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

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CNVT V



ch { }

2.5.4. Ɉɩɪɟɞɟɥɢɬɶ ɷɧɟɪɝɢɸ ɩɪɢ ɩɨɥɟ H=0. Ɋɚɫɫɦɨɬɪɟɬɶ

ɫɥɭɱɚɣ V>0 ɢ V<0 .

Ɉɬɜɟɬ

: ɨɛɚ ɫɥɭɱɚɹ ɫɨɜɩɚɞɚɸɬ:.ENV  th

2.5.5. ɇɚɣɬɢ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ V =

Ɉɬɜɟɬ:

FP EP

2TTexp( / )

2.5.

. Ɂɚɩɢɫɚɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ ɧɚ ɫɥɭɱɚɣ ɤɨɧɟɱ-

ɧɨɣ ɰɟɩɨɱɤɢ N >>1. Ɋɚɫɫɱɢɬɚɬɶ ɩɨɩɪɚɜɤɭ ɤ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ

ɨɞɧɨɦɟɪɧɨɣ ɰɟɩɨɱɤɢ ɡɚ ɫɱɟɬ ɤɨɧɟɱɧɨɫɬɢ ɫɢɫɬɟɦɵ.

Ⱦɜɭɦɟɪɧɚɹ ɦɨɞɟɥɶ ɂɡɢɧɝɚ; ɪɟɲɟɧɢɟ Ɉɧɡɚɝɟɪɚ.

ȼ 1944

ɝ. Ɉɧɡɚɝɟɪɨɦ ɛɵɥɨ ɩɪɟɞɥɨɠɟɧɨ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨ ɮɚɡɨ-

ɜɨɦ ɩɟɪɟɯɨɞɟ ɜ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɧɚ ɞɜɭɦɟɪɧɨɣ ɤɜɚɞɪɚɬɧɨɣ ɪɟ-

ɲɟɬɤɟ. Ɉɫɧɨɜɧɨɣ ɜɵɜɨɞ - ɫɭɳɟɫɬɜɭɟɬ ɩɟɪɟɯɨɞ ɜ ɮɟɪɪɨɦɚɝɧɢɬ-

ɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɩɪɢɱɟɦ Ɉɧɡɚɝɟɪɭ ɭɞɚɥɨɫɶ ɬɨɱɧɨ ɪɚɫɫɱɢɬɚɬɶ ɫɬɚ-

ɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ. Ⱦɥɹ ɩɨɞɪɨɛɧɨɝɨ ɨɡɧɚɤɨɦɥɟɧɢɹ ɫ ɷɬɢɦ ɢɡ-

ɜɟɫɬɧɵɦ ɪɟɡɭɥɶɬɚɬɨɦ ɦɵ ɨɬɫɵɥɚɟɦ ɱɢɬɚɬɟɥɟɣ ɤ ɤɧɢɝɚɦ [16,17].

Ɇɵ ɤɨɫɧɟɦɫɹ ɡɞɟɫɶ ɤɨɧɫɩɟɤɬɢɜɧɨ ɥɢɲɶ ɨɫɧɨɜɧɵɯ ɷɬɚɩɨɜ ɪɚɫ-

ɱɟɬɚ ɢ ɨɛɫɭɞɢɦ ɩɨɥɭɱɟɧɧɵɟ Ɉɧɡɚɝɟɪɨɦ ɬɨɱɧɵɟ pɟɡɭɥɶɬɚɬɵ.

Ɋɚɫɫɦɨɬɪɢɦ ɞɜɭɦɟɪɧɭɸ ɤɜɚɞɪɚɬɧɭɸ ɪɟɲɟɬɤɭ ɫ ɩɟɪɢɨ-

ɞɢɱɟɫɤɢɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɢ N ɭɡɥɚɦɢ, ɤɚɠɞɵɣ ɢɡ ɤɨ-

ɬɨɪɵɯ ɢɦɟɟɬ ɟɞɢɧɢɱɧɵɣ ɫɩɢɧ

. Ɍɨɝɞɚ ɭɞɨɛɧɨ ɜɜɟɫɬɢ ɧɭ-

ɦɟɪɚɰɢɸ ɭɡɥɨɜ ɩɨ ɢɯ ɤɨɨɪɞɢɧɚɬɚɦ x = ka, y = la, ɝɞɟ a - ɩɟɪɢɨɞ

ɪɟɲɟɬɤɢ; k,l - ɰɟɥɵɟ ɱɢɫɥɚ 1< k,l < L, L

=N. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɝɚ-

ɦɢɥɶɬɨɧɢɚɧ ɂɡɢɧɝɚ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ ɢ ɜ

ɨɬɫɭɬɫɬɜɢɢ ɜɧɟɲɧɟɝɨ ɩɨɥɹ ɧɚ ɬɚɤɨɣ ɪɟɲɟɬɤɟ ɢɦɟɟɬ ɜɢɞ:

kl kl kl k l

 



()

PP PP

 1

. (2.33)

ɋɬɚɬɢɫɬɢɱɟɫɤɚɹ ɫɭɦɦɚ ɬɚɤɨɣ ɫɢɫɬɟɦɵ:

kl kl kl k l







exp ( )

{}

TPP PP

, (2.34)

ɝɞɟ

V, ɚ {n} ɨɡɧɚɱɚɟɬ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɨ ɜɫɟɦ ɜɨɡɦɨɠɧɵɦ

ɫɨɫɬɨɹɧɢɹɦ ɫɢɫɬɟɦɵ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɬɟɦ ɮɚɤɬɨɦ, ɱɬɨ

=1.

Ɍɨɝɞɚ ɧɟɫɥɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫɥɟɞɭɸɳɟɟ ɬɨɠɞɟɫɬɜɨ:

exp{

k’l’

} = ch

4

k’l’

4

= ch

4

(1+

k’l’

). (2.35)

ɂɫɩɨɥɶɡɭɹ (2.35), ɩɪɟɞɫɬɚɜɢɦ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ ɫɥɟɞɭɸ-

ɳɢɦ ɨɛɪɚɡɨɦ:

th .

QxS



()1

(2.36)

Sxxx

kl kl kl k l

 





()(),

{}

PP PP T

(2.37)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɫɤɨɦɚɹ ɫɬɚɬɢɫɬɢɱɟɫɤɚɹ ɫɭɦɦɚ ɹɜɥɹɟɬɫɹ ɫɭɦ-

ɦɨɣ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɩɨɥɢɧɨɦɨɜ ɩɨ ɫɬɟɩɟɧɹɦ x ɢ

. ɉɨɫɤɨɥɶ-

ɤɭ ɤɚɠɞɵɣ ɭɡɟɥ (k,l) ɫɜɹɡɚɧ ɫ ɱɟɬɵɪɶɦɹ ɫɨɫɟɞɹɦɢ, ɬɨ ɤɚɠɞɨɟ

ɦɨɠɟɬ ɜɫɬɪɟɬɢɬɶɫɹ ɜ ɩɨɥɢɧɨɦɟ ɜ ɫɬɟɩɟɧɹɯ ɨɬ ɧɭɥɟɜɨɣ ɞɨ ɱɟɬ-

ɜɟɪɬɨɣ. ɉɨɫɥɟ ɫɭɦɦɢɪɨɜɚɧɢɹ ɩɨ ɜɫɟɦ

=rɱɥɟɧɵ, ɫɨɞɟɪɠɚ-

ɳɢɟ ɧɟɱɟɬɧɵɟ ɫɬɟɩɟɧɢ

, ɨɛɪɚɬɹɬɶɫɹ ɜ ɧɭɥɶ, ɬɚɤ ɱɬɨ ɧɟɧɭɥɟ-

ɜɨɣ ɜɤɥɚɞ ɞɚɞɭɬ ɬɨɥɶɤɨ ɱɥɟɧɵ, ɫɨɞɟɪɠɚɳɢɟ

ɜ ɫɬɟɩɟɧɹɯ 0, 2 ɢ

4. Ȼɨɥɟɟ ɬɨɝɨ, ɩɨɫɤɨɥɶɤɭ

=1, ɬɨ ɤɚɠɞɵɣ ɱɥɟɧ

ɩɨɥɢɧɨɦɚ, ɫɨɞɟɪɠɚɳɢɣ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ

ɜ ɱɟɬɧɵɯ ɫɬɟɩɟɧɹɯ,

ɞɚɫɬ ɜɤɥɚɞ ɜ ɫɭɦɦɭ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɣ ɩɨɥɧɨɦɭ ɱɢɫɥɭ ɤɨɧɮɢ-

ɝɭɪɚɰɢɣ 2

ɉɨɫɥɟ ɷɬɨɝɨ Ɉɧɡɚɝɟɪ ɩɪɟɞɥɨɠɢɥ, ɩɨɠɚɥɭɣ, ɨɞɢɧ ɢɡ ɩɟɪ-

ɜɵɯ ɜɚɪɢɚɧɬɨɜ ɞɢɚɝɪɚɦɦɧɨɣ ɬɟɯɧɢɤɢ ɞɥɹ ɫɭɦɦɢɪɨɜɚɧɢɹ (2.36)-

(2.37), ɤɨɬɨɪɚɹ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɬɟɨ-

ɪɢɢ ɤɨɧɞɟɧɫɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ. Ɉɧ ɡɚɦɟɬɢɥ, ɱɬɨ ɤɚɠɞɨɦɭ

ɱɥɟɧɭ ɩɨɥɢɧɨɦɚ ɦɨɠɧɨ ɨɞɧɨɡɧɚɱɧɨ ɩɨɫɬɚ

ɜɢɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɫɨɜɨɤɭɩɧɨɫɬɶ ɥɢɧɢɣ (ɝɪɚɮɢɤɨɜ), ɫɨɟɞɢ-

ɧɹɸɳɢɯ ɧɟɤɨɬɨɪɵɟ ɩɚɪɵ ɫɨɫɟɞɧɢɯ ɭɡɥɨɜ ɪɟɲɟɬɤɢ. Ɍɚɤ, ɢɡɨ-

ɛɪɚɠɟɧɧɵɦ ɧɚ ɪɢɫ.8 ɝɪɚɮɢɤɚɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɱɥɟɧɵ ɩɨɥɢɧɨ-

ɦɚ:

Ɋɢɫ.8. Ⱦɢɚɝɪɚɦɦɧɚɹ ɬɟɯɧɢ-

ɤɚ Ɉɧɡɚɝɟɪɚ ɧɚ ɩɥɨɫɤɨɣ

ɤɜɚɞɪɚɬɧɨɣ ɪɟɲɟɬɤɟ ɞɥɹ

ɦɨɞɟɥɢ ɂɡɢɧɝɚ

1) x

k+1l

k+1l-1

(2.38)

2) x

k+1l

k+1l-1

kl-1

kl-2

k-1l-1

k-1l-2

Ʉɚɠɞɨɣ ɥɢɧɢɢ ɝɪɚɮɢɤɚ ɫɨɩɨɫɬɚɜɥɹɟɬɫɹ ɦɧɨɠɢɬɟɥɶ x, ɚ ɤɚɠɞɨ-

ɦɭ ɟɟ ɤɨɧɰɭ - ɦɧɨɠɢɬɟɥɶ

. Ɍɚɤ ɤɚɤ ɨɬɥɢɱɧɵɣ ɨɬ ɧɭɥɹ ɜɤɥɚɞ ɜ

ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ ɞɚɸɬ ɥɢɲɶ ɱɥɟɧɵ ɩɨɥɢɧɨɦɚ, ɫɨɞɟɪɠɚ-

ɳɢɟ ɱɟɬɧɵɟ ɫɬɟɩɟɧɢ

, ɬɨ ɜ ɤɚɠɞɨɦ ɭɡɥɟ ɝɪɚɮɢɤɚ ɞɨɥɠɧɵ

ɨɤɚɧɱɢɜɚɬɶɫɹ ɥɢɛɨ ɞɜɟ, ɥɢɛɨ ɱɟɬɵɪɟ ɫɜɹɡɢ. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ,

ɫɭɦɦɢɪɨɜɚɧɢɟ ɜɟɞɟɬɫɹ ɬɨɥɶɤɨ ɩɨ ɡɚɦɤɧɭɬɵɦ ɝɪɚɮɢɤɚɦ, ɩɪɢɱɟɦ

ɞɨɩɭɫɤɚɟɬɫɹ ɫɚɦɨɩɟɪɟɫɟɱɟɧɢɟ.

Ɉɤɨɧɱɚɬɟɥɶɧɨ, ɫɭɦɦɚ Q ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ

ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:

. (2.39)

{}

ɝɞɟ g

- ɱɢɫɥɨ ɡɚɦɤɧɭɬɵɯ ɝɪɚɮɢɤɨɜ, ɫɨɫɬɚɜɥɟɧɧɵɯ ɢɡ ɱɟɬɧɨɝɨ

ɱɢɫɥɚ r ɫɜɹɡɟɣ.

Ⱦɚɥɟɟ Ɉɧɡɚɝɟɪ ɪɚɡɪɚɛɨɬɚɥ ɫɩɨɫɨɛ ɩɟɪɟɛɨɪɚ ɝɪɚɮɢɤɨɜ ɢɡ

ɫɭɦɦɵ (2.39) ɫ ɩɨɦɨɳɶɸ ɫɩɟɰɢɚɥɶɧɨɣ ɢɯ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɩɨ

ɞɥɢɧɟ ɫɜɹɡɟɣ r ɢ ɢɫɩɨɥɶɡɨɜɚɥ ɩɨɞɯɨɞ “ɛɥɭɠɞɚɸɳɟɣ” ɲɚɝ ɡɚ

ɲɚɝɨɦ ɩɨ ɪɟɲɟɬɤɟ ɬɨɱɤɢ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɩɟɪɟɯɨɞɚ, ɨɩɪɟɞɟɥɹɟ-

ɦɨɣ ɫɩɟɰɢɚɥɶɧɨɣ ɦɚɬɪɢɰɟɣ ɩɟɪɟɯɨɞɚ. ɂɡ-ɡɚ ɷɤɨɧɨɦɢɢ ɦɟɫɬɚ

ɦɵ ɩɪɢɜɟɞɟɦ

ɭɠɟ ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɪɟɡɭɥɶɬɚɬ, ɚ ɜɫɟ ɩɨɞɪɨɛɧɨɫɬɢ

ɪɚɫɱɟɬɚ ɢɡɥɨɠɟɧɵ ɜ ɤɧɢɝɟ [16]. Ɉɤɨɧɱɚɬɟɥɶɧɨɟ ɬɨɱɧɨɟ ɜɵɪɚ-

ɠɟɧɢɟ ɞɥɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɫɭɦɦɵ ɢɦɟɟɬ ɜɢɞ:





xxx

 u

u 





121

()

()(cos cos)

(2.40)

Ɂɚɞɚɱɢ

2.5.7. Ɋɚɫɫɱɢɬɚɬɶ ɫɜɨɛɨɞɧɭɸ ɷɧɟɪɝɢɸ ɞɜɭɦɟɪɧɨɣ ɦɨɞɟ-

ɥɢ ɂɡɢɧɝɚ F= - 1

E

lnQ.

Ɋɟɲɟɧɢɟ

. ɂɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ (2.40), ɧɚɯɨɞɢɦ:



FNT NT x

Txxx

   

 

ln ln( )

( )(cos cos ) .

121

ɉɟɪɟɣɞɟɦ ɨɬ ɫɭɦɦɢɪɨɜɚɧɢɹ ɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɸ ɜ ɩɪɟɞɟ-

ɥɟ

L, N

of

:



FNT NT x

xxx dd

   



³³

ln ln( )

()

ln ( ) (cos cos ) .

121

ZZZ

2.5.

8. ɂɫɫɥɟɞɨɜɚɬɶ ɫɜɨɛɨɞɧɭɸ ɷɧɟɪɝɢɸ ɜɛɥɢɡɢ ɤɪɢɬɢɱɟ-

ɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɪɚɫɫɱɢɬɚɬɶ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ.

Ɋɟɲɟɧɢɟ

. Ɏɭɧɤɰɢɹ F(T) ɢɦɟɟɬ ɨɫɨɛɭɸ ɬɨɱɤɭ ɩɪɢ ɬɨɦ

ɡɧɚɱɟɧɢɢ x (x = th (V/T) ), ɩɪɢ ɤɨɬɨɪɨɦ ɚɪɝɭɦɟɧɬ ɥɨɝɚɪɢɮɦɚ

ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɦɨɠɟɬ ɨɛɪɚɬɢɬɶɫɹ ɜ ɧɭɥɶ. Ʉɚɤ ɮɭɧɤɰɢɹ

ɷɬɨɬ ɚɪɝɭɦɟɧɬ ɦɢɧɢɦɚɥɟɧ ɩɪɢ cos

= =cos

=1, ɜ

ɷɬɨɦ ɫɥɭɱɚɟ ɨɧ ɪɚɜɟɧ (1+x

)

- 4x(1-x

). ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɢɦɟɟɬ

ɦɢɧɢɦɭɦ, ɜ ɤɨɬɨɪɨɦ ɨɧɨ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ x = x

= 2



ɩɪɢ ɷɬɨɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɬɟɦɩɟɪɚɬɭɪɚ T

(th V/T

) ɢ ɹɜɥɹɟɬɫɹ ɬɨɱɤɨɣ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ

ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɩɨɧɢɠɚɟɬɫɹ ɩɪɢ T<T

, ɧɟɩɪɟɪɵɜɧɚ ɩɪɢ

T=T

, ɚ ɬɟɩɥɨɟɦɤɨɫɬɶ ɢɫɩɵɬɵɜɚɟɬ ɥɨɝɚɪɢɮɦɢɱɟɫɤɭɸ ɨɫɨɛɟɧ-

ɧɨɫɬɶ ɜ ɬɨɱɤɟ ɩɟɪɟɯɨɞɚ, ɬ.ɟ. ɷɬɨ - ɬɢɩɢɱɧɵɣ ɬɟɪɦɨɞɢɧɚɦɢɱɟ-

ɫɤɢɣ ɩɟɪɟɯɨɞ ɜɬɨɪɨɝɨ ɪɨɞɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɞɜɭɦɟɪɧɨɣ ɦɨɞɟ-

ɥɢ ɂɡɢɧɝɚ ɧɚɛɥɸɞɚɟɬɫɹ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɩɚɪɚɦɚɝɧɟɬɢɤ-

ɮɟɪɪɨɦɚɝɧɟɬɢɤ ɩɪɢ ɤɨɧɟɱɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ.

2.5.

9. ɋɪɚɜɧɢɬɶ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ, ɩɨɥɭɱɟɧɧɭɸ

Ɉɧɡɚɝɟɪɨɦ, ɫ ɪɟɡɭɥɶɬɚɬɨɦ ɬɟɨɪɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ.

Ɋɟɲɟɧɢɟ

. ɂɫɯɨɞɹ ɢɡ ɭɪɚɜɧɟɧɢɹ thV/T

= 2



ɧɚɯɨɞɢɦ

=2V/ ln(1+ 2 ) |2J. Ɋɟɡɭɥɶɬɚɬ ɬɟɨɪɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ:

=ZV=4V. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɦɟɧɶɲɟ ɩɪɢɛɥɢ-

ɠɟɧɧɨɝɨ “ɫɪɟɞɧɟɩɨɥɟɜɨɝɨ” ɩɪɢɦɟɪɧɨ ɧɚ ɜɟɥɢɱɢɧɭ ~1/Z. ɂɧɬɟ-

ɪɟɫɧɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɫ ɷɬɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɜ ɨɞɧɨɦɟɪɧɨɦ ɫɥɭɱɚɟ

ɪɟɡɭɥɶɬɚɬ ɬɟɨɪɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ ɞɚɟɬ ɬɚɤɠɟ “ɡɚɜɵɲɟɧɧɵɣ ɪɟ-

ɡɭɥɶɬɚɬ” (T

=2V) ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɬɨɱɧɵɦ (T

=0).

2.5.

10. ɂɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ,

ɩɨɤɚɡɚɬɶ, ɱɬɨ ɜɛɥɢɡɢ ɤɪɢɬɢɱɟɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɜɢɫɢɦɨɫɬɶ

ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ ɧɟɩɪɟɪɵɜɧɚ, ɚ ɬɟɩɥɨɟɦɤɨɫɬɶ ɪɚɫɯɨɞɢɬɫɹ

ɥɨɝɚɪɢɮɦɢɱɟɫɤɢ.

Ɋɟɲɟɧɢɟ

. ɇɟɨɛɯɨɞɢɦɨ ɪɚɡɥɨɠɢɬɶɫɹ ɩɨ ɫɬɟɩɟɧɹɦ T-T

ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɩɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥɶɧɵɣ ɱɥɟɧ ɜ ɫɜɨɛɨɞ-

ɧɨɣ ɷɧɟɪɝɢɢ, ɢɦɟɸɳɢɣ ɨɫɨɛɟɧɧɨɫɬɢ. ɉɨɫɥɟ ɪɚɡɥɨɠɟɧɢɹ ɚɪɝɭ-

ɦɟɧɬɚ ɥɨɝɚɪɢɮɦɚ ɜɛɥɢɡɢ ɟɝɨ ɦɢɧɢɦɭɦɚ ɩɨ ɫɬɟɩɟɧɹɦ

1,2

ɢ T-

ɢɦɟɟɦ:

FAtB t

~ln ( ) ,

 

³³

ZZ ZZ

dd TT

ɝɞɟ A,B - ɩɨɫɬɨɹɧɧɵɟ. Ɋɚɫɱɢɬɵɜɚɹ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɩɪɢ t

ɧɚɯɨɞɢɦ:.FTT TTC TT

~ | | ln| |, ~ ln| |  

ɎɅɍɄɌɍȺɐɂɂ ȼ ɎȿɊɊɈɆȺȽɇȿɌɂɄȺɏ

Ɍɟɨɪɢɹ ɫɪɟɞɧɟɝɨ ɩɨɥɹ ɧɟɚɞɟɤɜɚɬɧɨ ɨɩɢɫɵɜɚɟɬ ɨɛɥɚɫɬɶ

ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ ɜ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɣ ɛɥɢɡɨɫɬɢ ɬɨɱɤɢ ɮɚɡɨɜɨɝɨ

ɩɟɪɟɯɨɞɚ ɢɡ-ɡɚ ɬɨɝɨ, ɱɬɨ ɧɟ ɭɱɢɬɵɜɚɟɬ ɮɥɭɤɬɭɚɰɢɢ, ɪɟɡɤɨ ɜɨɡ-

ɪɚɫɬɚɸɳɢɟ ɢɦɟɧɧɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ ɬɨɱɤɢ ɩɪɢ ɮɚɡɨɜɨɦ ɩɟ-

ɪɟɯɨɞɟ ɜɬɨɪɨɝɨ ɪɨɞɚ. ȼ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɭɱɬɟɦ ɮɥɭɤɬɭɚɰɢɢ ɢ ɩɨ-

ɤɚɠɟɦ, ɤɚɤ ɷɬɨ ɩɨɡɜɨɥɹɟɬ ɥɭɱɲɟ ɨɩɢɫɚɬɶ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ

ɞɚɧɧɵɟ ɩɨ ɬɟɩɥɨɟɦɤɨɫɬɢ ɢ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ ɜ ɮɟɪɪɨɦɚɝɧɟɬɢ-

ɤɚɯ.

3.1.

Ȼɥɢɠɧɢɣ ɢ ɞɚɥɶɧɢɣ ɩɨɪɹɞɨɤ

Ɏɭɧɤɰɢɹ ɤɨɪɪɟɥɹɰɢɢ

ȼɜɟɞɟɦ ɩɨɧɹɬɢɟ ɞɚɥɶɧɟɝɨ ɢ ɛɥɢɠɧɟɝɨ ɩɨɪɹɞɤɚ ɜ ɫɩɢɧɨ-

ɜɵɯ ɦɨɞɟɥɹɯ.

Ɍɢɩɢɱɧɵɣ ɞɚɥɶɧɢɣ ɩɨɪɹɞɨɤ ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɫɥɭɱɚɟ ɭɩɨ-

ɪɹɞɨɱɟɧɧɨɣ ɮɟɪɪɨɦɚɝɧɢɬɧɨɣ ɮɚɡɵ, ɤɨɝɞɚ <

0, ɧɟ ɡɚɜɢɫɢɬ

ɨɬ ɧɨɦɟɪɚ ɭɡɥɚ ɢ ɧɟ ɫɩɚɞɚɟɬ ɞɚɠɟ ɧɚ ɛɨɥɶɲɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ

ɭɡɥɚ j. Ⱦɚɥɶɧɢɣ ɩɨɪɹɞɨɤ ɫɩɨɧɬɚɧɧɨ ɢɫɱɟɡɚɟɬ ɜ ɬɨɱɤɟ ɩɟɪɟɯɨɞɚ.

Ȼɥɢɠɧɢɣ ɩɨɪɹɞɨɤ ɫɜɹɡɚɧ ɫ ɥɨɤɚɥɶɧɵɦɢ ɮɥɭɤɬɭɚɰɢɹɦɢ

ɫɩɢɧɨɜ ɧɚ ɛɥɢɡɤɢɯ ɪɚɫɫɬɨɹɧɢɹɯ. ȼ ɬɟɨɪɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ ɬɚɤɢɟ

ɮɥɭɤɬɭɚɰɢɢ ɧɟ ɭɱɢɬɵɜɚɸɬɫɹ, ɩɨɷɬɨɦɭ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ȼɟɣɫɫɚ ɫ

ɢɫɱɟɡɧɨɜɟɧɢɟɦ ɫɩɨɧɬɚɧɧɨɣ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ ɢɫɱɟɡɚɟɬ ɢ ɷɧɟɪ-

ɝɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɩɢɧɨɜ, ɬ.ɟ. ɛɥɢɠɧɢɣ ɩɨɪɹɞɨɤ:

>=<

>=0, ɟɫɥɢ <

>=<

>=0.

Ɉɞɧɚɤɨ ɟɫɥɢ ɬɚɤɨɟ ɭɩɨɪɹɞɨɱɟɧɢɟ ɢɦɟɟɬ ɦɟɫɬɨ, ɬɨ

z

ɞɚɠɟ ɟɫɥɢ <

>=0, ɬ.ɟ. ɤɨɪɪɟɥɹɰɢɢ ɜɨɡɦɨɠɧɵ ɢ ɡɚ

ɬɨɱɤɨɣ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. ɂɫɫɥɟɞɭɟɦ ɜɨɡɦɨɠɧɨɫɬɶ ɪɟɚɥɢɡɚ-

ɰɢɢ ɛɥɢɠɧɟɝɨ ɩɨɪɹɞɤɚ ɜ ɫɩɢɧɨɜɨɣ ɫɢɫɬɟɦɟ ɢ ɨɰɟɧɢɦ ɮɭɧɤɰɢɸ

ɤɨɪɪɟɥɹɰɢɢ <

>. Ɋɚɫɫɦɨɬɪɢɦ ɫɢɬɭɚɰɢɸ ɜɵɲɟ ɬɨɱɤɢ ɮɚɡɨɜɨ-

ɝɨ ɩɟɪɟɯɨɞɚ T >

. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɩɨɧɬɚɧɧɚɹ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ

>=0. ȼɵɞɟɥɢɦ ɞɜɚ ɫɩɢɧɚ ɢ ɨɰɟɧɢɦ ɫɧɚɱɚɥɚ ɤɨɪɪɟɥɹɰɢɸ

> ɜ ɩɪɟɧɟɛɪɟɠɟɧɢɢ ɮɥɭɤɬɭɚɰɢɹɦɢ ɨɫɬɚɥɶɧɵɯ ɫɩɢɧɨɜ.

ɉɭɫɬɶ ɜɧɟɲɧɟɟ ɩɨɥɟ ɪɚɜɧɨ ɧɭɥɸ. Ɍɨɝɞɚ









EPP

EEE E E

12 1212

12 1 2

12 12 12 12 12

 

 

Sp V

VVV V V

exp( )

.eee e e e th

ɍɱɬɟɦ, ɱɬɨ V

#

V - ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫ ɛɥɢɠɚɣɲɢɦɢ ɫɨɫɟɞɹɦɢ,

ɚ ɬɚɤɠɟ th[

V](T >

4

)

t

th [

@ . Ʉɪɨɦɟ ɬɨɝɨ,

VVVr

() (| |)0 

¦¦

r =

VZ,

ɝɞɟ

- ɱɢɫɥɨ ɛɥɢɠɚɣ-

ɲɢɯ ɫɨɫɟɞɟɣ, ɚ ɬɚɤɠɟ

V(0)



. Ɉɬɫɸɞɚ ɧɟɫɥɨɠɧɨ ɨɰɟɧɢɬɶ

ɦɚɫɲɬɚɛ ɛɥɢɠɧɢɯ ɤɨɪɪɟɥɹɰɢɣ: <

>~th1/Z~ 1/Z ɩɪɢ Z>>1.

Ɂɧɚɱɢɬ, ɱɟɦ ɛɨɥɶɲɟ ɱɢɫɥɨ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ, ɬɟɦ ɫɥɚɛɟɟ

ɤɨɪɪɟɥɹɰɢɢ, ɬɟɦ ɥɭɱɲɟ ɪɚɛɨɬɚɟɬ ɩɪɢɛɥɢɠɟɧɢɟ ɫɪɟɞɧɟɝɨ ɩɨɥɹ.

Ɉɬɦɟɬɢɦ, ɱɬɨ ɝɪɚɧɢɱɧɨɟ ɡɧɚɱɟɧɢɟ Z

, ɤɨɝɞɚ ɤɨɪɪɟɥɹɰɢɢ

ɧɚɱɢɧɚɸɬ ɫɭɳɟɫɬɜɟɧɧɨ ɪɚɡɪɭɲɚɬɶ ɞɚɥɶɧɢɣ ɩɨɪɹɞɨɤ, ɡɚɜɢɫɢɬ ɨɬ

ɦɨɞɟɥɢ. Ʉɚɤ ɦɵ ɜɢɞɟɥɢ ɜ ɩɪɟɞɵɞɭɳɟɣ ɝɥɚɜɟ, ɜ ɨɞɧɨɦɟɪɧɨɣ

ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɩɪɢ Z=2 ɩɟɪɟɯɨɞ ɨɬɫɭɬɫɬɜɭɟɬ, ɚ ɜ ɞɜɭɦɟɪɧɨɣ

ɪɟɲɟɬɤɟ ɫ Z=4 ɮɥɭɤɬɭɚɰɢɢ ɭɠɟ ɧɟ ɪɚɡɪɭɲɚɸɬ ɞɚɥɶɧɢɣ ɩɨɪɹ-

ɞɨɤ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɨɰɟɧɢɬɶ, ɱɬɨ ɞɥɹ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɫ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɬɨɥɶɤɨ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ ɤɪɢɬɢɱɟɫɤɨɟ

ɡɧɚɱɟɧɢɟ Z

~ 3.

3.2. Ɏɭɧɤɰɢɹ ɤɨɪɪɟɥɹɰɢɢ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ

ɜ ɦɨɞɟɥɢ ɂɡɢɧɝɚ

Ɉɩɪɟɞɟɥɢɦ ɛɨɥɟɟ ɬɨɱɧɨ ɮɭɧɤɰɢɸ ɩɚɪɧɨɣ ɤɨɪɪɟɥɹɰɢɢ

>, ɤɨɬɨɪɚɹ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɧɟ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ

ɢɫɱɟɡɧɨɜɟɧɢɢ ɞɚɥɶɧɟɝɨ ɩɨɪɹɞɤɚ. Ɇɵ ɭɠɟ ɨɰɟɧɢɥɢ ɟɟ ɩɨɪɹɞɨɤ

ɜɟɥɢɱɢɧɵ <

> ~ 1/Z. ɑɬɨɛɵ ɩɨɥɭɱɢɬɶ ɤɨɥɢɱɟɫɬɜɟɧɧɵɣ ɪɟ-

ɡɭɥɶɬɚɬ, ɢɫɩɨɥɶɡɭɟɦ ɬɭ ɠɟ ɢɞɟɸ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɝɨ ɩɨɥɹ.

Ɋɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ ɂɡɢɧɝɚ ɩɪɢ T>

, T

. ɉɭɫɬɶ ɢɦɟɟɬɫɹ

ɜɵɞɟɥɟɧɧɵɣ ɦɨɦɟɧɬ ɫɨ ɡɧɚɱɟɧɢɟɦ

=+1. Ɍɨɝɞɚ ɮɭɧɤɰɢɹ ɤɨɪ-

ɪɟɥɹɰɢɢ ɟɫɬɶ ɩɪɨɫɬɨ ɫɪɟɞɧɟɟ <

> ɜ ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɦ ɷɬɨɦɭ

ɭɫɥɨɜɢɸ ɚɧɫɚɦɛɥɟ:

!



Sp H

eff

exp( )

. (3.1)

Ɂɞɟɫɶ H

eff

- ɷɮɮɟɤɬɢɜɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ

ɷɬɨɦɭ ɚɧɫɚɦɛɥɸ.

ɉɨɥɚɝɚɹ ɜɧɟɲɧɟɟ ɩɨɥɟ ɧɭɥɟɜɵɦ, ɭɱɬɟɦ, ɱɬɨ ɞɟɣɫɬɜɭɸ-

ɳɟɟ ɧɚ ɫɩɢɧ 2 ɜɧɭɬɪɟɧɧɟɟ ɩɨɥɟ, ɫɨɡɞɚɜɚɟɦɨɟ ɫɩɢɧɨɦ 1, ɚ ɬɚɤɠɟ

ɢ ɨɫɬɚɥɶɧɵɦɢ ɫɩɢɧɚɦɢ, ɪɚɜɧɨ

HV V

221 2



. (3.2)

ɋɪɟɞɧɟɟ ɩɨɥɟ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɢɦɟɟɬ ɜɢɞ

!  !  

HV V V V

221 2

21 2 1

)

. (3.3)

ɍɱɢɬɵɜɚɹ ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ (1.11), ɧɟɦɟɞɥɟɧɧɨ ɩɨɥɭɱɚɟɦ

! !  



PP E E PP

12 2 21 2 1

th thHVV

. (3.4)

Ɉɛɨɡɧɚɱɢɦ ɮɭɧɤɰɢɸ ɤɨɪɪɟɥɹɰɢɢ . ɉɟɪɟ-

ɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (3.4):

! 

gR R(

gRR VRR VRRgRR

() ()()(

oo oo o o oo

   



12 12 2 1

. (3.5)

ȼ ɩɪɢɛɥɢɠɟɧɢɢ ȼɟɣɫɫɚ ɮɭɧɤɰɢɹ = 0. ɂɳɟɦ ɧɟɬɪɢ-

ɜɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɢ ɩɨɥɨɠɢɦ

gR R

(



gR R

(



<<1, VR <<

()



ɉɨɫɥɟɞɧɟɟ ɧɟɪɚɜɟɧɫɬɜɨ ɫɩɪɚɜɟɞɥɢɜɨ ɜ ɩɪɟɞɟɥɟ ɛɨɥɶɲɨɝɨ ɱɢɫ-

ɥɚ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ 1/Z <<1 (

=ZV). Ɉɬɫɸɞɚ:

)gRR VRR VRRgRR

()()()(

oo oo o o oo

   



12 12 2 1

. (3.6)

ɍɪɚɜɧɟɧɢɟ (3.6) ɪɟɲɚɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ:

gq iqR

() ()exp( )

gq gR R iqR R

() ( )exp( [ ])

ooo

oo o

 

, (3.7)

Vq VR R iqR R

() ( )exp( [ ])

ooo

oo o

 

ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɭɪɚɜɧɟɧɢɟ (3.6) ɫ ɦɧɨɠɢɬɟɥɟɦ

exp(-iqR) ɩɨ ɤɨɨɪɞɢɧɚɬɟ, ɩɨɥɭɱɚɟɦ ɞɥɹ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɵ

ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ:

gq Vq g qVq() () ( )()

ooo





. (3.8)

ȼɜɢɞɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɢ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ

gR g R() (||)

VR() (||)

ɨɬɤɭɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ (ɷɬɨ ɧɟɫɥɨɠɧɨ ɩɨɤɚɡɚɬɶ ɩɪɹ-

ɦɨ ɢɡ (3.7)). Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ:

gq g q() ( )



T()

()



. (3.9)

ȼ ɞɥɢɧɧɨɜɨɥɧɨɜɨɦ ɩɪɟɞɟɥɟ ɮɭɧɤɰɢɹ V(q)= V(0) =

. ȼ

ɫɜɨɸ ɨɱɟɪɟɞɶ

T()



of o

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɢɞɧɨ, ɱɬɨ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɚ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ

ɩɪɢɛɥɢɠɟɧɢɢ ɤ ɬɨɱɤɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. ɂɫɫɥɟɞɭɟɦ ɩɪɨ-

ɫɬɪɚɧɫɬɜɟɧɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɮɭɧɤɰɢɢ ɤɨɪɪɟɥɹɰɢɢ:

iqR

()

exp( )



. (3.10)

Ɋɚɫɫɦɨɬɪɢɦ ɞɥɢɧɧɨɜɨɥɧɨɜɵɣ ɩɪɟɞɟɥ q o0 (ɬ.ɟ. ɢɫɫɥɟ-

ɞɭɟɦ ɫɢɬɭɚɰɢɸ ɜ ɩɪɟɞɟɥɟ ɛɨɥɶɲɢɯ ɤɨɪɪɟɥɹɰɢɨɧɧɵɯ ɞɥɢɧ

ɨɤɨɥɨ ɬɨɱɤɢ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. Ɍɨɝɞɚ ɮɭɪɶɟ-

ɤɨɦɩɨɧɟɧɬɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɪɢɛɥɢɠɟɧɧɨ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞ-

ɫɬɚɜɥɟɧɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

Vq V r iqr

Vq Vr

() (||)(

()

)

() , (||) .

ooo





(3.11)

ɉɚɪɚɦɟɬɪ

ɩɨ ɩɨɪɹɞɤɭ ɜɟɥɢɱɢɧɵ ɪɚɜɟɧ

~V(0)/r

ɯɚɪ

ɝɞɟ r

ɯɚɪ

- ɯɚɪɚɤɬɟɪɧɚɹ ɞɥɢɧɚ, ɧɚ ɤɨɬɨɪɨɣ ɞɟɣɫɬɜɭɟɬ ɜɡɚɢɦɨɞɟɣ-

ɫɬɜɢɟ (ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɟɞɫɬɚɜɥɟɧɢɟ (3.11) ɩɪɢɜɨɞɢɬ ɤ

ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɦɭ ɫɩɚɞɭ

V(r) ~ (1/r)exp (- r/r

ɯɚɪ

) - ɫɦ.ɡɚɞɚɱɭ 3.2.1).

ɉɨɞɫɬɚɜɥɹɹ (3.11) ɜ (3.10), ɧɚɯɨɞɢɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɭɸ

ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ ɜ ɩɪɟɞɟɥɟ

Do

0 (ɢɧɬɟ-

ɝɪɚɥ (3.10) ɛɟɪɟɬɫɹ ɜ ɫɮɟɪɢɱɟɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ ɩɨ q, ɬɚɤ ɱɬɨ

³³³



()

cos

dd d

) :

()



exp[ ]

(3.12)

ɝɞɟ d - ɯɚɪɚɤɬɟɪɧɚɹ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɞɥɢɧɚ;

- ɨɛɴɟɦ ɷɥɟɦɟɧ-

ɬɚɪɧɨɣ ɹɱɟɣɤɢ.