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Физика плазмы

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Жданов С.К Цветков И.В - Основы физических процессов в плазме и в плазменных установках (2000)

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§ 29. ɉɨɩɟɪɟɱɧɵɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɜ ɧɟɡɚɦɚɝɧɢɱɟɧɧɨɣ ɩɥɚɡɦɟ

ȼɨɨɪɭɠɢɜɲɢɫɶ ɩɨɥɭɱɟɧɧɵɦɢ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɚɪɚɝɪɚɮɟ ɪɟɡɭɥɶɬɚɬɚɦɢ, ɪɚɫɫɦɨɬɪɢɦ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɜ ɩɥɚɡɦɟ. Ʉɚɤ ɦɵ ɭɠɟ ɡɧɚɟɦ (ɫɦ. § 27)

ɡɚɞɚɱɚ ɫɜɨɞɢɬɫɹ ɤ ɪɟɲɟɧɢɸ ɞɢɫɩɟɪɫɢɨɧɧɵɯ ɭɪɚɜɧɟɧɢɣ

εl = 0 ,

ɢɥɢ

εtr = N 2

ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɤɚɤɨɣ ɤɨɧɤɪɟɬɧɨ ɬɢɩ ɜɨɥɧɵ ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ: ɱɢɫɬɨ ɩɪɨɞɨɥɶɧɵɟ ɢɥɢ ɱɢɫɬɨ ɩɨɩɟɪɟɱɧɵɟ ɜɨɥɧɵ. ɉɨɫɥɟɞɧɢɣ ɫɥɭɱɚɣ ɨɫɨɛɟɧɧɨ ɩɪɨɫɬ. ɋ ɧɟɝɨ ɢ ɧɚɱɧɟɦ ɨɛɫɭɠɞɟɧɢɟ ɩɥɚɡɦɟɧɧɵɯ ɜɨɥɧ.

ɉɨɫɤɨɥɶɤɭ ɞɥɹ ɩɨɩɟɪɟɱɧɵɯ ɜɨɥɧ ɞɢɷɥɟɤɬɪɢɱɟɫɤɭɸ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɩɥɚɡɦɵ ɨɩɪɟɞɟɥɹɟɬ ɫɨɨɬɧɨɲɟɧɢɟ (4.33), ɬɨ ɞɢɫɩɟɪɫɢɨɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɛɭɞɟɬ ɬɚɤɢɦ:

1 −	ω p2	= N 2	≡	k 2c	2	.
	ω 2			ω 2

Ɋɟɲɟɧɢɟɦ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɨɱɟɜɢɞɧɨ, ɹɜɥɹɟɬɫɹ

ω = ω p2 + c2 k 2 .

Ɋɢɫ. 4.1. Ɂɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ. ɉɭɧɤɬɢɪ - ω = kc

(4.36)

ɗɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɢ ɨɩɪɟɞɟɥɹɟɬ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ (ɪɢɫ. 4.1). Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ,

ɱɬɨ ɞɥɹ ɤɨɪɨɬɤɢɯ ɜɨɥɧ, ɤɨɝɞɚ k→∞, ɩɨɥɭɱɚɟɦ ω→kc, ɬɚɤ ɱɬɨ ɜ ɷɬɨɦ ɩɪɟɞɟɥɟ ɜɨɥɧɚ (4.36) ɫɬɚɧɨɜɢɬɫɹ ɨɛɵɱɧɨɣ

ɫɜɟɬɨɜɨɣ ɜɨɥɧɨɣ. Ⱦɥɹ ɞɥɢɧɧɵɯ ɜɨɥɧ, ɤɨɝɞɚ k→0, ɩɪɢɛɥɢɠɟɧɧɨ

§c2 k 2 ·

ω≈ ω p ¨¨©1 + 2ω p2 ¸¸¹ .

Ɂɚɦɟɬɢɦ ɬɚɤɠɟ, ɱɬɨ ɞɥɹ ɮɚɡɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɨɥɧɵ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ (4.36) ɞɚɟɬ ɡɧɚɱɟɧɢɟ

ω 2

v = c 1 + 2 0 2 > c , ɮ k c

ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ. ɉɨɷɬɨɦɭ ɞɥɹ ɬɚɤɢɯ ɜɨɥɧ ɧɟɫɭɳɟɫɬɜɟɧɧɵ ɪɟɡɨɧɚɧɫɧɵɟ ɷɮɮɟɤɬɵ. ɉɪɨɫɬɨ ɩɨɬɨɦɭ, ɱɬɨ ɢɯ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɡɚɜɟɞɨɦɨ ɩɪɟɜɵɲɚɟɬ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɩɥɚɡɦɵ, ɜɫɟɝɞɚ ɦɟɧɶɲɢɯ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ. ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɝɪɭɩɩɨɜɚɹ ɫɤɨɪɨɫɬɶ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ, ɨɬɜɟɱɚɸɳɚɹ ɡɚ ɩɟɪɟɧɨɫ ɜɨɥɧɨɜɨɣ ɷɧɟɪɝɢɢ,

vɝɪ	=	∂ω	=	c	< c ,
		∂k	1	ω02
			1	+ k 2c2

ɨɤɚɡɵɜɚɟɬɫɹ, ɤɚɤ ɷɬɨ ɢ ɞɨɥɠɧɨ ɛɵɬɶ, ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɜ ɜɚɤɭɭɦɟ.

§ 30. əɜɥɟɧɢɟ ɨɬɫɟɱɤɢ ɧɢɡɤɨɱɚɫɬɨɬɧɨɣ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ

Ʉɚɤ ɦɵ ɜɢɞɢɦ ɢɡ ɮɨɪɦɭɥɵ (4.36), ɱɚɫɬɨɬɚ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ ɜɫɟɝɞɚ ɛɨɥɶɲɟ ɩɥɚɡɦɟɧɧɨɣ ɱɚɫɬɨɬɵ, ɩɨɷɬɨɦɭ ɩɨɩɟɪɟɱɧɵɟ ɜɨɥɧɵ, ɱɚɫɬɨɬɚ ɤɨɬɨɪɵɯ ɦɟɧɶɲɟ ɩɥɚɡɦɟɧɧɨɣ ɱɚɫɬɨɬɵ, ɧɟ ɦɨɝɭɬ ɜ ɧɟɣ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɚɞɚɸɳɚɹ ɢɡ ɜɚɤɭɭɦɚ ɧɚ ɝɪɚɧɢɰɭ ɩɥɚɡɦɵ ɩɨɩɟɪɟɱɧɚɹ ɜɨɥɧɚ ɫ ɦɚɥɨɣ ɱɚɫɬɨɬɨɣ ɞɨɥɠɧɚ ɨɬɪɚɠɚɬɶɫɹ. ɂɦɟɟɬ ɦɟɫɬɨ, ɤɚɤ ɝɨɜɨɪɹɬ ɹɜɥɟɧɢɟ ɨɬɫɟɱɤɢ ɜɨɥɧɵ (ɜ ɚɧɝɥɢɣɫɤɨɣ ɥɢɬɟɪɚɬɭɪɟ - cut off). Ʉɪɢɬɢɱɟɫɤɚɹ ɱɚɫɬɨɬɚ − ɱɚɫɬɨɬɚ ɨɬɫɟɱɤɢ,

ωɤ ɪ = ω p ≡	4πn0e	2	§	1	+	·	,	(4.37)
ωɤ ɪ = ω p ≡	4πn0e		¨	1	+	Zme ¸	,	(4.37)
	me		©			mi ¹

ɡɚɜɢɫɢɬ ɨɬ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɥɚɡɦɵ. Ɍɚɤ ɱɬɨ, ɢɡɦɟɪɹɹ ɤɪɢɬɢɱɟɫɤɭɸ ɱɚɫɬɨɬɭ, ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɨɧɰɟɧɬɪɚɰɢɸ ɩɥɚɡɦɵ. ɗɬɨ ɨɞɢɧ ɢɡ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɦɟɬɨɞɨɜ ɞɢɚɝɧɨɫɬɢɤɢ ɩɥɚɡɦɵ.

ɗɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɧɢɡɤɨɱɚɫɬɨɬɧɨɣ ɜɨɥɧɵ ɱɚɫɬɢɱɧɨ ɜɫɟ ɠɟ ɩɪɨɧɢɤɚɟɬ ɜ ɩɥɚɡɦɭ, ɧɨ ɟɝɨ ɚɦɩɥɢɬɭɞɚ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɭɦɟɧɶɲɚɟɬɫɹ ɜɝɥɭɛɶ ɩɥɚɡɦɵ. Ƚɥɭɛɢɧɚ ɩɪɨɧɢɤɧɨɜɟɧɢɹ ɜ ɩɥɚɡɦɭ ɩɨɥɹ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ ɫ ɧɢɡɤɨɣ ɱɚɫɬɨɬɨɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɳɢɧɨɣ ɜɚɤɭɭɦɧɨɝɨ ɫɤɢɧ-ɫɥɨɹ, ɤɨɬɨɪɚɹ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɥɚɡɦɟɧɧɨɣ ɱɚɫɬɨɬɟ:

δ	=	c	.	(4.38)

ɜɚɤ		ω p

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɝɥɭɛɢɧɚ ɩɪɨɧɢɤɧɨɜɟɧɢɹ ɜɨɥɧɵ ɜ ɩɥɚɡɦɭ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɧɟɪɰɢɟɣ ɟɟ ɱɚɫɬɢɰ, ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ - ɷɥɟɤɬɪɨɧɨɜ. ȼ ɩɪɟɧɟɛɪɟɠɟɧɢɢ ɢɧɟɪɰɢɟɣ ɝɥɭɛɢɧɚ ɩɪɨɧɢɤɧɨɜɟɧɢɹ ɩɨɥɹ ɛɵɥɚ ɛɵ ɧɭɥɟɜɨɣ.

ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦ ɫɤɚɡɚɧɧɨɟ ɩɪɨɫɬɵɦ ɩɪɢɦɟɪɨɦ. ɉɭɫɬɶ ɢɡ ɜɚɤɭɭɦɚ ɧɚ ɩɥɨɫɤɭɸ ɝɪɚɧɢɰɭ ɩɥɚɡɦɵ ɩɚɞɚɟɬ ɧɢɡɤɨɱɚɫɬɨɬɧɚɹ ɜɨɥɧɚ, ɫɥɟɜɚ ɧɚɩɪɚɜɨ, ɤɚɤ ɷɬɨ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 4.2. ɋɥɟɜɚ ɢ ɫɩɪɚɜɚ ɨɬ ɝɪɚɧɢɰɵ ɪɚɡɞɟɥɚ ɡɚɤɨɧɵ ɞɢɫɩɟɪɫɢɢ ɜɨɥɧɵ ɪɚɡɧɵɟ:

						2	, x	> 0,
	ω 2 − k2c2				ω		, x	> 0,
	ω 2 − k2c2		= ®			p
				¯ 0, x < 0.
	ɗɬɢ ɫɨɨɬɧɨɲɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦ ɜɢɞɟ.
	ɉɭɫɬɶ ɱɚɫɬɨɬɚ ɜɨɥɧɵ ɮɢɤɫɢɪɨɜɚɧɚ, ɡɚɦɟɧɢɜ
	k → −i∂x ,
	ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ
		ω p2		− ω		2	f ,	x > 0,
		°
	∂ 2 f	°		c2
	∂ 2 f	= ®°		c2				(4.39)
	x	°	−	ω 2		f ,		x < 0,
		°	−	c	2	f ,		x < 0,
		¯		c
	ɝɞɟ ɮɭɧɤɰɢɹ f ɡɚɞɚɟɬ ɩɨɥɟ ɜɨɥɧɵ: ɧɚɩɪɢɦɟɪ, ɷɬɨ ɦɨɠɟɬ ɛɵɬɶ
Ɋɢɫ. 4.2. Ɉɬɫɟɱɤɚ ɧɢɡɤɨɱɚɫɬɨɬɧɨɣ	ɤɨɦɩɨɧɟɧɬɚ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ. ɇɚ ɝɪɚɧɢɰɟ
Ɋɢɫ. 4.2. Ɉɬɫɟɱɤɚ ɧɢɡɤɨɱɚɫɬɨɬɧɨɣ	ɪɚɡɞɟɥɚ ɩɨɬɪɟɛɭɟɦ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ:
ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ ɧɚ ɝɪɚɧɢɰɟ ɩɥɚɡɦɵ	ɪɚɡɞɟɥɚ ɩɨɬɪɟɛɭɟɦ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ:
	∂x f \|x=+0 = ∂x f \|x=−0 ,

f \|x=+0 =	f \|x=−0 .	(4.40)

ɇɟɬɪɭɞɧɨ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (4.39), (4.40), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɷɬɢɦ ɭɫɥɨɜɢɹɦ. ɉɪɟɞɥɚɝɚɟɦ ɱɢɬɚɬɟɥɸ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɬɚɤɨɜɵɦ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ ɜ ɨɛɥɚɫɬɢ ɜɚɤɭɭɦɚ ɩɨɥɟ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɩɨɥɹ ɩɚɞɚɸɳɟɣ ɢ ɨɬɪɚɠɟɧɧɨɣ ɜɨɥɧɵ, ɚ ɜ ɨɛɥɚɫɬɢ ɩɥɚɡɦɵ ɜɨɥɧɨɜɨɟ ɩɨɥɟ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɡɚɬɭɯɚɟɬ:

°	§	ω	·	§	−
f = f0 ®	exp¨i	c	x¸	+ α exp¨	−
f = f0 ®	©	c	¹	©
°			β exp(− κx),
¯			β exp(− κx),

ɝɞɟ ɨɛɨɡɧɚɱɟɧɨ

κ =	ω p2	− ω 2	> 0
		c2

i	ω	·	, x < 0,
	c	x¸		(4.41)
		¹

x > 0,

- ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɡɚɬɭɯɚɧɢɹ ɩɨɥɹ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ, f0 − ɚɦɩɥɢɬɭɞɚ ɩɚɞɚɸɳɟɣ ɧɚ ɝɪɚɧɢɰɭ ɪɚɡɞɟɥɚ ɜɨɥɧɵ. Ⱥɦɩɥɢɬɭɞɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ α ɞɥɹ ɨɬɪɚɠɟɧɧɨɣ ɜɨɥɧɵ ɢ β ɞɥɹ ɩɨɥɹ ɜ ɩɥɚɡɦɟ, ɤɚɤ ɷɬɨ ɜɵɬɟɤɚɟɬ ɢɡ ɭɫɥɨɜɢɣ ɧɟɩɪɟɪɵɜɧɨɣ ɫɲɢɜɤɢ (4.40), ɨɤɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ:

α =	ω − iκc	,	β =	2ω
	ω + iκc				.
				ω + iκc

Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ, ɱɬɨ ɱɢɫɥɢɬɟɥɶ ɢ ɡɧɚɦɟɧɚɬɟɥɶ ɩɟɪɜɨɣ ɮɨɪɦɭɥɵ ɹɜɥɹɸɬɫɹ ɤɨɦɩɥɟɤɫɧɨ-ɫɨɩɪɹɠɟɧɧɵɦɢ.

ɉɨɷɬɨɦɭ ɩɨɥɭɱɚɟɦ

|α|= 1 ,

ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɚɦɩɥɢɬɭɞɵ ɩɚɞɚɸɳɟɣ ɜɨɥɧɵ ɢ ɨɬɪɚɠɟɧɧɨɣ ɫɨɜɩɚɞɚɸɬ. ɗɬɨ ɢ ɨɡɧɚɱɚɟɬ ɧɚɥɢɱɢɟ ɩɨɥɧɨɝɨ ɨɬɪɚɠɟɧɢɹ ɩɚɞɚɸɳɟɣ ɧɚ ɩɥɚɡɦɭ ɜɨɥɧɵ.

ȼ ɩɪɟɞɟɥɟ ɫɨɜɫɟɦ ɧɢɡɤɢɯ ɱɚɫɬɨɬ, ɤɨɝɞɚ ω → 0 , ɩɨɥɭɱɚɟɦ ɩɪɢɛɥɢɠɟɧɧɨ

α ≈ −1 − 2i	ω	→ −1, β = −2i	ω	→ 0, κ ≈ δ −1 ,

	ω p		ω p
				ɜɚɤ

ɢ ɞɥɢɧɚ ɡɚɬɭɯɚɧɢɹ ɩɨɥɹ ɜ ɩɥɚɡɦɟ ɫɨɜɩɚɞɚɟɬ ɫ ɞɥɢɧɨɣ ɜɚɤɭɭɦɧɨɝɨ ɫɤɢɧ-ɫɥɨɹ.

§ 31. Ʌɟɧɝɦɸɪɨɜɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɢ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ. ɉɥɚɡɦɨɧɵ

Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɯ ɩɪɨɞɨɥɶɧɵɯ ɩɥɚɡɦɟɧɧɵɯ ɜɨɥɧ ɫ ɱɚɫɬɨɬɨɣ ɜ ɨɛɥɚɫɬɢ ɥɟɧɝɦɸɪɨɜɫɤɨɣ ɱɚɫɬɨɬɵ. Ɉɧɢ ɢɡɜɟɫɬɧɵ ɤɚɤ ɥɟɧɝɦɸɪɨɜɫɤɢɟ ɜɨɥɧɵ ɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɜɚɠɧɟɣɲɢɣ ɬɢɩ ɜɨɡɦɭɳɟɧɢɣ, ɫɩɨɫɨɛɧɵɯ ɫɭɳɟɫɬɜɨɜɚɬɶ ɢ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɜ ɩɥɚɡɦɟ.

Ɂɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɩɪɨɞɨɥɶɧɵɯ ɜɨɥɧ ɨɩɪɟɞɟɥɹɟɬ, ɤɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɭɪɚɜɧɟɧɢɟ

εl = 0 ,

ɜ ɤɨɬɨɪɨɟ ɫɥɟɞɭɟɬ ɩɨɞɫɬɚɜɢɬɶ ɩɪɨɞɨɥɶɧɭɸ ɤɨɦɩɨɧɟɧɬɭ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɢ. ȿɫɥɢ ɩɥɚɡɦɭ ɫɱɢɬɚɬɶ ɯɨɥɨɞɧɨɣ, ɬɨ ɞɢɷɥɟɤɬɪɢɱɟɫɤɭɸ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɹɬɶ ɩɨ ɮɨɪɦɭɥɟ (4.19), ɢ ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɭɪɚɜɧɟɧɢɸ

1 −	ω p2	= 0 .
	ω 2
Ɉɧɨ ɢɦɟɟɬ ɞɜɚ ɪɟɲɟɧɢɹ, ɨɬɥɢɱɚɸɳɢɟɫɹ ɡɧɚɤɨɦ. ɉɨɥɨɠɢɬɟɥɶɧɵɣ ɤɨɪɟɧɶ ɪɚɜɟɧ
ω = ω p		(4.42)

Ʉɚɤ ɦɵ ɜɢɞɢɦ, ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ ɱɚɫɬɨɬɚ ɜɨɥɧɵ ɫɨɜɩɚɞɚɟɬ ɫ ɥɟɧɝɦɸɪɨɜɫɤɨɣ ɱɚɫɬɨɬɨɣ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɟɥɢɱɢɧɵ ɜɨɥɧɨɜɨɝɨ ɱɢɫɥɚ. Ɏɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɬɚɤɢɯ ɜɨɥɧ

vɮ ≡	ω	=	ω p		(4.43)
	k			k

ɭɦɟɧɶɲɚɟɬɫɹ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɜɨɥɧɨɜɨɝɨ ɱɢɫɥɚ, ɚ ɝɪɭɩɩɨɜɚɹ ɫɤɨɪɨɫɬɶ ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ ɧɭɥɸ:

&		∂ω		∂ω p
vɝɪ	≡	&	=	&	≡ 0 .	(4.44)
		∂k		∂k

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɯɨɥɨɞɧɨɣ ɩɥɚɡɦɟ ɥɟɧɝɦɸɪɨɜɫɤɢɟ ɜɨɥɧɵ ɧɟ ɦɨɝɭɬ ɩɟɪɟɧɨɫɢɬɶ ɷɧɟɪɝɢɸ: ɮɚɤɬɢɱɟɫɤɢ ɷɬɨ ɨɛɵɱɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɥɨɬɧɨɫɬɢ ɡɚɪɹɞɚ, ɜɨɡɧɢɤɚɸɳɢɟ ɜɫɥɟɞɫɬɜɢɟ ɧɚɪɭɲɟɧɢɹ ɤɜɚɡɢɧɟɣɬɪɚɥɶɧɨɫɬɢ ɩɥɚɡɦɵ. ȿɫɥɢ ɠɟ ɦɵ ɭɱɬɟɦ ɬɟɩɟɪɶ ɬɟɩɥɨɜɨɟ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰ ɩɥɚɡɦɵ, ɬɨ ɫɢɬɭɚɰɢɹ ɢɡɦɟɧɢɬɫɹ ɤɚɪɞɢɧɚɥɶɧɨ. Ⱦɢɷɥɟɤɬɪɢɱɟɫɤɭɸ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬ ɬɟɩɟɪɶ ɮɨɪɦɭɥɚ (4.32) ɢ ɞɢɫɩɟɪɫɢɨɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɩɪɨɞɨɥɶɧɵɯ ɜɨɥɧ ɫɬɚɧɨɜɢɬɫɹ ɬɚɤɢɦ:

εl = 1 − ¦

ω p2α

= 0 ,

− k

ɢɥɢ

α =e,iω

csα

ω pe2

ω pi2

1 −

−

= 0,

= γ

e,i

(4.45)

ω 2

− k 2c2

ω 2

− k 2c2

se,i

e,i m

e,i

ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɧɟɫɥɨɠɧɨ ɪɟɲɢɬɶ ɜ ɨɛɳɟɦ ɜɢɞɟ. ɇɨ ɜ ɢɧɬɟɪɟɫɭɸɳɟɣ ɧɚɫ ɫɟɣɱɚɫ ɜɵɫɨɤɨɱɚɫɬɨɬɧɨɣ ɨɛɥɚɫɬɢ ɫɥɟɞɭɟɬ ɭɱɟɫɬɶ, ɱɬɨ ɢɨɧɵ ɩɥɚɡɦɵ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɧɟɩɨɞɜɢɠɧɵɦɢ, ɚ ɩɨɬɨɦɭ ɢɯ ɜɤɥɚɞ ɜ ɞɢɷɥɟɤɬɪɢɱɟɫɤɭɸ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɛɭɞɟɬ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɵɦ. Ɏɨɪɦɚɥɶɧɨ ɷɬɨ ɨɬɜɟɱɚɟɬ ɩɪɟɞɟɥɭ mi→∞, ɢ ɭɪɚɜɧɟɧɢɟ (4.45) ɭɩɪɨɳɚɟɬɫɹ:

		ω pe2						T
1 −				= 0,	c2	= γ		e	.
	ω 2	− k	2c2
					se		e m
			se					e

Ɍɟɩɟɪɶ ɟɝɨ ɭɠɟ ɧɟ ɫɥɨɠɧɨ ɪɟɲɢɬɶ, ɢ ɦɵ, ɜɧɨɜɶ ɜɵɛɢɪɚɹ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɤɨɪɟɧɶ, ɩɨɥɭɱɚɟɦ:

ω = ω pe2 + k2cse2 .

(4.46)

ɗɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɢ ɨɩɪɟɞɟɥɹɟɬ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɥɟɧɝɦɸɪɨɜɫɤɨɣ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ ɫ ɤɨɧɟɱɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ.

Ʌɸɛɨɩɵɬɧɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɷɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɩɨ ɜɢɞɭ ɨɤɚɡɵɜɚɟɬɫɹ ɜɩɨɥɧɟ ɚɧɚɥɨɝɢɱɧɵɦ ɢɡɜɟɫɬɧɨɣ ɮɨɪɦɭɥɟ, ɨɩɪɟɞɟɥɹɸɳɟɣ ɫɜɹɡɶ ɷɧɟɪɝɢɢ ɢ ɢɦɩɭɥɶɫɚ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫɬɢɰɵ:

ε = (mc2 )2 + p2c2 .

ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɨ ɡɚɤɨɧɟ ɞɢɫɩɟɪɫɢɢ (4.46) ɝɨɜɨɪɹɬ ɤɚɤ ɨ «ɱɚɫɬɢɰɟ-ɩɨɞɨɛɧɨɦ», ɚ ɥɟɧɝɦɸɪɨɜɫɤɢɟ ɜɨɥɧɵ ɜ ɷɬɨɦ ɩɥɚɧɟ ɹɜɥɹɸɬɫɹ «ɤɜɚɡɢɱɚɫɬɢɰɚɦɢ», ɤɨɬɨɪɵɟ ɩɪɢɧɹɬɨ

ɧɚɡɵɜɚɬɶ ɩɥɚɡɦɨɧɚɦɢ.

ɉɨɥɟɡɧɨ ɨɬɦɟɬɢɬɶ ɬɚɤɠɟ, ɱɬɨ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ (4.46) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:

ω =ω	pe	1 +γ	e	k 2r2 .	(4.47)
	pe		e	De

ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɩɨɞ ɤɨɪɧɟɦ ɛɭɞɟɬ ɛɨɥɶɲɟ ɢɥɢ ɩɨɪɹɞɤɚ ɟɞɢɧɢɰɵ, ɤɨɝɞɚ ɞɥɢɧɚ ɜɨɥɧɵ ɛɭɞɟɬ ɦɟɧɶɲɟ ɞɟɛɚɟɜɫɤɨɝɨ ɪɚɞɢɭɫɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɥɟɧɝɦɸɪɨɜɫɤɚɹ ɜɨɥɧɚ ɫɢɥɶɧɨ ɩɨɝɥɨɳɚɟɬɫɹ ɡɚ ɫɱɟɬ ɦɟɯɚɧɢɡɦɚ ɛɟɫɫɬɨɥɤɧɨɜɢɬɟɥɶɧɨɝɨ ɩɨɝɥɨɳɟɧɢɹ Ʌɚɧɞɚɭ, ɬɚɤ ɤɚɤ ɨɤɚɡɵɜɚɟɬɫɹ ɪɟɡɨɧɚɧɫɧɨɣ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɷɥɟɤɬɪɨɧɚɦ ɩɥɚɡɦɵ,

vɮ ~ vTe .

ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɥɟɧɝɦɸɪɨɜɫɤɢɟ ɜɨɥɧɵ ɦɨɝɭɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɜ ɩɥɚɡɦɟ ɛɟɡ ɫɭɳɟɫɬɜɟɧɧɨɝɨ

ɩɨɝɥɨɳɟɧɢɹ ɥɢɲɶ ɜ ɨɛɪɚɬɧɨɦ ɩɪɟɞɟɥɟ, ɤɨɝɞɚ ɢɯ ɞɥɢɧɚ ɜɨɥɧɵ ɦɟɧɶɲɟ ɞɟɛɚɟɜɫɤɨɝɨ ɪɚɞɢɭɫɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜ (4.47) ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɩɨɞ ɤɨɪɧɟɦ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɦɚɥɵɦ ɢ

ɪɚɡɥɨɠɢɬɶ ɩɨ ɷɬɨɣ ɦɚɥɨɫɬɢ:


ω ≈ω	§	1	+	γ	e k 2r2		·	, k 2r2	<<1 .
ω ≈ω	¨	1	+		e k 2r2		¸	, k 2r2	<<1 .
	pe ©			2		De ¹		De

Ⱥɧɚɥɨɝɢɹ ɫ ɷɧɟɪɝɢɟɣ ɱɚɫɬɢɰɵ ɨɩɹɬɶ ɨɫɬɚɟɬɫɹ ɜ ɫɢɥɟ, ɧɨ ɬɟɩɟɪɶ ɜ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɦ ɩɪɟɞɟɥɟ, ɤɨɝɞɚ ɷɧɟɪɝɢɹ ɫɜɹɡɚɧɚ ɫ ɢɦɩɭɥɶɫɨɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

ε ≈ mc2 + p2 . 2m

ȼ ɨɛɥɚɫɬɢ ɱɚɫɬɨɬ ɥɟɧɝɦɸɪɨɜɫɤɢɯ ɜɨɥɧ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ, ɫɥɟɞɫɬɜɢɟɦ ɤɨɬɨɪɨɝɨ ɮɚɤɬɢɱɟɫɤɢ ɹɜɥɹɟɬɫɹ ɡɚɤɨɧ (4.47), ɛɭɞɟɬ ɚɞɟɤɜɚɬɧɵɦ ɩɪɢ ɜɵɛɨɪɟ

γ e = 3 .

ɉɨɞɫɬɚɜɢɜ ɷɬɨ ɡɧɚɱɟɧɢɟ ɜ (4.47), ɩɨɥɭɱɢɦ ɨɤɨɧɱɚɬɟɥɶɧɨ
ω ≈ω	§		+	3	·			<<1 .
	¨	1			k 2r2 ¸	,	k 2r2		(4.48)
				2
	pe ©				De ¹		De

Ɋɢɫ.4.3 Ɂɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɥɟɧɝɦɸɪɨɜɫɤɨɣ ɜɨɥɧɵ

ɂɦɟɧɧɨ ɨɛ ɷɬɨɦ ɫɨɨɬɧɨɲɟɧɢɢ ɢ ɝɨɜɨɪɹɬ ɨɛɵɱɧɨ ɤɚɤ ɨ ɡɚɤɨɧɟ ɞɢɫɩɟɪɫɢɢ ɥɟɧɝɦɸɪɨɜɫɤɢɯ ɜɨɥɧ ɜ ɩɥɚɡɦɟ. ɋɬɪɨɝɨ ɝɨɜɨɪɹ, ɨɧ ɫɩɪɚɜɟɞɥɢɜ, ɤɚɤ ɦɵ ɜɢɞɟɥɢ, ɥɢɲɶ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɫɢɥɶɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ k2rDe2 <<1. Ɉɞɧɚɤɨ ɤɚɱɟɫɬɜɟɧɧɨ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ (4.48) ɨɫɬɚɟɬɫɹ ɜ ɫɢɥɟ ɢ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɛɨɥɟɟ ɦɹɝɤɨɝɨ ɭɫɥɨɜɢɹ, ɤɨɝɞɚ ɞɥɢɧɚ ɜɨɥɧɵ ɫɨɫɬɚɜɥɹɟɬ ɧɟɫɤɨɥɶɤɨ ɞɟɛɚɟɜɫɤɢɯ ɪɚɞɢɭɫɨɜ. ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɫɤɨɛɤɚɯ ɜ ɮɨɪɦɭɥɟ (4.48) ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ ɬɟɩɥɨɜɨɣ ɩɨɩɪɚɜɤɨɣ. ɍɱɟɬ ɷɬɨɣ ɩɨɩɪɚɜɤɢ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ,

ɱɬɨ ɝɪɭɩɩɨɜɚɹ ɫɤɨɪɨɫɬɶ ɥɟɧɝɦɸɪɨɜɫɤɨɣ ɜɨɥɧɵ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɫɥɭɱɚɹ ɯɨɥɨɞɧɨɣ ɩɥɚɡɦɵ, ɫɬɚɧɨɜɢɬɫɹ ɧɟɧɭɥɟɜɨɣ (ɫɦ. ɪɢɫ.4.3):

v&ɝɪ	=	∂ω&	&	= 3vTe krDe ,	(4.49)
			= 3ω pe krDe2 , vɝɪ
		∂k

ɮɚɡɨɜɚɹ ɠɟ ɫɤɨɪɨɫɬɶ ɩɪɢɛɥɢɠɟɧɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ

vɮ	=	ω	≈	ω pe	=	v	Te .	(4.50)
		k		k
						3krDe

ɉɪɢ ɭɱɟɬɟ ɬɟɩɥɨɜɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɥɟɧɝɦɸɪɨɜɫɤɢɟ ɜɨɥɧɵ ɩɨɥɭɱɚɸɬ ɜɨɡɦɨɠɧɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɜ ɩɥɚɡɦɟ, ɩɟɪɟɧɨɫɹ ɷɧɟɪɝɢɸ.

§ 32. ɂɨɧɧɵɟ ɥɟɧɝɦɸɪɨɜɫɤɢɟ ɜɨɥɧɵ. ɂɨɧɧɨ-ɡɜɭɤɨɜɵɟ ɜɨɥɧɵ ɜ ɩɥɚɡɦɟ

ȼɟɪɧɟɦɫɹ ɜɧɨɜɶ ɤ ɞɢɫɩɟɪɫɢɨɧɧɨɦɭ ɭɪɚɜɧɟɧɢɸ (4.45). Ⱦɥɹ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜɵɲɟ ɥɟɧɝɦɸɪɨɜɫɤɢɯ ɜɨɥɧ ɝɪɭɩɩɨɜɚɹ ɢ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɧɟɪɚɜɟɧɫɬɜɭ

vɝɪ < vTe < vɮ .

Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɜɨɡɦɨɠɧɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜ ɩɥɚɡɦɟ ɜɨɥɧ, ɮɚɡɨɜɚɹ

ɫɤɨɪɨɫɬɶ ɤɨɬɨɪɵɯ ɡɧɚɱɢɬɟɥɶɧɨ ɦɟɧɶɲɟ ɬɟɩɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɷɥɟɤɬɪɨɧɨɜ: vɮ << vTe .

ȿɫɥɢ ɷɬɨ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɟɧɨ, ɬɨ ɜ ɭɪɚɜɧɟɧɢɢ (4.45) ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜɬɨɪɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɦɨɠɧɨ ɨɩɭɫɬɢɬɶ ω 2 ɢ ɬɨɝɞɚ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɪɢɜɨɞɢɬɫɹ ɤ ɜɢɞɭ

	ω pe2				ω pi2					T
1 +			−			= 0,	c2	= γ		e,i	.
	k 2c	2		ω 2	− k 2c2
							se,i		e,i m
		se			si					e,i

Ɍɟɩɟɪɶ ɭɠɟ ɧɟ ɫɥɨɠɧɨ ɧɚɣɬɢ ɢɧɬɟɪɟɫɭɸɳɟɟ ɧɚɫ ɪɟɲɟɧɢɟ:

ω 2 = k 2c2	+	ω pi2		.
			ω pe2
si		1 +
			k 2cse2

ɍɱɬɟɦ ɬɟɩɟɪɶ, ɱɬɨ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɥɢɱɢɧ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɧɨɲɟɧɢɟ:

≡

≡ γ

r2 .

ω 2

4πe2 n

ei e

Ɍɨɝɞɚ ɩɨɥɭɱɟɧɧɵɣ ɧɚɦɢ ɪɟɡɭɥɶɬɚɬ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ

ω 2

= k 2csi2

ω pi2

(4.51)

1 +

k 2r2

Ⱦɥɹ

ɤɨɪɨɬɤɢɯ

ɜɨɥɧ, ɤɨɝɞɚ ɞɥɢɧɚ ɜɨɥɧɵ ɦɟɧɶɲɟ

ɷɥɟɤɬɪɨɧɧɨɝɨ

ɞɟɛɚɟɜɫɤɨɝɨ ɪɚɞɢɭɫɚ, ɡɧɚɦɟɧɚɬɟɥɶ ɜɨ ɜɬɨɪɨɦ ɫɥɚɝɚɟɦɨɦ ɩɪɢɦɟɪɧɨ ɪɚɜɟɧ ɟɞɢɧɢɰɟ, ɢ ɦɵ ɩɨɥɭɱɚɟɦ:

ω 2 k 2c2

+ ω 2

≡ ω 2

γ iTi

k 2r2

<< 1 .

(4.52)

ɑɚɫɬɨɬɚ ɷɬɢɯ ɜɨɥɧ ɨɤɚɡɵɜɚɟɬɫɹ ɩɨɪɹɞɤɚ ɢɨɧɧɨɣ ɥɟɧɝɦɸɪɨɜɫɤɨɣ ɱɚɫɬɨɬɵ. ɉɨ ɚɧɚɥɨɝɢɢ ɫ (4.46), ɷɬɢ ɜɨɥɧɵ ɧɚɡɵɜɚɸɬ ɢɨɧɧɵɦɢ ɥɟɧɝɦɸɪɨɜɫɤɢɦɢ ɜɨɥɧɚɦɢ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɟɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɢɨɧɨɜ ɧɟ ɦɚɥɚ, ɨɧɢ ɫɢɥɶɧɨ ɡɚɬɭɯɚɸɬ ɜ ɩɥɚɡɦɟ, ɬɚɤ ɤɚɤ ɨɤɚɡɵɜɚɸɬɫɹ ɪɟɡɨɧɚɧɫɧɵɦɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɢɨɧɚɦ.

ȼ ɨɛɪɚɬɧɨɦ ɩɪɟɞɟɥɟ ɞɥɢɧɧɵɯ ɜɨɥɧ, ɞɥɢɧɚ ɜɨɥɧɵ ɤɨɬɨɪɵɯ ɩɪɟɜɵɲɚɟɬ ɷɥɟɤɬɪɨɧɧɵɣ ɞɟɛɚɟɜɫɤɢɣ ɪɚɞɢɭɫ, ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜɬɨɪɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɜ ɮɨɪɦɭɥɟ (4.51) ɝɥɚɜɧɵɦ, ɧɚɩɪɨɬɢɜ, ɹɜɥɹɟɬɫɹ ɜɬɨɪɨɣ ɱɥɟɧ, ɢ ɦɵ ɩɨɥɭɱɚɟɦ:

ω 2

2c2

+ γ

k 2r2

ω 2

≡ k

2c2

k2r2

<< 1 .

(4.53)

ɝɞɟ ɨɛɨɡɧɚɱɟɧɨ

Zγ eTe + γ iTi

≡ c2

+ γ

ω 2

(4.54)

e De

ɋɪɚɜɧɢɜ (4.54) ɫ ɬɨɱɧɵɦ ɪɟɡɭɥɶɬɚɬɨɦ ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ [18], ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɞɢɚɩɚɡɨɧɟ ɱɚɫɬɨɬ ɤɢɧɟɬɢɤɚ ɢ ɝɢɞɪɨɞɢɧɚɦɢɤɚ, ɢɫɩɨɥɶɡɨɜɚɧɧɚɹ ɧɚɦɢ, ɞɚɸɬ ɫɨɜɩɚɞɚɸɳɢɟ ɪɟɡɭɥɶɬɚɬɵ ɩɪɢ ɜɵɛɨɪɟ

γ e = 1, γ i = 3 ,

ɬɚɤ ɱɬɨ (4.54) ɫɥɟɞɭɟɬ ɡɚɩɢɫɵɜɚɬɶ ɜ ɜɢɞɟ:
cs2 =	ZTe + 3Ti	.	(4.55)

	mi
ȼɵɬɟɤɚɸɳɢɣ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ (4.53) ɡɚɤɨɧ
ɞɢɫɩɟɪɫɢɢ
ω = kcs ,			(4.56)
ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɦɭ ɱɚɫɬɨɬɚ ɜɨɥɧɵ ɨɤɚɡɵɜɚɟɬɫɹ
ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɜɨɥɧɨɜɨɦɭ ɱɢɫɥɭ,
ɬɢɩɢɱɟɧ ɞɥɹ ɡɜɭɤɨɜɵɯ ɜɨɥɧ (ɧɚɩɨɦɧɢɦ, ɱɬɨ ɦɵ
ɨɛɫɭɠɞɚɟɦ ɫɟɣɱɚɫ ɩɪɨɞɨɥɶɧɵɟ ɜɨɥɧɵ!).
ɇɚɩɪɢɦɟɪ, ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɡɜɭɤɚ ɜ ɨɛɵɱɧɨɦ

Ɋɢɫ. 4.4. ɂɨɧɧɨ-ɡɜɭɤɨɜɵɟ ɜɨɥɧɵ ɜ ɝɚɡɟ		∂p		γT
ɩɥɚɡɦɟ	ω = kcs , cs =		=		,
	ω = kcs , cs =		=		,
		∂ρ		M

ɝɞɟ Ɍ - ɬɟɦɩɟɪɚɬɭɪɚ, ɚ Ɇ - ɦɚɫɫɚ ɦɨɥɟɤɭɥ ɝɚɡɚ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɜɨɥɧɵ ɫ ɡɚɤɨɧɨɦ ɞɢɫɩɟɪɫɢɢ (4.55), (4.56) ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ ɢɨɧɧɨ-ɡɜɭɤɨɜɵɦɢ ɜɨɥɧɚɦɢ. ɇɚɪɹɞɭ ɫ

ɥɟɧɝɦɸɪɨɜɫɤɢɦɢ ɜɨɥɧɚɦɢ, ɷɬɨ ɜɚɠɧɟɣɲɢɣ ɬɢɩ ɫɩɨɫɨɛɧɵɯ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɜ ɩɥɚɡɦɟ ɜɨɥɧ (ɫɦ. ɪɢɫ.4.4).

Ɉɱɟɜɢɞɧɨ, ɮɚɡɨɜɚɹ ɢ ɝɪɭɩɩɨɜɚɹ ɫɤɨɪɨɫɬɢ ɢɨɧɧɨ-ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɫɨɜɩɚɞɚɸɬ:

vɮ = vɝɪ = cs =	ZTe + 3Ti .	(4.57)
	mi

ȼɟɥɢɱɢɧɚ ɷɬɢɯ ɫɤɨɪɨɫɬɟɣ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɢɬ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪ ɤɨɦɩɨɧɟɧɬ ɩɥɚɡɦɵ. ɉɪɢ ɷɬɨɦ ɟɫɥɢ

Ti ≥ Te ,

ɬɨ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɢɨɧɧɨ-ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɛɭɞɟɬ ɩɨ ɜɟɥɢɱɢɧɟ ɩɨɪɹɞɤɚ ɬɟɩɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɢɨɧɨɜ. Ɍɚɤɢɟ ɜɨɥɧɵ ɞɨɥɠɧɵ ɫɢɥɶɧɨ ɩɨɝɥɨɳɚɬɶɫɹ ɜ ɩɥɚɡɦɟ, ɬɚɤ ɤɚɤ ɨɧɢ ɫɬɚɧɨɜɹɬɫɹ ɪɟɡɨɧɚɧɫɧɵɦɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɢɨɧɚɦ. ɉɨɷɬɨɦɭ ɢɨɧɧɨ-ɡɜɭɤɨɜɚɹ ɜɨɥɧɚ ɦɨɠɟɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɬɨɥɶɤɨ ɜ ɫɢɥɶɧɨ ɧɟɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɥɚɡɦɟ, ɤɨɝɞɚ ɷɥɟɤɬɪɨɧɧɚɹ ɤɨɦɩɨɧɟɧɬɚ ɫɢɥɶɧɨ «ɩɟɪɟɝɪɟɬɚ» ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɢɨɧɧɨɣ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɤɥɚɞ ɬɟɦɩɟɪɚɬɭɪɵ ɢɨɧɨɜ ɜ ɮɨɪɦɭɥɟ (4.57) ɹɜɥɹɟɬɫɹ ɦɚɥɵɦ, ɢ ɩɨɷɬɨɦɭ ɫɤɨɪɨɫɬɶ ɢɨɧɧɨ-ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɨɣ ɷɥɟɤɬɪɨɧɨɜ:

c	s	ZTe , T	>> T .	(4.58)
	s	e	i
		mi

ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɩɨɫɤɨɥɶɤɭ ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ ɜ ɤɚɱɟɫɬɜɟ ɦɟɪɵ ɬɟɩɥɨɜɨɣ ɷɧɟɪɝɢɢ ɜɯɨɞɢɬ ɷɥɟɤɬɪɨɧɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ, ɚ ɜ ɤɚɱɟɫɬɜɟ ɢɧɟɪɰɢɨɧɧɨɝɨ ɮɚɤɬɨɪɚ ɜɯɨɞɢɬ ɦɚɫɫɚ ɢɨɧɨɜ, ɩɪɢɧɹɬɨ ɷɬɢ ɜɨɥɧɵ ɧɚɡɵɜɚɬɶ «ɢɨɧɧɵɦ ɡɜɭɤɨɦ ɫ ɷɥɟɤɬɪɨɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ». ɋɨɛɢɪɚɹ ɜɦɟɫɬɟ ɜɫɟ ɭɤɚɡɚɧɧɵɟ ɜɵɲɟ ɧɟɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɢɦ ɨɛɥɚɫɬɶ ɮɚɡɨɜɵɯ ɫɤɨɪɨɫɬɟɣ, ɜ ɤɨɬɨɪɨɣ ɜɨɡɦɨɠɧɨ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɢɨɧɧɨ-ɡɜɭɤɨɜɵɯ ɜɨɥɧ:

vTi =	3Ti << vɮ ~ cs	ZTe << vTe =	3Te .	(4.59)
	mi	mi	me

ȼɡɹɜ ɬɟɩɟɪɶ ɷɬɢ ɧɟɪɚɜɟɧɫɬɜɚ ɜ ɤɚɱɟɫɬɜɟ ɨɬɩɪɚɜɧɨɣ ɬɨɱɤɢ, ɦɨɠɧɨ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɫɬɢɬɶ ɜɵɜɨɞ ɡɚɤɨɧɚ ɞɢɫɩɟɪɫɢɢ ɞɥɹ ɢɨɧɧɨɝɨ ɡɜɭɤɚ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɨɫɤɨɥɶɤɭ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɨɥɧɵ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɦɚɥɨɣ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɬɟɩɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɷɥɟɤɬɪɨɧɨɜ, ɢɧɟɪɰɢɹ ɩɨɫɥɟɞɧɢɯ

ɫɬɚɧɨɜɢɬɫɹ ɧɟɫɭɳɟɫɬɜɟɧɧɨɣ, ɩɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɷɥɟɤɬɪɨɧɧɚɹ ɩɨɞɫɢɫɬɟɦɚ ɦɨɠɟɬ ɫɱɢɬɚɬɶɫɹ ɤɜɚɡɢɪɚɜɧɨɜɟɫɧɨɣ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɟɣɫɬɜɭɸɳɢɟ ɜ ɷɬɨɣ ɩɨɞɫɢɫɬɟɦɟ ɫɢɥɵ − ɫɢɥɚ ɫɨ ɫɬɨɪɨɧɵ ɩɨɥɹ ɜɨɥɧɵ ɢ ɫɢɥɚ, ɨɛɭɫɥɨɜɥɟɧɧɚɹ ɝɪɚɞɢɟɧɬɨɦ ɷɥɟɤɬɪɨɧɧɨɝɨ ɞɚɜɥɟɧɢɹ, − ɞɨɥɠɧɵ ɛɵɬɶ ɭɪɚɜɧɨɜɟɲɟɧɵ. ɉɨɫɤɨɥɶɤɭ ɩɪɨɞɨɥɶɧɚɹ ɜɨɥɧɚ ɜɫɟɝɞɚ ɹɜɥɹɟɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ, ɜɜɟɞɟɦ ɩɨɬɟɧɰɢɚɥ ɩɨɥɹ ɜɨɥɧɵ ɫɨɝɥɚɫɧɨ

~& = − ϕ

E .

Ɍɨɝɞɚ ɛɚɥɚɧɫ ɭɤɚɡɚɧɧɵɯ ɫɢɥ, ɩɪɟɞɩɨɥɚɝɚɹ ɷɥɟɤɬɪɨɧɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɩɨɫɬɨɹɧɧɨɣ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ:

|e|ne ϕ − Te ne = 0 ,

(4.60)

ɢ, ɤɚɤ ɦɵ ɜɢɞɢɦ, ɤɨɧɰɟɧɬɪɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɩɨɬɟɧɰɢɚɥɚ ɩɨɥɹ ɜɨɥɧɵ. Ɇɨɠɧɨ ɫɤɚɡɚɬɶ ɢ ɬɚɤ: «ɛɟɡɵɧɟɪɰɢɨɧɧɵɟ» ɷɥɟɤɬɪɨɧɵ ɦɝɧɨɜɟɧɧɨ ɩɨɞɫɬɪɚɢɜɚɸɬɫɹ ɩɨɞ ɩɪɨɮɢɥɶ ɩɨɥɹ, ɫɨɡɞɚɜɚɟɦɵɣ ɜɨɥɧɨɣ, ɫɤɚɩɥɢɜɚɹɫɶ ɜ ɬɟɯ ɨɛɥɚɫɬɹɯ, ɝɞɟ ɩɨɬɟɧɰɢɚɥ ɩɨɥɹ ɛɨɥɶɲɟ:

§|e|ϕ · ne = ne0 exp¨ T ¸ .

Ɂɞɟɫɶ ne0 - ɧɟɜɨɡɦɭɳɟɧɧɚɹ ɩɨɥɟɦ ɜɨɥɧɵ ɤɨɧɰɟɧɬɪɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ (ɜ ɨɛɥɚɫɬɢ ɧɭɥɟɜɨɝɨ

ɩɨɬɟɧɰɢɚɥɚ). ɉɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ, ɤɚɤ ɦɵ ɜɢɞɢɦ, ɫɨɜɩɚɞɚɟɬ ɫ ɢɡɜɟɫɬɧɨɣ ɮɨɪɦɭɥɨɣ Ȼɨɥɶɰɦɚɧɚ ɞɥɹ ɪɚɜɧɨɜɟɫɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɱɚɫɬɢɰ, ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɷɥɟɤɬɪɨɧɨɜ, ɜɨ ɜɧɟɲɧɟɦ ɩɨɥɟ.

Ⱦɥɹ ɢɨɧɧɨɣ ɩɨɞɫɢɫɬɟɦɵ ɩɥɚɡɦɵ ɫɢɬɭɚɰɢɹ ɩɪɨɬɢɜɨɩɨɥɨɠɧɚɹ. ɉɨɫɤɨɥɶɤɭ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɨɥɧɵ ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɵɲɚɟɬ ɬɟɩɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɢɨɧɨɜ, ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɤɨɧɟɱɧɨɫɬɶɸ ɢɨɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ, ɫɱɢɬɚɹ ɢɨɧɵ ɯɨɥɨɞɧɵɦɢ, ɧɨ ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɭɱɟɫɬɶ ɢɧɟɪɰɢɸ ɢɨɧɨɜ, ɨɝɪɚɧɢɱɢɜɚɸɳɭɸ ɱɚɫɬɨɬɭ ɤɨɥɟɛɚɧɢɣ. Ⱦɥɹ ɯɨɥɨɞɧɵɯ Z-ɤɪɚɬɧɨ ɢɨɧɢɡɨɜɚɧɧɵɯ ɢɨɧɨɜ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɜɵɝɥɹɞɢɬ ɬɚɤ:

m	dv&i	= −Z\|e\| ϕ .	(4.61)

i	dt

ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɩɨɥɹ ɜɨɥɧɵ ɢɨɧɵ ɩɥɚɡɦɵ ɩɪɢɯɨɞɹɬ ɜ ɞɜɢɠɟɧɢɟ, ɢ ɢɯ ɤɨɧɰɟɧɬɪɚɰɢɹ ɧɚɱɢɧɚɟɬ ɢɡɦɟɧɹɬɶɫɹ, ɬɚɤ ɤɚɤ ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɸ ɧɟɩɪɟɪɵɜɧɨɫɬɢ

∂ni + div(n v& )= 0 .

(4.62)

∂t	i i

ȼ ɤɚɱɟɫɬɜɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ ɭɫɥɨɜɢɹ, ɡɚɦɵɤɚɸɳɟɝɨ ɫɢɫɬɟɦɭ (4.60) − (4.61), ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɬɪɟɛɨɜɚɧɢɟɦ ɤɜɚɡɢɧɟɣɬɪɚɥɶɧɨɫɬɢ ɩɥɚɡɦɵ

ne = Zni ,

(4.63)

ɤɨɬɨɪɨɟ, ɨɱɟɜɢɞɧɨ, ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ, ɬɚɤ ɤɚɤ ɞɥɢɧɚ ɜɨɥɧɵ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɵɲɚɸɳɟɣ ɷɥɟɤɬɪɨɧɧɵɣ ɞɟɛɚɟɜɫɤɢɣ ɪɚɞɢɭɫ. ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜɨɥɧɵ ɤɨɪɨɬɤɢɟ, ɭɫɥɨɜɢɟ (4.63) ɫɥɟɞɭɟɬ ɡɚɦɟɧɢɬɶ ɭɪɚɜɧɟɧɢɟɦ ɉɭɚɫɫɨɧɚ

∆ϕ = 4π|e|(ne − Zni ).

ɇɟɥɢɧɟɣɧɵɟ ɭɪɚɜɧɟɧɢɹ (4.60) − (4.63) ɫɩɪɚɜɟɞɥɢɜɵ, ɜ ɪɚɦɤɚɯ ɜɵɫɤɚɡɚɧɧɵɯ ɜɵɲɟ ɩɪɟɞɩɨɥɨɠɟɧɢɣ, ɞɥɹ ɢɨɧɧɨ-ɡɜɭɤɨɜɵɯ ɜɨɥɧ ɥɸɛɨɣ ɚɦɩɥɢɬɭɞɵ. Ɉɝɪɚɧɢɱɢɦɫɹ ɜɨɥɧɚɦɢ ɦɚɥɨɣ

ɚɦɩɥɢɬɭɞɵ. ɉɨɥɚɝɚɟɦ	~	& ~&
~
ϕ = ϕ , ne,i	= ne,i0 + ne,i , vi = vi ,

ɝɞɟ ɡɧɚɤɨɦ ɬɢɥɶɞɚ ɩɨɦɟɱɟɧɵ ɦɚɥɵɟ ɜɨɡɦɭɳɟɧɢɹ. ɉɨɞɫɬɚɜɢɦ ɷɬɨ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜ ɭɪɚɜɧɟɧɢɹ (4.60) − (4.63), ɪɚɡɥɨɠɢɦ ɡɚɬɟɦ ɩɨ ɚɦɩɥɢɬɭɞɟ ɢ, ɨɩɭɫɬɢɜ ɧɟɥɢɧɟɣɧɵɟ ɫɥɚɝɚɟɦɵɟ, ɜ ɪɟɡɭɥɶɬɚɬɟ

ɩɨɥɭɱɢɦ		~	~
		~	~
\|e\|ne0 ϕ − Te ne = 0 ,
	~&		~
mi	∂vi		~
mi	∂t	= −Z\|e\| ϕ ,		(4.64)
~			v& )= 0 ,
∂ni + div(n			v& )= 0 ,
∂t		i0	i
~		~
ne	= Zni , ne0 = Zni0 .

ȼɵɪɚɡɢɜ ɬɟɩɟɪɶ ɜɫɟ ɜɟɥɢɱɢɧɵ ɱɟɪɟɡ ɨɞɧɭ ɢɡ ɧɢɯ, ɧɚɩɪɢɦɟɪ, ɱɟɪɟɡ ɩɨɬɟɧɰɢɚɥ ɩɨɥɹ ɜɨɥɧɵ, ɩɨɥɭɱɢɦ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ

	∂ 2					ZT
		~	2	~	2	e	,	(4.65)
	∂t2 ϕ		− cs	∆ϕ = 0,	cs =	m	,	(4.65)
						i
ɫɥɟɞɫɬɜɢɟɦ ɤɨɬɨɪɨɝɨ, ɤɚɤ ɧɟɬɪɭɞɧɨ ɩɪɨɜɟɪɢɬɶ, ɹɜɥɹɟɬɫɹ ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ (4.56), ɩɪɢɱɟɦ
ɜɟɥɢɱɢɧɚ ɜɯɨɞɹɳɟɣ ɜ ɧɟɝɨ ɫɤɨɪɨɫɬɢ ɡɜɭɤɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ (4.58).

Ɇɵ ɪɚɫɫɦɨɬɪɟɥɢ ɫɚɦɵɟ ɩɪɨɫɬɵɟ ɞɢɫɩɟɪɫɢɨɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ

ɧɟɡɚɦɚɝɧɢɱɟɧɧɨɣ ɩɥɚɡɦɵ. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɱɢɬɚɬɟɥɟɣ, ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɟ ɢɡ ɧɢɯ

ɫɜɟɞɟɧɵ ɜ ɧɢɠɟɫɥɟɞɭɸɳɭɸ ɬɚɛɥɢɰɭ 4.1:

Ɍɚɛɥɢɰɚ 4.1

Ɍɢɩ ɜɨɥɧɵ

Ɂɚɤɨɧ ɞɢɫɩɟɪɫɢɢ

Ɏɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ

Ƚɪɭɩɩɨɜɚɹ

ɉɪɢɦɟɱɚɧɢɟ

ɫɤɨɪɨɫɬɶ

ɗɥɟɤɬɪɨɧɧɚɹ

ωLe

krDe → 0

ɥɟɧɝɦɸɪɨɜɫɤɚɹ ɜɨɥɧɚ

ɜ ɯɨɥɨɞɧɨɣ ɩɥɚɡɦɟ

ɗɥɟɤɬɪɨɧɧɚɹ

3 2 2 ·

≈ ωLe k

v (kv ω )

krDe << 1

ɥɟɧɝɦɸɪɨɜɫɤɚɹ ɜɨɥɧɚ

¨1 +

k r

Te Te

ɜ «ɬɟɩɥɨɣ» ɩɥɚɡɦɟ

Le ©

De ¹

ɂɨɧɧɨ-ɡɜɭɤɨɜɚɹ

kcs

ZTe , Te >> Ti

ɜɨɥɧɚ

krDe << 1

ɉɨɩɟɪɟɱɧɚɹ

+ k

ɩɥɚɡɦɟɧɧɚɹ ɜɨɥɧɚ

ωLe

c 1 + ωLe

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