨ ɞɨɫɬɚɬɨɱɧɨ

ɭɱɟɫɬɶ ɜ (8.34) ɩɟɪɟɫɤɨɤɢ ɬɨɥɶɤɨ ɧɚ ɛɥɢɠɚɣɲɢɟ ɫɨɫɟɞɢ.

ɉɨɞɫɬɚɜɢɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɨɩɟɪɚɬɨɪɨɜ ɤ ɭɡɟɥɶɧɨɦɭ ɜɢ-

ɞɭ ɜ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɱɚɫɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧɚ, ɩɨɥɭɱɢɦ, ɱɬɨ ɜ

ɭɡɟɥɶɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɨɧɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:

HVaaaa

V ipr pr pr pr

ijkl i j l k

ijkl

ijkl p p p p

pp p p

ijlk

int

exp[ { }].





oo o o

12 21

121 2

VVV V



(8.35)

ɉɨɥɚɝɚɹ, ɱɬɨ ɝɥɚɜɧɵɣ ɜɤɥɚɞ ɞɚɟɬ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɧɚ ɨɞɧɢɯ ɢ

ɬɟɯ ɠɟ ɭɡɥɚɯ ɫ i=j=k=l (V

ijkl

{

iiii

= V(|r|=0)=U), ɚ ɧɚ ɛɨɥɶɲɢɯ

ɪɚɫɫɬɨɹɧɢɹɯ V ɦɧɨɝɨ ɦɟɧɶɲɟ, ɩɨɥɭɱɚɟɦ ɨɤɨɧɱɚɬɟɥɶɧɨ ɝɚɦɢɥɶ-

ɬɨɧɢɚɧ ɏɚɛɛɚɪɞɚ

H taa Uaaa a

ij i j

iii i









¦¦

VVVV

. (8.36)

Ɂɞɟɫɶ ɩɟɪɜɵɣ ɱɥɟɧ ɝɚɦɢɥɶɬɨɧɢɚɧɚ (ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ) ɨɩɢ-

ɫɵɜɚɟɬ ɩɟɪɟɫɤɨɤɢ ɧɚ ɫɨɫɟɞɧɢɟ ɭɡɥɵ ɫ ɚɦɩɥɢɬɭɞɨɣ t

. ȼɬɨɪɨɣ

ɱɥɟɧ ɨɩɢɫɵɜɚɟɬ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ ɧɚ ɭɡɥɟ ɫ ɩɨɬɟɧɰɢɚ-

ɥɨɦ U ɢ ɭɱɢɬɵɜɚɟɬ, ɱɬɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɚ ɭɡɥɟ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶ-

ɫɹ ɱɚɫɬɢɰɵ ɬɨɥɶɤɨ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɫɩɢɧɨɦ. Ƚɚɦɢɥɶɬɨɧɢɚɧ

141

ɡɚɩɢɫɵɜɚɸɬ ɜ ɛɨɥɟɟ ɭɞɨɛɧɨɣ ɮɨɪɦɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɪɢ-

ɛɥɢɠɟɧɢɹ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ Z

= - t, i,j = Z)

H t aa Unn n aa

 



¦¦

. (8.37)

ȼɨ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ ɱɚɫɬɢ (8.37) ɤɨɷɮɮɢɰɢɟɧɬ 1/2 ɫɤɨɦ-

ɩɟɧɫɢɪɨɜɚɥɫɹ ɡɚ ɫɱɟɬ ɫɭɦɦɢɪɨɜɚɧɢɹ ɩɨ ɫɩɢɧɚɦ. Ɍɚɤɢɦ ɨɛɪɚ-

ɡɨɦ, ɜ ɦɨɞɟɥɢ ɜɫɟɝɨ ɞɜɚ ɩɚɪɚɦɟɬɪɚ: ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɩɟɪɟ-

ɫɤɨɤɚ ɧɚ ɫɨɫɟɞɧɢɣ ɭɡɟɥ t ɢ ɤɭɥɨɧɨɜɫɤɨɟ ɨɬɬɚɥɤɢɜɚɧɢɟ ɧɚ ɭɡɥɟ

U. Ȼɨɥɟɟ ɬɨɝɨ, ɟɫɥɢ ɨɬɫɱɢɬɵɜɚɬɶ ɜɫɟ ɷɧɟɪɝɢɢ ɜ ɟɞɢɧɢɰɚɯ t, ɬɨ

ɨɫɬɚɟɬɫɹ ɟɞɢɧɫɬɜɟɧɧɵɣ ɩɚɪɚɦɟɬɪ t/U.

Ɂɧɚɤ ɩɟɪɟɫɤɨɤɚ (ɦɢɧɭɫ) ɜɵɛɢɪɚɟɬɫɹ ɢɡ ɭɞɨɛɫɬɜɚ ɨɬɫɱɟɬɚ

ɩɨɥɭɱɚɸɳɢɯɫɹ ɡɨɧ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬ ɰɟɧɬɪɚ ɡɨɧɵ Ȼɪɢɥɥɸɷɧɚ.

Ɍɚɤɚɹ ɜɨɥɶɧɨɫɬɶ ɜ ɜɵɛɨɪɟ ɡɧɚɤɚ ɫɜɹɡɚɧɚ ɫɨ ɫɥɟɞɭɸɳɢɦ ɫɜɨɣɫɬ-

ɜɨɦ ɝɚɦɢɥɶɬɨɧɢɚɧɚ (8.37): ɡɚɦɟɧɚ t

-t ɧɟ ɦɟɧɹɟɬ ɫɩɟɤɬɪ ɦɨɞɟ-

ɥɢ ɏɚɛɛɚɪɞɚ, ɟɫɥɢ ɩɟɪɟɫɤɨɤ ɩɪɨɢɫɯɨɞɢɬ ɬɨɥɶɤɨ ɦɟɠɞɭ ɛɥɢ-

ɠɚɣɲɢɦɢ ɫɨɫɟɞɹɦɢ (ɫɦ. ɡɚɞɚɱɢ ɤ ɩɚɪɚɝɪɚɮɭ).

Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ ɬɚɤɠɟ, ɱɬɨ ɝɚɦɢɥɶɬɨɧɢɚɧ ɤɨɦɦɭɬɢɪɭɟɬ

ɫ ɨɩɟɪɚɬɨɪɨɦ ɩɨɥɧɨɝɨ ɱɢɫɥɚ ɱɚɫɬɢɰ N ɢ ɩɨɥɧɨɣ ɩɪɨɟɤɰɢɢ ɫɩɢ-

ɧɚ ɧɚ ɨɫɶ z - S

N aa S aa aa n n











¦¦ ¦

,( )(

 ), (8.38)

ɬ.ɟ. ɡɚɞɚɱɭ ɦɨɠɧɨ ɪɟɲɚɬɶ ɩɪɢ ɤɨɧɤɪɟɬɧɨɦ ɡɚɩɨɥɧɟɧɢɢ ɢ ɩɪɨ-

ɟɤɰɢɢ ɫɩɢɧɚ ɧɟɡɚɜɢɫɢɦɨ, ɬɚɤ ɤɚɤ ɨɧɢ ɫɨɯɪɚɧɹɸɬɫɹ ɜ ɦɨɞɟɥɢ

ɏɚɛɛɚɪɞɚ.

ɉɭɫɬɶ ɱɢɫɥɨ ɭɡɥɨɜ ɪɚɜɧɨ N

. ɉɪɢ ɩɨɥɨɜɢɧɧɨɦ ɡɚɩɨɥɧɟ-

ɧɢɢ, ɬ.ɟ. ɤɨɝɞɚ N=N

, ɢ ɛɨɥɶɲɨɦ ɤɭɥɨɧɨɜɫɤɨɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ

U/t>>1 ɜɫɟ ɷɥɟɤɬɪɨɧɵ ɩɪɚɤɬɢɱɟɫɤɢ “ɡɚɩɟɪɬɵ” ɧɚ ɫɜɨɢɯ ɭɡɥɚɯ,

ɨɛɪɚɡɭɹ ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɟ ɭɩɨɪɹɞɨɱɟɧɢɟ:

npnpnpnpnpn

142

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɝɚɦɢɥɶɬɨɧɢɚɧ ɏɚɛɛɚɪɞɚ ɷɤɜɢɜɚɥɟɧɬɟɧ ɝɟɣɡɟɧ-

ɛɟɪɝɨɜɫɤɨɦɭ ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɦɭ ɝɚɦɢɥɶɬɨɧɢɚɧɭ (ɫɦ. ɫɨɨɬ-

ɜɟɬɫɬɜɭɸɳɭɸ ɡɚɞɚɱɭ ɧɢɠɟ).

Ɂɚɦɟɬɢɦ, ɱɬɨ ɦɨɞɟɥɶ ɏɚɛɛɚɪɞɚ ɫɢɦɦɟɬɪɢɱɧɚ ɨɬɧɨɫɢ-

ɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɷɥɟɤɬɪɨɧ-ɞɵɪɤɚ (a

®b, a®b

). ɉɨɷɬɨ-

ɦɭ, ɩɨɥɶɡɭɹɫɶ ɫɜɨɣɫɬɜɨɦ (t®-t) ɢ ɷɬɢɦɢ ɩɪɟɨɛɪaɡɨɜɚɧɢɹɦɢ,

ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ.

ɉɭɫɬɶ

- ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɜɟɪɯ;

’

- ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɧɢɡ,

M+M’=N. Ɍɨɝɞɚ ɢɦɟɸɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ (ɜɩɟɪ-

ɜɵɟ ɩɨɥɭɱɟɧɧɵɟ Ʌɢɛɨɦ ɢ ȼɭ ɜ 1968 ɝ. [19]):

E(M,M’, U) = -(N

-M-M’)U + E(N

-M, N

-M’,U),

E(M,M’, U) = MU + E(M, N

-M’,-U), (8.39)

E(M,M’, U) = M’U + E(N

-M, M’,-U).

ɉɨɥɭɱɢɦ, ɧɚɩɪɢɦɟɪ, ɩɟɪɜɨɟ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ (8.39). ɉɨɞɫɬɚɜɢɜ

ɜ (8.37) ɞɵɪɨɱɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ a

®b, a®b

, ɢɦɟɟɦ:

H t bb Ubbbb

UUbb Ubb



 



¦¦

¦¦¦





(8.40)

ɍɱɢɬɵɜɚɹ, ɱɬɨ ɱɢɫɥɨ ɞɵɪɨɤ ɫɨ ɫɩɢɧɨɦ ɜɜɟɪɯ

bb N M





ɚ ɱɢɫɥɨ ɞɵɪɨɤ ɫɨ ɫɩɢɧɨɦ ɜɧɢɡ

bb N M



',

ɩɨɥɭɱɚɟɦ ɩɟɪɜɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (8.39) (E=<H>).

Ɂɚɞɚɱɢ

8.3.1.

1. Ⱦɨɤɚɡɚɬɶ ɨɩɟɪɚɬɨɪɧɵɟ ɬɨɠɞɟɫɬɜɚ:

exp{ } exp{ } exp( )



iaa a iaa a i

ii i ii i

VV V VV V

exp{ } exp{ } exp( )

  

iaa a iaa a i

ii i ii i

VV V VV V

ɍɤɚɡɚɧɢɟ

: ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɥɟɜɭɸ ɱɚɫɬɶ ɬɨɠɞɟɫɬɜ

ɩɨ ɩɟɪɟɦɟɧɧɨɣ

143

8.3.1.2. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɡɚɦɟɧɚ

o

-t ɧɟ ɦɟɧɹɟɬ ɫɩɟɤɬɪ

ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ, ɟɫɥɢ ɩɟɪɟɫɤɨɤ ɩɪɨɢɫɯɨɞɢɬ ɬɨɥɶɤɨ ɦɟɠɞɭ

ɛɥɢɠɚɣɲɢɦɢ ɫɨɫɟɞɹɦɢ.

Ɋɟɲɟɧɢɟ

. Ɋɚɡɨɛɴɟɦ ɭɡɥɵ ɧɚ ɞɜɟ ɩɨɞɪɟɲɟɬɤɢ

, ɜɥɨ-

ɠɟɧɧɵɟ ɨɞɧɚ ɜ ɞɪɭɝɭɸ, ɬɚɤ ɱɬɨ ɛɥɢɠɚɣɲɢɦ ɫɨɫɟɞɨɦ ɭɡɥɚ

ɨɛɹɡɚɬɟɥɶɧɨ ɹɜɥɹɟɬɫɹ ɭɡɟɥ

ɢ ɧɚɨɛɨɪɨɬ. Ⱦɥɹ ɜɫɟɯ ɭɡɥɨɜ

ɫɞɟ-

ɥɚɟɦ ɭɧɢɬɚɪɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ



¦¦

exp{ }

ɤɨɬɨɪɨɟ ɦɟɧɹɟɬ ɡɧɚɤ ɨɩɟɪɚɬɨɪɨɜ

. ɗɬɨ

ɫɥɟɞɭɟɬ ɢɡ ɨɩɟɪɚɬɨɪɧɨɝɨ ɬɨɠɞɟɫɬɜɚ:

aaa

iiii

VVV



o o, a

exp{ } exp{ } exp( )



ia a a ia a a i

ii i ii i

VV V VV V

ɤɨɬɨɪɨɟ ɧɟɫɥɨɠɧɨ ɞɨɤɚɡɚɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨ-

ɜɚɧɢɟɦ ɩɨ ɩɚɪɚɦɟɬɪɭ

M

(ɫɦ.ɩɪɟɞɵɞɭɳɭɸ ɡɚɞɚɱɭ). ɉɪɢɦɟɧɢɦ

ɨɩɟɪɚɬɨɪ U ɤ ɝɚɦɢɥɶɬɨɧɢɚɧɭ ɏɚɛɛɚɪɞɚ:

1

. Ɋɚɫɫɦɨɬɪɢɦ

ɩɟɪɟɫɤɨɤɨɜɵɣ ɱɥɟɧ. Ɍɨɝɞɚ ɡɚɦɟɬɢɦ, ɱɬɨ ɜɫɟɝɞɚ ɢɦɟɟɦ ɜ ɧɟɦ

ɤɨɦɛɢɧɚɰɢɸ a

ɢɥɢ a

+Ȼ

. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɭɧɢɬɚɪɧɨɟ ɩɪɟ-

ɨɛɪɚɡɨɜɚɧɢɟ, ɢɡɦɟɧɹɹ ɡɧɚɤ ɬɨɥɶɤɨ ɨɩɟɪɚɬɨɪɚ Ⱥ, ɢɡɦɟɧɢɬ ɢ ɜɟɫɶ

ɡɧɚɤ ɩɟɪɟɞ ɱɥɟɧɨɦ. Ʌɟɝɤɨ ɩɨɧɹɬɶ, ɱɬɨ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɱɥɟɧɟ

ɱɢɫɥɨ ɢɡɦɟɧɟɧɢɣ ɡɧɚɤɨɜ ɜɫɟɝɞɚ ɱɟɬɧɨ:

-1

U=U

-1

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɣ ɱɥɟɧ ɢɧɜɚɪɢɚɧɬɟɧ ɨɬɧɨ-

ɫɢɬɟɥɶɧɨ ɬɚɤɨɝɨ ɭɧɢɬɚɪɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ. Ⱥ ɬɚɤ ɤɚɤ

U HUU U U EUU U EU U

UHU E U U

  





11 11 1

<<<,,



ɬɨ ɫɩɟɤɬɪ E ɧɟ ɢɡɦɟɧɢɥɫɹ , ɧɨ ɩɪɢ ɷɬɨɦ t ɢɡɦɟɧɢɥɫɹ ɧɚ -t, ɱɬɨ

ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ.

8.3.1.

3. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɝɚɦɢɥɶɬɨɧɢɚɧ ɏɚɛɛɚɪɞɚ (8.37)

ɤɨɦɦɭɬɢɪɭɟɬ ɫ ɨɩɟɪɚɬɨɪɨɦ ɩɨɥɧɨɝɨ ɱɢɫɥɚ ɱɚɫɬɢɰ ɢ ɩɨɥɧɨɣ

ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɧɚ ɨɫɶ z.

144

8.3.1.4. ɉɭɫɬɶ N

- ɱɢɫɥɨ ɭɡɥɨɜ; M - ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɜɟɪɯ;

M’ - ɱɢɫɥɨ ɫɩɢɧɨɜ ɜɧɢɡ; N= M+M’ - ɱɢɫɥɨ ɷɥɟɤɬɪɨɧɨɜ. Ⱦɨɤɚ-

ɡɚɬɶ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɷɧɟɪɝɢɢ ɜ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ:

E(M,M’,U) = MU + E(M, N

-M’,-U),

E(M,M’ U) = M’U + E(N

-M, M’,-U).

ɍɤɚɡɚɧɢɟ

: ɩɟɪɟɣɬɢ ɤ ɞɵɪɨɱɧɨɦɭ ɩɪɟɞɫɬɚɜɥɟɧɢɸ ɞɥɹ

ɷɥɟɤɬɪɨɧɨɜ ɫɨ ɫɩɢɧɨɦ ɜɧɢɡ ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɢ ɫɨ ɫɩɢɧɨɦ ɜɜɟɪɯ

ɜɨ ɜɬɨɪɨɦ.

8.3.1.

5. ɑɟɦɭ ɪɚɜɧɚ ɷɧɟɪɝɢɹ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɦɨɞɟ-

ɥɢ ɏɚɛɛɚɪɞɚ ɩɪɢ ɩɨɥɨɜɢɧɧɨɦ ɡɚɩɨɥɧɟɧɢɢ N = N

ɢ ɜ ɩɪɟɞɟɥɟ

of

8.3.1.

6. Ɋɚɫɫɱɢɬɚɬɶ ɱɢɫɥɨ ɜɨɡɦɨɠɧɵɯ ɭɡɟɥɶɧɵɯ ɫɨɫɬɨɹ-

ɧɢɣ (ɪɚɡɦɟɪ ɝɢɥɶɛɟɪɬɨɜɚ ɩɪɨɫɬɪɚɧɫɬɜɚ) ɜ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ

ɩɪɢ:

1)ɱɢɫɥɟ ɭɡɥɨɜ N

=12, ɱɢɫɥɟ ɷɥɟɤɬɪɨɧɨɜ ɫɨ ɫɩɢɧɨɦ ɜɜɟɪɯ

M=4, ɫɨ ɫɩɢɧɨɦ ɜɧɢɡ M’=3;

2) N

=12, M=3, M’=3;

3) N

=8, M=4, M’=4;

4) N

=8, M=2, M’=3.

8.3.1.

7. ɉɨɤɚɡɚɬɶ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɶ ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɣ

ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ ɢ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ ɜɛɥɢɡɢ ɩɨɥɨɜɢɧɧɨɝɨ

ɡɚɩɨɥɧɟɧɢɹ ɩɪɢ U/t>>1.

Ɋɟɲɟɧɢɟ

. Ɋɚɡɞɟɥɢɦ ɝɚɦɢɥɶɬɨɧɢɚɧ ɏɚɛɛɚɪɞɚ ɧɚ ɧɟɫɤɨɥɶ-

ɤɨ ɫɥɚɝɚɟɦɵɯ:

HT T T V

dhmix

 , (8.41)

ɝɞɟ

145

146

 ..),VUnnTtnaankc

Tt naa n kc

Tt naan

naa n kc

d i ijj

hiijj

mix i i j j

iij j



   

  





















¦¦

{( ) ( ) . },

{( )

().}.

VV V V

VVV V

VV V V

(8.42)

ɨɩɢɫɵɜɚɟɬ ɤɨɪɪɟɥɢɪɨɜɚɧɧɵɟ ɩɟɪɟɫɤɨɤɢ ɫ ɭɡɥɚ ɧɚ ɭɡɟɥ ɫɨ

ɫɩɢɧɨɦ

, ɤɨɝɞɚ ɭɡɥɵ ɡɚɧɹɬɵ ɷɥɟɤɬɪɨɧɚɦɢ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠ-

ɧɵɦ ɫɩɢɧɨɦ -

, ɱɥɟɧ T

, ɧɚɨɛɨɪɨɬ, ɨɩɢɫɵɜɚɟɬ ɩɟɪɟɫɤɨɤɢ ɫ ɭɡ-

ɥɚɦɢ, ɧɟɡɚɩɨɥɧɟɧɧɵɦɢ ɷɥɟɤɬɪɨɧɚɦɢ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɫɩɢ-

ɧɨɦ, ɚ T

mix

- ɩɪɨɰɟɫɫɵ ɩɟɪɟɫɤɨɤɚ, ɤɨɝɞɚ ɥɢɛɨ ɨɞɢɧ, ɥɢɛɨ ɞɪɭ-

ɝɨɣ (ɧɨ ɧɟ ɞɜɚ ɨɞɧɨɜɪɟɦɟɧɧɨ) ɭɡɥɚ, ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɩɟɪɟɫɤɨɤɟ,

ɡɚɧɹɬɵ ɷɥɟɤɬɪɨɧɨɦ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɫɩɢɧɨɦ.

ȼ ɩɪɟɞɟɥɟ U/t >>1 ɫɚɦɵɣ ɦɚɥɵɣ ɜɤɥɚɞ ɞɚɟɬ ɱɥɟɧ T

, ɬɚɤ ɤɚɤ

ɷɬɨɬ ɩɪɨɰɟɫɫ ɫɜɹɡɚɧ ɫ ɛɨɥɶɲɨɣ ɞɨɛɚɜɤɨɣ ɤ ɷɧɟɪɝɢɢ ɷɥɟɤɬɪɨɧɚ

~U.

Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɨɟ ɭɧɢɬɚɪɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɝɚ-

ɦɢɥɶɬɨɧɢɚɧɚ, ɧɟ ɦɟɧɹɸɳɟɟ ɟɝɨ ɫɩɟɤɬɪɚ:

HiSH

eff

exp( ) exp( )iS .

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ S ɦɚɥɨ ɩɨ ɩɚɪɚɦɟɬɪɭ t/U, ɢ ɜ ɩɟɪɜɨɦ ɩɪɢ-

ɛɥɢɠɟɧɢɢ:

HiSHiSHiS

eff

  ()() [,11 H ]

. (8.43)

Ɉɩɪɟɞɟɥɢɦ ɷɮɮɟɤɬɢɜɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ ɢ ɭɧɢɬɚɪɧɨɟ

ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ. ɂɳɟɦ S, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɸ

iS T T V T

dh m

[, ] . (8.44)

Ɉɬɫɸɞɚ

. (8.45) HTTViST

eff d h mix

[, ]

ɑɥɟɧɨɦ V ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɬɚɤ ɤɚɤ ɩɪɢ ɛɨɥɶɲɢɯ U ɜɟɪɨɹɬ-

ɧɨɫɬɶ ɧɚɯɨɠɞɟɧɢɹ ɧɚ ɨɞɧɨɦ ɭɡɥɟ ɞɜɭɯ ɷɥɟɤɬɪɨɧɨɜ ɦɚɥɚ. ɉɨ

ɷɬɨɣ ɠɟ ɩɪɢɱɢɧɟ ɩɪɟɧɟɛɪɟɠɟɦ ɢ T

. Ɍɨɝɞɚ ɢɦɟɟɦ

HTiST

eff h mix

#[ , ]. (8.46)

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ |n> - ɫɨɛɫɬɜɟɧɧɵɟ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɨɩɟ-

ɪɚɬɨɪɚ T

+V, ɚ

- ɫɨɛɫɬɜɟɧɧɵɟ ɷɧɟɪɝɢɢ. Ɍɨɝɞɚ ɢɡ (8.44)

ɩɨɥɭɱɚɟɦ:

inST T Vm inT T VSm

nT m

inSm inSm nT m

nSm

nT m

dh dh

mix

mnmix

mix

!!

 ! 

 !  !  !

 !

!



|( )| |( )|

|| || | |

()

(8.47)

ɉɪɨɫɭɦɦɢɪɭɟɦ ɩɨɫɥɟɞɧɟɟ ɜɵɪɚɠɟɧɢɟ ɜ (8.47) ɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ

ɩɨɥɧɨɬɨɣ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ, ɬɨɝɞɚ ɞɥɹ S ɩɨɥɭɱɢɦ

||| ||

()

|| |

()

nnSmm n

nT m

nn S n

nT m

mix

! ! !

!





!  !

!





¦¦

(8.48)

ȼɛɥɢɡɢ ɩɨɥɨɜɢɧɧɨɝɨ ɡɚɩɨɥɧɟɧɢɹ ɩɪɢ ɛɨɥɶɲɨɦ U ɥɸɛɵɟ ɜɨɡ-

ɛɭɠɞɟɧɢɹ ɫɢɫɬɟɦɵ ɧɚɞ ɨɫɧɨɜɧɵɦ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ t/U ɛɭɞɭɬ

U. Ɉɬɫɸɞɚ

nnTmm

mix

!!





! !



||| |

!

(8.49)

Ɂɞɟɫɶ T

mix

ɢ T

mix

- ɱɚɫɬɢ ɨɩɟɪɚɬɨɪɚ T

mix

, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ

ɭɜɟɥɢɱɟɧɢɸ (ɭɦɟɧɶɲɟɧɢɸ) ɷɧɟɪɝɢɢ.

147

148

)

Ɉɩɪɟɞɟɥɢɦ ɩɪɚɜɢɥɶɧɨ ɡɧɚɤɢ ɜ ɜɵɪɚɠɟɧɢɢ (8.49). ȼɜɵ-

ɪɚɠɟɧɢɢ ɞɥɹ T

mix

(8.42) ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ

[~ ] ɩɨɧɢɠɚɟɬ ɷɧɟɪɝɢɸ ɧɚ U (ɬ.ɟ. ɫɨɜ-

ɩɚɞɚɟɬ ɫ T

()1 







naan

iijj

VVV

mix

), ɬɚɤ ɤɚɤ ɷɥɟɤɬɪɨɧ ɭɯɨɞɢɬ ɫ ɞɜɚɠɞɵ ɡɚɩɨɥɧɟɧɧɨ-

ɝɨ ɭɡɥɚ j ɧɚ ɩɭɫɬɨɣ ɭɡɟɥ i. ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ

[],ɧɚɨɛɨɪɨɬ, ɩɨɜɵɲɚɟɬ ɷɧɟɪɝɢɸ ɧɚ U

(ɬ.ɟ. ɫɨɜɩɚɞɚɟɬ ɫ T

naa n

iij j





VV V V

mix

). ɇɟɫɥɨɠɧɨ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɷɪɦɢɬɨɜɨ-

ɫɨɩɪɹɠɟɧɧɚɹ ɱɚɫɬɶ T

mix

ɫɨɜɩɚɞɚɟɬ ɫ ɩɟɪɜɨɣ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɡɚ-

ɦɟɧɵ T

<->T

ɢ ɞɨɩɨɥɧɹɟɬ ɞɨ ɩɨɥɧɨɝɨ ɫɭɦɦɢɪɨɜɚɧɢɹ

(n<m + n>m = n,m) ɜ (8.49). ɉɨɷɬɨɦɭ ɜ ɢɬɨɝɟ ɢɡ (8.49) ɢɦɟɟɦ,

ɢɫɩɨɥɶɡɭɹ ɩɨɥɧɨɬɭ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ:

naa n

naan

iij j

iijj

  













{(

() }.

VV V V

VVV V

)

(8.50)

ɍɱɢɬɵɜɚɹ (8.46), ɧɚɣɞɟɦ ɤɨɦɦɭɬɚɬɨɪ i[S, T

mix

]. ȿɫɥɢ ɨɛɨɡɧɚ-

ɱɢɬɶ

Atnaa n

Bt naan

iij j

i ijj

 











VV V V

VVVV

()

(8.51)

ɬɨ ɫɨɝɥɚɫɧɨ (8.42) ɢ (8.50) ɧɚɯɨɞɢɦ:

TABS

iST

naa n n aan

mix

iij j

i ijj

 



  











¦¦

,(),

[, ] [, ]

[(),()

VV V V

VVVV

(8.52)

ȼɜɟɞɟɦ ɬɟɩɟɪɶ ɫɩɢɧɨɜɵɟ ɨɩɟɪɚɬɨɪɵ ɫ ɩɨɥɧɨɣ ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ

1/2 ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

SaaSaaS nn

SSiSSSiS











 

,,(

(8.53)

Ʌɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɜɜɟɞɟɧɧɵɟ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɨɩɟɪɚɬɨɪɵ

ɹɜɥɹɸɬɫɹ ɨɛɵɱɧɵɦɢ ɨɩɟɪɚɬɨɪɚɦɢ ɫɩɢɧɚ 1/2, ɩɨɞɱɢɧɹɸɬɫɹ

ɜɫɟɦ ɤɨɦɦɭɬɚɰɢɨɧɧɵɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɢ ɜɵɪɚɠɚɸɬɫɹ ɱɟɪɟɡ

ɦɚɬɪɢɰɵ ɉɚɭɥɢ: S



= 1/2

, S



= 1/2

, S



= 1/2

ɂɫɩɨɥɶɡɭɹ (8.53), ɩɨɫɥɟ ɝɪɨɦɨɡɞɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɢɡ

(8.52) ɩɨɥɭɱɚɟɦ:

iS T

SS nn

mix i j i j

[, ] ( )  

. (8.54)

Ɉɤɨɧɱɚɬɟɥɶɧɨ, ɭɱɢɬɵɜɚɹ (8.46), ɧɚɯɨɞɢɦ ɷɮɮɟɤɬɢɜɧɵɣ ɝɚ-

ɦɢɥɶɬɨɧɢɚɧ

Ht naankc

SS nn

eff i i j j

ij ij

    









{( ) ( ) . }

().

VVV V

(8.55)

Ƚɚɦɢɥɶɬɨɧɢɚɧ (8.55) - ɩɪɟɞɟɥɶɧɵɣ ɫɥɭɱɚɣ ɝɚɦɢɥɶɬɨɧɢɚ-

ɧɚ ɏɚɛɛɚɪɞɚ ɩɪɢ ɛɨɥɶɲɢɯ U ɜɛɥɢɡɢ ɩɨɥɨɜɢɧɧɨɝɨ ɡɚɩɨɥɧɟɧɢɹ.

Ɋɨɜɧɨ ɩɪɢ ɩɨɥɨɜɢɧɧɨɦ ɡɚɩɨɥɧɟɧɢɢ, ɤɨɝɞɚ ɧɚ ɤɚɠɞɵɣ ɭɡɟɥ

ɩɪɢɯɨɞɢɬɫɹ ɨɞɢɧ ɷɥɟɤɬɪɨɧ, ɩɟɪɟɫɤɨɤɨɜɵɣ ɱɥɟɧ ɡɚɧɭɥɹɟɬɫɹ, ɢ

ɦɨɞɟɥɶ (8.55) ɫɬɚɧɨɜɢɬɫɹ ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɣ ɦɨɞɟɥɶɸ Ƚɟɣ-

ɡɟɧɛɟɪɝɚ (ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɧɟɫɭɳɟɫɬɜɟɧɧɨɣ ɩɨɫɬɨɹɧɧɨɣ):

eff i j

() (8.56)

ɫ ɷɮɮɟɤɬɢɜɧɵɦ ɨɛɦɟɧɧɵɦ ɢɧɬɟɝɪɚɥɨɦ J = t

/U. Ɂɚɦɟɬɢɦ, ɱɬɨ

ɜɟɥɢɱɢɧɚ J ~ 0.1-0.5 ɷȼ (ɬɚɤ ɤɚɤ t ~ 1 ɷȼ, U ~ 3-10 ɷȼ), ɱɬɨ

ɜɩɨɥɧɟ ɫɨɝɥɚɫɭɟɬɫɹ ɫ ɨɰɟɧɤɚɦɢ ɨɛɦɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ. Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ, ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ Ƚɟɣɡɟɧɛɟɪɝɚ ɹɜ-

ɥɹɟɬɫɹ ɩɪɟɞɟɥɶɧɵɦ ɫɥɭɱɚɟɦ ɦɨɞɟɥɢ ɏɚɛɛɚɪɞɚ ɩɪɢ ɛɨɥɶɲɢɯ U

ɢ ɩɨɥɨɜɢɧɧɨɦ ɡɚɩɨɥɧɟɧɢɢ.

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