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

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ɧɢɬɧɨɣ ɢɧɞɭɤɰɢɢ

(ɢɫɬɢɧɧɨɝɨ ɜɧɭɬɪɟɧɧɟɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ)

ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ:

&& & & &

BH M H H   { 414 4

SSFPPS

(); 1

Ɂɞɟɫɶ

- ɦɚɝɧɢɬɧɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɢ

 1

(ɦɧɨɠɢɬɟɥɶ 4

ɜɜɨɞɢɬɫɹ ɞɥɹ ɭɞɨɛɫɬɜɚ).

ɉɪɢɧɹɬɚ ɫɥɟɞɭɸɳɚɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɜɟɳɟɫɬɜ ɩɨ ɢɯ ɦɚɝ-

ɧɢɬɧɵɦ ɫɜɨɣɫɬɜɚɦ

1) ɩɚɪɚɦɚɝɧɟɬɢɤɢ:

! 1  ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɜɧɭɬɪɢ

ɭɫɢɥɢɜɚɟɬɫɹ;

2) ɞɢɚɦɚɝɧɟɬɢɤɢ:

 1,

 0  ɨɫɥɚɛɥɟɧɢɟ ɦɚɝɧɢɬ-

ɧɨɝɨ ɩɨɥɹ;

3) ɮɟɪɪɨɦɚɝɧɟɬɢɤɢ: ɜ ɨɬɫɭɬɫɬɜɢɢ ɜɧɟɲɧɟɝɨ ɦɚɝɧɢɬɧɨɝɨ

ɩɨɥɹ

, ɬ.ɟ.

ɩɪɢ

B z 0 B z 0

f f, ;

4) ɢɞɟɚɥɶɧɵɣ ɞɢɚɦɚɝɧɟɬɢɡɦ: ɩɨɥɧɚɹ ɷɤɪɚɧɢɪɨɜɤɚ ɜɧɟɲ-

ɧɟɝɨ ɩɨɥɹ:

HBz  00 0

Ɍɚɤ, ɢɞɟɚɥɶɧɵɦ ɞɢɚɦɚɝɧɟɬɢɤɨɦ ɹɜɥɹɟɬɫɹ ɫɜɟɪɯɩɪɨɜɨɞɧɢɤ;

5) ɧɟɦɚɝɧɢɬɧɵɟ ɜɟɳɟɫɬɜɚ:

10,

1.2. Ɇɚɝɧɢɬɧɵɟ ɫɜɨɣɫɬɜɚ ɫɢɫɬɟɦɵ ɥɨɤɚɥɶɧɵɯ

ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɛɟɡ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ

Ɋɚɫɫɦɨɬɪɢɦ ɤɪɢɫɬɚɥɥ ɜɨ ɜɧɟɲɧɟɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɫ

ɭɡɥɚɦɢ, ɢɦɟɸɳɢɦɢ ɦɚɝɧɢɬɧɵɟ ɦɨɦɟɧɬɵ. ɉɭɫɬɶ ɬɟɦɩɟɪɚɬɭɪɚ

0, ɢ ɦɚɝɧɢɬɧɵɟ ɦɨɦɟɧɬɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɬɨɥɶɤɨ ɫ ɜɧɟɲ-

ɧɢɦ ɩɨɥɟɦ. ɉɪɢ ɜɤɥɸɱɟɧɧɨɦ ɜɧɟɲɧɟɦ ɩɨɥɟ ɷɧɟɪɝɢɹ ɜɡɚɢɦɨ-

ɞɟɣɫɬɜɢɹ

V  H

, ɢ ɩɨɥɟ ɫɬɪɟɦɢɬɫɹ ɪɚɡɜɟɪɧɭɬɶ ɦɨɦɟɧɬɵ ɜ

ɫɜɨɸ ɫɬɨɪɨɧɭ (ɚ ɬɟɦɩɟɪɚɬɭɪɚ - ɧɚɨɛɨɪɨɬ, ɪɚɡɭɩɨɪɹɞɨɱɢɬɶ). Ɍɨ-

ɝɞɚ: , ɝɞɟ S - ɫɩɢɧ (ɰɟɥɵɣ ɢɥɢ ɩɨɥɭɰɟ-

ɥɵɣ).

VH HS

 

/ S

ɑɢɫɥɨ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ ɫɭɦɦɚɪɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ

ɪɚɜɧɨ 2S

+1, ɩɪɢ ɷɬɨɦ ɩɪɢ S

=1/2 ɱɢɫɥɨ ɡɧɚɱɟɧɢɣ (ɫɬɟɩɟɧɟɣ

ɫɜɨɛɨɞɵ) ɥɨɤɚɥɢɡɨɜɚɧɧɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɦɨɦɟɧɬɚ ɪɚɜɧɨ 2 (ɜɞɨɥɶ

ɢ ɩɪɨɬɢɜ ɩɨɥɹ). ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɷɧɟɪɝɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɨɬ-

ɞɟɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɫ ɜɧɟɲɧɢɦ ɩɨɥɟɦ ɢɦɟɟɬ ɬɨɥɶɤɨ ɞɜɚ ɡɧɚɱɟ-

ɧɢɹ:



(ɜɞɨɥɶ ɩɨɥɹ),



n ; (1.1)

 H

(ɩɪɨɬɢɜ ɩɨɥɹ),



Ɋɚɫɫɱɢɬɚɟɦ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ

. Ⱦɥɹ ɷɬɨɝɨ ɪɚɫɫɦɨɬ-

ɪɢɦ ɩɨɥɧɭɸ ɫɪɟɞɧɸɸ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ ɫɢɫɬɟɦɵ:

¦¦

PPP

ɩɪɢ ɷɬɨɦ ɭɝɥɨɜɵɟ ɫɤɨɛɤɢ ɛɭɞɭɬ ɨɡɧɚɱɚɬɶ ɤɜɚɧɬɨɜɨ-

ɦɟɯɚɧɢɱɟɫɤɨɟ ɭɫɪɟɞɧɟɧɢɟ ɩɨ ɚɧɫɚɦɛɥɸ Ƚɢɛɛɫɚ. Ɍɨɝɞɚ ɩɨ ɨɩɪɟ-

ɞɟɥɟɧɢɸ ɢɦɟɟɦ



Sp H kT

exp[ / ]

nHkT

nHkTn

| exp[ / ]|

|exp[ / ]|



ɝɞɟ “

Sp” ɨɡɧɚɱɚɟɬ ɲɩɭɪ (ɫɭɦɦɭ ɞɢɚɝɨɧɚɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ);

ɝɚɦɢɥɶɬɨɧɢɚɧ ɫɢɫɬɟɦɵ;

k - ɩɨɫɬɨɹɧɧɚɹ Ȼɨɥɶɰɦɚɧɚ (ɟɟ ɜ ɞɚɥɶ-

ɧɟɣɲɟɦ ɩɨɥɚɝɚɟɦ ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ).

ɍɱɢɬɵɜɚɹ, ɱɬɨ ɦɵ ɩɨɧɢɦɚɟɦ ɩɨɞ ɫɨɫɬɨɹɧɢɟɦ |

n> ɩɪɟɞ-

ɫɬɚɜɥɟɧɢɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɝɚɦɢɥɶɬɨɧɢɚɧɚ (ɧɚɩɪɢɦɟɪ,

ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɝɧɨɜɟɧɧɨɝɨ ɪɚɫɩɨɥɨɠɟɧɢɹ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧ-

ɬɨɜ ɧɚ ɭɡɥɚɯ, ɩɪɢ ɷɬɨɦ ɬɚɤɢɯ ɫɨɫɬɨɹɧɢɣ ɛɭɞɟɬ 2

(N - ɱɢɫɥɨ

ɭɡɥɨɜ) - ɪɚɡɦɟɪɧɨɫɬɶ Ƚɢɥɶɛɟɪɬɨɜɚ ɩɪɨɫɬɪɚɧɫɬɜɚ), ɚ ɬɚɤɠɟ ɢɯ

ɩɨɥɧɨɬɭ ɢ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɨɫɬɶ, ɢɦɟɟɦ

exp[ / ]| exp[ / ]| , | ;

|exp[ / ]| exp[ / ],

 

¦¦

HTn E Tn nn

nHTn ET

ɝɞɟ E

- ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɜ ɫɨɫɬɨɹɧɢɢ |n> . Ʉɪɨɦɟ ɬɨɝɨ, ɭɱɢɬɵ-

ɜɚɹ, ɱɬɨ

nHTnnETn E

nnn

PPP

exp[ / ]| exp[ / ]| exp[ / ],  T

ɩɨɥɭɱɚɟɦ



exp[ / ]

. (1.2)

ɇɨɪɦɢɪɨɜɨɱɧɵɣ ɡɧɚɦɟɧɚɬɟɥɶ ɡɞɟɫɶ - ɫɬɚɬɢɫɬɢɱɟɫɤɚɹ ɫɭɦɦɚ

ɍɱɢɬɵɜɚɹ, ɱɬɨ ɦɚɝɧɢɬɧɵɟ ɦɨɦɟɧɬɵ ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭ-

ɸɬ, ɦɨɠɧɨ ɧɟɡɚɜɢɫɢɦɨ ɪɚɫɫɱɢɬɚɬɶ ɜɵɪɚɠɟɧɢɟ (1.2) ɞɥɹ ɤɚɠɞɨ-

ɝɨ ɦɨɦɟɧɬɚ, ɢɦɟɹ ɜɜɢɞɭ ɬɨɥɶɤɨ ɟɝɨ ɞɜɚ ɫɨɫɬɨɹɧɢɹ (1.1):

PP P P





00 0 0

exp[ /] exp[ /]

HT HT

ɇɚɩɨɦɧɢɦ, ɱɬɨ ɡɞɟɫɶ

H - ɜɧɟɲɧɟɟ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ. Ɉɬɫɸɞɚ

ɢɦɟɟɦ



PP P

th HT/



. (1.3)

ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ. ȿɫɥɢ ɜɧɟɲ-

ɧɟɟ ɩɨɥɟ ɭɫɬɪɟɦɢɬɶ ɤ ɧɭɥɸ, ɬɨ ɩɨɥɭɱɚɟɦ ɢɡ (1.3) ɫɥɟɞɭɸɳɟɟ

o# 00

! . (1.4)

ɉɪɢ ɷɬɨɦ ɟɫɥɢ ɩɨɥɨɠɢɬɶ ɜɧɟɲɧɟɟ ɩɨɥɟ ɬɨɱɧɨ ɧɭɥɸ, ɬɨ ɩɪɢ

ɥɸɛɨɣ ɬɟɦɩɟɪɚɬɭɪɟ

H {00,,

M 0

. (1.5)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ (1.4)-(1.5) ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɫɢɫɬɟɦɚ ɥɨɤɚɥɢ-

ɡɨɜɚɧɧɵɯ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɩɪɨɹɜ-

ɥɹɟɬ ɬɢɩɢɱɧɵɟ ɩɚɪɚɦɚɝɧɢɬɧɵɟ ɫɜɨɣɫɬɜɚ. Ȼɨɥɟɟ ɬɨɝɨ, ɨɧɚ ɧɢɤɨ-

ɝɞɚ ɧɟ ɢɫɩɵɬɵɜɚɟɬ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ (ɪɢɫ.3) - ɞɥɹ ɷɬɨɝɨ ɧɟɨɛ-

ɯɨɞɢɦɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ. Ɍɟɦɩɟɪɚɬɭɪɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɜɨɫɩɪɢ-

ɢɦɱɢɜɨɫɬɢ (~ 1/T) , ɩɨɤɚɡɚɧɧɚɹ ɧɚ ɪɢɫ. 3, ɧɚɡɵɜɚɟɬɫɹ ɡɚɤɨɧɨɦ

Ʉɸɪɢ.

Ɋɢɫ.3. Ɍɟɦɩɟɪɚɬɭɪɧɚɹ ɡɚɜɢɫɢ-

ɦɨɫɬɶ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ

- ɡɚ-

ɤɨɧ Ʉɸɪɢ (a1/T) ɞɥɹ ɧɟɜɡɚɢɦɨ-

ɞɟɣɫɬɜɭɸɳɢɯ ɦɨɦɟɧɬɨɜ

Ɂɚɞɚɱɢ

1.2.

1. ɇɚɣɬɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɫɭɦɦɵ Q

ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɛɟɡ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ.

Ɋɟɲɟɧɢɟ

. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɦɨɞɟɥɢ:

HHSS

iii

 r

PPP

. Ɍɚɤ ɤɚɤ ɦɨɦɟɧɬɵ ɧɟ

ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ, ɬɨ :

QSH



exp( ) ( )

EP EP

2ch .

1.2.2. ɇɚɣɬɢ ɷɧɟɪɝɢɸ E ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɛɟɡ ɜɡɚɢɦɨɞɟɣɫɬ-

ɜɢɹ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ T.

Ɉɬɜɟɬ

EQNH   H

PEP

ln [ ]

th .

1.2.

3. ɇɚɣɬɢ ɷɧɬɪɨɩɢɸ S(T) ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɛɟɡ ɜɡɚɢɦɨ-

ɞɟɣɫɬɜɢɹ.

Ɋɟɲɟɧɢɟ. ȼɟɪɨɹɬɧɨɫɬɶ n-ɝɨ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɫ ɷɧɟɪ-

ɝɢɟɣ E

: PEQ

exp( ) /

. ɉɨɷɬɨɦɭ

SPPQ E EQ

QEN H H H

   

 

¦¦

ln exp( )ln[exp( ) / ]

ln ln{ [ ]} [ ] .

000

EEPEPEP

ch th

1.2.

4. ɇɚɣɬɢ ɷɧɬɪɨɩɢɸ S ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɛɟɡ ɜɡɚɢɦɨɞɟɣ-

ɫɬɜɢɹ ɜ ɦɢɤɪɨɤɚɧɨɧɢɱɟɫɤɨɦ ɚɧɫɚɦɛɥɟ ɩɪɢ ɡɚɞɚɧɧɨɣ ɷɧɟɪɝɢɢ

Ɋɟɲɟɧɢɟ. Ɉɛɨɡɧɚɱɢɦ W - ɫɬɚɬɢɫɬɢɱɟɫɤɢɣ ɜɟɫ, ɱɢɫɥɨ

ɫɩɨɫɨɛɨɜ, ɤɨɬɨɪɵɦɢ ɦɨɠɧɨ ɪɟɚɥɢɡɨɜɚɬɶ ɫɨɫɬɨɹɧɢɟ ɫ ɡɚɞɚɧɧɨɣ

ɷɧɟɪɝɢɟɣ. ɗɧɟɪɝɢɹ

E ɡɚɞɚɟɬɫɹ ɱɢɫɥɨɦ ɫɩɢɧɨɜ ɜɜɟɪɯ N

ɢ ɩɨɥ-

ɧɵɦ ɱɢɫɥɨɦ ɫɩɢɧɨɜ

N. Ɍɨɝɞɚ:

WC NN NN







![ !( !)].

ɉɨɥɨɠɢɦ

=xN, 0<x<1. ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ ɋɬɢɪɥɢɧɝɚ ɞɥɹ

of (lnN! = NlnN - N), ɩɨɥɭɱɚɟɦ ɷɧɬɪɨɩɢɸ:

SNxxNx x   



ln ( )ln( )11.

ɍɱɢɬɵɜɚɹ,ɱɬɨ ɷɧɟɪɝɢɹ

EHNNNHx





21() ()

,ɢɦɟɟɦ:

SN z z z zN       / ( )ln( ) ( )ln( ) ln21 1 1 1 2

, z=E/N

1.2.

5. ɍɫɬɚɧɨɜɢɬɶ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɷɧɬɪɨɩɢɟɣ ɜ ɤɚ-

ɧɨɧɢɱɟɫɤɨɦ ɢ ɦɢɤɪɨɤɚɧɨɧɢɱɟɫɤɨɦ ɚɧɫɚɦɛɥɟ ɜ ɧɟɜɡɚɢɦɨɞɟɣɫɬ-

ɜɭɸɳɟɣ ɦɨɞɟɥɢ ɂɡɢɧɝɚ.

ɍɤɚɡɚɧɢɟ

: ɢɫɩɨɥɶɡɨɜɚɬɶ ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɷɧɬɪɨɩɢɢ,

ɩɨɥɭɱɟɧɧɨɦ ɜ ɡɚɞɚɱɟ 1.2.4, ɷɧɟɪɝɢɸ ɞɥɹ ɤɚɧɨɧɢɱɟɫɤɨɝɨ ɚɧ-

ɫɚɦɛɥɹ (ɡɚɞɚɱɚ 1.2.2).

1.2.

6. ɇɚɣɬɢ ɫɜɨɛɨɞɧɭɸ ɷɧɟɪɝɢɸ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ

ɫɢɫɬɟɦɵ ɫɩɢɧɨɜ.

Ɉɬɜɟɬ

FETS TN H TQ   ln [ ] ln2

1.2.

7. ɇɚɣɬɢ ɫɪɟɞɧɟɟ ɱɢɫɥɨ ɫɩɢɧɨɜ,ɩɨɜɟɪɧɭɬɵɯ ɜɜɟɪɯ N

(T).

Ɉɬɜɟɬ:

.)2exp(1

)}2(exp{/1)(











NNHCNQTN

1.2.

8. ɉɨɥɭɱɢɬɶ ɬɟɩɥɨɟɦɤɨɫɬɶ ɫɢɫɬɟɦɵ ɧɟɜɡɚɢɦɨɞɟɣɫɬ-

ɜɭɸɳɢɯ ɢɡɢɧɝɨɜɫɤɢɯ ɫɩɢɧɨɜ. ɂɫɫɥɟɞɨɜɚɬɶ ɜ ɩɪɟɞɟɥɶɧɵɯ ɫɥɭ-

ɱɚɹɯ

0 ɢ T

. ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤ C(T).

Ɉɬɜɟɬ:

CT E T T F T N HT HT() / ( ) [ ] 



dd ch

ww P P

1.2.

9. ɇɚɣɬɢ ɫɪɟɞɧɸɸ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ ɫɢɫɬɟɦɵ ɩɪɢ

ɬɟɦɩɟɪɚɬɭɪɟ

Ɉɬɜɟɬ: MNN NNN  H



! 



 

000

2()() [th

1.2.

10. ɇɚɣɬɢ ɭɞɟɥɶɧɭɸ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ

ɫɢɫɬɟɦɵ.

Ɉɬɜɟɬ

(/ ) |1

NMH T

dd .

1.2.

11. ɇɚɣɬɢ ɫɜɹɡɶ ɦɟɠɞɭ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶɸ, ɦɚɝɧɢɬ-

ɧɨɣ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶɸ ɢ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɟɣ ɦɨɞɟɥɢ ɂɡɢɧɝɚ

ɛɟɡ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ.

Ɉɬɜɟɬ

: MQHFH F  

wwEwwFww

ln ,

H .

1.3. Cɢɫɬɟɦɚ ɥɨɤɚɥɶɧɵɯ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ

cɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ.

ɗɮɮɟɤɬɢɜɧɨɟ ɩɨɥɟ ȼɟɣɫɫɚ.

ɉɭɫɬɶ ɦɚɝɧɢɬɧɵɟ ɦɨɦɟɧɬɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ

ɞɪɭɝɨɦ, ɩɪɢ ɷɬɨɦ ɷɧɟɪɝɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ

(1.6)



OP P

Ɂɞɟɫɶ

- ɛɟɡɪɚɡɦɟɪɧɵɣ ɩɚɪɚɦɟɬɪ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɦɨ-

ɦɟɧɬɚɦɢ. ɋɭɦɦɚɪɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ ɢɦɟɟɬ ɜɢɞ

ij i j

 

oo o o

12/.

OPP P

H

(1.7)

ȼ ɷɬɨɦ ɩɚɪɚɝɪɚɮɟ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɜɨɡɦɨɠɧɭɸ ɨɪɢɟɧɬɚɰɢɸ

ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɬɨɥɶɤɨ ɩɨ (+

) ɢ ɩɪɨɬɢɜ (-

) ɩɨɥɹ. ɋɨ-

ɝɥɚɫɧɨ (1.6)-(1.7), ɦɚɝɧɢɬɧɵɟ ɦɨɦɟɧɬɵ ɫɬɪɟɦɹɬɫɹ ɭɩɨɪɹɞɨɱɢɬɶ

ɞɪɭɝ ɞɪɭɝɚ. Ɍɨɝɞɚ ɧɚ ɜɵɞɟɥɟɧɧɵɣ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ ɛɭɞɟɬ

ɞɟɣɫɬɜɨɜɚɬɶ ɜɧɟɲɧɟɟ ɩɨɥɟ

H ɢ ɩɨɥɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ (1.6) ɨɬ

ɜɫɟɯ ɨɫɬɚɥɶɧɵɯ ɦɨɦɟɧɬɨɜ

iijj

. (1.8)

ɍɫɪɟɞɧɟɧɧɨɟ (ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɢ) ɫɭɦɦɚɪɧɨɟ ɩɨɥɟ ɛɭɞɟɬ

ɬɨɝɞɚ ɪɚɜɧɨ

iijj

 (1.9)

ȼ ɫɢɥɭ ɬɪɚɧɫɥɹɰɢɨɧɧɨɣ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ (<

> ɧɟ ɡɚɜɢɫɢɬ ɨɬ j)

ɜ ɫɪɟɞɧɟɦ ɧɚ ɜɵɞɟɥɟɧɧɵɣ ɫɩɢɧ ɞɟɣɫɬɜɭɟɬ ɪɟɡɭɥɶɬɢɪɭɸɳɟɟ

ɷɮɮɟɤɬɢɜɧɨɟ ɩɨɥɟ

HH N

eff i

 !  ! 

. (1.10)

Ɂɞɟɫɶ ɨɛɨɡɧɚɱɟɧɨ

, ɬɚɤ ɱɬɨ Oɢɦɟɟɬ ɫɦɵɫɥ ɛɟɡ-

ɪɚɡɦɟɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ, ɩɪɢɯɨɞɹɳɟɝɨɫɹ ɧɚ

ɨɞɢɧ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ. ɉɪɢɛɥɢɠɟɧɢɟ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɩɨɥɹ

ȼɟɣɫɫɚ ɫɨɫɬɨɢɬ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɢɫɬɢɧɧɨɟ ɫɭɦɦɚɪɧɨɟ

ɩɨɥɟ ɜ

i-ɦ ɭɡɥɟ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɪɟɞɧɢɦ ɩɨɥɟɦ H

eff

ɢ ɧɟ ɡɚɜɢɫɢɬ

ɨɬ ɨɪɢɟɧɬɚɰɢɢ

i-ɝɨ ɫɩɢɧɚ

. Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɷɬɨ ɞɟɣɫɬɜɢ-

ɬɟɥɶɧɨ ɩɪɢɛɥɢɠɟɧɢɟ, ɬɚɤ ɤɚɤ ɩɪɢ ɧɚɩɪɚɜɥɟɧɧɨɦ ɜɜɟɪɯ i-ɦɫɩɢ-

ɧɟ ɜɟɪɨɹɬɧɨɫɬɶ ɞɥɹ ɫɨɫɟɞɟɣ ɛɵɬɶ ɧɚɩɪɚɜɥɟɧɧɵɦɢ ɜɜɟɪɯ ɛɨɥɶɲɟ

ɫɪɟɞɧɟɣ, ɬ.ɟ. ɜ ɩɪɢɛɥɢɠɟɧɢɢ ȼɟɣɫɫɚ ɦɵ ɩɪɟɧɟɛɪɟɝɚɟɦ ɮɥɭɤ-

ɬɭɚɰɢɹɦɢ. ɗɬɨ ɥɭɱɲɟ ɜcɟɝɨ ɜɵɩɨɥɧɹɟɬɫɹ ɩɪɢ ɛɨɥɶɲɟɦ ɱɢɫɥɟ

ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ

Z. Ȼɨɥɟɟ ɬɨɝɨ, ɩɪɢ Zof ɩɪɢɛɥɢɠɟɧɢɟ

ȼɟɣɫɫɚ ɞɚɟɬ

ɬɨɱɧɵɣ ɪɟɡɭɥɶɬɚɬ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɱɢɬɵɜɚɹ, ɱɬɨ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɧɚɯɨɞɢɬ-

ɫɹ ɜ ɷɮɮɟɤɬɢɜɧɨɦ ɫɪɟɞɧɟɦ ɩɨɥɟ

eff

(1.10), ɢɡ (1.3) ɦɨɠɧɨ ɩɨ-

ɥɭɱɢɬɶ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ

 !

. (1.11)

ɍɪɚɜɧɟɧɢɟ (1.11) - ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ. Ɋɚɫɫɦɨɬɪɢɦ ɩɪɟɞɟɥɶɧɵɣ

ɫɥɭɱɚɣ

o

ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ T ɢɡ

(1.11) ɩɨɥɭɱɚɟɦ :

 !

(1.12)



. (1.13)

Ɉɛɨɡɧɚɱɢɦ

NO =

- ɬɟɦɩɟɪɚɬɭɪɭ Ʉɸɪɢ-ȼɟɣɫɫɚ. Ɂɚɤɨɧ Ʉɸ-

ɪɢ-ȼɟɣɫɫɚ ɞɥɹ ɦɨɞɟɥɢ ɫ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ

ɜɢɞ:



. (1.14)

ɂɬɚɤ, ɩɪɢ ɛɨɥɶɲɨɣ ɬɟɦɩɟɪɚɬɭɪɟ

Ɍ >

F

> 0, ɬ.ɟ.

ɪɟɚɥɢɡɭɟɬɫɹ ɩɚɪɚɦɚɝɧɢɬɧɨɟ ɫɨɫɬɨɹɧɢɟ. Ʉɚɤ ɜɢɞɧɨ ɢɡ (1.14), ɜ

ɬɨɱɤɟ

Ɍ =

ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ

ɢ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ

ɪɚɫɯɨ-

ɞɹɬɫɹ (ɪɢɫ.4), ɢ ɞɚɥɟɟ (ɩɪɢ

Ɍ <

) ɜɨɡɦɨɠɟɧ ɩɟɪɟɯɨɞ ɜ ɮɟɪɪɨ-

ɦɚɝɧɢɬɧɨɟ ɫɨɫɬɨɹɧɢɟ. ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɬɨɱɧɨɟ ɭɪɚɜɧɟɧɢɟ

(1.11) ɩɪɢ ɩɨɥɟ

H=0. ȼɜɟɞɟɦ ɛɟɡɪɚɡɦɟɪɧɵɣ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ

ɧɚ ɭɡɥɟ

R = <

Ɋɢɫ.4. Ɂɚɤɨɧ Ʉɸɪɢ - ȼɟɣɫɫɚ

(

~|1 T |

) ɞɥɹ ɦɚɝ-

ɧɢɬɧɨɣ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ ɜ

ɪɚɦɤɚɯ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ

ɦɨɞɟɥɢ ɂɡɢɧɝɚ

Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ -1 < R < +1. Ɋɚɫɫɦɨɬɪɢɦ ɩɨɥɨɠɢɬɟɥɶɧɭɸ

ɨɛɥɚɫɬɶ ɡɧɚɱɟɧɢɣ

R. ɍɪɚɜɧɟɧɢɟ (1.11) ɩɪɢɧɢɦɚɟɬ ɜɢɞ :

. (1.15)

ȿɫɥɢ ɨɛɨɡɧɚɱɢɬɶ x = R

4

/ T, ɬɨ xT /

= th x. Ɍɨɝɞɚ ɨɱɟɜɢɞɧɨ,

ɱɬɨ ɩɪɢ T >

4

ɢɦɟɟɬɫɹ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ R=0 ɢɧɟɬɭɩɨ-

ɪɹɞɨɱɟɧɢɹ. ȼ ɫɥɭɱɚɟ T <

4

ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ ɢɦɟɟɬ ɧɟɬɪɢɜɢ-

ɚɥɶɧɨɟ ɪɟɲɟɧɢɟ R

z

0, ɨɩɢɫɵɜɚɸɳɟɟ ɮɟɪɪɨɦɚɝɧɢɬɧɨɟ ɫɨɫɬɨɹ-

ɧɢɟ.

Ɉɛɳɢɣ ɜɢɞ ɡɚɜɢɫɢɦɨɫɬɢ R(T) ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ.1,ɚ.

ȼɛɥɢɡɢ ɨɛɥɚɫɬɢ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɧɟɫɥɨɠɧɨ ɩɨɥɭɱɢɬɶ

= 3(

4

/T - 1), ɨɬɤɭɞɚ ɢ ɫɥɟɞɭɟɬ, ɱɬɨ ɩɪɢ T >

4

R=0, ɚɜɫɥɭ-

ɱɚɟ T <

4

0, ɬ.ɟ. ɪɟɲɟɧɢɟ, ɩɨɞɯɨɞɹɳɟɟ ɤ ɬɨɱɤɟ ɩɟɪɟɯɨɞɚ ɫ

ɛɟɫɤɨɧɟɱɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ (dR/dT ~

1/ /

T  1). Ʉɚɤ ɞɚɥɶɲɟ

ɛɭɞɟɬ ɩɨɤɚɡɚɧɨ, ɬɚɤɨɟ ɛɟɫɤɨɧɟɱɧɨɟ ɜɨɡɪɚɫɬɚɧɢɟ ɫɤɨɪɨɫɬɢ ɧɚ-

ɦɚɝɧɢɱɟɧɧɨɫɬɢ ɫɜɹɡɚɧɨ ɫ ɪɚɫɯɨɞɢɦɨɫɬɶɸ ɮɥɭɤɬɭɚɰɢɣ ɢ ɤɨɪɪɟ-

ɥɹɰɢɨɧɧɵɯ ɞɥɢɧ ɩɪɢ ɩɪɢɛɥɢɠɟɧɢɢ ɤ ɬɨɱɤɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ

ɜɬɨɪɨɝɨ ɪɨɞɚ.

ȼ ɞɪɭɝɨɦ ɩɪɟɞɟɥɶɧɨɦ ɫɥɭɱɚɟ T

o





ɩɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ R

o

1 ɩɨ ɡɚɤɨɧɭ R = 1 - 2exp(-2

/T). Ɂɚɦɟɬɢɦ, ɱɬɨ ɷɤɫɩɨɧɟɧɰɢ-

ɚɥɶɧɨ ɦɚɥɨɟ ɨɬɤɥɨɧɟɧɢɟ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ ɨɬ ɩɨɥɧɨɝɨ ɧɚɫɵɳɟ-

ɧɢɹ ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɹɜɥɹɟɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɬɨɝɨ, ɱɬɨ

ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɩɨɱɬɢ ɜɫɟ ɫɩɢɧɵ ɨɪɢɟɧɬɢɪɨɜɚɧɵ

ɜɜɟɪɯ(ɩɨ ɩɨɥɸ). ɗɧɟɪɝɢɹ, ɧɟɨɛɯɨɞɢɦɚɹ ɞɥɹ ɩɨɜɨɪɨɬɚ ɨɞɧɨɝɨ

ɫɩɢɧɚ, ɪɚɜɧɚ

44 

, ɬɚɤ ɤɚɤ ɫɨɡɞɚɜɚɟɦɨɟ ɨɫɬɚɥɶɧɵɦɢ

ɫɩɢɧɚɦɢ ɩɨɥɟ ɪɚɜɧɨ

. ɉɪɢɛɥɢɠɟɧɢɟ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɩɨɥɹ

ɩɪɢɜɨɞɢɬ ɤ ɬɨɱɧɨɦɭ ɪɟɡɭɥɶɬɚɬɭ ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ, ɤɨ-

ɝɞɚ ɫɩɢɧɵ ɛɨɥɶɲɟ ɧɟ ɮɥɭɤɬɭɢɪɭɸɬ. ɉɪɢ ɷɬɨɦ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨ-

ɜɨɪɨɬɚ ɜɬɨɪɨɝɨ ɫɩɢɧɚ, ɞɚɸɳɟɝɨ ɜɤɥɚɞ ɜ ɩɨɥɟ, ɩɪɟɧɟɛɪɟɠɢɦɨ

ɦɚɥɚ, ɬɚɤ ɤɚɤ ɨɧ ɭɞɟɪɠɢɜɚɟɬɫɹ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɧɚ ɦɟɫɬɟ ɫɜɨɢ-

ɦɢ ɫɨɫɟɞɹɦɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɜɨɡɛɭ-

ɠɞɟɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɨɬɞɟɥɟɧɵ ɨɬ ɨɫɧɨɜɧɨɝɨ ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ

ɳɟɥɶɸ 2

Ɉɰɟɧɢɦ ɦɚɫɲɬɚɛ ɬɟɦɩɟɪɚɬɭɪɵ Ʉɸɪɢ-ȼɟɣɫɫɚ. ɂɡɜɟɫɬɧɨ

ɢɡ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɱɬɨ

ɷɤɫɩ

=300-1000K (ɬɚɤ, ɞɥɹ ɠɟɥɟɡɚ

=1043K, ɞɥɹ ɧɢɤɟɥɹ

=627K, ɞɥɹ ɝɚɞɨɥɢɧɢɹ

=293K). ȼɫɢɫ-

ɬɟɦɟ ɋȽɋ ɦɚɝɧɟɬɪɨɧ Ȼɨɪɚ

~ 10

-20

. Ɉɬɫɸɞɚ, ɩɨɥɚɝɚɹ ɛɟɡɪɚɡ-

ɦɟɪɧɭɸ ɤɨɧɫɬɚɧɬɭ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ

ɩɨɪɹɞɤɚ ɟɞɢɧɢɰɵ, ɚɱɢɫ-

ɥɨ ɚɬɨɦɨɜ N~ 10

, ɢɦɟɟɦ

ɬɟɨɪ

~0.1K. Ɍɚɤɨɟ ɪɚɡɥɢɱɢɟ ɫɜɹɡɚɧɨ ɫ

ɬɟɦ, ɱɬɨ ɪɟɚɥɶɧɵɣ ɦɚɫɲɬɚɛ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɭɳɟɫɬɜɟɧɧɨ

ɛɨɥɶɲɟ (

~ 10

), ɬɚɤ ɤɚɤ ɜ ɫɩɢɧ-ɫɩɢɧɨɜɨɦ ɭɩɨɪɹɞɨɱɟɧɢɢ ɢɝɪɚ-

ɟɬ ɨɫɧɨɜɧɭɸ ɪɨɥɶ ɫɩɟɰɢɮɢɱɟɫɤɨɟ ɨɛɦɟɧɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ,

ɤɨɬɨɪɨɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɜ ɫɥɟɞɭɸɳɟɦ ɩɚɪɚɝɪɚɮɟ.

1.4. ɉɪɢɪɨɞɚ ɨɛɦɟɧɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ.

Ɇɨɞɟɥɢ ɂɡɢɧɝɚ ɢ Ƚɟɣɡɟɧɛɟɪɝɚ

ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ ɦɟɠɞɭ ɦɚɝɧɢɬɧɵɦɢ ɦɨɦɟɧɬɚɦɢ ɧɨɫɢɬ

ɱɢɫɬɨ ɤɜɚɧɬɨɜɵɣ ɯɚɪɚɤɬɟɪ - ɷɬɨ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ ɨɛɦɟɧɧɨɟ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ. Ɇɵ ɩɨɤɚɠɟɦ, ɱɬɨ ɤɨɧɫɬɚɧɬɚ ɨɛɦɟɧɧɨɝɨ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɹ ɞɨɫɬɢɝɚɟɬ

~ 10

- 10

, ɱɬɨ ɫɨɝɥɚɫɭɟɬɫɹ ɫ ɷɤɫɩɟ-

ɪɢɦɟɧɬɨɦ.

Ⱦɥɹ ɚɧɫɚɦɛɥɹ ɨɞɢɧɚɤɨɜɵɯ ɤɜɚɧɬɨɜɵɯ ɱɚɫɬɢɰ ɞɨɥɠɟɧ

ɜɵɩɨɥɧɹɬɶɫɹ ɩɪɢɧɰɢɩ ɬɨɠɞɟɫɬɜɟɧɧɨɫɬɢ - ɨɧɢ ɞɨɥɠɧɵ ɛɵɬɶ

ɧɟɪɚɡɥɢɱɢɦɵ ɜ ɫɢɥɭ ɩɪɢɧɰɢɩɚ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ: p x

ʄ.

ȿɫɥɢ ɢɦɟɸɬɫɹ ɜɫɟɝɨ ɞɜɟ ɱɚɫɬɢɰɵ, ɬɨ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ, ɩɨɥɭ-

ɱɚɸɳɢɟɫɹ ɞɪɭɝ ɢɡ ɞɪɭɝɚ ɩɪɨɫɬɨ ɩɟɪɟɫɬɚɧɨɜɤɨɣ ɨɛɟɢɯ ɱɚɫɬɢɰ,

ɞɨɥɠɧɵ ɛɵɬɶ ɮɢɡɢɱɟɫɤɢ ɩɨɥɧɨɫɬɶɸ ɷɤɜɢɜɚɥɟɧɬɧɵ. ɗɬɨ ɡɧɚɱɢɬ,

ɱɬɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɚɤɨɣ ɩɟɪɟɫɬɚɧɨɜɤɢ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɫɢɫɬɟ-

ɦɵ ɦɨɠɟɬ ɢɡɦɟɧɢɬɶɫɹ ɬɨɥɶɤɨ ɧɚ ɧɟɫɭɳɟɫɬɜɟɧɧɵɣ ɮɚɡɨɜɵɣ

ɦɧɨɠɢɬɟɥɶ: ɟɫɥɢ |

(1,2)|

(2,1)|

, ɬɨ

(1,2) = exp(i

)

(2,1), ɝɞɟ

- ɧɟɤɨɬɨɪɚɹ ɜɟɳɟɫɬɜɟɧɧɚɹ ɩɨɫɬɨɹɧɧɚɹ. ɉɨɜɬɨɪɧɚɹ

ɩɟɪɟɫɬɚɧɨɜɤɚ ɫ ɧɟɨɛɯɨɞɢɦɨɫɬɶɸ ɩɪɢɜɨɞɢɬ ɤ ɭɫɥɨɜɢɸ exp(2i

)

= 1, ɬ.ɟ.

(1,2) = r

(2,1). ɉɨɷɬɨɦɭ ɟɫɬɶ ɜɫɟɝɨ ɞɜɟ ɜɨɡɦɨɠɧɨ-

ɫɬɢ: ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɥɢɛɨ ɫɢɦɦɟɬɪɢɱɧɚ (ɷɬɨ ɫɬɚɬɢɫɬɢɤɚ Ȼɨ-

ɡɟ), ɥɢɛɨ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɚ (ɷɬɨ ɫɬɚɬɢɫɬɢɤɚ Ɏɟɪɦɢ).

Ⱦɚɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ɫɩɪɚɜɟɞɥɢɜ ɢ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɱɢɫ-

ɥɚ ɱɚɫɬɢɰ.

Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɞɜɟ ɜɵɞɟɥɟɧɧɵɟ ɱɚɫɬɢɰɵ, ɢɦɟɸɳɢɟ

ɤɜɚɧɬɨɜɭɸ ɫɬɚɬɢɫɬɢɤɭ ɢ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɧɟ ɜɡɚɢɦɨɞɟɣ-

ɫɬɜɭɸɳɢɟ (ɤɭɥɨɧɨɜɫɤɨɟ ɢ ɞɪɭɝɢɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɭɱɬɟɧɵ ɧɟ-

ɤɢɦ ɫɪɟɞɧɢɦ ɩɨɥɟɦ). ɉɨɥɧɭɸ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ ɫɢɫɬɟɦɵ

ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɨɞɧɨɱɚɫɬɢɱɧɵɟ ɮɭɧɤɰɢɢ

(1) ɢ

(2)

ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

(1,2) = 1/

{

(1)

<

(2) r

<

(2)

<

(1) }.

ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɛɨɡɨɧɚɦ ɨɬɜɟɱɚɟɬ ɡɧɚɤ “+”, ɚ ɮɟɪɦɢɨɧɚɦ “-”,

ɪɟɚɥɢɡɭɸɳɢɟ ɫɢɦɦɟɬɪɢɱɧɭɸ ɢ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɭɸ ɫɢɬɭɚɰɢɢ.

ȼɢɞɧɨ, ɱɬɨ ɟɫɥɢ ɫɨɫɬɨɹɧɢɟ a ɪɚɜɧɨ b, ɬɨ ɞɥɹ ɡɧɚɤɚ “-” ɜɨɥɧɨ-