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

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4.1.8. ɂɫɫɥɟɞɨɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɬɟɩɥɨɟɦɤɨɫɬɢ ɩɪɢ T

Ɉɬɜɟɬ

: CN TN F T T/[/] /()   

ww T

52 20 7

4.1.

9. ɇɚɣɬɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɞɨɥɶɧɨɣ ɧɚɦɚɝɧɢɱɟɧɧɨ-

ɫɬɢ ɢɫɯɨɞɹ ɢɡ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ.

Ɉɬɜɟɬ

: MFHN  R

4.1.

10. ȼɵɪɚɡɢɬɶ ɩɪɨɞɨɥɶɧɭɸ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢ-

ɜɨɫɬɶ

F

ɱɟɪɟɡ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɮɭɧɤɰɢɢ Ʌɚɧɠɟɜɟɧɚ

Fx F x

'( ) dd.

Ɉɬɜɟɬ

PEP



ddRH

SF HS

TSZJF H

()'( )

() '(

)

4.1.

11. ɇɚɣɬɢ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ ɩɪɢ T>

Ɉɬɜɟɬ

()

.RH



4.1.

12. ɇɚɣɬɢ ɩɪɨɞɨɥɶɧɭɸ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ

ɩɪɢ

T o

- 0.

Ɋɟɲɟɧɢɟ

. ɉɨɥɶɡɭɹɫɶ ɚɫɢɦɩɬɨɬɚɦɢ ɮɭɧɤɰɢɢ Ʌɚɧɠɟɜɟɧɚ

(ɫɦ.ɡɚɞɚɱɭ 1.4.

3), ɩɨɥɭɱɚɟɦ:

FP T

()(

6ST).

4.1.

13. ɇɚɣɬɢ ɩɪɨɞɨɥɶɧɭɸ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ

ɩɪɢ

T o 0.

Ɉɬɜɟɬ

| TZJ S()

4.1.

14. ɂɫɫɥɟɞɨɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɩɚɪɚɦɟɬɪɚ ɩɨɪɹɞɤɚ R(T)

ɩɪɢ

T o

- 0 ɢ T o 0.

Ɉɬɜɟɬ

ST T

RZJyy y yTT

o

  o



TTTT

223

13 0

547449 0

(), ;

[//],(), .

4.1.

. ȼɵɪɚɡɢɬɶ ɬɟɩɥɨɟɦɤɨɫɬɶ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɨɞɟɥɢ

Ƚɟɣɡɟɧɛɟɪɝɚ C ɱɟɪɟɡ ɩɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ R, ɢɫɩɨɥɶɡɭɹ

C= -T

w

. Ɋɚɫɫɦɨɬɪɟɬɶ ɩɪɟɞɟɥɶɧɵɟ ɫɥɭɱɚɢ T

0 ɢ T

Ɉɬɜɟɬ

CN T T

//,

//[()/],

//,.



 o

 o

521 8 7

13 0

Ⱥɧɚɥɢɡ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɫɱɟɬɨɜ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞ-

ɧɟɝɨ ɩɨɥɹ ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ ɩɨ ɡɚɞɚɱɚɦ 4.1.2 - 4.1.15. Ʉɚɤ

ɦɵ ɜɢɞɢɦ, ɩɪɢɛɥɢɠɟɧɢɟ ɫɪɟɞɧɟɝɨ ɩɨɥɹ ɯɨɪɨɲɨ ɨɩɢɫɵɜɚɟɬ ɮɚ-

ɡɨɜɵɣ ɩɟɪɟɯɨɞ ɢ ɡɚɤɨɧ Ʉɸɪɢ-ȼɟɣɫɫɚ ɞɥɹ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ

(ɡɚɞɚɱɢ 4.1.

11, 4.1.12). Ɉɞɧɚɤɨ ɜ ɨɬɥɢɱɢɟ ɨɬ ɦɨɞɟɥɢ ɂɡɢɧɝɚ

ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ ɥɢɧɟɣɧɨ ɫɩɚɞɚɟɬ ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ

(ɡɚɞɚɱɚ 4.1.

13), ɱɬɨ ɫɜɹɡɚɧɨ ɫ ɨɬɫɭɬɫɬɜɢɟɦ ɳɟɥɢ ɜ ɫɩɟɤɬɪɟ ɡɚ

ɫɱɟɬ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɭ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɫɩɢɧɚ

ɢ ɜɵɪɨɠɞɟɧɢɹ ɩɨ ɭɝɥɭ ɜɪɚɳɟɧɢɹ

ɜɨɤɪɭɝ ɨɫɢ z). Ȼɨɥɟɟ ɬɨɝɨ,

ɬɟɩɥɨɟɦɤɨɫɬɶ ɧɟ ɫɬɪɟɦɢɬɶɫɹ ɤ ɧɭɥɸ ɩɪɢ T

0 (ɡɚɞɚɱɚ 4.1.15),

ɱɬɨ ɹɜɥɹɟɬɫɹ ɧɚɪɭɲɟɧɢɟɦ ɬɟɨɪɟɦɵ ɇɟɪɧɫɬɚ ɢ ɫɜɹɡɚɧɨ ɫ ɭɩɨɦɹ-

ɧɭɬɵɦ ɜɵɪɨɠɞɟɧɢɟɦ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ (ɧɚɩɨɦɧɢɦ, ɱɬɨ

ɬɟɨɪɟɦɚ ɇɟɪɧɫɬɚ ɫɩɪɚɜɟɞɥɢɜɚ ɞɥɹ ɧɟɜɵɪɨɠɞɟɧɧɨɝɨ ɨɫɧɨɜɧɨɝɨ

ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ). ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɞɚɥɟɟ ɩɨɤɚɠɟɦ, ɱɬɨ

ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɜɫɟ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɭ ɫɩɢɧɚ “ɜɵ-

ɦɟɪɡɚɸɬ”, ɤɪɨɦɟ ɞɜɭɯ ɢɡɢɧɝɨɜɫɤɢɯ: ɩɨ ɩɨɥɸ ɢ ɩɪɨɬɢɜ, ɬɚɤ ɱɬɨ

ɯɨɪɨɲɢɦ ɩɪɢɛɥɢɠɟɧɢɟɦ

ɞɥɹ Ƚɟɣɡɟɧɛɟɪɝɨɜɫɤɨɝɨ ɮɟɪɪɨɦɚɝɧɟ-

ɬɢɤɚ ɹɜɥɹɟɬɫɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɞɟɥɶ ɂɡɢɧɝɚ. ɉɨɷɬɨɦɭ ɩɪɢ T

ɧɟɥɶɡɹ ɤɥɚɫɫɢɱɟɫɤɢɦ ɩɨɞɯɨɞɨɦ ɚɞɟɤɜɚɬɧɨ ɨɩɢɫɚɬɶ ɩɨɜɟɞɟɧɢɟ

ɫɢɫɬɟɦɵ ɢ ɧɟɨɛɯɨɞɢɦɨ ɤɨɪɪɟɤɬɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɜɚɧɬɨɜɨɟ

ɨɩɢɫɚɧɢɟ, ɱɟɦɭ ɢ ɛɭɞɭɬ ɩɨɫɜɹɳɟɧɵ ɫɥɟɞɭɸɳɢɟ ɩɚɪɚɝɪɚɮɵ.

4.2. Ɉɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ ɮɟɪɪɨɦɚɝɧɟɬɢɤɚ

ɜ ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ

Ɋɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ Ƚɟɣɡɟɧɛɟɪɝɚ ɩɪɢ ɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚ-

ɬɭɪɟ. ɉɭɫɬɶ ɨɛɦɟɧɧɵɣ ɢɧɬɟɝɪɚɥ J > 0, ɬ.ɟ. ɢɫɫɥɟɞɭɟɦ ɮɟɪɪɨ-

ɦɚɝɧɢɬɧɭɸ ɫɢɬɭɚɰɢɸ. ɉɨɥɨɠɢɦ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɢɡɥɨɠɟɧɢɹ

ɜɧɟɲɧɟɟ ɩɨɥɟ ɪɚɜɧɵɦ ɧɭɥɸ. Ɉɛɨɡɧɚɱɢɦ |S’>

ɜɨɥɧɨɜɭɸ ɮɭɧɤ-

ɰɢɸ ɫɩɢɧɚ ɧɚ ɭɡɥɟ l ɫ ɩɪɨɟɤɰɢɟɣ S’ ɧɚ ɨɫɶ z. Ɍɟɩɟɪɶ ɡɧɚɱɟɧɢɟ

ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ S’ ɧɚ ɭɡɥɟ (ɜ ɨɬɥɢɱɢɟ ɨɬ ɦɨɞɟɥɢ ɂɡɢɧɝɚ) ɦɨɠɟɬ

ɩɪɢɧɢɦɚɬɶ ɰɟɥɵɣ ɪɹɞ ɡɧɚɱɟɧɢɣ. ȼ ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ ɜɫɟ

ɫɩɢɧɵ ɧɚɩɪɚɜɥɟɧɵ ɜ ɨɞɧɭ ɫɬɨɪɨɧɭ (ɧɚɩɪɢɦɟɪ, ɩɨ ɨɫɢ z) ɫ ɦɚɤ-

ɫɢɦɚɥɶɧɵɦ ɡɧɚɱɟɧɢɟɦ ɩɪɨɟɤɰɢɢ. Ɍɨɝɞɚ ɨɩɟɪɚɬɨɪ z-ɣ ɩɪɟɤɰɢɢ

ɫɩɢɧɚ: ɫɨɯɪɚɧɹɟɬ ɫɩɢɧ ɧɚ ɭɡɥɟ, ɢ ɟɝɨ ɫɨɛɫɬ-

ɜɟɧɧɨɟ ɱɢɫɥɨ S= S

SS SS

||! !

max

, ɬ.ɟ. ɦɨɞɭɥɸ ɩɨɥɧɨɝɨ ɫɩɢɧɚ (ɨɛɵɱɧɨ S

max

~ 2-3).

nnnnnnnnnnnn

ȼɜɟɞɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɩɨɥɧɭɸ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ

ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ:

| | | ......| |...........0

! ! ! ! !SS S S

. (4.6)

Ɉɬɦɟɬɢɦ, ɱɬɨ ɨɩɟɪɚɬɨɪ z-ɣ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɤɨɦɦɭɬɢɪɭɟɬ ɫ ɝɚ-

ɦɢɥɶɬɨɧɢɚɧɨɦ:

HS S H

 0

, ɬ.ɟ. ɩɨɥɧɚɹ ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ z

ɫɨɯɪɚɧɹɟɬɫɹ.

ɉɨɤɚɠɟɦ, ɱɬɨ

HE||0

! !0

, ɝɞɟ E

- ɷɧɟɪɝɢɹ ɨɫɧɨɜɧɨ-

ɝɨ ɫɨɫɬɨɹɧɢɹ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɫ ɩɨɜɟɪɧɭɬɵɦɢ ɜ ɫɬɨɪɨɧɭ ɨɫɢ z

ɫɩɢɧɚɦɢ, ɢ ɪɚɫɫɱɢɬɚɟɦ E

. ɂɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɹ (4.5), ɩɟɪɟ-

ɩɢɲɟɦ ɝɚɦɢɥɶɬɨɧɢɚɧ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:

HJSSSSS

ij i

ij ij

  

 

/{ ( S )}

. (4.7)

Ɋɚɫɫɱɢɬɚɟɦ ɩɪɟɠɞɟ ɜɫɟɝɨ ɞɟɣɫɬɜɢɟ ɨɩɟɪɚɬɨɪɨɜ S

ɢ S

ɧɚ |0>.

ɉɨɤɚɠɟɦ, ɱɬɨ ɨɧɢ ɭɜɟɥɢɱɢɜɚɸɬ (ɭɦɟɧɶɲɚɸɬ) z-ɩɪɨɟɤɰɢɸ ɫɩɢ-

ɧɚ ɧɚ ɭɡɥɟ ɧɚ ɟɞɢɧɢɰɭ. Ⱦɥɹ ɷɬɨɝɨ, ɭɱɢɬɵɜɚɹ, ɱɬɨ

SS SS

||! !, ɩɨɥɭɱɢɦ ɞɥɹ ɤɚɤɨɝɨ-ɥɢɛɨ ɭɡɥɚ (ɟɝɨ ɢɧɞɟɤɫ ɧɟ

ɛɭɞɟɦ ɩɢɫɚɬɶ) ɫɥɟɞɭɸɳɟɟ:

SS S SS iS S iS S S S S

ZXYXY 

!    !  !|' ['( ) ]|' (' ) |'1 , (4.8)

ɬ.ɟ. z-ɩɪɨɟɤɰɢɹ ɫɨɫɬɨɹɧɢɹ S

|S’> ɪɚɜɧɚ S’+1. Ɉɬɫɸɞɚ

|S’>=|S’+1>. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ S

|S’>=|S’-1>.

Ɉɬɫɸɞɚ ɧɟɦɟɞɥɟɧɧɨ ɫɥɟɞɭɟɬ, ɱɬɨ ɫɨɫɬɨɹɧɢɟ S

|S’>=0 ɩɪɢ

S’=S=S

max

, ɬɚɤ ɤɚɤ ɛɨɥɶɲɟ ɡɧɚɱɟɧɢɟ ɫɩɢɧɚ ɛɵɬɶ ɧɟ ɦɨ-

ɠɟɬ.Ɍɨɝɞɚ ɩɪɢ ɞɟɣɫɬɜɢɢ ɧɚ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ |0> ɝɚɦɢɥɶɬɨ-

ɧɢɚɧɚ (4.7) ɜɤɥɚɞ ɞɚɫɬ ɬɨɥɶɤɨ ɱɥɟɧ ɫ z-ɩɪɨɟɤɰɢɟɣ:

HJSE

||;

!  ! 

¦¦

JS. (4.9)

Ɇɵ ɩɨɥɭɱɢɥɢ ɫɨɝɥɚɫɧɨ (4.9) ɷɧɟɪɝɢɸ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ,

ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɫɨɛɨɣ ɫɢɫɬɟɦɭ ɫɨɧɚɩɪɚɜɥɟɧɧɵɯ (ɩɨ ɨɫɢ z)

ɫɩɢɧɨɜ.

4.3. ȼɨɡɛɭɠɞɟɧɢɹ ɜ ɝɟɣɡɟɧɛɟɪɝɨɜɫɤɨɦ

ɮɟɪɪɨɦɚɝɧɟɬɢɤɟ. Ɇɚɝɧɨɧɵ

Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɧɢɡɲɢɟ ɜɨɡɛɭɠɞɟɧɧɵɟ (ɧɚɞ ɨɫɧɨɜ-

ɧɵɦ) ɫɨɫɬɨɹɧɢɹ ɮɟɪɪɨɦɚɝɧɟɬɢɤɚ. ɉɭɫɬɶ ɬɟɦɩɟɪɚɬɭɪɚ T

0, ɢ

T<<

. ȼ ɷɬɨɦ ɫɥɭɱɚɟ z-ɩɪɨɟɤɰɢɹ ɫɩɢɧɚ ɧɚ ɭɡɥɟ ɭɠɟ ɧɟ ɨɛɹɡɚ-

ɬɟɥɶɧɨ ɪɚɜɧɚ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸ S. Ɉɩɟɪɚɬɨɪɵ S

ɢ S

ɬɟɩɟɪɶ ɦɨɝɭɬ “ɩɟɪɟɦɟɳɚɬɶ” ɫɩɢɧɵ (ɜɟɪɧɟɟ, ɭɦɟɧɶɲɚɬɶ ɢ ɭɜɟ-

ɥɢɱɢɜɚɬɶ ɧɚ ɟɞɢɧɢɰɭ) ɫɭɡɥɚɧɚɭɡɟɥ, ɧɟ ɜɵɯɨɞɹ ɡɚ ɩɪɟɞɟɥɶɧɨɟ

ɡɧɚɱɟɧɢɟ. Ɍɚɤ, ɟɫɥɢ ɧɚ ɭɡɥɟ l ɛɵɥɚ ɩɪɨɟɤɰɢɹ S’, ɚɧɚɭɡɥɟl’ -

ɩɪɨɟɤɰɢɹ S, ɬɨ

SS S S S S

ll l







ɋɩɢɧɵ ɩɟɪɟɦɟɳɚɸɬɫɹ ɫ ɭɡɥɚ l’ ɧɚ ɭɡɟɥ l. Ⱦɥɹ ɬɚɤɨɝɨ “ɞɜɢɠɟ-

ɧɢɹ” (ɫɩɢɧɨɜɨɣ ɜɨɥɧɵ) ɦɨɠɧɨ ɜɜɟɫɬɢ ɷɥɟɦɟɧɬɚɪɧɵɟ ɜɨɡɛɭɠɞɟ-

ɧɢɹ ɦɚɝɧɢɬɧɨɝɨ ɦɨɦɟɧɬɚ - ɦɚɝɧɨɧɵ. ɗɬɢ ɜɨɡɛɭɠɞɟɧɢɹ ɦɨɠɧɨ

ɫɱɢɬɚɬɶ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɦɢ ɩɪɢ ɢɯ ɦɚɥɨɣ ɩɥɨɬɧɨɫɬɢ

ɜɛɥɢɡɢ ɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭɪɵ. ɍɞɨɛɧɨ ɧɨɜɵɟ ɤɜɚɡɢɱɚɫɬɢɰɵ

(ɦɚɝɧɨɧɵ) ɜɜɟɫɬɢ ɤɚɤ ɪɚɡɧɢɰɭ ɦɟɠɞɭ ɜɨɡɛɭɠɞɟɧɧɵɦ ɢ ɧɟɜɨɡ-

ɛɭɠɞɟɧɧɵɦ ɫɨɫɬɨɹɧɢɟɦ ɭɡɥɚ, ɬ.ɟ. ɢɯ ɩɥɨɬɧɨɫɬɶ n

=S - S

, ɩɪɢ

ɷɬɨɦ n

<< 1. ɇɚ ɹɡɵɤɟ ɜɨɡɛɭɠɞɟɧɢɣ ɨɩɟɪɚɬɨɪ S

ɬɟɩɟɪɶ ɭɧɢɱ-

ɬɨɠɚɟɬ ɜɨɡɛɭɠɞɟɧɢɟ ɧɚ ɭɡɥɟ, S

- ɪɨɠɞɚɟɬ ɟɝɨ. ɋɨɨɬɜɟɬɫɬɜɟɧ-

ɧɨ, ɢ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ ɦɨɠɧɨ ɨɬɫɱɢɬɵɜɚɬɶ ɨɬ ɨɫɧɨɜɧɨɝɨ

ɫɨɫɬɨɹɧɢɹ, ɬ.ɟ. ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɱɢɫɥɨ ɜɨɡɛɭɠɞɟɧɢɣ ɧɚ

ɭɡɥɚɯ:

| | | ......| ...........

! ! ! ! !nn n n

lN12

. (4.10)

ȼɜɟɞɟɦ ɨɩɟɪɚɬɨɪɵ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ ɧɚ ɭɡɥɚɯ:

, ɝɞɟ A - ɧɟɤɚɹ ɧɨɪɦɢɪɨɜɨɱɧɚɹ ɩɨɫɬɨɹɧɧɚɹ,

ɬɪɟɛɭɸɳɚɹ ɨɩɪɟɞɟɥɟɧɢɹ. Ʉɪɨɦɟ ɬɨɝɨ, ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ ɞɟɣɫɬɜɢɟ

ɷɬɢɯ ɨɩɟɪɚɬɨɪɨɜ ɧɚ ɭɡɟɥɶɧɭɸ ɮɭɧɤɰɢɸ:

aASaAS

lll





?an an



! ! |?,|

Ɋɚɫɫɦɨɬɪɢɦ ɤɨɦɦɭɬɚɬɨɪ:

aa aa

SS SS

SS S S

YX XY

   

  

()( )

. (4.11)

ɂɡ (4.11) ɫɥɟɞɭɟɬ, ɱɬɨ ɟɫɥɢ ɩɨɥɨɠɢɬɶ

2 , ɬɨ ɩɨɥɭɱɚɟɦ

ɫɨɨɬɧɨɲɟɧɢɟ ɤɨɦɦɭɬɚɰɢɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɨɩɟɪɚɬɨɪɚɦ ɫɬɚ-

ɬɢɫɬɢɤɢ Ȼɨɡɟ:

, ɟɫɥɢ ɭɱɟɫɬɶ ɬɚɤɠɟ, ɱɬɨ ɩɪɨɟɤɰɢɹ

ɧɚ ɨɫɶ z - ɯɨɪɨɲɟɟ ɤɜɚɧɬɨɜɨɟ ɱɢɫɥɨ, ɬɨ ɩɪɢ ɭɫɥɨɜɢɢ n

aa a a



 1

<< 1 ɨɧɚ

ɪɚɜɧɚ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸ S, ɬ.ɟ. S

S. ɇɟɦɟɞɥɟɧɧɨ ɫɥɟ-

ɞɭɟɬ ɬɚɤɠɟ ɩɨɥɟɡɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɨɩɟɪɚɬɨɪɨɦ ɱɢɫɥɚ

ɜɨɡɛɭɠɞɟɧɢɣ n

ɢ ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ ɧɚ ɨɫɶ z:

SSaaSn

ll l

 



Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɜɜɟɥɢ ɧɨɜɵɟ ɛɨɡɟɜɫɤɢɟ ɜɨɡɛɭɠɞɟɧɢɹ - ɦɚɝ-

ɧɨɧɵ:

aa n S S

ll l l

Z







,, . (4.12)

Ɉɩɟɪɚɬɨɪɵ (4.12) ɞɟɣɫɬɜɭɸɬ ɧɚ ɭɡɟɥɶɧɵɟ ɮɭɧɤɰɢɢ ɨɛɵɱɧɵɦ

ɨɛɪɚɡɨɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɚɬɢɫɬɢɤɨɣ Ȼɨɡɟ:

an nn an n n aan nn

ll ll ll l l lll ll

 

!  ! !   ! ! !||,| |,||111 .

(4.13)

ɉɨɤɚɠɟɦ, ɱɬɨ ɜ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɦɚɝɧɨɧɨɜ ɩɪɢ ɩɟɪɟɯɨɞɟ ɢɡ ɥɨ-

ɤɚɥɶɧɵɯ ɨɩɟɪɚɬɨɪɨɜ ‘S’ ɤ ɨɩɟɪɚɬɨɪɚɦ ɫɩɢɧɨɜɵɯ ɜɨɥɧ ‘a’ ɝɚ-

ɦɢɥɶɬɨɧɢɚɧ ɞɢɚɝɨɧɚɥɢɡɭɟɬɫɹ, ɬ.ɟ. ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ ɝɚ-

ɦɢɥɶɬɨɧɢɚɧɚ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ. Ɍɚɤɨɟ ɩɪɟɞɫɬɚɜ-

ɥɟɧɢɟ ɨɛɹɡɚɬɟɥɶɧɨ ɞɨɥɠɧɨ ɛɵɬɶ, ɟɫɥɢ ɦɚɝɧɨɧɵ - ɢɫɬɢɧɧɵɟ ɤɜɚ-

ɡɢɱɚɫɬɢɰɵ ɧɚɞ ɨɫɧɨɜɧɵɦ ɫɨɫɬɨɹɧɢɟɦ. ɉɨɞɫɬɚɜɢɦ ɧɨɜɵɟ ɨɩɟɪɚ-

ɬɨɪɵ ɜ ɝɚɦɢɥɶɬɨɧɢɚɧ (4.7):

HJSaaSaaSaaaa

ll ll ll ll ll

    

 

''''

()()(

)

. (4.14)

Ɋɚɫɤɪɵɜɚɹ ɫɤɨɛɤɢ ɜ (4.14) ɢ ɭɱɢɬɵɜɚɹ ɫɨɨɬɧɨɲɟɧɢɹ ɤɨɦɦɭɬɚ-

ɰɢɢ ɢ ɦɚɥɨɫɬɶ ɱɢɫɥɚ ɜɨɡɛɭɠɞɟɧɢɣ (ɬ.ɟ. ɩɪɟɧɟɛɪɟɝɚɹ ɱɥɟɧɚɦɢ

a << 1), ɚ ɬɚɤɠɟ ɬɨ, ɱɬɨ , ɩɨɥɭɱɚɟɦ:JJrr J

ll l l l l''

(| |) 

H E S J aa aa

ll ll ll

 



. (4.15)

ȼ (4.15) ɦɵ ɜɵɞɟɥɢɥɢ ɷɧɟɪɝɢɸ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ (4.9). Ⱦɚ-

ɥɟɟ ɢɫɩɨɥɶɡɭɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ ɨɩɟɪɚɬɨɪɨɜ:

aiqR

exp( )

aiqR





exp( )

ɉɨɞɫɬɚɜɢɜ ɷɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜ (4.15), ɢɭɱɢɬɵɜɚɹ, ɱɬɨ

exp( [ '] )

iq q R N



ooo

ɧɚɯɨɞɢɦ ɩɨɫɥɟ ɧɟɤɨɬɨɪɵɯ ɩɟɪɟɨɛɨɡɧɚɱɟɧɢɣ:

HE aa SJR iqR

qq q

 



¦¦

1!!

, (| |){ exp( )}

.(4.16)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɟɫɥɢ

=a a



- ɨɩɟɪɚɬɨɪ ɱɢɫɥɚ ɜɨɡɛɭɠɞɟɧɢɣ ɫ

ɜɨɥɧɨɜɵɦ ɜɟɤɬɨɪɨɦ

q, ɚ !

- ɫɩɟɤɬɪ ɜɨɡɛɭɠɞɟɧɢɣ, ɬɨ ɝɚɦɢɥɶ-

ɬɨɧɢɚɧ ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɢ ɝɚɦɢɥɶɬɨɧɢɚɧɚ

ɝɚɡɚ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɛɨɡɨɧɧɵɯ ɜɨɡɛɭɠɞɟɧɢɣ - ɦɚɝɧɨ-

ɧɨɜ:

HE n



Ɍɚɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɭɞɨɛɧɨ ɞɥɹ ɜɵɱɢɫɥɟɧɢɣ.

Ɂɚɞɚɱɢ

4.3.

1. Ɂɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɞɥɹ ɫɩɢɧɨɜɵɯ ɨɩɟ-

ɪɚɬɨɪɨɜ ɜ ɩɪɟɞɫɬɚɜɥɟɧɢɢ Ƚɟɣɡɟɧɛɟɪɝɚ, ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɜ

ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (4.1), ɜ ɧɭɥɟɜɨɦ ɜɧɟɲɧɟɦ ɩɨ-

ɥɟ.

Ɋɟɲɟɧɢɟ

. ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ:

HS XYZ

[, ], ,,.

Ɉɬɫɸɞɚ

JSS JSS

il i



[]

ɨɫɬɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɸɬɫɹ ɰɢɤɥɢɱɟɫɤɨɣ ɩɟɪɟɫɬɚɧɨɜɤɨɣ.

4.3.

2. Ɂɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɜ ɩɪɢɛɥɢɠɟɧɢɢ

ɫɪɟɞɧɟɝɨ ɩɨɥɹ, ɩɨɥɚɝɚɹ

=S ɱɢɫɥɨɦ, ɪɚɜɧɵɦ ɦɨɞɭɥɸ ɫɩɢɧɚ, ɚ

ɨɫɬɚɥɶɧɵɟ ɨɩɟɪɚɬɨɪɵ - ɦɚɥɵɦɢ. ɍɱɟɫɬɶ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɬɨɥɶɤɨ

ɫ ɛɥɢɠɚɣɲɢɦɢ ɫɨɫɟɞɹɦɢ.

Ɋɟɲɟɧɢɟ. Ɋɚɡɥɨɠɢɜ ɩɪɨɢɡɜɟɞɟɧɢɹ ɨɩɟɪɚɬɨɪɨɜ

SS SSSS S S S

li lili l

DE D E D E

| !   ! | ! ,,

ɢ ɩɨɥɚɝɚɹ

! ɦɚɥɵɦɢ, ɩɨɥɭɱɚɟɦ:SS



[].

4.3.

3. ɂɫɯɨɞɹ ɢɡ ɩɨɥɭɱɟɧɧɵɯ ɜ ɡɚɞɚɱɟ 4.3.2 ɭɪɚɜɧɟɧɢɣ

ɞɜɢɠɟɧɢɹ ɧɚɣɬɢ ɫɩɟɤɬɪ ɦɚɝɧɨɧɨɜ.

Ɋɟɲɟɧɢɟ

. ɉɟɪɟɩɢɲɟɦ ɫɧɚɱɚɥɚ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ,

ɜɜɟɞɹ ɨɩɟɪɚɬɨɪɵ

ɢ S

= S

+ iS

ɢ S



= S

- iS

ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɜɢɞ:

iSJ





 



[]



ɂɫɩɨɥɶɡɭɹ ɮɭɪɶɟ-ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ, ɢɳɟɦ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ

. Ɍɨɝɞɚ ɢɦɟɟɦ

SS iqrit

exp( )



iSJ

r }

ɝɞɟ . Ɉɤɨɧɱɚɬɟɥɶɧɨ ɫɩɟɤɬɪ ɦɚɝɧɨ-

ɧɨɜ . Ɂɚɦɟɬɢɦ, ɱɬɨ ɪɟɡɭɥɶɬɚɬ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɩɟɤ-

ɬɪɨɦ (4.16), ɩɨɥɭɱɟɧɧɵɦ ɞɢɚɝɨɧɚɥɢɡɚɰɢɟɣ ɝɚɦɢɥɶɬɨɧɢɚɧɚ (ɫ

ɬɨɱɧɨɫɬɶɸ ɞɨ ɧɚɲɢɯ ɩɟɪɟɨɛɨɡɧɚɱɟɧɢɣ

Aiqr

 

oo o

{exp[( )]1

SJA

J).

4.4. ɋɜɨɣɫɬɜɚ ɦɚɝɧɨɧɨɜ. ɋɩɟɤɬɪ ɜɨɡɛɭɠɞɟɧɢɣ.

ɗɮɮɟɤɬɢɜɧɚɹ ɦɚɫɫɚ

Ɋɚɫɫɱɢɬɚɟɦ ɫɧɚɱɚɥɚ ɷɧɟɪɝɢɸ ɦɚɝɧɨɧɨɜ !

. Ɋɚɫɫɦɨɬ-

ɪɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɬɨɥɶɤɨ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ ɢ ɢɫɫɥɟɞɭɟɦ

ɩɪɨɫɬɟɣɲɢɣ ɫɥɭɱɚɣ ɬɪɟɯɦɟɪɧɨɣ ɤɭɛɢɱɟɫɤɨɣ ɪɟɲɟɬɤɢ. ȼɤɭɛɢ-

ɱɟɫɤɨɦ ɤɪɢɫɬɚɥɥɟ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ ɜɫɟɝɨ ɲɟɫɬɶ:

) =

(

) = J(

) = J, ɝɞɟ a - ɩɟɪɢɨɞ ɪɟɲɟɬɤɢ. Ɍɨɝɞɚ ɢɦɟɟɦ:

SJ R i q R

SJ q a q a q a



  

ooo

(| |){ exp( )}

{cos cos cos}

(4.17)

Ɋɚɫɫɦɨɬɪɢɦ ɞɥɢɧɧɨɜɨɥɧɨɜɵɣ ɩɪɟɞɟɥ

|q|a<<1. ȼ ɷɬɨɦ

ɫɥɭɱɚɟ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ (4.17) ɢɦɟɟɬ ɜɢɞ

, ɫɨɜɩɚ-

ɞɚɸɳɢɣ ɩɨ ɮɨɪɦɟ ɫ ɡɚɤɨɧɨɦ ɞɢɫɩɟɪɫɢɢ ɞɥɹ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢ-

ɰɵ ().

SJa q

qm h

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɝɨɜɨɪɢɬɶ ɨ ɦɚɫɫɟ ɦɚɝɧɨɧɚ (ɤɨ-

ɧɟɱɧɨ, ɤɚɤ ɨɛ ɷɮɮɟɤɬɢɜɧɨɣ ɦɚɫɫɟ

m*):

SJa



. (4.18)

ɂɫɯɨɞɹ ɢɡ (4.18) ɨɰɟɧɢɦ ɦɚɫɲɬɚɛ ɷɮɮɟɤɬɢɜɧɨɣ ɦɚɫɫɵ.

ɉɨɥɚɝɚɹ

S ~ 3, a~ 3·10

-8

ɫɦ, J ~ k

4

~ 300Ʉ, ɩɨɥɭɱɚɟɦ m*

|

5 m

- ɨɤɨɥɨ ɩɹɬɢ ɷɥɟɤɬɪɨɧɧɵɯ ɦɚɫɫ, ɬ.ɟ. ɦɚɝɧɨɧɵ - ɞɨɫɬɚɬɨɱ-

ɧɨ ɬɹɠɟɥɵɟ ɥɨɤɚɥɢɡɨɜɚɧɧɵɟ ɤɜɚɡɢɱɚɫɬɢɰɵ.

Ɍɚɤ ɤɚɤ ɦɚɝɧɨɧɵ ɩɨɞɱɢɧɹɸɬɫɹ ɫɬɚɬɢɫɬɢɤɟ Ȼɨɡɟ, ɬɨ ɢɯ

ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɟɫɬɶ ɨɛɵɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ȼɨɡɟ-

ɗɣɧɲɬɟɣɧɚ:

nT n aa

qqqq

()

exp([ ] )

 !  !





, (4.19)

ɝɞɟ

- ɯɢɦɢɱɟɫɤɢɣ ɩɨɬɟɧɰɢɚɥ. Ɍɚɤ ɤɚɤ ɱɢɫɥɨ ɛɨɡɨɧɨɜ ɧɟɮɢɤ-

ɫɢɪɨɜɚɧo, ɬɨ ɯɢɦɢɱɟɫɤɢɣ ɩɨɬɟɧɰɢɚɥ

=0. Ɍɚɤɢɦ ɨɛ-

ɪɚɡɨɦ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɦɚɝɧɨɧɨɜ

()

exp( )



. (4.20)

ȼ ɡɚɤɥɸɱɟɧɢɟ ɨɬɦɟɬɢɦ, ɱɬɨ ɦɚɝɧɨɧɵ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ

ɩɪɢ ɪɚɫɫɟɹɧɢɢ ɧɟɣɬɪɨɧɨɜ, ɬɚɤ ɤɚɤ ɧɟɣɬɪɨɧɵ ɨɛɥɚɞɚɸɬ ɦɚɝɧɢɬ-

ɧɵɦ ɦɨɦɟɧɬɨɦ. ɂɫɩɨɥɶɡɭɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɢ ɢɦ-

ɩɭɥɶɫɚ ɩɪɢ ɨɞɧɨɦɚɝɧɨɧɧɨɦ ɪɚɫɫɟɹɧɢɢ ɧɟɣɬɪɨɧɚ

, ɝɞɟ

Epp qE E

oo o

  ',

ɢ p - ɷɧɟɪɝɢɹ ɢ ɢɦɩɭɥɶɫ ɩɚ-

ɞɚɸɳɟɝɨ ɧɟɣɬɪɨɧɚ,

p’

ɢ p’ - ɪɚɫɫɟɹɧɧɨɝɨ, ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ

ɷɧɟɪɝɢɸ ɢ ɢɦɩɭɥɶɫ ɦɚɝɧɨɧɚ.

Ɂɚɞɚɱɢ

4.4.

1. ɇɚɣɬɢ ɫɩɟɤɬɪ ɦɚɝɧɨɧɨɜ ɜ ȽɐɄ ɪɟɲɟɬɤɟ ɜ ɩɪɢɛɥɢ-

ɠɟɧɢɢ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ.

Ɉɬɜɟɬ

ZX ZY XY

qa q a qa qa qa qa

  { [cos cos cos cos cos cos ]}12 4

22 22 22

ɝɞɟ

a - ɩɟɪɢɨɞ ɪɟɲɟɬɤɢ. ɉɪɢ q

0 , ɬ.ɟ. ɪɟɡɭɥɶɬɚɬ

ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɥɭɱɚɟɦ ɩɪɨɫɬɨɣ ɤɭɛɢɱɟɫɤɨɣ ɪɟɲɟɬɤɨɣ.

JSa q

4.4.

2. ɇɚɣɬɢ ɤɪɢɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɮɟɪɪɨɦɚɝɧɟɬɢɤɚ

ɢɡ ɭɫɥɨɜɢɹ ɛɨɡɟ-ɤɨɧɞɟɧɫɚɰɢɢ ɦɚɝɧɨɧɨɜ ɢ ɫɪɚɜɧɢɬɶ ɫ ɤɪɢɬɢɱɟ-

ɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ.

Ɋɟɲɟɧɢɟ

. Ⱦɥɹ ɢɞɟɚɥɶɧɨɝɨ ɛɨɡɟ-ɝɚɡɚ (ɫɦ. ɝɥɚɜɭ 9) ɬɟɦɩɟ-

ɪɚɬɭɪɚ ɤɨɧɞɟɧɫɚɰɢɢ

331 .

()

. Ⱦɥɹ ɦɚɝɧɨɧɨɜ