Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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ɇɟɜɨɡɦɭɳɟɧɧɵɟ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ
ln
r
,
H ɨɛɥɚɞɚɸɬ (12
l
)-ɤɪɚɬɧɵɦ
ɜɵɪɨɠɞɟɧɢɟɦ ɩɨ ɦɚɝɧɢɬɧɨɦɭ ɤɜɚɧɬɨɜɨɦɭ ɱɢɫɥɭ . ɉɨɷɬɨɦɭ ɬɟɩɟɪɶ
ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɧɟɞɢɚɝɨɧɚɥɶɧɵɟ ɩɨ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɜɨɡɦɭɳɟɧɢɹ ɢ
ɜɵɹɫɧɢɬɶ, ɧɟ ɫɥɟɞɭɟɬ ɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɟɬɨɞɵ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɫ
ɜɵɪɨɠɞɟɧɢɟɦ. ɍɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ, ɫɜɹɡɵɜɚɸɳɢɟ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɞɟɤɚɪɬɨɜɵ ɢ
ɫɮɟɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ,
m
m
MT cossin
r
x
, MT sinsin
r
y , T cos
r
z , (6)
ɡɚɦɟɬɢɦ, ɱɬɨ
1,1
3/8sin
r
Mr
S T r Yreriyx
i
B ,
0,1
3/4 Yrz S . (7)
ȼ ɫɢɥɭ (7) ɭɝɥɨɜɚɹ ɱɚɫɬɶ ɮɭɧɤɰɢɢ
mln
r
z \ ɪɚɜɧɚ ɢ ɟɟ ɦɨɠɧɨ ɮɨɪɦɚɥɶɧɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɜɟɥɢɱɢɧɭ, ɚɧɚɥɨɝɢɱɧɭɸ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɩɪɨɢɡɜɟɞɟɧɢɹ
ɫɨɫɬɨɹɧɢɣ ɜɢɞɚ
ml
YY
0,1
mlml ,;0,1,0,1 { .
ɉɨɞɨɛɧɚɹ ɚɧɚɥɨɝɢɹ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɬɨɦ, ɱɬɨ ɭɤɚɡɚɧɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ
ɫɨɫɬɨɹɧɢɹ ɢ ɮɭɧɤɰɢɹ ɨɛɥɚɞɚɸɬ ɨɞɢɧɚɤɨɜɵɦ ɩɨɜɟɞɟɧɢɟɦ ɩɨ ɨɬɧɨɲɟɧɢɸ
ɤ ɨɩɟɪɚɬɨɪɚɦ ɜɪɚɳɟɧɢɣ. Ɍɨɝɞɚ ɞɨɥɠɧɨ ɢɦɟɬɶ ɦɟɫɬɨ ɫɥɟɞɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ
mln
r
z \
~
ˆ
mlnzmln
rr
ccc
mlml ,;0,1,
cc
. (8)
ɋɨɝɥɚɫɧɨ ɩɪɚɜɢɥɭ ɫɥɨɠɟɧɢɹ ɦɨɦɟɧɬɨɜ ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ
ml,;0,1 ɹɜɥɹɟɬɫɹ
ɨɞɧɢɦ ɢɡ ɛɚɡɢɫɧɵɯ ɜɟɤɬɨɪɨɜ ɜ ɪɚɡɥɨɠɟɧɢɢ ɫɨɫɬɨɹɧɢɣ
ml
c
c
, ɫ ɫɭɦɦɚɪɧɵɦ
ɦɨɦɟɧɬɨɦ
l
c
, ɬɨ ɟɫɬɶ – ɫɨɫɬɨɹɧɢɣ ɫ ɩɪɨɟɤɰɢɟɣ ɦɨɦɟɧɬɚ
21
mmm
c
(9)
ɢ ɜɟɥɢɱɢɧɨɣ ɦɨɦɟɧɬɚ
212121
,...,1, lllllll
c
. (10)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɫɬɨɹɧɢɟ
ml,;0,1,ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɞɨɩɭɫɤɚɟɬ ɪɚɡɥɨɠɟɧɢɟ ɩɨ
ɫɨɫɬɨɹɧɢɹɦ
ml
cc
, ɫɨ ɡɧɚɱɟɧɢɹɦɢ
l
c
ɢ m
c
, ɭɤɚɡɚɧɧɵɦɢ ɜ (10) ɢ (9). ɉɨɥɨɠɢɜ
,,0
1
mmm
2
1
1
l ,,ɦɵ ɡɚɤɥɸɱɚɟɦ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨ ɩɪɚɜɚɹ
ɫɬɨɪɨɧɚ ɫɨɨɬɧɨɲɟɧɢɹ (8) ɦɨɠɟɬ ɛɵɬɶ ɨɬɥɢɱɧɚ ɨɬ ɧɭɥɹ ɬɨɥɶɤɨ ɩɪɢ
ll
2
mm
c
ɢ
. ɗɬɨɬ ɜɵɜɨɞ ɫɞɟɥɚɧ ɥɢɲɶ ɫ ɭɱɟɬɨɦ ɫɜɨɣɫɬɜ ɫɨɫɬɨɹɧɢɣ ɩɨ1,,1
c
llll
110
ɨɬɧɨɲɟɧɢɸ ɤ ɜɪɚɳɟɧɢɹɦ, ɧɨ ɟɳɟ ɫɥɟɞɭɟɬ ɩɪɢɧɹɬɶ ɜɨ ɜɧɢɦɚɧɢɟ ɱɟɬɧɨɫɬɶ
ɫɨɫɬɨɹɧɢɣ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɭɸ ɢɯ ɩɨɜɟɞɟɧɢɟ ɩɪɢ ɢɧɜɟɪɫɢɢ ɤɨɨɪɞɢɧɚɬ (ɨɩɟɪɚɰɢɹ
ɢɧɜɟɪɫɢɢ ɹɜɥɹɟɬɫɹ ɞɢɫɤɪɟɬɧɨɣ ɢ ɧɟ ɫɜɨɞɢɬɫɹ ɤ ɤɚɤɨɣ-ɥɢɛɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
ɜɪɚɳɟɧɢɣ). ɑɟɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ
ml
cc
, ɢ
22112211
,;,,, mlmlmlml
ɟɫɬɶ
ɢ , ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɋɨɫɬɨɹɧɢɹ ɫ ɪɚɡɥɢɱɧɨɣ ɱɟɬɧɨɫɬɶɸ ɜɡɚɢɦɧɨ
ɨɪɬɨɝɨɧɚɥɶɧɵ, ɩɨɷɬɨɦɭ ɩɪɚɜɚɹ ɫɬɨɪɨɧɚ (8) ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ
l
c
)1(
21
)1(
ll
l
l
c
.
Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ (8) ɦɨɠɟɬ ɛɵɬɶ ɧɟ ɪɚɜɟɧ ɧɭɥɸ ɬɨɥɶɤɨ
ɞɥɹ ɫɨɫɬɨɹɧɢɣ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɱɟɬɧɨɫɬɶɸ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɢɦ
ɩɪɚɜɢɥɚɦ ɨɬɛɨɪɚ ɞɥɹ
ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɨɩɟɪɚɬɨɪɚ ɤɨɨɪɞɢɧɚɬɵ :
z
z
: mm
c
,1,1
c
lll . (11)
Ɂɚɨɞɧɨ ɨɬɦɟɬɢɦ, ɱɬɨ ɭɝɥɨɜɚɹ ɱɚɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɹ
mln
r
iyx \r )( ɪɚɜɧɚ .
ɉɨɷɬɨɦɭ ɚɧɚɥɨɝɢɱɧɵɦ ɩɭɬɟɦ ɥɟɝɤɨ ɩɪɢɞɬɢ ɢ ɤ ɩɪɚɜɢɥɚɦ ɨɬɛɨɪɚ ɞɥɹ ɦɚɬɪɢɱɧɵɯ
ɷɥɟɦɟɧɬɨɜ
ml
YY
1,1 r
mlyixml ,)
ˆˆ
(, r
cc
:
iy
x
:1
c
mm ,1,1
c
lll , (12)
iy
x
:1
c
mm ,1,1
c
lll . (13)
ɉɪɚɜɢɥɚ ɨɬɛɨɪɚ – ɷɬɨ ɧɟɨɛɯɨɞɢɦɵɟ ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ
ɦɨɝɭɬ ɛɵɬɶ ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ [1,10], ɱɬɨ ɩɪɚɜɢɥɚ ɨɬɛɨɪɚ (11)–
(13) ɜɟɪɧɵ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɨɩɟɪɚɬɨɪɨɜ ɤɨɨɪɞɢɧɚɬ, ɧɨ ɢ ɞɥɹ ɨɩɟɪɚɬɨɪɨɜ ɥɸɛɨɣ
ɜɟɤɬɨɪɧɨɣ ɜɟɥɢɱɢɧɵ, ɩɪɨɟɤɰɢɢ ɤɨɬɨɪɨɣ ɩɪɢ ɩɨɜɨɪɨɬɚɯ ɢ ɢɧɜɟɪɫɢɢ
ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɤɨɦɩɨɧɟɧɬɚɦ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ (ɬɚɤɢɟ ɜɟɤɬɨɪɵ
ɧɚɡɵɜɚɸɬ ɩɨɥɹɪɧɵɦɢ).
r
ɋɨɝɥɚɫɧɨ (11) ɧɟɞɢɚɝɨɧɚɥɶɧɵɟ ɩɨ
ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɜɨɡɦɭɳɟɧɢɹ (2)
ɪɚɜɧɵ ɧɭɥɸ, ɢ ɩɨɷɬɨɦɭ ɜɵɪɨɠɞɟɧɢɟ ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ
m
ln
r
,
H ɩɨ ɜ ɞɚɧɧɨɣ
ɡɚɞɚɱɟ ɧɟ ɜɟɞɟɬ ɤ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɟɨɪɢɸ ɜɨɡɦɭɳɟɧɢɣ ɫ
ɜɵɪɨɠɞɟɧɢɟɦ. Ʉɪɨɦɟ ɬɨɝɨ, ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɨɬɫɭɬɫɬɜɢɟ
ɜɵɪɨɠɞɟɧɢɹ ɩɨ
m
l
. ɋ ɭɱɟɬɨɦ (5) ɦɵ ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ
ɩɨɩɪɚɜɤɢ ɤ ɷɧɟɪɝɢɢ ɪɚɜɧɵ ɧɭɥɸ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡɦɟɧɟɧɢɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ ɫɥɟɞɭɟɬ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜɨ ɜɬɨɪɨɦ ɩɨɪɹɞɤɟ ɩɨ ɷɥɟɤɬɪɢɱɟɫɤɨɦɭ ɩɨɥɸ. ɉɨɩɪɚɜɤɢ ɤ ɷɧɟɪɝɢɢ
ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɮɨɪɦɭɥɨɣ
ln
r
,
H
¦
z
c
c
HH
c
H
kk
k
k
k
kVk
)0()0(
2
)2(
ˆ
2
2
1
E
k
E{ , (14)
ɝɞɟ
111
¦
z
c
c
HH
c
E
kk
k
k
k
kzk
q
)0()0(
2
2
ˆ
2 , (15)
k ɨɡɧɚɱɚɟɬ ɫɨɜɨɤɭɩɧɨɫɬɶ ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ ,
r
n
l
,;ɪɨɥɶ ɧɟɜɨɡɦɭɳɟɧɧɵɯ
ɡɧɚɱɟɧɢɣ ɷɧɟɪɝɢɢ ɢɝɪɚɸɬ ɭɪɨɜɧɢ
m
)0(
k
H
ln
r
,
H
. Ʉɨɷɮɮɢɰɢɟɧɬ ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɨɛɨɣ
ɩɨɥɹɪɢɡɭɟɦɨɫɬɶ
k
E
k -ɝɨ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɞɟɫɶ
ɚɬɨɦɧɨɣ ɫɢɫɬɟɦɵ (ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ )(
r
U
, ɫɨɡɞɚɜɚɟɦɨɦ ɡɚɤɪɟɩɥɟɧɧɵɦ ɜ ɧɚɱɚɥɟ
ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɨɦ ɩɪɢɬɹɠɟɧɢɹ). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɨɩɪɟɞɟɥɢɦ ɞɢɩɨɥɶɧɵɣ
ɷɥɟɤɬɪɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɢɫɬɟɦɵ ɮɨɪɦɭɥɨɣ
rd q , ɬɚɤ ɱɬɨ
zqd
z
ˆ
ˆ
(16)
ɢ
z
dEHH
ˆ
ˆˆ
0
. (17)
Ɍɨɝɞɚ
z
d
E
H
ˆ
ˆ
w
w
,
ɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (11) ɡɚɞɚɱɢ 6.8 ɢɦɟɟɦ:
Ekdk
kz
wHw /
ˆ
. (18)
ɉɨɞɫɬɚɜɢɜ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ (18) ɧɚɣɞɟɧɧɨɟ ɜɨ ɜɬɨɪɨɦ ɩɨɪɹɞɤɟ ɬɟɨɪɢɢ
ɜɨɡɦɭɳɟɧɢɣ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
k
H ɝɚɦɢɥɶɬɨɧɢɚɧɚ
H
ˆ
k
H
2
)0(
2
1
E
k
k
EH , (19)
ɧɚɣɞɟɦ, ɱɬɨ ɜ
k -ɨɦ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ ɢɧɞɭɰɢɪɭɟɦɵɣ ɧɚɩɪɚɜɥɟɧɧɵɦ
ɜɞɨɥɶ ɩɨɥɟɦ ɭɫɪɟɞɧɟɧɧɵɣ ɞɢɩɨɥɶɧɵɣ ɦɨɦɟɧɬ ɫɢɫɬɟɦɵ ɟɫɬɶ
z E
Ed
kz
E . (20)
ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɜɟɥɢɱɢɧɭ
k
E ɤɚɤ ɩɨɥɹɪɢɡɭɟɦɨɫɬɶ ɫɢɫɬɟɦɵ.
Ⱦɥɹ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ (0
r
n ,0
l
,0 m ) ɡɧɚɦɟɧɚɬɟɥɶ ɜ (15)
ɨɬɪɢɰɚɬɟɥɟɧ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɹɪɢɡɭɟɦɨɫɬɶ
0
E ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
ɩɨɥɨɠɢɬɟɥɶɧɚ, ɭɪɨɜɟɧɶ ɷɧɟɪɝɢɢ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜɨ ɜɧɟɲɧɟɦ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɩɨɥɟ ɫɦɟɳɚɟɬɫɹ ɜɧɢɡ. Ʉɨɧɟɱɧɨ, ɩɨɩɪɚɜɤɚ ɤ ɷɧɟɪɝɢɢ ɨɫɧɨɜɧɨɝɨ
ɫɨɫɬɨɹɧɢɹ ɜɨ ɜɬɨɪɨɦ ɩɨɪɹɞɤɟ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɨɬɪɢɰɚɬɟɥɶɧɚ ɩɪɢ ɥɸɛɨɦ
ɜɢɞɟ ɜɨɡɦɭɳɟɧɢɹ.
112
ȼ ɜɵɪɚɠɟɧɢɢ (15) ɞɥɹ ɩɨɥɹɪɢɡɭɟɦɨɫɬɢ ɫɨɫɬɨɹɧɢɹ ɫ ɤɜɚɧɬɨɜɵɦɢ ɱɢɫɥɚɦɢ
,
r
n
l
, ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ, ɫɨɝɥɚɫɧɨ (11), ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵm
mlnzmln
rr
,,
ˆ
,1, r
c
(ɟɫɥɢ 0
l
, ɬɨ ɞɨɩɭɫɬɢɦɵɦ ɛɭɞɟɬ ɥɢɲɶ 1
c
l
). Ɉɧɢ
ɡɚɜɢɫɹɬ ɨɬ ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɩɨɷɬɨɦɭ ɨɬ ɷɬɢɯ
ɱɢɫɟɥ, ɜ ɱɚɫɬɧɨɫɬɢ ɨɬ , ɡɚɜɢɫɹɬ ɩɨɥɹɪɢɡɭɟɦɨɫɬɶ ɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɫɞɜɢɝ (14).
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɧɟɲɧɟɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɫɧɢɦɚɟɬ ɜɵɪɨɠɞɟɧɢɟ
ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ ɩɨ . Ɉɞɧɚɤɨ ɬɚɤɨɟ ɫɧɹɬɢɟ ɜɵɪɨɠɞɟɧɢɹ ɧɟ ɦɨɠɟɬ ɛɵɬɶ
ɩɨɥɧɵɦ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɝɚɦɢɥɶɬɨɧɢɚɧ ɫ ɩɨɬɟɧɰɢɚɥɨɦ
m
E
m
zq
E
r
U
)( ɢɧɜɚɪɢɚɧɬɟɧ
ɤ ɨɬɪɚɠɟɧɢɸ ɜ ɥɸɛɨɣ ɩɥɨɫɤɨɫɬɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɨɫɶ . ɉɨɷɬɨɦɭ
ɫɬɚɰɢɨɧɚɪɧɵɟ ɫɨɫɬɨɹɧɢɹ, ɩɨɥɭɱɚɸɳɢɟɫɹ ɞɪɭɝ ɢɡ ɞɪɭɝɚ ɭɤɚɡɚɧɧɵɦ ɨɬɪɚɠɟɧɢɟɦ,
ɮɢɡɢɱɟɫɤɢ ɷɤɜɢɜɚɥɟɧɬɧɵ ɢ ɞɨɥɠɧɵ ɢɦɟɬɶ ɨɞɧɭ ɢ ɬɭ ɠɟ ɷɧɟɪɝɢɸ. ɇɨ ɩɨɫɤɨɥɶɤɭ
ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɹɜɥɹɟɬɫɹ ɚɤɫɢɚɥɶɧɵɦ ɜɟɤɬɨɪɨɦ, ɚ ɧɟ ɩɨɥɹɪɧɵɦ, ɬɚɤɢɟ
ɫɨɫɬɨɹɧɢɹ ɨɛɥɚɞɚɸɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ
z
ml
z
{ .
Ɂɧɚɱɢɬ, ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ ɜɨɡɦɭɳɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɜɨɡɧɢɤɚɸɳɢɟ ɩɪɢ
ɪɚɫɳɟɩɥɟɧɢɢ ɜɵɪɨɠɞɟɧɧɨɝɨ ɩɨ ɭɪɨɜɧɹ
m
ln
r
,
H ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɪɢɥɨɠɟɧɧɨɝɨ
ɢɡɜɧɟ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ, ɞɥɹ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɡɧɚɱɟɧɢɣ
m ɫɨɜɩɚɞɚɸɬ.
ɂɡɦɟɧɟɧɢɟ ɷɧɟɪɝɢɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ ɚɬɨɦɧɨɣ ɫɢɫɬɟɦɵ ɩɨɞ
ɜɥɢɹɧɢɟɦ ɜɧɟɲɧɟɝɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɩɨɥɭɱɢɥɨ ɧɚɡɜɚɧɢɟ
ɷɮɮɟɤɬɚ
ɒɬɚɪɤɚ
.
ȼ ɡɚɞɚɱɟ 3.4 ɭɩɨɦɢɧɚɥɨɫɶ ɨ ɩɪɢɬɹɠɟɧɢɢ ɧɚ ɛɨɥɶɲɨɦ ɪɚɫɫɬɨɹɧɢɢ
R
ɦɟɠɞɭ
ɚɬɨɦɨɦ ɜɨɞɨɪɨɞɚ
H
ɢ ɢɨɧɨɦ
H
(ɩɪɨɬɨɧɨɦ), ɤɨɬɨɪɨɟ ɜɨɡɧɢɤɚɟɬ ɡɚ ɫɱɟɬ
ɢɧɞɭɰɢɪɭɟɦɨɝɨ ɭ ɚɬɨɦɚ
H
ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɞɢɩɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ. Ɍɟɩɟɪɶ ɦɵ
ɦɨɠɟɦ ɨɛɫɭɞɢɬɶ ɷɬɨɬ ɜɨɩɪɨɫ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɨɥɭɱɟɧɧɵɯ ɜɵɲɟ ɮɨɪɦɭɥ (15)
ɢ (19). ɉɪɟɠɞɟ ɜɫɟɝɨ ɡɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɜɨɞɨɪɨɞɨɩɨɞɨɛɧɵɯ ɚɬɨɦɨɜ (ɬɨ ɟɫɬɶ
ɞɥɹ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ )(
r
U
) ɢɦɟɟɬɫɹ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ ɫɥɭɱɚɣɧɨɟ
ɜɵɪɨɠɞɟɧɢɟ
ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ ɩɨ ɤɜɚɧɬɨɜɨɦɭ ɱɢɫɥɭ
l
, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
ɞɥɹ ɷɬɢɯ ɭɪɨɜɧɟɣ ɦɨɠɧɨ ɨɠɢɞɚɬɶ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɥɢɧɟɣɧɨɝɨ ɩɨ ɷɥɟɤɬɪɢɱɟɫɤɨɦɭ
ɩɨɥɸ ɷɮɮɟɤɬɚ ɒɬɚɪɤɚ (ɫɦ. ɡɚɞɚɱɭ 9.5). Ʉɪɨɦɟ ɬɨɝɨ, ɩɪɢ ɭɱɟɬɟ ɫɩɢɧ-
ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜ ɪɚɫɱɟɬɟ ɷɮɮɟɤɬɚ ɒɬɚɪɤɚ ɫɥɟɞɭɟɬ ɩɪɢɧɢɦɚɬɶ ɜɨ
ɜɧɢɦɚɧɢɟ ɬɨɧɤɭɸ ɫɬɪɭɤɬɭɪɭ ɭɪɨɜɧɟɣ. Ɉɞɧɚɤɨ ɷɬɢ ɡɚɦɟɱɚɧɢɹ ɧɟ ɚɤɬɭɚɥɶɧɵ ɞɥɹ
ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ (
s
1-ɫɨɫɬɨɹɧɢɟ), ɩɨɫɤɨɥɶɤɭ
s
1-ɭɪɨɜɟɧɶ ɧɟ
ɢɦɟɟɬ ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɵɪɨɠɞɟɧɢɹ ɢ ɧɟ ɢɫɩɵɬɵɜɚɟɬ ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ
ɪɚɫɳɟɩɥɟɧɢɹ.
ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ
2
/ ReE , (21)
ɫɨɡɞɚɜɚɟɦɨɟ ɨɤɨɥɨ ɩɪɟɛɵɜɚɸɳɟɝɨ ɜ ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ ɚɬɨɦɚ
H
ɧɚɯɨɞɹɳɢɦɫɹ ɧɚ ɛɨɥɶɲɨɦ ɪɚɫɫɬɨɹɧɢɢ ()ɢɨɧɨɦ
B
aR !!
H
ɦɨɠɧɨ ɫɱɢɬɚɬɶ
ɩɪɚɤɬɢɱɟɫɤɢ ɨɞɧɨɪɨɞɧɵɦ. ɉɨɞɫɬɚɜɢɜ (21) ɜ (19), ɞɥɹ ɷɧɟɪɝɢɢ )(
R
u
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
H
ɢ
H
, ɩɪɟɞɫɬɚɜɥɹɸɳɟɣ ɫɨɛɨɣ ɩɨɩɪɚɜɤɭ ɤ ɷɧɟɪɝɢɢ ɚɬɨɦɚ
113
ɜɨɞɨɪɨɞɚ ɜ ɩɨɥɟ (21), ɩɨɥɭɱɢɦ:
4
2
2
)(
R
e
Ru
E
, (22)
ɝɞɟ – ɩɨɥɹɪɢɡɭɟɦɨɫɬɶ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ ɜ ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ. ȼ ɫɢɥɭ
ɧɟɪɚɜɟɧɫɬɜɚ 0ɷɧɟɪɝɢɹ (22) ɨɬɪɢɰɚɬɟɥɶɧɚ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɩɪɢɬɹɠɟɧɢɸ ɦɟɠɞɭ
E
!E
H
ɢ
H
. Ɏɨɪɦɭɥɚ (15) ɫ eq
ɞɥɹ ɩɨɥɹɪɢɡɭɟɦɨɫɬɢ ɚɬɨɦɚ
ɜɨɞɨɪɨɞɚ ɜ ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ ɩɪɢɜɨɞɢɬ ɤ ɡɧɚɱɟɧɢɸ
3
2
9
B
a E . (23)
ȼɵɜɨɞ ɪɟɡɭɥɶɬɚɬɚ (23) ɢ ɩɨɞɪɨɛɧɵɣ ɚɧɚɥɢɡ ɷɮɮɟɤɬɚ ɒɬɚɪɤɚ ɜ ɚɬɨɦɟ ɜɨɞɨɪɨɞɚ
ɡɚɢɧɬɟɪɟɫɨɜɚɧɧɵɣ ɱɢɬɚɬɟɥɶ ɧɚɣɞɟɬ ɜ [1].
9.5. Ʉɪɚɬɧɨɫɬɶ ɜɵɪɨɠɞɟɧɢɹ ɭɪɨɜɧɹ
2
H
n
ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ ɛɟɡ ɭɱɟɬɚ ɫɩɢɧɚ
ɷɥɟɤɬɪɨɧɚ ɪɚɜɧɚ ɱɟɬɵɪɟɦ: ɷɬɨɦɭ ɭɪɨɜɧɸ ɩɪɢɧɚɞɥɟɠɚɬ ɬɪɢ
p
2-ɫɨɫɬɨɹɧɢɹ (1
l
)
ɢɨɞɧɨ
s
2-ɫɨɫɬɨɹɧɢɟ (0
l
). ɉɪɨɧɭɦɟɪɭɟɦ ɭɤɚɡɚɧɧɵɟ ɫɨɫɬɨɹɧɢɹ, ɧɚɩɪɢɦɟɪ,
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
0,0,21 mln , 0,1,22 mln ,
1,1,23 mln , 1,1,24 mln . (1)
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɩɪɚɜɨɤ ɩɟɪɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɤ ɷɧɟɪɝɢɢ ɫɨɫɬɨɹɧɢɣ (1)
ɩɪɢ ɧɚɥɢɱɢɢ ɜɨɡɦɭɳɟɧɢɹ
)1(
H
zEqV
ˆ
ˆ
(2)
ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɫ ɜɵɪɨɠɞɟɧɢɟɦ ɫɥɟɞɭɟɬ ɧɚɣɬɢ
ɦɚɬɪɢɰɭ ɨɩɟɪɚɬɨɪɚ (2) ɜ ɛɚɡɢɫɟ (1) ɢ ɪɟɲɢɬɶ ɫɟɤɭɥɹɪɧɨɟ ɭɪɚɜɧɟɧɢɟ
kk
V
c
0)(det
)1(
GH
cc
kkkk
V . (3)
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɚɜɢɥɚɦɢ ɨɬɛɨɪɚ, ɩɪɢɜɟɞɟɧɧɵɦɢ ɜ ɡɚɞɚɱɟ 9.4. ɋɨɝɥɚɫɧɨ
ɮɨɪɦɭɥɟ (11) ɭɤɚɡɚɧɧɨɣ ɡɚɞɚɱɢ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɧɟ ɪɚɜɧɵɦɢ ɧɭɥɸ ɦɨɝɭɬ
ɛɵɬɶ ɥɢɲɶ ɞɜɚ ɢɡ ɲɟɫɬɧɚɞɰɚɬɢ ɢɧɬɟɪɟɫɭɸɳɢɯ ɧɚɫ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ:
1
ˆ
2 z
02
ˆ
1 z z . (4)
ȼɵɱɢɫɥɢɦ ɜɟɥɢɱɢɧɵ (4). ɇɚɩɨɦɧɢɦ ɱɢɬɚɬɟɥɸ ɜɢɞ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɜɵɯ
ɪɚɞɢɚɥɶɧɵɯ ɮɭɧɤɰɢɣ )ɜ ɡɚɞɚɱɟ ɨ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɹɯ ɜ ɞɢɫɤɪɟɬɧɨɣ(
rR
ln
114
ɱɚɫɬɢ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ:
s
1
U
eaR )/2(
3
10
,
s
2
2/3
20
)2/1()2/1(
U
U eaR ,
p
2 U
U 2/3
21
)62/1( eaR ,
ɝɞɟ – ɪɚɞɢɭɫ Ȼɨɪɚ,
B
aa a
r
/
U . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ
ɫɨɫɬɨɹɧɢɹ
1 ɪɚɜɧɚ
0020
YR
2/3
)2/1()8/1(
U
US ea . (5)
ȼɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɞɥɹ
2
ˆ
z ɫ ɭɱɟɬɨɦ ɪɚɜɟɧɫɬɜɚ
10
3/4cos Yrrz S T ɢɦɟɟɬ
ɜɢɞ
S
102110
3/4 YRYr
2
10
22/
)(18/ Yea US
U
. (6)
ɉɟɪɟɦɧɨɠɢɜ (5) ɢ (6) ɢ ɢɧɬɟɝɪɢɪɭɹ ɩɨɥɭɱɢɜɲɟɟɫɹ ɜɵɪɚɠɟɧɢɟ ɩɨ ɪɚɞɢɚɥɶɧɨɣ ɢ
ɭɝɥɨɜɵɦ ɤɨɨɪɞɢɧɚɬɚɦ, ɩɨɥɭɱɢɦ:
4
0
12
2
ˆ
1
UU
³
f
d
a
z )2
/
1( U
2
10
)(Yde
³
:
U
. (7)
ȼ ɫɢɥɭ ɜɟɳɟɫɬɜɟɧɧɨɫɬɢ ɜɟɥɢɱɢɧɵ ɢ ɭɫɥɨɜɢɹ
10
Y
mmllmlml
YYd
cc
cc
GG :
³
(8)
ɢɧɬɟɝɪɚɥ ɩɨ ɭɝɥɚɦ ɜ (7) ɪɚɜɟɧ ɟɞɢɧɢɰɟ. Ɋɚɞɢɚɥɶɧɵɣ ɢɧɬɟɝɪɚɥ ɜ (7) ɜɵɱɢɫɥɹɟɬɫɹ
ɫ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥɵ
k
d UU
³
f
0
U
e !)1( kk
* ; (9)
ɨɧ ɪɚɜɟɧ . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,36
1
ˆ
2 z az 32
ˆ
1 (10)
ɢ
eEaVV 3
1221
, ɨɫɬɚɥɶɧɵɟ . (11)0
c
kk
V
Ɍɨɬ ɮɚɤɬ, ɱɬɨ ɜɫɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ, ɫɜɹɡɚɧɧɵɟ
ɫ ɫɨɫɬɨɹɧɢɹɦɢ
3 ɢ 4,ɪɚɜɧɵ ɧɭɥɸ, ɨɡɧɚɱɚɟɬ, ɱɬɨ ɷɬɢ ɞɜɚ ɫɨɫɬɨɹɧɢɹ ɭɠɟ
115
ɹɜɥɹɸɬɫɹ ɩɪɚɜɢɥɶɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ, ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɢɦ ɡɧɚɱɟɧɢɟ ɷɧɟɪɝɢɢ
()ɧɟ ɢɫɩɵɬɵɜɚɟɬ ɢɡɦɟɧɟɧɢɹ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ ɷɥɟɤɬɪɢɱɟɫɤɨɦɭ
ɩɨɥɸ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɩɪɚɜɨɤ
2
H
n
)1(
H ɤ ɷɧɟɪɝɢɢ ɫɨɫɬɨɹɧɢɣ 1 ɢ 2 ɪɟɲɚɟɦ
ɭɪɚɜɧɟɧɢɟ
0
3
3
det
)1(
)1(
¸
¸
¹
·
¨
¨
©
§
H
H
eEa
eEa
. (12)
Ɉɧɨ ɢɦɟɟɬ ɞɜɚ ɤɨɪɧɹ:
)1(
H e
E
a3r . (13)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɧɟɲɧɟɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɪɚɫɳɟɩɥɹɟɬ ɱɟɬɵɪɟɯɤɪɚɬɧɨ
ɜɵɪɨɠɞɟɧɧɵɣ ɭɪɨɜɟɧɶ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ ɫ ɝɥɚɜɧɵɦ ɤɜɚɧɬɨɜɵɦ ɱɢɫɥɨɦ 2
n ɧɚ
ɬɪɢ ɭɪɨɜɧɹ. Ɉɞɢɧ ɢɡ ɷɬɢɯ ɭɪɨɜɧɟɣ ɞɜɭɤɪɚɬɧɨ ɜɵɪɨɠɞɟɧ ɢ ɜ ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ
ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɪɚɜɟɧ ɫɜɨɟɦɭ ɧɟɜɨɡɦɭɳɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ; ɞɜɚ ɞɪɭɝɢɯ
ɭɪɨɜɧɹ ɨɬɞɟɥɟɧɵ ɨɬ ɧɟɝɨ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ ɢɧɬɟɪɜɚɥɚɦɢ (13),
ɭɜɟɥɢɱɢɜɚɸɳɢɦɢɫɹ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɩɪɢɥɨɠɟɧɧɨɝɨ ɢɡɜɧɟ
ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ (ɥɢɧɟɣɧɵɣ ɷɮɮɟɤɬ ɒɬɚɪɤɚ).
ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɩɪɢɦɟɧɢɦ ɞɥɹ ɫɥɭɱɚɹ ɧɟ ɫɥɢɲɤɨɦ ɫɥɚɛɨɝɨ ɩɨɥɹ, ɤɨɝɞɚ
ɢɧɬɟɪɜɚɥɵ (13) ɧɚɦɧɨɝɨ ɩɪɟɜɵɲɚɸɬ
ɷɧɟɪɝɢɸ ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ ɪɚɫɳɟɩɥɟɧɢɹ,
ɬɚɤ ɱɬɨ ɩɨɫɥɟɞɧɟɣ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. ɇɨ ɩɨɥɟ ɧɟ ɞɨɥɠɧɨ ɛɵɬɶ ɢ ɫɥɢɲɤɨɦ
ɫɢɥɶɧɵɦ, ɩɨɫɤɨɥɶɤɭ ɬɟɨɪɢɹ ɜɨɡɦɭɳɟɧɢɣ ɩɪɢɦɟɧɢɦɚ ɬɨɥɶɤɨ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ
ɨɩɪɟɞɟɥɹɟɦɵɟ ɟɸ ɩɨɩɪɚɜɤɢ ɤ ɭɪɨɜɧɹɦ ɷɧɟɪɝɢɢ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
ɢɫɯɨɞɧɵɦɢ ɢɧɬɟɪɜɚɥɚɦɢ ɦɟɠɞɭ ɭɪɨɜɧɹɦɢ. Ʉɪɨɦɟ ɬɨɝɨ, ɜ ɨɱɟɧɶ ɫɢɥɶɧɨɦ ɩɨɥɟ
ɤɚɪɬɢɧɚ ɷɮɮɟɤɬɚ ɒɬɚɪɤɚ ɭɫɥɨɠɧɹɟɬɫɹ ɹɜɥɟɧɢɟɦ ɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ ɡɚ ɫɱɟɬ
ɬɭɧɧɟɥɶɧɨɝɨ ɷɮɮɟɤɬɚ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɨɬɟɧɰɢɚɥ ɜɢɞɚ ɜ ɨɞɧɨɦ ɢɡ
ɧɚɩɪɚɜɥɟɧɢɣ ɜɞɨɥɶ ɨɫɢ ɢɦɟɟɬ ɮɨɪɦɭ ɛɚɪɶɟɪɚ. ɋ ɧɟɤɨɬɨɪɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ,
ɢɫɱɟɡɚɸɳɟ ɦɚɥɨɣ ɩɪɢ 0
zeEre /
2
z
o
E
, ɷɥɟɤɬɪɨɧ ɦɨɠɟɬ ɩɪɨɧɢɤɧɭɬɶ ɫɤɜɨɡɶ ɛɚɪɶɟɪ, ɬɚɤ
ɱɬɨ ɩɪɢ ɧɚɥɢɱɢɢ ɜɧɟɲɧɟɝɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ (0
z
E
) ɤɚɠɞɵɣ ɞɢɫɤɪɟɬɧɵɣ
ɭɪɨɜɟɧɶ ɚɬɨɦɚ, ɫɬɪɨɝɨ ɝɨɜɨɪɹ, ɨɤɚɡɵɜɚɟɬɫɹ ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɵɦ ɢ
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɧɟɤɨɬɨɪɨɣ ɲɢɪɢɧɨɣ
Ƚ, ɦɚɥɨɣ ɜ ɦɟɪɭ ɦɚɥɨɫɬɢ ɬɭɧɧɟɥɶɧɨɣ
ɩɪɨɧɢɰɚɟɦɨɫɬɢ ɛɚɪɶɟɪɚ. Ɇɚɥɨɣ ɲɢɪɢɧɟ ɭɪɨɜɧɹ ɨɬɜɟɱɚɟɬ ɛɨɥɶɲɨɟ ɜɪɟɦɹ ɠɢɡɧɢ
(
/
~ =W Ƚ) ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɩɨɷɬɨɦɭ ɫ ɩɪɚɤɬɢɱɟɫɤɨɣ ɬɨɱɤɢ
ɡɪɟɧɢɹ ɫɨɫɬɨɹɧɢɹ ɚɬɨɦɚ ɜ ɧɟ ɫɥɢɲɤɨɦ ɫɢɥɶɧɨɦ ɩɨɥɟ ɦɨɠɧɨ ɫ ɯɨɪɨɲɟɣ
ɬɨɱɧɨɫɬɶɸ ɫɱɢɬɚɬɶ ɫɬɚɰɢɨɧɚɪɧɵɦɢ. Ɉɞɧɚɤɨ ɫ ɪɨɫɬɨɦ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɜɧɟɲɧɟɝɨ
ɩɨɥɹ
E
ɜɵɫɨɬɚ ɢ ɲɢɪɢɧɚ ɩɨɬɟɧɰɢɚɥɶɧɨɝɨ ɛɚɪɶɟɪɚ ɭɦɟɧɶɲɚɸɬɫɹ, ɢ ɜɟɪɨɹɬɧɨɫɬɶ
ɜɵɯɨɞɚ ɷɥɟɤɬɪɨɧɚ ɢɡ ɚɬɨɦɚ ɩɨɫɪɟɞɫɬɜɨɦ ɬɭɧɧɟɥɶɧɨɝɨ ɷɮɮɟɤɬɚ ɫɭɳɟɫɬɜɟɧɧɨ
ɭɜɟɥɢɱɢɜɚɟɬɫɹ; ɩɪɨɧɢɤɧɭɜ ɫɤɜɨɡɶ ɛɚɪɶɟɪ, ɷɥɟɤɬɪɨɧ ɛɭɞɟɬ ɭɫɤɨɪɹɬɶɫɹ ɜɧɟɲɧɢɦ
ɩɨɥɟɦ
E
ɢ ɭɞɚɥɢɬɫɹ ɨɬ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɢɨɧɚ.
116
9.6. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɚɬɨɦɧɨɣ ɫɢɫɬɟɦɵ ɜ ɩɪɟɧɟɛɪɟɠɟɧɢɢ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɷɥɟɤɬɪɨɧɚ ɢ «ɢɨɧɚ», ɫɨɡɞɚɸɳɟɝɨ
ɩɨɬɟɧɰɢɚɥ )(
r
U
, ɢɦɟɟɬ ɜɢɞ
BsslA
P
D
¸
¹
·
¨
©
§
ˆˆ
ˆ
)()(
2
1
ˆ
2
s
rrU
c
e
i
m
H
e
= . (1)
Ɂɞɟɫɶ ɦɚɫɫɚ , ɫɩɢɧ 2m
/
1
s
, ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ
Be
P P 00116.1 ɢ
ɷɥɟɤɬɪɢɱɟɫɤɢɣ ɡɚɪɹɞ 0ɨɬɧɨɫɹɬɫɹ ɤ ɷɥɟɤɬɪɨɧɭ.e
ɇɚɩɪɚɜɢɦ ɨɫɶ ɜɞɨɥɶ B . ɉɪɢ ɷɬɨɦ ɜɟɤɬɨɪɧɵɣ ɩɨɬɟɧɰɢɚɥ ɨɞɧɨɪɨɞɧɨɝɨ
ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ (
z A
AB u ) ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɜɵɛɪɚɬɶ ɜ ɮɨɪɦɟ
2/yBA
x
, 2/xBA
y
, 0
z
A . (2)
Ɍɚɤɨɣ ɜɵɛɨɪ ɜɟɤɬɨɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɩɨɡɜɨɥɹɟɬ ɜɵɞɟɥɢɬɶ ɜ ɝɚɦɢɥɶɬɨɧɢɚɧɟ (1)
ɫɥɚɝɚɟɦɨɟ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɟ ɨɩɟɪɚɬɨɪɭ ɩɪɨɟɤɰɢɢ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɧɚ
)//(
ˆ
xyyxil
z
wwww . (3)
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɪɚɫɤɪɵɜ ɜ (1) ɫɤɨɛɤɢ ɢ ɭɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ (2),(3), ɩɨɥɭɱɢɦ:
2
2
22
2
ˆ
2
ˆ
2
ˆ
ˆ
)()(
2
ˆ
Asl
mc
e
sBlB
mc
e
rrU
m
H
zez
PD'
==
. (4)
ɉɨɫɥɟɞɧɟɟ ɫɥɚɝɚɟɦɨɟ ɜ (4) ɜ ɫɥɭɱɚɟ ɞɨɫɬɚɬɨɱɧɨ ɫɥɚɛɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ,
ɨɱɟɜɢɞɧɨ, ɫɥɟɞɭɟɬ ɬɪɚɤɬɨɜɚɬɶ, ɤɚɤ ɤɜɚɞɪɚɬɢɱɧɨɟ ɩɨ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ
ɜɨɡɦɭɳɟɧɢɟ,
)(
82
22
2
22
2
2
2
yx
mc
Be
mc
e
A , (5)
ɚ ɞɜɚ ɩɪɟɞɲɟɫɬɜɭɸɳɢɯ ɱɥɟɧɚ ɜ (4) ɛɭɞɭɬ ɢɝɪɚɬɶ ɪɨɥɶ ɜɨɡɦɭɳɟɧɢɹ, ɥɢɧɟɣɧɨɝɨ
ɩɨ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ. ɍɱɢɬɵɜɚɹ, ɱɬɨ ɡɚɪɹɞ ɷɥɟɤɬɪɨɧɚ ɨɬɪɢɰɚɬɟɥɟɧ (
ee ) ɢ
ɱɬɨ
Be
mce P{|P 2/= , ɥɢɧɟɣɧɵɣ ɩɨ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɨɩɟɪɚɬɨɪ ɜɨɡɦɭɳɟɧɢɹ
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɮɨɪɦɟ
¸
¸
¹
·
¨
¨
©
§
P
P
P
z
B
e
zB
slBV
ˆ
2
ˆ
ˆ
)
ˆ
2
ˆ
(
zzB
slB P| . (6)
ɋɪɚɜɧɢɦ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɜɟɥɢɱɢɧɵ ɨɬɞɟɥɶɧɵɯ ɫɥɚɝɚɟɦɵɯ ɝɚɦɢɥɶɬɨɧɢɚɧɚ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɫɩɟɤɬɪ ɭɪɨɜɧɟɣ ɧɟɜɨɡɦɭɳɟɧɧɨɝɨ ɝɚɦɢɥɶɬɨɧɢɚɧɚ,
ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɫɨɛɨɣ ɫɭɦɦɭ ɩɟɪɜɵɯ ɬɪɟɯ ɱɥɟɧɨɜ ɜ (4), ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ
117
ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ ɦɚɫɲɬɚɛɨɦ
11
10~~
Ry ɷɪɝ. ɉɭɫɬɶ
5
10
~
B
Ƚɫ – ɩɨɪɹɞɨɤ
ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɞɨɫɬɭɩɧɵɯ ɜ ɨɛɵɱɧɵɯ ɥɚɛɨɪɚɬɨɪɧɵɯ
ɭɫɥɨɜɢɹɯ ɩɪɢ ɢɡɦɟɪɟɧɢɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɫɩɟɤɬɪɨɜ ɚɬɨɦɨɜ. Ɍɨɝɞɚ
ɷɪɝ
15
10~
P B
B
Ry
4
10~
, ɩɪɢɱɟɦ ɜɟɥɢɱɢɧɚ ɤɜɚɞɪɚɬɢɱɧɨɝɨ ɩɨ ɩɨɥɸ ɱɥɟɧɚ (5)
ɩɪɢ
222
~
B
ayx ɫɨɫɬɚɜɢɬ ɜɫɟɝɨ
20
10
ɷɪɝ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɩɟɪɚɬɨɪ (6)
ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɦɚɥɨɟ ɜɨɡɦɭɳɟɧɢɟ, ɚ ɤɜɚɞɪɚɬɢɱɧɵɦ ɩɨ
ɩɨɥɸ ɫɥɚɝɚɟɦɵɦ (5) ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɭɫɥɨɜɢɹɯ ɫɥɟɞɭɟɬ ɜɨɨɛɳɟ ɩɪɟɧɟɛɪɟɱɶ.
Ɂɚɦɟɬɢɦ ɜ ɬɨ ɠɟ ɜɪɟɦɹ, ɱɬɨ ɧɚɣɞɟɧɧɵɣ ɩɨɪɹɞɨɤ ɜɟɥɢɱɢɧɵ ɷɧɟɪɝɢɢ (6)
ɨɤɚɡɵɜɚɟɬɫɹ ɫɨɩɨɫɬɚɜɢɦɵɦ ɫ ɦɚɫɲɬɚɛɨɦ ɢɧɬɟɪɜɚɥɨɜ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ
(),ɨɛɭɫɥɨɜɥɟɧɧɨɣ ɪɟɥɹɬɢɜɢɫɬɫɤɢɦɢ ɩɨɩɪɚɜɤɚɦɢ ɤ
ɭɪɨɜɧɹɦ ɷɧɟɪɝɢɢ ɚɬɨɦɚ. ȼ ɝɚɦɢɥɶɬɨɧɢɚɧɟ (4) ɬɚɤɢɟ ɩɨɩɪɚɜɤɢ ɩɪɟɞɫɬɚɜɥɟɧɵ
ɨɩɟɪɚɬɨɪɨɦ ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ . ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ
ɧɟɨɛɯɨɞɢɦɨ ɪɚɡɥɢɱɚɬɶ ɞɜɚ ɩɪɟɞɟɥɶɧɵɯ ɫɥɭɱɚɹ: ɚ) ɩɪɢ ɫɥɚɛɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ,
ɤɨɝɞɚ ɷɧɟɪɝɢɹ ɦɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɢɧɬɟɪɜɚɥɚɦɢ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ
ɭɪɨɜɧɟɣ, ɧɟɜɨɡɦɭɳɟɧɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ ɛɭɞɭɬ ɫɨɫɬɨɹɧɢɹ
RyceRy
422
10~)/(
=
sl
ˆ
ˆ
)( D r
B
B
P
mj, ɫ
ɨɩɪɟɞɟɥɟɧɧɵɦ ɩɨɥɧɵɦ ɦɨɦɟɧɬɨɦ , ɪɚɜɧɵɦ ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɭɦɦɟ
ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
j
l
ɢ ɫɩɢɧɚ ɷɥɟɤɬɪɨɧɚ 2
/
1
s
; ɛ) ɜ ɫɢɥɶɧɵɯ ɦɚɝɧɢɬɧɵɯ
ɩɨɥɹɯ, ɤɨɝɞɚ ɷɧɟɪɝɢɹ ɜɟɥɢɤɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɢɧɬɟɪɜɚɥɚɦɢ ɬɨɧɤɨɣ
ɫɬɪɭɤɬɭɪɵ, ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɵɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɢ ɜ ɤɚɱɟɫɬɜɟ
ɧɟɜɨɡɦɭɳɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɨɫɬɨɹɧɢɹ
B
B
P
sl
msml ,2/1, , ɝɞɟ
ɢ – ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɷɥɟɤɬɪɨɧɚ ɢ ɟɝɨ ɫɩɢɧɚ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɉɛɫɭɞɢɦ ɞɥɹ ɨɛɨɢɯ ɫɥɭɱɚɟɜ ɯɚɪɚɤɬɟɪ ɩɨɩɪɚɜɨɤ ɤ ɷɧɟɪɝɢɢ,
ɜɨɡɧɢɤɚɸɳɢɯ ɜ ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ ɩɨ ɜɨɡɦɭɳɟɧɢɸ (6).
l
m
s
m z
ɚ) ɋɥɚɛɨɟ ɩɨɥɟ: B
B
P
'jnjn
HH , ɝɞɟ
'jnjn
HH – ɪɚɡɧɨɫɬɶ ɫɨɫɟɞɧɢɯ
ɡɧɚɱɟɧɢɣ ɷɧɟɪɝɢɢ ɜ ɫɟɦɟɣɫɬɜɟ ɭɪɨɜɧɟɣ ɫ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɪɚɞɢɚɥɶɧɵɦ
ɤɜɚɧɬɨɜɵɦ ɱɢɫɥɨɦ , ɪɚɫɳɟɩɥɟɧɧɵɯ ɡɚ ɫɱɟɬ ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ (ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɬɚɤɚɹ ɪɚɡɧɨɫɬɶ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɧɶɲɟ
ɢɧɬɟɪɜɚɥɚ ɦɟɠɞɭ ɭɪɨɜɧɹɦɢ ɫ ɪɚɡɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ). Ɇɚɝɧɢɬɧɨɟ ɩɨɥɟ
ɧɚɪɭɲɚɟɬ ɫɮɟɪɢɱɟɫɤɭɸ ɫɢɦɦɟɬɪɢɸ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ (ɨɫɬɚɟɬɫɹ ɥɢɲɶ
ɚɤɫɢɚɥɶɧɚɹ ɫɢɦɦɟɬɪɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ), ɩɨɷɬɨɦɭ ɫɥɟɞɭɟɬ ɨɠɢɞɚɬɶ, ɱɬɨ
ɤɚɠɞɵɣ )-ɤɪɚɬɧɨ ɜɵɪɨɠɞɟɧɧɵɣ ɩɨ ɭɪɨɜɟɧɶ ɢɫɩɵɬɵɜɚɟɬ
ɪɚɫɳɟɩɥɟɧɢɟ, ɦɚɥɨɟ ɜ ɦɟɪɭ ɦɚɥɨɫɬɢ
n
n
z
12( jm
jn
H
B
B
P , ɢ ɜɵɪɨɠɞɟɧɢɟ ɩɨ ɫɧɢɦɚɟɬɫɹ.
ɂɧɬɟɪɟɫɭɹɫɶ ɢɦɟɧɧɨ ɷɬɢɦ ɪɚɫɳɟɩɥɟɧɢɟɦ ɭɪɨɜɧɹ
m
jn
H , ɦɵ ɞɨɥɠɧɵ ɧɚɣɬɢ
ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ (6) ɩɨ )12( j ɫɨɫɬɨɹɧɢɹɦ
mj, ɫ ɡɚɞɚɧɧɵɦ
ɩɨɥɧɵɦ ɦɨɦɟɧɬɨɦ ɢ ɜɫɟɜɨɡɦɨɠɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɩɪɨɟɤɰɢɢ ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ
, ɩɪɢɱɟɦ ɡɚɞɚɧɧɵɦɢ ɹɜɥɹɸɬɫɹ ɬɚɤɠɟ ɨɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ
j
sl
mmm
l
ɢ ɫɩɢɧ
118
2
/
1
s
. Ʉɚɠɞɨɟ ɬɚɤɨɟ ɫɨɫɬɨɹɧɢɟ ɢɦɟɟɬ ɜɢɞ ɫɭɦɦɵ
¦
s
sl
m
sl
mj
msml
msmlCmj ,,, (7)
ɩɨ ɬɟɦ ɡɧɚɱɟɧɢɹɦ (ɢ ), ɞɥɹ ɤɨɬɨɪɵɯ
s
m
l
m
sl
mm ɪɚɜɧɨ ɡɧɚɱɟɧɢɸ ,
ɭɤɚɡɚɧɧɨɦɭ ɜ ɥɟɜɨɣ ɫɬɨɪɨɧɟ ɪɚɜɟɧɫɬɜɚ (7).
m
Ɂɚɩɢɲɟɦ ɩɪɢɫɭɬɫɬɜɭɸɳɭɸ ɜ (6) ɤɨɦɛɢɧɚɰɢɸ ɨɩɟɪɚɬɨɪɨɜ ɜ ɮɨɪɦɟ
zz
sl
ˆ
2
ˆ
zz
sJ
ˆ
ˆ
, (8)
ɬɚɤ ɱɬɨ
BV
B
P
ˆ
)
ˆ
ˆ
(
zz
sJ . (9)
Ɇɚɬɪɢɰɚ ɨɩɟɪɚɬɨɪɚ ɩɪɨɟɤɰɢɢ ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ
ɜ ɛɚɡɢɫɟ (7)
ɞɢɚɝɨɧɚɥɶɧɚ:
zzz
slJ
ˆ
ˆ
ˆ
mmz
mmjJmj
c
G
c
,
ˆ
, . (10)
ɉɨɤɚɠɟɦ, ɱɬɨ ɦɚɬɪɢɰɚ ɨɩɟɪɚɬɨɪɚ , ɨɩɪɟɞɟɥɹɟɦɚɹ ɩɨ ɫɨɫɬɨɹɧɢɹɦ (7), ɬɚɤɠɟ
ɞɢɚɝɨɧɚɥɶɧɚ (ɬɨ ɟɫɬɶ ɨɩɟɪɚɬɨɪ ɦɨɠɟɬ ɢɦɟɬɶ ɧɟɞɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ
ɬɨɥɶɤɨ ɞɥɹ ɫɨɫɬɨɹɧɢɣ ɫ ɪɚɡɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ , ɧɨ ɧɟ ɞɥɹ ɫɨɫɬɨɹɧɢɣ ɫ
ɨɞɢɧɚɤɨɜɵɦɢ ɢ ɪɚɡɧɵɦɢ ). ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɢɜ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɫɬɨɪɨɧɵ
ɪɚɜɟɧɫɬɜɚ
z
s
ˆ
z
s
ˆ
j
jm
¦
s
sl
m
szl
mj
msml
z
mssmlCmjs ,
ˆ
,,
ˆ
¦
s
sl
m
sls
mj
msml
msmlmC ,,
ɧɚ ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ
¦
c
c
cc
c
c
c
s
sl
m
sl
mj
msml
msmlCmj ,,, , (11)
ɝɞɟ , ɩɨɥɭɱɢɦmmm
sl
c
c
c
c
mjsmj
z
,
ˆ
,
¦¦
c
c
cc
s
sl
s
m
mj
msml
m
C
mj
msml
sl
C
cc
sslls
msmsmlmlm ,,,,
¦¦
c
c
cc
s
sl
s
m
mj
msml
m
C
mj
msml
sl
C
ssll
mmmms
m
cc
GG . (12)
ȿɫɥɢ , ɬɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɞɟɥɶɬɚ-ɫɢɦɜɨɥɨɜ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (12) ɧɟ ɦɨɝɭɬ
ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɧɭɥɹ, ɢ ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ (12) ɢɫɱɟɡɚɟɬ. ɉɪɢ ɧɟ ɪɚɜɧɵɟ
ɧɭɥɸ ɫɥɚɝɚɟɦɵɟ ɫ
mm z
c
mm
c
ll
mm
c
ɢ
ss
mm
c
ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (12) ɫɭɳɟɫɬɜɭɸɬ, ɢɦɵ
119