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

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110

HH G

pp p

NV Vpp f   

10/(()(|'|)

. (6.15)

ȼ (6.15) ɭɱɬɟɧɚ ɧɨɪɦɢɪɨɜɤɚ ɧɚ 1 ɷɥɟɤɬɪɨɧ (ɦɧɨɠɢɬɟɥɶ 1/N).

ɉɪɟɧɟɛɪɟɠɟɦ ɚɞɞɢɬɢɜɧɨɣ ɩɨɫɬɨɹɧɧɨɣ (ɫɥɚɝɚɟɦɵɦ ɫ V(0)), ɟɟ

ɦɨɠɧɨ ɭɛɪɚɬɶ ɜ ɨɬɫɱɟɬ ɷɧɟɪɝɢɢ. ɍɱɬɟɦ, ɱɬɨ ɤɭɥɨɧɨɜɤɨɟ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɟ ɫɢɥɶɧɨ ɷɤɪɚɧɢɪɨɜɚɧɨ, ɢ ɩɨɷɬɨɦɭ ɩɪɢ ɦɚɥɵɯ ɢɦ-

ɩɭɥɶɫɚɯ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɡɚɜɢɫɢɦɨɫɬɶɸ ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧ-

ɬɚ ɨɬ ɢɦɩɭɥɶɫɚ:

V(| |)



const

ɝɞɟ k - ɢɦɩɭɥɶɫ ɨɛɪɟɡɚɧɢɹ Ɍɨɦɚɫɚ-Ɏɟɪɦɢ (



keE

Ɍɨɝɞɚ ɨɤɨɧɱɚɬɟɥɶɧɨ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ

pp p

VN f 

. (6.16)

ȼɜɟɞɟɦ ɫɪɟɞɧɸɸ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ ɫɢɫɬɟɦɵ:

NN N f NNN

 

{}, ,

(6.17)

Ɍɟɩɟɪɶ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ R

ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

HH HH

   

2/ , / . (6.18)

ɍɛɢɪɚɹ ɜ ɨɬɫɱɟɬ ɷɧɟɪɝɢɢ ɚɞɞɢɬɢɜɧɭɸ ɤɨɧɫɬɚɧɬɭ V/2, ɩɨɞɫɬɚɜɢɜ

(6.18) ɜ (6.17), ɧɚɯɨɞɢɦ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɋɬɨɧɟɪɚ

ɞɥɹ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ ɫ ɤɭɥɨɧɨɜɫɤɢɦ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɟɦ:





fx x









EH P EH P

,, , ,, ,

, , , {exp[ ( )] } .

(6.19)

Ʉɪɨɦɟ ɬɨɝɨ, ɞɥɹ ɩɨɥɧɨɣ ɡɚɦɤɧɭɬɨɫɬɢ ɭɪɚɜɧɟɧɢɹ (6.19) ɢɦɟɟɬ

ɦɟɫɬɨ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ - ɧɨɪɦɢɪɨɜɤɚ ɧɚ ɱɢɫɥɨ ɱɚɫɬɢɰ:



1exp ( )

EH P

. (6.20)

ɂɡ (6.19) ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫɥɭɱɚɣ ɩɨɥɧɨɣ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤ-

ɬɪɨɧɨɜ (ɬ.ɟ. ɥɨɤɚɥɢɡɨɜɚɧɧɵɟ ɫɩɢɧɵ - ɦɨɞɟɥɶ ɂɡɢɧɝɚ). ȿɫɥɢ

ɩɨɥɨɠɢɬɶ ɷɮɮɟɤɬɢɜɧɭɸ ɦɚɫɫɭ ɛɨɥɶɲɨɣ ɢ (

o0 (ɥɨɤɚɥɢɡɚ-

ɰɢɹ ɨɤɨɥɨ ɭɪɨɜɧɹ Ɏɟɪɦɢ ɩɨ ɷɧɟɪɝɢɢ), ɬɨ ɢɡ (6.19)-(6.20) ɩɪɢ

7

ɩɨɥɭɱɚɟɦ: RVR th{ / }

4,ɬ.ɟ. ɨɛɵɱɧɨɟ ɭɪɚɜɧɟ-

ɧɢɟ ȼɟɣɫɫɚ ɞɥɹ ɩɚɪɚɦɟɬɪɚ ɩɨɪɹɞɤɚ ɜ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɜ ɧɭɥɟɜɨɦ

ɩɨɥɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɠɟ ɜ ɷɬɨɦ ɩɪɟɞɟɥɟ ɜ ɦɨɞɟɥɢ ɋɬɨɧɟɪɚ

ɫɭɳɟɫɬɜɭɟɬ ɤɪɢɬɢɱɟɫɤɚɹ ɬɟɦɩɟɪɚɬɭɪɚ

4

= V/4 ɢ ɮɚɡɨɜɵɣ ɩɟɪɟ-

ɯɨɞ ɜ ɭɩɨɪɹɞɨɱɟɧɧɭɸ ɮɟɪɪɨɦɚɝɧɢɬɧɭɸ ɮɚɡɭ.

ɂɫɫɥɟɞɭɟɦ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɭɫɥɨɜɢɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɮɚ-

ɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. Ɋɚɡɥɨɠɢɦ (6.19) ɜ ɩɪɟɞɟɥɟ R

o

0 ɜɛɥɢɡɢ ɢɫ-

ɤɨɦɨɣ ɬɨɱɤɢ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. ɉɪɢ ɪɚɡɥɨɠɟɧɢɢ ɞɨ ɩɟɪɜɨɣ

ɩɪɨɢɡɜɨɞɧɨɣ ɮɟɪɦɢɟɜɫɤɨɣ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɨ ɷɧɟɪɝɢɢ

ɢɦɟɟɦ

1 

. (6.21)

ɍɱɢɬɵɜɚɹ, ɱɬɨ ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ

 

wwHGHP

f (),

ɢɡ (6.21) ɩɨɥɭɱɚɟɦ:

1 V

()

. (6.22)

ȼɵɪɚɠɟɧɢɟ (6.22) ɦɨɠɧɨ ɭɬɨɱɧɢɬɶ, ɢ ɩɪɢɜɟɫɬɢ ɤ ɭɫɥɨɜɢɸ ɋɬɨ-

ɧɟɪɚ ɞɥɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɩɪɢ ɧɟɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɫɩɨɧɬɚɧ-

ɧɨɣ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ R

z

()! 1

. (6.23)

111

Ɉɛɨɫɧɭɟɦ ɟɝɨ ɛɨɥɟɟ ɤɨɪɪɟɤɬɧɨ. Ɋɚɡɥɨɠɢɦ ɭɪɚɜɧɟɧɢɟ ɋɬɨɧɟɪɚ

(6.19) ɞɨ ɜɬɨɪɨɣ (ɨɧɚ ɫɨɤɪɚɳɚɟɬɫɹ) ɢ ɬɪɟɬɶɟɣ ɩɪɨɢɡɜɨɞɧɵɯ

ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɨ ɷɧɟɪɝɢɢ:

 

¦¦

() (

)

. (6.24)

Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ ɢɫɯɨɞɹ ɢɡ ɜɢɞɚ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɱɬɨ ɜ

ɧɚɢɛɨɥɟɟ ɫɭɳɟɫɬɜɟɧɧɨɣ ɨɛɥɚɫɬɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ

H

ɗɬɨ ɫɬɚɧɨɜɢɬɫɹ ɩɨɧɹɬɧɨ ɢɡ ɜɢɞɚ ɬɪɟɬɶɟɣ ɩɪɨɢɡɜɨɞɧɨɣ



([])

[]

ɩɪɢ

H

. Ɉɤɨɧɱɚɬɟɥɶɧɨ ɭɫɥɨɜɢɟ

Ȼɥɨɯɚ

ɋɬɨɧɟɪɚ

ɞɥɹ ɫɭɳɟ-

ɫɬɜɨɜɚɧɢɹ ɩɪɢ ɧɟɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɫɩɨɧɬɚɧɧɨɣ ɧɚɦɚɝɧɢ-

ɱɟɧɧɨɫɬɢ ɢɦɟɟɬ ɜɢɞ:

10

()

ɢɥɢ

! 1

. (6.25)

ɂɡ ɭɫɥɨɜɢɹ (6.25) ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɩɨɥɭɱɚɟɦ (6.23),

ɨɬɤɭɞɚ ɫ ɭɱɟɬɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ (6.6)-(6.7)

ɧɚɯɨɞɢɦ ɭɫɥɨɜɢɟ ɧɚ ɩɚɪɚɦɟɬɪ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ

. (6.26)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɟɪɪɨɦɚɝɧɟɬɢɡɦ ɋɬɨɧɟɪɚ ɜɨɡɧɢɤɚɟɬ

ɬɨɥɶɤɨ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɦ ɩɚɪɚɦɟɬɪɟ ɤɭɥɨɧɨɜɫɤɨɝɨ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ. ɉɪɢɱɢɧɚ ɬɚɤɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ ɫɧɢɡɭ ɜ ɬɨɦ, ɱɬɨ

ɞɥɹ ɩɨɞɚɜɥɟɧɢɹ ɮɥɭɤɬɭɚɰɢɣ, ɫɨɡɞɚɜɚɟɦɵɯ ɞɟɥɨɤɚɥɢɡɚɰɢɟɣ

ɷɥɟɤɬɪɨɧɨɜ ɫ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ ɦɚɫɲɬɚɛɚ E

, ɧɟɨɛɯɨɞɢɦɨ

112

ɬɚɤɨɝɨ ɠɟ ɦɚɫɲɬɚɛɚ (ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ) ɪɚɫɳɟɩɥɟɧɢɟ ɷɧɟɪɝɢɢ

ɩɪɢ ɪɚɡɥɢɱɧɨɣ ɨɪɢɟɧɬɚɰɢɢ ɫɩɢɧɚ ɡɚ ɫɱɟɬ ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɹ ɞɥɹ ɪɟɚɥɢɡɚɰɢɢ ɩɪɨɬɢɜɨɞɟɣɫɬɜɭɸɳɟɝɨ ɮɥɭɤɬɭɚɰɢ-

ɹɦ ɮɟɪɪɨɦɚɝɧɢɬɧɨɝɨ ɭɩɨɪɹɞɨɱɟɧɢɹ.

Ɂɚɞɚɱɢ

6.3.

. ɉɨɥɭɱɢɬɶ ɩɨɩɪɚɜɤɭ ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ ɤ ɯɢɦɢɱɟɫɤɨ-

ɦɭ ɩɨɬɟɧɰɢɚɥɭ

ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ

(

ETE

), ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɭɪɚɜɧɟɧɢɟɦ ɧɨɪɦɢ-

ɪɨɜɤɢ (6.20).

ɍɤɚɡɚɧɢɟ

: ɩɪɢ ɪɚɫɱɟɬɟ ɩɪɟɧɟɛɪɟɱɶ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɦɢ

ɩɨɩɪɚɜɤɚɦɢ ~ exp(-

/T).

6.3.

. ɇɚɣɬɢ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɜ ɦɨɞɟɥɢ ɋɬɨɧɟ-

ɪɚ (T

<< E

), ɭɱɢɬɵɜɚɹ ɬɟɦɩɟɪɚɬɭɪɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɯɢɦɩɨ-

ɬɟɧɰɢɚɥɚ

(ɫɦ. ɩɪɟɞɵɞɭɳɭɸ ɡɚɞɚɱɭ).

ɍɤɚɡɚɧɢɟ

: ɪɚɡɥɨɠɢɬɶ ɩɪɚɜɭɸ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ ɋɬɨɧɟɪɚ

ɩɪɢ R

Ɉɬɜɟɬ

: TE EV

cf f

24 1 2 3

().

6.3.

. ɉɨɥɭɱɢɬɶ ɬɨɱɧɨɟ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚ

ɩɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ R ɜ ɦɨɞɟɥɢ ɋɬɨɧɟɪɚ ɩɪɢ T =0 ɢ ɧɚɣɬɢ ɟɝɨ,

ɩɨɥɚɝɚɹ R <<E

/V.

Ɉɬɜɟɬ

RE V E V

[]46 1 2 3

ɩɪɢ ɭɫɥɨɜɢɢ R<<E

/V.

113

114

7. ɄɈɇȾȿɇɋȺɐɂə ɂ ɈɌȼȿɊȾȿȼȺɇɂȿ

ɂɫɫɥɟɞɨɜɚɧɢɹ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɩɪɟɞɫɬɚɜɥɟɧɧɵɟ ɜ ɩɪɟɞɵ-

ɞɭɳɢɯ ɝɥɚɜɚɯ, ɩɨɡɜɨɥɹɸɬ ɝɥɭɛɠɟ ɪɚɡɜɢɬɶ ɬɟɨɪɢɸ ɬɚɤɢɯ ɮɚɡɨ-

ɜɵɯ ɩɟɪɟɯɨɞɨɜ, ɤɚɤ ɤɨɧɞɟɧɫɚɰɢɹ ɢ ɨɬɜɟɪɞɟɜɚɧɢɟ (ɤɪɢɫɬɚɥɥɢɡɚ-

ɰɢɹ), ɜɜɟɫɬɢ ɜ ɧɢɯ ɩɨɧɹɬɢɟ ɮɥɭɤɬɭɚɰɢɣ, ɩɨɥɭɱɢɬɶ ɨɫɧɨɜɧɵɟ

ɭɪɚɜɧɟɧɢɹ ɫɨɫɬɨɹɧɢɹ. ȼ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɮɚɡɨɜɵɟ

ɩɪɟɜɪɚɳɟɧɢɹ ɜɟɳɟɫɬɜɚ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɩɟɪɟɫɬɪɨɣɤɟ ɩɪɨɫɬɪɚɧ-

ɫɬɜɟɧɧɨɝɨ ɭɩɨɪɹɞɨɱɟɧɢɹ ɚɬɨɦɨɜ ɫ ɢɡɦɟɧɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ.

Ⱦɨ ɫɢɯ ɩɨɪ ɦɵ ɢɫɫɥɟɞɨɜɚɥɢ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɮɚɡɨɜɵɟ ɩɟɪɟ-

ɯɨɞɵ ɜɬɨɪɨɝɨ ɪɨɞɚ. Ɂɞɟɫɶ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪɵ ɮɚɡɨɜɵɯ

ɩɟɪɟɯɨɞɨɜ ɩɟɪɜɨɝɨ ɪɨɞɚ, ɩɨɡɧɚɤɨɦɢɦɫɹ ɫ ɬɢɩɢɱɧɵɦɢ ɦɨɞɟɥɹ-

ɦɢ, ɨɩɢɫɵɜɚɸɳɢɦɢ ɢɯ - ɫ ɦɨɞɟɥɶɸ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ ɢ ɪɟɲɟ-

ɬɨɱɧɨɝɨ ɝɚɡɚ.

7.1. Ɏɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɝɚɡ - ɠɢɞɤɨɫɬɶ. Ʉɨɧɞɟɧɫɚɰɢɹ.

ɍɪɚɜɧɟɧɢɟ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ

Ɉɫɧɨɜɧɨɟ ɨɬɥɢɱɢɟ ɠɢɞɤɨɣ ɮɚɡɵ ɜɟɳɟɫɬɜɚ ɨɬ ɝɚɡɨɨɛɪɚɡ-

ɧɨɣ - ɟɟ ɩɪɚɤɬɢɱɟɫɤɢ ɢɞɟɚɥɶɧɚɹ ɧɟɫɠɢɦɚɟɦɨɫɬɶ, ɬ.ɟ. ɝɚɡ ɡɚɩɨɥ-

ɧɹɟɬ ɜɟɫɶ ɩɪɟɞɨɫɬɚɜɥɟɧɧɵɣ ɟɦɭ ɨɛɴɟɦ, ɚ ɠɢɞɤɨɫɬɶ - ɬɨɥɶɤɨ

ɱɚɫɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɟɟ ɩɥɨɬɧɨɫɬɶɸ. Ⱦɥɹ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ɟɝɨ

ɫɠɢɦɚɟɦɨɫɬɶ ɩɨɞ ɜɧɟɲɧɢɦ ɞɚɜɥɟɧɢɟɦ P ɢ ɬɟɦɩɟɪɚɬɭɪɨɣ T

ɨɩɢɫɵɜɚɟɬɫɹ ɢɡɜɟɫɬɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫɨɫɬɨɹɧɢɹ:

P V = R T ,

R=Nk, k - ɩɨɫɬɨɹɧɧɚɹ Ȼɨɥɶɰɦɚɧɚ. Ɉɬɫɸɞɚ ɞɚɜɥɟɧɢɟ:

P = k T N / V = k T

ɉɪɢ ɞɚɜɥɟɧɢɢ P=0 ɩɥɨɬɧɨɫɬɶ

ɬɚɤɠɟ ɪɚɜɧɚ ɧɭɥɸ, ɢ ɜɫɟ ɦɨɥɟ-

ɤɭɥɵ ɪɚɡɥɟɬɚɸɬɫɹ, ɪɚɜɧɨɦɟɪɧɨ ɡɚɩɨɥɧɹɹ ɨɛɴɟɦ. Ɉɞɧɚɤɨ ɞɥɹ

ɨɩɢɫɚɧɢɹ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɢɡ ɝɚɡɨɨɛɪɚɡɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ

ɤɨɧɞɟɧɫɢɪɨɜɚɧɧɭɸ ɮɚɡɭ (ɠɢɞɤɨɫɬɶ) ɧɟɨɛɯɨɞɢɦɨ ɭɱɟɫɬɶ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɟ ɱɚɫɬɢɰ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɭɞɟɪɠɚɬɶ ɱɚɫɬɢɰɵ ɞɪɭɝ

ɨɤɨɥɨ ɞɪɭɝɚ, ɩɪɟɩɹɬɫɬɜɭɹ ɢɯ ɪɚɡɥɟɬɭ ɧɚ ɛɨɥɶɲɨɟ ɪɚɫɫɬɨɹɧɢɟ.

Ɍɨɝɞɚ ɜ ɭɪɚɜɧɟɧɢɢ ɫɨɫɬɨɹɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɜɜɟɫɬɢ ɩɨɧɹɬɢɟ

ɜɧɭɬɪɟɧɧɟɝɨ ɞɚɜɥɟɧɢɹ, ɫɜɹɡɚɧɧɨɝɨ ɫ ɷɬɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ, ɢ

ɭɱɟɫɬɶ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɤɨɧɟɱɧɨɝɨ ɨɛɴɟɦɚ, ɦɟɧɶɲɟ ɤɨɬɨɪɨɝɨ

ɫɢɫɬɟɦɚ ɧɟ ɦɨɠɟɬ ɡɚɧɢɦɚɬɶ.

7.1.1. Ɍɟɨɪɢɹ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ

ȼɜɟɞɟɦ ɩɚɪɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɱɚɫɬɢɰ V(|r

-r

|) ɜ ɜɢɞɟ,

ɩɨɤɚɡɚɧɧɨɦ ɧɚ ɪɢɫ. 11. Ʉɚɤ ɜɢɞɧɨ ɢɡ ɪɢɫɭɧɤɚ, ɧɚ ɦɚɥɵɯ ɪɚɫ-

ɫɬɨɹɧɢɹɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɨɬɬɚɥ-

ɤɢɜɚɸɳɟɟ, ɩɪɟɩɹɬɫɬɜɭɸɳɟɟ

ɜɡɚɢɦɨɩɪɨɧɢɤɧɭɬɶ ɱɚɫɬɢɰɚɦ ɞɪɭɝ

ɜɞɪɭɝɚ, ɦɨɞɟɥɢɪɭɸɳɟɟ ɬɜɟɪɞɵɟ

ɲɚɪɵ, ɡɚɧɢɦɚɸɳɢɟ ɤɨɧɟɱɧɵɣ

ɨɛɴɟɦ (ɫ ɪɚɞɢɭɫɨɦ C, ɫɦ.ɪɢɫ.11).

Ɋɨɥɶ ɬɜɟɪɞɵɯ ɲɚɪɨɜ ɫɨɫɬɨɢɬ ɜ

ɬɨɦ, ɱɬɨ ɨɧɢ ɩɪɢɜɨɞɹɬ ɤ ɧɚɫɵ-

ɳɟɧɧɨɣ ɠɢɞɤɨɫɬɢ, ɨɛɴɟɦ ɤɨɬɨɪɨɣ

ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɦɟɧɶɲɟ ɨɛɴɟɦɚ

ɩɥɨɬɧɨɣ ɭɩɚɤɨɜɤɢ ɬɜɟɪɞɵɯ ɹɞɟɪ.

ɇɚ ɛɨɥɶɲɢɯ ɪɚɫɫɬɨɹɧɢɹɯ ɢɦɟɟɬ

ɦɟɫɬɨ ɨɛɥɚɫɬɶ ɩɪɢɬɹɠɟɧɢɹ, ɱɬɨ

ɧɟ ɞɚɟɬ ɪɚɡɨɣɬɢɫɶ ɱɚɫɬɢɰɚɦ ɧɚ

ɛɨɥɶɲɨɟ ɪɚɫɫɬɨɹɧɢɟ. ɉɪɢɱɢɧɨɣ

ɤɨɧɞɟɧɫɚɰɢɢ ɤɚɤ ɪɚɡ ɢ ɹɜɥɹɟɬɫɹ

ɬɚ ɱɚɫɬɶ ɩɨɬɟɧɰɢɚɥɚ, ɤɨɬɨɪɚɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɢɬɹɠɟɧɢɸ.

Ɋɢɫ.11. ɉɚɪɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬ-

ɜɢɟ ɱɚɫɬɢɰ ɩɪɢ ɨɩɢɫɚɧɢɢ ɤɨɧ-

ɞɟɧɫɚɰɢɢ (ɬɨɱɤɚ |r|=C ɫɨɨɬ-

ɜɟɬɫɬɜɭɟɬ ɞɢɚɦɟɬɪɭ ɱɚɫɬɢɰ ɜ

ɦɨɞɟɥɢ ɬɜɟɪɞɵɯ ɲɚɪɨɜ)

Ɍɟɨɪɢɹ ɤɨɧɞɟɧɫɚɰɢɢ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ, ɤɨɬɨɪɭɸ ɦɵ ɪɚɫ-

ɫɦɨɬɪɢɦ ɜ ɷɬɨɦ ɩɚɪɚɝɪɚɮɟ, ɢɝɪɚɟɬ ɬɭ ɠɟ ɪɨɥɶ ɜ ɨɛɳɟɣ ɬɟɨɪɢɢ

ɤɨɧɞɟɧɫɚɰɢɢ, ɱɬɨ ɢ ɬɟɨɪɢɹ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɩɨɥɹ ȼɟɣɫɫɚ ɜ ɨɛ-

ɳɟɣ ɬɟɨɪɢɢɢ ɮɟɪɪɨɦɚɝɧɟɬɢɡɦɚ.

Ɋɚɫɫɦɨɬɪɢɦ ɫɪɟɞɧɟɟ ɩɨɥɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ N ɱɚɫɬɢɰ ɜ

ɤɚɧɨɧɢɱɟɫɤɨɦ ɚɧɫɚɦɛɥɟ:

115

!



oo o

V V dr dr dr

Vdrdr dr

exp[ ] ....

(7.1)

ɝɞɟ

VVVVrr

ij i j

,(| |).

Ɍɚɤ ɤɚɤ ɜɫɟ ɱɥɟɧɵ ɷɬɨɣ ɫɭɦɦɵ ɢɞɟɧɬɢɱɧɵ, ɭɱɢɬɵɜɚɹ, ɱɬɨ ɢɯ

ɱɢɫɥɨ ɪɚɜɧɨ N(N-1)/2, ɧɟɫɥɨɠɧɨ ɩɨɥɭɱɢɬɶ

!



oo o

V V dr dr dr

Z V dr dr dr

()

exp[ ] ....

exp[ ] .... .

12 1 2

(7.2)

ȼɜɟɞɟɦ ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɧɚɪɭɠɢɬɶ ɱɚɫɬɢɰ 1 ɢ 2 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ

ɷɥɟɦɟɧɬɚɪɧɵɯ ɨɛɴɟɦɚɯ dr

ɢ dr

()

(, )

exp[ ] ....

Vdrdr dr

oo o



. (7.3)

ȼɟɥɢɱɢɧɚ

(2)

ɢɦɟɟɬ ɩɨɪɹɞɨɤ 1/

, ɝɞɟ

- ɨɛɴɟɦ. ɂɡ-ɡɚ ɬɪɚɧɫ-

ɥɹɰɢɨɧɧɨɣ ɫɢɦɦɟɬɪɢɢ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɪɚɡɧɨɫɬɢ

. ȼɜɟɞɟɦ ɩɚɪɧɭɸ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɮɭɧɤɰɢɸ

()

(, )



()

(, ) ( )

rr gr r

oo o o



. (7.4)

116

Ɋɢɫ.12. ɉɚɪɧɚɹ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ

ɮɭɧɤɰɢɹ ɜ ɦɨɞɟɥɢ ɜɡɚɢɦɨɞɟɣɫɬ-

ɜɭɸɳɢɯ ɬɜɟɪɞɵɯ ɲɚɪɨɜ:

ɚ - ɨɛɳɢɣ ɜɢɞ; ɛ - ɢɞɟɚɥɶɧɵɟ

ɬɜɟɪɞɵɟ ɲɚɪɵ

ȼɟɥɢɱɢɧɚ g ɡɚɜɢɫɢɬ ɨɬ ɜɡɚɢ-

ɦɨɞɟɣɫɬɜɢɹ. ȼ ɢɞɟɚɥɶɧɨɦ ɝɚɡɟ ɩɪɢ V = 0 (ɤɥɚɫɫɢɱɟɫɤɢɟ ɬɜɟɪ-

ɞɵɟ ɲɚɪɵ) g = 1 ɩɪɢ |r| > C, ɢ g = 0 ɩɪɢ |r| < C. ȼ ɨɛɳɟɦ ɫɥɭ-

ɱɚɟ ɜɢɞ g ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ.12. ɉɨɞɫɬɚɜɢɜ (7.4) ɜ (7.2), ɜɵɪɚɡɢɦ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɱɟɪɟɡ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɮɭɧɤɰɢɸ:

!





o o oo oo

oo o

³³



dr dr g r r V r r

drg r V r

rrr

()

()()

()() .

1 2 12 12

(7.5)

Ⱦɥɹ ɰɟɧɬɪɚɥɶɧɵɯ ɫɢɥ ɤɨɪɪɟɥɹɬɨɪ g ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ

ɦɨɞɭɥɹ r:

. ɉɪɢ r

of

ɮɭɧɤɰɢɹ g

1 (ɫɦ. ɪɢɫ.12).

Ʉɪɨɦɟ ɬɨɝɨ, ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɮɭɧɤɰɢɹ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɞɨɥɠɧɚ

ɡɚɜɢɫɟɬɶ ɨɬ ɩɥɨɬɧɨɫɬɢ

. ȼ ɬɟɨɪɢɢ ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ ɷɬɨɣ ɡɚɜɢ-

ɫɢɦɨɫɬɶɸ ɩɪɟɧɟɛɪɟɝɚɸɬ. Ɍɨɝɞɚ ɨɤɨɧɱɚɬɟɥɶɧɨ ɡɚɩɢɫɵɜɚɟɦ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ:

gr g r() (||)

117

!  

 !

VN a a N

adrgrVr

//,/

() () .

UXX

: 1

(7.6)

ȼ (7.6) ɤɨɧɫɬɚɧɬɚ a > 0 (ɢɦɟɸɳɚɹ ɪɚɡɦɟɪɧɨɫɬɶ ɷɧɟɪɝɢɢ, ɭɦɧɨ-

ɠɟɧɧɨɣ ɧɚ ɨɛɴɟɦ) ɜ ɫɢɥɭ ɬɨɝɨ, ɱɬɨ g = 0 ɩɪɢ |r| < C ɢ V(r) <

0, g ~ 1 ɩɪɢ |r| > C (ɬ.ɟ. ɮɚɤɬɢɱɟɫɤɢ ɦɵ ɜɵɛɢɪɚɟɦ ɩɨɱɬɢ ɢɞɟ-

ɚɥɶɧɭɸ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɮɭɧɤɰɢɸ (ɫɦ. ɪɢɫ.12,ɛ) ɢ ɫɨɨɬɜɟɬɫɬ-

ɜɭɸɳɢɣ ɩɨɬɟɧɰɢɚɥ).

ɍɪɚɜɧɟɧɢɟ (7.6) - ɷɬɨ ɮɚɤɬɢɱɟɫɤɢ ɝɢɩɨɬɟɡɚ ɨ ɦɨɥɟɤɭ-

ɥɹɪɧɨɦ ɩɨɥɟ, ɞɟɣɫɬɜɭɸɳɟɦ ɧɚ ɱɚɫɬɢɰɭ ɜ ɬɨɱɤɟ r ɨɬ ɞɪɭɝɢɯ ɱɚɫ-

ɬɢɰ ɜ ɬɨɱɤɚɯ r’: . ȿɫɥɢ ɩɨɥɨɠɢɬɶ ɩɥɨɬ-

ɧɨɫɬɶ

U

ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ ɤɨɨɪɞɢɧɚɬɵ, ɬɨ ɫɪɟɞɧɢɣ ɩɨɬɟɧɰɢɚɥ

ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ

dr r V r r'(')( ')



drV r r a'( ') /

ooo

|

2 . (7.7)

Ʉɚɤ ɜɢɞɢɦ, “ɫɪɟɞɧɟɩɨɥɟɜɨɟ”ɜɵɪɚɠɟɧɢɟ (7.7) ɩɨɥɧɨɫɬɶɸ

ɢɞɟɧɬɢɱɧɨ (ɫ ɭɱɟɬɨɦ ɦɧɨɠɢɬɟɥɹ 2) ɫɨɨɬɧɨɲɟɧɢɸ ȼɚɧ-ɞɟɪ-

ȼɚɚɥɶɫɚ (7.6). ɂɡ ɭɪɚɜɧɟɧɢɹ (7.6) ɜɵɬɟɤɚɟɬ ɫɭɳɟɫɬɜɨɜɚɧɢɟ

“ɜɧɭɬɪɟɧɧɟɝɨ” ɞɚɜɥɟɧɢɹ, ɤɨɬɨɪɨɟ ɫɬɪɟɦɢɬɫɹ ɫɠɚɬɶ ɝɚɡ:

PV a

int

// !

. (7.8)

ɉɨɥɧɨɟ ɞɚɜɥɟɧɢɟ ɪɚɜɧɨ ɫɭɦɦɟ ɜɧɭɬɪɟɧɧɟɝɨ ɢ ɜɧɟɲɧɟɝɨ:

PP Pa 

int

. (7.9)

ɋɭɦɦɚɪɧɨɟ ɞɚɜɥɟɧɢɟ (7.9) ɭɪɚɜɧɨɜɟɲɢɜɚɟɬɫɹ ɬɟɩɥɨɜɵɦ

ɞɚɜɥɟɧɢɟɦ

kT /

, ɤɨɬɨɪɨɟ ɫɬɪɟɦɢɬɫɹ ɪɚɫɲɢɪɢɬɶ ɝɚɡ. ȼ ɬɟɨɪɢɢ

ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ ɜ ɬɟɩɥɨɜɨɦ ɞɚɜɥɟɧɢɢ ɧɟɨɛɯɨɞɢɦɨ ɭɱɟɫɬɶ ɫɭ-

ɳɟɫɬɜɨɜɚɧɢɟ ɦɢɧɢɦɚɥɶɧɨɝɨ ɨɛɴɟɦɚ ɲɚɪɨɜ. ȿɫɥɢ ɨɛɨɡɧɚɱɢɬɶ

ɨɛɴɟɦ ɨɞɧɨɝɨ ɲɚɪɚ b, ɬɨ ɷɮɮɟɤɬɢɜɧɵɣ ɨɛɴɟɦ, ɩɪɢɯɨɞɹɳɢɣɫɹ

ɧɚ ɨɞɢɧ ɲɚɪ:

eff

Nb : / . (7.10)

Ɉɤɨɧɱɚɬɟɥɶɧɨ, ɭɱɢɬɵɜɚɹ (7.9) ɢ (7.10), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ

ȼɚɧ-ɞɟɪ-ȼɚɚɥɶɫɚ

118

akT





. (7.11)

ɍɪɚɜɧɟɧɢɟ (7.11) ɨɩɢɫɵɜɚɟɬ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɩɟɪɜɨɝɨ

ɪɨɞɚ “ɝɚɡ - ɠɢɞɤɨɫɬɶ” ɢ ɢɦɟɟɬ “ɩɟɬɥɸ” ɧɚ ɢɡɨɬɟɪɦɚɯ ɩɪɢ ɬɟɦ-

ɩɟɪɚɬɭɪɚɯ, ɦɟɧɶɲɢɯ ɤɪɢɬɢɱɟ-

ɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɤɨɧɞɟɧɫɚɰɢɢ

(ɪɢɫ.13). Ɋɟɚɥɶɧɨ ɩɟɬɥɹ ɧɟɭɫ-

ɬɨɣɱɢɜɚ, ɢ ɜ ɨɛɜɟɞɟɧɧɨɣ ɩɭɧɤ-

ɬɢɪɧɨɣ ɥɢɧɢɟɣ ɨɛɥɚɫɬɢ ɢɦɟɟɬ

ɦɟɫɬɨ ɪɚɜɧɨɜɟɫɢɟ ɠɢɞɤɨɣ ɢ

ɝɚɡɨɨɛɪɚɡɧɨɣ ɮɚɡ ɫ ɩɨɫɬɨɹɧɧɵɦ

ɞɚɜɥɟɧɢɟɦ, ɬ.ɟ. ɪɟɚɥɢɡɭɟɬɫɹ

ɨɬɦɟɱɟɧɧɚɹ ɧɚ ɪɢɫ.13 ɥɢɧɢɹ

P=const.

ɂɡ ɭɪɚɜɧɟɧɢɹ (7.11) ɢ

ɪɢɫ.13 ɫɥɟɞɭɟɬ, ɱɬɨ ɬɨɱɤɚ ɩɟɪɟ-

ɯɨɞɚ T

ɢɦɟɟɬ ɦɟɫɬɨ, ɤɨɝɞɚ ɧɚ

ɢɡɨɬɟɪɦɚɯ ɬɨɱɤɚ ɩɟɪɟɝɢɛɚ ɫɨɜ-

ɩɚɞɚɟɬ ɫ ɷɤɫɬɪɟɦɭɦɨɦ ɡɚɜɢɫɢ-

ɦɨɫɬɢ P(V), ɢ ɷɬɚ ɬɪɢɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ, ɡɚ ɤɨɬɨɪɨɣ ɩɪɢ ɛɨɥɶ-

ɲɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɨɫɬɚɟɬɫɹ ɬɨɥɶɤɨ ɝɚɡɨɨɛɪɚɡɧɚɹ ɮɚɡɚ, ɨɩɪɟɞɟ-

ɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ

Ɋɢɫ.13. ɂɡɨɬɟɪɦɵ ȼɚɧ-ɞɟɪ-

ȼɚɚɥɶɫɚ.

- ɤɪɢɬɢɱɟɫɤɢɣ

ɨɛɴɟɦ

wwX w wX

PP/,/ 0

0 . (7.12)

ɂɫɩɨɥɶɡɭɹ (7.11) ɢ (7.12), ɩɨɥɭɱɢɦ, ɱɬɨ ɤɪɢɬɢɱɟɫɤɚɹ ɬɟɦɩɟɪɚ-

ɬɭɪɚ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɟɣ ɦɢɧɢɦɚɥɶɧɵɣ ɟɞɢɧɢɱɧɵɣ ɨɛɴɟɦ,

ɡɚɧɢɦɚɟɦɵɣ ɠɢɞɤɨɣ ɮɚɡɨɣ, ɪɚɜɧɵ:

. (7.13)

Ɂɚɞɚɱɚ

119