h k g h < u n b a b q ? k d b o i j h p ? k k h < < i e...
TRANSCRIPT
- ( )
. . . . . . . .
2000
533.9 (075) 22.333
42
. ., . ., . , . .. :
, 2000
,,
, ,,
, . « », « »
« »,,
.
,, - ,
, ,, ,
- ,.
, . - . . . .
. . , . . , . . , . . , 2000.
- , 2000.
1.
§1.
§2. ,
§ 3. ,
§ 4.
§ 5.
§ 6.
§ 7.
§ 8.
§ 9.
§ 10.
§ 11.
A 2.
§ 12.
§ 13.
§ 14. .
§ 15.
§ 16.
§17.
§ 18.
§ 19.
§ 20.
3.
§ 21. .
§ 22.
§ 23.
§ 24.
§ 25. § 26.
4. .
§ 27.
§ 28. .
§ 29.
§ 30. § 31. . § 32. .
- . § 33. § 34.
§ 35.
§ 36. § 37.
5. § 38. § 39.
§ 40.
§ 41. -
6.
§ 42.
§ 43.
7.
§ 44.
§ 45.
§ 46.
§ 47.
§ 48.
8.
§ 49.
§ 50.
§ 51.
§ 52.
§ 53. ,
, ,. ,
« » ,,
, ,,
.,
, - , ,
,.
« », «» « »,
,.
,,
,, ,
, ,
- ,.
.. 1
,, ,
. 2 , . 3
. 4
,,
. , ,,
, [1,2]. ,
5 6. 7 , , 8
.,
.
1
- , ,. -
, : - - - , (Sir
William Croocus , 1879). ,
( , ), ,. .1.
;
- -b.-b
,
b ( 13,9 )
( 3643 ).b ,
:
,
. b-c " " .
c-d ,. c-d
. d ,. d-c ,
. d- .
, .
, . ( ,
387 90 . .).
. ,.
( ,),
. −, .,
− ( ) ( )., ,
. (
. .1.
2
) ,. (
), , ,, . . .2
dn/ndv,dv, v. −
. ,, , ,
., ,
,, (
),
,
.,
( ,
: =mv2/2; , ).,
, ., −
, , ( ),- . 1 3 103 - 105 ,
2.7⋅1019 .,
. (
) ;.
.,
. :− , − .
. , , ,.
., ,
, , ( - -
) .,
, ( )
.
. .2. .
3
- - ,. : ,
. . “ ” 1928 .,
,. , -
: ,, ,. ,
, ,
, . ( ), , .
. ,,
- , .: “ ”
,. ,
,, ?
,?
, ,.
, ( ) .
( ). , ,
, - , - “ ”
, , ., , ,
« » , [3,4].
. ., , ,
, .,
. , . .,
( ). , - , ,
., , . " " -
.
4
, - .
. - ,
..
, .,
, , . ., , ,
.,
,.
( ) ., ,
. , - .
- - ,
. ,: -
,.
,.
.,
,,
.,
., ,, ,
, ,,
, “ ” - .
, ,, ,
, , “ ” ( 2.7 ), “ ”
, - .
1
1
§1.
,.
. ,, . ,
, - , , ,
,. - - ,
., , . . ,
, , ,E=(v/c)B. ,
.
, ( ) CD:
1) + CD → + D - ;2) + CD → * + D - . *
( * ). CD* * D* .
. ;3) + D → + + D - .
+ CD → + +D ; + D → + + + D
4) + D → + + D + - . ( ) + D → + CD+ + “ ” . + D → A + + + + 2
, , :, , . ,
,. “ ”
, , . , - .
, , ., :
, . “ ”,
, .
2
,, - ,
- ,.
,
, ,. ,
, - .
ν, ν, , , j r( rotation), D j ( D r).
, ,, ( ),
( ). ,
- .: , , (
), , ,, ,
, ( ), - .
1928 .. , - , :
, (). ,
, . .“ ” , ,
, . ,
mp = 1.67⋅10-24 , me = 0.91⋅10-27 , mp/me≅1836.
.me
2≅511 , m 2≅938 .
σ .
, −. , ,
B = 0.529⋅10-8 ,σ ~10-16 2.
.,
, ,.
3
, [5] :
σ πi Bla
RI
nl
u= åçæõ÷ö +
22
2 1Φ( ), (1.1)
R=( )≅13.6 – , I- , nl -
, l - , E -
, u=(E-I)/I. (u) ,,
(u>1),
( )Φ uu
u> =
++
10 57
11
0 012.
ln.
. (1.2)
. 1.1. -
.
[5]:
σπ α ω
ωκ κ
πκphB tha
Zarctg
= åçæõ÷ö
−− −
23
41 2
92 2
2
4 exp( )exp( )
, κω
ω ω=
− th, (1.3)
α = 1137 - , ω - , ,
ωth - , . ,, , 109678,758 -1.
, , ,, , , .
C ,, [6]:
σ λph [ ] . .[ ]/2 7 223 8=
nn−5.
. , () E ~ 108 / ,
E ~ 106 / .
. 1.1.
§2. ,
,.
, , ,. ,
.,
( .1.2, , ,),
. x=4πnex. n - , -
( ). ,, ; d
:
A eEdxe n
dd
= =⋅
ñ4
2
2
0
2π, (1.4)
,, , (1/2) ( ,
, k T, , ,).
dTne
=4 2π
. (1.5)
, d, ;. , d,
. - ,d,
. d ( .).
n≅1014 -3, ≅104 , d≅5⋅10-3 ..
, , ( . . 1.2).
m x eE ne xe ,= − = −4 2π (1.6)
.,
ωπ
pe
nem
=4 2
. (1.7)
, , (ω ω0) (
ωLe). , .
. 1.2.
n≅1014 -3
ωp≅6⋅1011c-1.
§ 3. ,
. ϕ
.
n neT
= −åçæõ÷ö0 exp ,
ϕ (1.8)
n0 – , ϕ - .
( ), :
14
2
2r rr e Zn ni e
∂∂
ϕ π( ) ( )= − − ,
ni,e – , Z – . ,
.
. , ,,
- .,
(1.8), , e i.
, . . |eϕ|<<Te,i,. , . .
Znoi=noe, :∂∂
ϕπ
ϕ ϕ2
2 2
4r r
rZe n T n T
T T de e i i
e i
( )( )
=+
≡ , (1.9)
dT T
Ze T n T ne i
e e i i=
+4 2π ( ) (1.10)
- - .
(1.9) ( – ) :
ϕ = −qr
e r d , (1.11)
q - “ ” .Z, q=Z|e|.
, r<<d,, (ϕ≈q/r), r>>d
. ,.
(1.9) .
, ϕ, x>>d ( .1.3).
, e= i,Z=1,
noi=noe=no. (1.8) :
dEdx
ddx
e n n en sheTi e= − = − = − åçæõ÷ö
2
2 04 8ϕ
π πϕ
( ) . (1.12)
, .,
eϕ/T<<1, :E = Eoe-x/d,
Eo - ,=0 [7]. ,
, ,.
:
dTne
=8 2π
. (1.13)
, ( , , . . 1.4).
,,
, - –
. , [8],
,. 1.4. ,
,.
(1.10). , ,
, “ ” .
., ,
e Tmm
TTe
i
e
e
iϕ0
12
=åçæ
õ÷öln . (1.14)
.,
,.
j n vi Ti=14 0 ,
“ ”
.1.3.
. 1.4. .
j n veTe Te
e= −
å
çææ
õ
÷öö
14 0
0expϕ
,
, (1.14). (1.10). , ne=ni Te=Ti,
(1.13). , ,Ti>> e, , (1.5) ( c e
), , (1.13), 2.
.
. ,, -
..
,.
, , , ,, (1.7), :
ω ω ω ωπ
ωπ
p Le Li Lee
eLi
i
i
e nm
Z e nm
= + = =2 22 2 24 4
, , ,
ωLe,Li « » « » . , (1.10) ,
:1 1 1
4 42 2 2 2 2d r rr
Te n
rT
Z e nDe DiDe
e
eDi
i
i= + = =, , ,
π πrDe,i – « » « » . ,
. :
,. « »
., ,
,, . –
, .« » , ,
. -. , 1/ωp
.
§ 4.
,.
- - , - .
, R , e2/R.n R∼n−1/3,
T,. ,
:
e n T2 1 3/ << ,
γ = <<e nT
6
3 1 , (1.15)
γ - ..
,
feT
~2
.
, , (
)f R n<< −~ ,/1 3
(1.15). ,
. ,, ND, :
N nrD D= −43
3 3 2π γ~ / . (1.16)
(1.15), , ,. ND
.. ,
, ( . §2), ND~108 >>1,. ,
~104 , , ~1019 -3,ND~0.1. ,
. , ,, . ,
.
§ 5.
, - :, , . ,
- (), (
), ( ).:
,,
. ..
- , , ( ).
,, - .
, ,, .
δ , δ ν
mM
, δ j :mM
,
M - , m - . , 2
= 4,7 , ν= 0,54 ,j= 7,6⋅10-3 .
,.
( 2, N2, O2 . .) .1.5., .
,
.
. ,
( ),.
(~m/ ) ,, ,
- . - - ( -
10-5-10-9). ( ).
- , . ..
.1.5.
- -. -
( , - 90% ),
. — (),
( )..
, ( ), .
( ),.
.
, .
" " .
( , , ) ,.
, ( . .),
,. , t
" " , . ,, t >>t . t =a/v, a -
, a v - . t:
t E= δ ,δE - .
a v E/ >> δ
a E vδ / >>1,
. , δ ,. ( , ,
, )., :
) * → + γ - ( );
) * + → + - ,,
.;
) * + → + + - . .;
) * + → + + + - ; ,* .
) * + → * + - , ( ).
+ +
, ,. . .
.,
. , , -, ,
. ,. , , , , - .
(24,5 ), Cs (3,9 ). 2-3 , : Li (5,4 75,6
).
( , , , ),, ,. ,
,. ,
., , -
,.
,. ,
( ). ( ), -
( ). ,.
( , ).
, .,
, - , ( ) . ,
. , - : + + → + +;
*., . . + 2 → + , + + → + , + 2 → 2 + .
() , ( , ).
(, , 2 2 2).
- .: ( )
( ).
, ,, : ,
+ → + +
+ + → + ,
.
§ 6.
, ..
,.
, ,.
, .
. ,.
., , , -
,.
,
. ( ), ( ).
,.
,
ρ (, . 1.6), θ,
µ,Z v:
tg(θ/2) = ρ⊥/ρ, ρ⊥ =Z Z e
v1 2
2
2µ (1.17)
ρ⊥ - , π/2.
µ . (1.17)
( )dd
dd
σ ρθ
ρθ
ρθΩ
= =å
çææ
õ
÷öö
⊥
sin sin2 22
2
. (1.18)
, δv = v(1-cosθ),
∆v = v⋅sinθ. : ∆v -, δv - . ,
δv, . δvθ ( ρ),
n dx,:
.1.6.
( )dv nvdx d= − −ñ2 1π θ ρ ρcos , (1.18) ,
( )sin22
2 22θρ
ρ ρ=
+⊥
⊥
,
:
dv nvdxd
= −+⊥
⊥
∞
ñ4 22 2
0
π ρρ ρ
ρ ρ. (1.19)
, .,
. - d,
0 < ρ < d.dv = −4πnvρ⊥
2dxLc, (1.20)
Lc = ln(d/ρ⊥) (1.21) - . Lc ,Lc ≈ 10÷20 .
λ σc:dvv
dx= −
λ,
λσ
=1
n c, (1.22)
σc = 4πρ⊥2Lc.
, σc , ( ):
σc E~
12 ∼
12T
. (1.23)
, :
τλ
σccv n v
= =1
. (1.24)
,τc ∼ T3/2. (1.25)
,.
( . 1.7): - .
:+ + ↔ + +.
( ): A ≡ B - : .
.1.7.
≠ , ,.
, , ,. ,
,,
( ).
§ 7.
,( , , , , ).
, .— (
, , ), ,, -
. (
:, . .;
). ,, .
( / )a+e→i+2e ( ) wi = kinane
i+2e→a+e ( ) wr = krnine2
na, ni, ne - , ; ki, kr - . , ,
:dndt
w wii r= − . (1.26)
,wi = wr. K,
– , :
Kkk
n nn
i
r
e i
a= = . (1.27)
,.
( / )a+γ→i+e ( ) ′ = ′w k n ji i a
i+e→a+γ ( ) ′ = ′w k n np p i e
na, ni, ne - , , ′ ′k j ki p, - .
′ = ′w wi p , (1.28) , , ,
Kk jk
n nn
i
p
i e
a=
′′
= , (1.29)
( ),
Kn nni e
a= . (1.30)
α () - :
α =+
=n
n nnn
i
i a
i
o, n n ni a0 = + , (1.31)
n0 - ( ).K [9].
, ni = n ., (1.27), (1.30) (1.31),
( )n Kn K n ni a i2
0= = − ,ni. (1.31),
:
α = − +åçæ
õ÷ö +
Kn
Kn
Kno o2 2
2
0. (1.32)
, , ni << no, n Kni ≈ 0 ,
α = ≈nn
Kn
i
o0
, (1.33)
. . ,.
, , ni >> n ,α→1.
,. , . .
, ,.
,:
( / )a+e→i+2e ( ) w k n ni i a e=i+e→a+γ ( ) ′ = ′w k n np p i e
,:
′ = =′
Knn
kk
i
a
i
p, (1.34)
,
α =′
+ ′K
K1 (1.35)
., ,
, [9], :
Kn nn
g gg
m Tee i
a
i e
a
e I T= =′å
çæõ÷ö
−
2 2
3 2
π
// ,
gi, ge, ga — , ; I -, ′ = +m m m m me e i e i/ ( ) - .
α, ,α
α π
2
2 2
3 2
1 2−=
′åçæ
õ÷ö
−g gg
m T Tp
ei e
a
e I T/
/ , (1.36)
= (ne+ni+na)T - ,. , ,
,.
(1.36) - - : (
meme/mi, mi - ), ( , , ),
.: ,
., , ,
, n, 2n2. ( ) ,
, ., ,
: , -. , , ,
. ,. ,
. , –
. - , " "
., ,
., , , ,
, .; " " .
( , . .) , - ,
, .
§ 8.
, :, , (
mp/me≅1836).:
,, .
, : ( , , ,
Z- . .), ( , -, θ- , . .).
,,
, . -, , , , ,
, , ..
(§ 51), ~0.01.
,.
1-10 / , 1-10 ., ,
,.
. , ,,
σi~10-14 2, σi~10-
15 2. λ σe i a e in, ,~ 1 , na
, λi~10-4-10-3 λ ~10-3-10-2
., . . 0.03 ,
~1 ..
,
∆ i = eEλi, (1.37)
∆ e = eEλe. (1.38) , − , .
∆ i ≅10-4-10-3 , ∆ ≅10-3-10-2 .,
,
< >≅∆ueEmi
iiτ , < >≅ −∆u
eEme
eeτ . (1.39)
(1.39)
τλ ε
e ie i
e ie i
e i
e ivv
m,,
,,
,
,,=
< >< >=
< >2, (1.40)
εe i, - . ( - ),
., ,
∆ i,, , ,
. , , ,, ( )m me a eε ,
εe ..
, ,:
mm
m u m eEm
e
ae
e e e
eeε τ≅ =
åçæ
õ÷ö
∆2 2
2 2. (1.41)
τ , (1.40), , , (1.41) εe ,
ε λe ea
eeE
mm
≅12
. (1.42)
(1/2 0.43) , [20]. (1.42)
(1.38), , (
) ,. “ ”
, ,,
. (1.42)
. ,, , :
., , . . ,
,« » . ,
,7000 – 8000 . ,
- , ,,
.,
. ,,
: “ ”
, . 2- ∼25%, NO N2 O2 6-7 .
., , . ,
,.
§ 9.
, . . ,.
, ., , ,
[8].
.. ,
- - ., ,
, .
τσee
Te c
Knv
=1
. (1.43)
σc , vTe – .σc (1.23) v T mTe e e= 3 , :
τπee
c
eeK
e Lmn
T=3 3
4 43 2/ . (1.44)
, <2.
τπii
c
iiK
Z Z e Lmn
T=3 3
4 12
22 4
3 2/ , (1.45)
Z1,2 - . τee τii: Te=Ti , Z1=Z2=1:
ττ
ee
ii
e
i
mm
= .
τee <<τii., ,
. [11] ,“α“
“β“ ()
ταβ = 3/4(2π)-1/2[Tα3/2/(eα
2eβ2Lcnβ)]mαµαβ
−1/2, µαβ = mβmα/( mβ+mα), (1.46) eα,eβ - , mα ,mβ - , µαβ - .
α β ., - ,- - . ,
:τei = 3/4(2π)−1/2[Te
3/2/(e4Lcn)]me1/2, τee = 21/2 τei, τii = (2mi/me)1/2(Ti/Te)3/2τei. (1.47)
,,
. , τei
- ( ,) , τie
. ,. ,
;, . ,
, , ,, .
- , ,.
, − ,
. ,Te 2 .
− ., Ti,
m mi e (~50 ) .Te Ti .
- -, ,
, ,, -
, , me/mi..
[10]:
τπε =
+38 2
3 2
2( )
( )
/m T m Tm m n e e Le i i e
e i i e c. (1.48)
, , ,,
, - () () .
“ ” . (1.48) i
. ,? , .
:dTdt
T T dTdt
T Te e i i e i= −−
=−
τ τε ε, .
., ,
,T T T T conste i e i+ = + =0 0 .
,, = i= . =( 0+ i0)/2.
τν = τ−1.
[12]:
λ σ
τ ν
eie
c
eei
c
ee
eie
c
eei
c
e e
TnL
Tn
LT
T
TnL
Tn
nLT
nT
= ⋅ ≈ ⋅ = ⋅ ≈ ⋅
= ≈ ⋅ = ≈
− − −
−
4 5 10 3 10 2 10 3 10
0 67 4 5 10 1 5 22
52
42
62
5 2
3 22
3 2
3 2 3 2
. , ,
. . , . ./ /
/ /
(1.49)
Lc=15,. ,
,T=108K, n=1014 -3,
λei ≈ 3⋅106 , σei ≈ 3⋅10-22 2,,
, c τei ≈ 4.5⋅10-4, τee ≈ 6.4⋅10-4, τii ≈ 0.04, τε ≈ 0.8.
§ 10.
,, , . .
, . ,, ,
. , ,, , ,
- .,
D ∼( )∆x 2 / τ , (1.50)
∆x - , τ -. ∆x λ,
,D = 1/3λvT , (1.51)
vTmT = 3 - .
η κ :η ~ mnD (1.52) κ ~ nD (1.53)
, (1.51)-(1.53). ,,
σ~π 2.λ~1/(nσ)~1/(π 2n). , (0° , 1 .)
n≅2.7⋅1019 -3, λ 10-6 , =5 A .,
. , ,. , .
– , , - ,
vx>0 λvx/v vx>0. ,
/λvx/v/. ,. ,
, , 1/6 vT., ,
( ) ( )j n x v n x v vnxx T T T≅ − − + ≅ −
16
16
13
λ λ λ∂∂
.
, . (1.51). , ,
, , , 3/2 , – , ,
( ) ( )q nv T x T x n vTx
Tx
n vx T T T= − − +åçæ
õ÷ö ≅ − ≡ − ≈
16
32
32
12
12
λ λ λ∂∂
κ∂∂
κ λ, ,
(1.53). , .. , Vy
, , -, - [5]
π η∂∂
η τyxyv
xp= , ~ ,
– , τ - . p=nT, τ=λ/vT,
η τλ λ
λ~ ~p nTv
mnTm T m
mn vT
T= =3
13
,
(1.52).
, , , ,.
.
, ,.
, ,.
(1.51) λ (1.22)
vTmT = 3 ,
DT
e L n mc
=3 3
4
5 2
4
/
π. (1.54)
, , ( ) (
D ∼ T ). .
, ,, ,
., .
( ),
., ,
, :
j Ddndx
bddxi i
ii= − −
ϕ, (1.55)
bi - , ϕ - .:
j Ddndx
bddxe e
ee= − +
ϕ, (1.56)
be - . , ,
ne=ni=ndndx
dndx
dndx
i e= = ,ddxϕ
(1.55) (1.56), ,
:
jD b D b
b bdndx
Ddndx
i e e i
i ea= −
++
= − . (1.57)
:
DD b D b
b bai e e i
i e=
++
. (1.58)
, , ,
be,i = (|e|D/T)e,i, (1.59) , , , (1.54), De>>Di , ,
,
D DTTa i
e
i= +åçæ
õ÷ö1 . (1.60)
, Di<Da<<De, ,:
, ,; -
- ., Te = i ,
Da = 2Di., ,
« » ,. « » ,
. ,,
.,
., , .
(1924) , .
κ ~ nD ~ T5/2 / L mc . (1.61)
( ). ,
~108 K, , , ,, , ,
. ,.
( . ),. (1.61), -
() ,
. [13] :
κi ≅ 3.9niTiτii/mi, κe ≅ 3.16neTeτei/me. (1.62)
η = mnD ~ mT 5 2/ /Lc. (1.63)
, ,,
. [13] :
ηi≅0.96niTiτii, ηe≅0.73neTeτei, (1.64)
.
( ),
. ( ). j
E
j Eddx
= = −σ σϕ
, (1.65)
σ - , ϕ - . (1.65) , , ,
- U = IR.
,. ,
, , = const. ( )
= −dϕ/dx , ,
, ue>>ui. . ,τei
ue, . . meue = Fτei,, ~mevTe,
., , F = e ,
mue/τei, - :eE = mue/τei . (1.66)
(1.66) ue:
j = neue, (1.67)
j = ne2τeiE/m. (1.68) (1.65), :
σ = ne2τei/me . (1.69) .
τei (1.47), :
σ ∝TLe
c
3 2/
∼ Te3 2/ . (1.70)
, , Lc,,
. , ,.
[13]:
σ σ σ= ≅⋅ − −1 96
0 9 10101
3 21
131 3 2. ,
.( / )
,/ /TL
cec
. (1.71)
ρ=σ -1.,
. ( ,, j2/σ) .
( ,)
. . ,, ~1 .
. , ,, .
, ,, ( ,
n T2 ). , [12], ,
..
(1.69) ,
. ,.
. ,, , ,
, . (~v3),
,, , .
:|e|Eτ > mevTe. (1.72)
, , « »,
.,
. " " " " . , τ≈τei,
,, " ", . .
[11]:E > E ≈ 0.214Lce/rDe
2. (1.73)
, « » > 0.1E .
§ 11.
, ., ,
. ( . §3), , ,:
ddx rDe
2
2 2ϕ ϕ
= . (1.74)
rDe – . ,( )ϕ ϕ ω~ expk ikx i t− . -
ϕ , , :
kk rDe
k2
2 211
0( ) .+ =ϕ (1.75)
dDdx
= 0 . (1.76)
- :k k k
2 0ε ϕ = , (1.77) εk - ,
. (1.75) (1.77),
εkDek r
= +11
2 2 . (1.78)
, , ,, :
,, , .
, , ω −.
− ., ,
: -,
. . ,,
., :
~ .E E ei t= 0ω (1.79)
∆
m x eE eE e
xe
mE
ei t~ ,
~.
= =
= −
0
2
ω
ω∆
(1.80)
~ ~ ~ ,D E E P= = +ε πω 4 (1.81)
= ∆x - ,. , :
εωω
ωπ
ω = −åçæõ÷ö =1
42
22
pp
e
nem
, , (1.82)
.
, ( . 3):
N 2 = ε , (1.83) N= /ω – . (1.82) ,
ω > ωp , N2>0. ,ω<ωp (1.82) ,
,.
- , .,
. ,:
εω
ωωkp
Dek r, = −−
12
2 2 2 , (1.84)
, , .
2
§ 12.
, ,,
. « » , . .
.,
, , ,- .
, , « »,
. ,- ( . §18),
,, ,
,.
:
mr qEqc
v B F= + × + , (2.1)
, ( ),
, . (2.1) ,
. , ,, ,
. , . ,, , ,
.. , . ..
§ 13.
, , (2.1)
mvqc
v B= × . (2.2)
(2.2) , ,, (2.2),
mvvddt
mv≡ =
2
20 .
, :mv
const2
2= . (2.3)
,. , , ,
.: v vB B|| ( ) /=
v v v B B⊥ = − || / ,:
mvec
BB
v B( )|| = × ≡ 0 ,
- ,,
v vB B const|| ( ) /= = . (2.4)
mvqc
v B⊥ ⊥= × , (2.5)
, , ., ,
.ρ ω.
v⊥ , ,−
,, −
m vmv q
cvB m| |⊥
⊥= = =2
2
ρρω .
ω ω= ≡qBmc
; (2.6)
ρ ρω
= ≡ =⊥ ⊥v mcvqB
(2.7)
ρ , ω - (
, , “B”: ρ ω ,,
). (2.6) (2.7) ,, ,
,, ,
. ,, ( ,
) - .j = qω/2π,
,:
µ =jc
S . (2.8)
S - ,, S = πρ2. j S (2.8),
µ µ µ= − = ⊥BB
mvB
,/2 2
. (2.9)
, , ( . 2.1).
.
« »,,
,,
.
−
..
« » :, ,
, . .. ,
,, « » . ,
,, , .
, − , , , « » .
.2.1. (
)
§ 14. .
, ,, ,
, ., , .
, , ( - guiding center approximation). , ,
, , ,. ,
rotEc
Bt
= −1 ∂
∂,
.,
,,
. « »
,. , ,
,,
:∆ ∆t T l,>> >> ρ .
:∂ ∂ω
B tB
<< 1 , ∂ ∂ω
E tE
<< 1 ,
ρB
B∇
<< 1 , ρE
E∇
<< 1 . (2.10)
, ,,
,. ,
E ., ,
δω
vqEm
TqE
mc
EB
v~ ~ ≡ << .
Evc
B<< . (2.11)
(2.10) (2.11)
. ,, .
, -
. , ,
.
§ 15.
, , F ,:
mvqc
v B F= × + . (2.12)
v F, , ,
mv F|| ||= mv Fqc
v B⊥ ⊥ ⊥= + × . (2.13)
, ,. ,
, ,. ,
, , ( )
. ,
v v vd r⊥ = + , (2.14) vd — , vr — .
, (!), . (2.14)
(2.13)
mv mv Fqc
v Bqc
v Br d r⊥ ⊥≡ = + × + × . (2.15)
− ,.
Fqc
v Bd⊥ + × = 0 .
:
vcq
F BBd =×
2 . (2.16)
(2.15)
mvec
v Br r= × , (2.17)
, (2.5). ,
, ,.
, , , :
v v vr d= −⊥| |0 , (2.18)
v⊥0 - ., v⊥ =0 0
v vcFqBr d= = ⊥ . (2.19)
:
ρ =mcvqB
r . (2.20)
,(2.16).
:
r r v tv v B
Bt
v vtd
d d⊥ ⊥
⊥ ⊥= + +− ×
− +−
00 01
( )( cos ) sin
ωω
ωω . (2.21)
.
. 2.2: , ,
:v vd⊥ =0 , (2.22)
, ,.
.
F ,
F : ()
. (2.16) , ,
, . ,, , (
!) ,.
,,
, .
.2.2.
(−
)
§ 16.
,, ( . . 2.3),
2> 1. , , ,, ,
( .2.3).
,,
.
« ». ,
, ,,
:.2.3
,,
.
.. ,
∆t T T= + = +1 21 2
πω
πω
, (2.23)
ω 1 21 2
,,=
qBmc
,
,.
∆xv v
= − = −2 21 21 2
( ) ( )ρ ρω ω
. (2.24)
v − ,. ,
vxt
v B BB B
v BBd = =
−+
≡< >
∆∆
∆2 2 1
2 1π π, (2.25)
∆ = 2− 1 − , < >=( 2+ 1)/2 −. , − ,
, :
.2.3.
, , ,!
, , ,. −
, , ,.
,, ( . .2.4).
., ,
,,
rot Bc
j=4π
, .
, .
, − ,, ,
. ,.
. – ,
« » ,,, .2.4.
,, ,
:
vxt T
vvd = = = =
∆∆
4 42
2ρ ωπ ω π
//
. (2.26)
« », , , ,, ! ,
.
.2.4.
§17.
§15
. , ,, , , ,
., ,
. (2.16). ,
,. , , .,
v vd T<< , (2.27) , ,
.,
.
F ,. , F eE= ,
v cE B
BE =×⊥2 . (2.28)
- .
(2.11), . ,
,. ,
:, , ,
. " ".
( ∇B ≠0)
,.
.• . ,
R,
Fmv
Rn= − ||
2
, (2.29)
,
vce
mvRB
vR
v BB
= = =∇|| || || | |2 2 21
ω ω, (2.30)
( , . . 2.5):
vv B B
B=
∇|| [ ]2
2ω, (2.31)
v - , ω − .
• .
,.
.
,:
F B= − ∇µ . (2.32) ,
, :∇
=⊥ BB
nR
, (2.33)
R, , - ., ,
FBR
n⊥ = −µ , (2.34)
n - .F⊥ (2.16),
vv B B
B[ ]
=∇⊥
2
22ω. (2.35)
,:
, . ,.
( ):
vv v B B
B Rv v bb =
+ ∇= +⊥
⊥
22
12
22 2
22 2||
||[ ]
( )ω ω
, (2.36)
b — ( . .2.5). (
), ,.
, ,v Bb⊥ v Bb⊥∇ .
( ∇ ≠B 0 ), , :j nevb
e i=ä
,, (2.37)
..
. 2.5.
, ( ∇ ≠p 0 ).
,F mv= − ,
(2.17). ,, .
,
~~
v cE B
Bn =×⊥2 ,
~vn - ; ~E⊥ - ,. .
~E⊥~vn , . . , , , F mvu n= − .
vce
mEBn = ⊥
2
2
~. (2.38)
vn ( ), ,.
~E⊥ - -
. - :
j nevcB
En n m= = ⊥ρ2
2 , (2.39)
ρm - .ε⊥ ,
. jn
j E= ⊥
14π
(
),
j j jcB
Em= + = + ⊥1
41 4
2
2ππ ρ( ) . (2.40)
,
j E=επ4
, (2.41)
ε πρ⊥ = +1 42
2mcB
. (2.42)
.
1/R ( .2.6)., ,
2π(R-r) , 2π(R+r), r - ( . .2.6).
, ( .2.6 - ), ( .2.6 - ) , , ,
. ,.
⊥ ,.
,
., ,, ,
.ϕ,
z,
,.
( ) ., ,
qBB
aR
z=ϕ
, (2.43)
, . . - .
, , ,,
,. :
, ,,
, .,
,,
.,
(1/r (r !)
).
,R.
, .
.
. 2.6. .
.2.7.
( «») ,
. , ( ), ,
, .,
..
, . 2.7. ( ) :
) – (). ,
.-
⊥ ,;
) - ().
« » , , :, “ ”
,
.:
( .2.7, ) ∇ ,vd ∼ B B× ∇ , ,
. . « » ( .2.7, ),
∇ ,, vd .
− ,, . . .
( ),,
, , ( -
) . ( .2.7, ).
., ,, .
:
., (
), ().
§ 18.
, ,, , , ,
, (
, .2.8)., ,
W, ., ,
. ,
,-
( , ,),
.,
..
:J W T~ < > . (2.44)
, - ( , , ,
T l g= 2π / ,), .
− .
.µ
,,
. ,
( ) .,
z ( ..2.9), B t B t ez( ) ( )= , ez - .
− = ≡1 1c t
B rot E Ez z∂∂ ρ
∂∂ρ
ρ ϕ , (2.45)
Ec
B tϕρ
= −12
( ) . (2.46)
, ,
.2.8. « »
.2.9.( - )
10
01
0
c tE rot B
div B
div E E
z∂∂
ρ∂
∂ ϕ
ϕ ϕ
ϕ
= ≡
≡
= ≡
,
,
.
(2.47)
, :
B t B B t( ) = +0 0 , Ec
Bϕρ
= −12 0 (2.48)
B0 - ( ), (), ( ); B0 - .
− , :, .
, ,, ( , , )
.
B t B t ez( ) ( )= , Ec
B t e= −12
ρϕ( ) (2.49)
(t),∆t~L/c, L - , .
µ −−
, ,− .
.2.10., = 0(1+εcosΩt); ω - , µ0 -
mv eEϕ ϕ= , mvec
Bϕ ρ= − (2.50)
, ,ddt
v vB
dBdt
ϕ ϕ2 2
2 2å
çææõ
÷öö = . (2.51)
,vB
constϕ2
= (2.52)
. vϕ→v⊥,mv
Bconst⊥ = =
2
2µ . (2.53)
, , . ( ., [11]) µ
. « » µ.2.10
.
µ
, , ( .2.11):
FBz
mdvdtz = − =µ
∂∂
|| ; (2.54)
vdzdt|| = . (2.55)
− =å
çææ
õ
÷ööµ
dBdt
ddt
mv||2
2. (2.56)
mv mvconst||
2 2
2 2+ =⊥ ,
− = −åçæ
õ÷ö −⊥µ µ
dBdt
ddt
mv ddt
B2
2( ) ; (2.57)
,µ=const. (2.58)
, , ,. . , . .
(2.10). (2.53) (2.58), ,
..
mvB
m v ce B
emc
B const B const const⊥ ⊥= = ⋅ = ⋅ =2 2 2 2
2 2
2
22
2 2ρ Φ (2.59)
.2.11.
.
, , , .,
:
ρ ~1B
, (2.60)
, .:
mv const⊥ =ρ , (2.61) .
v||⋅l ( .2.12).
, , v||,U<<v||. « »
, .
δv = 2U ( ).
δ tl
v=
2
||, (2.62)
dvdt
vt
Ul
v||||= =
δδ
22
. (2.63)
Udldt
= − ,
dvv
dll
||
||+ = 0 (2.64)
v l const|| = . (2.65) .
.2.12..
§ 19.
( )
.-
, ..2.13, .
,v ,
α(0≤α≤π).
(2.53), , v v⊥ = sinα ,
:sin2
2 2α µ
B mv= , (2.66)
- « » .
, (2.66)
– µ mv2/2,,
, α. , (2.66)
sin2α. ,
, , , v⊥ .
, , v⊥=v,, v||=0.
- ( .2.13,a), , ,
, , ,. , “ ”
( - mirrors, mirror machine). ,, ,
,, ( ) .
m 0 ( ..2.13, ),
R B Bm= >/ 0 1 ,, , ,
,sin /α ≥ 1 R . (2.67)
.2.13..
sin α ,. ,
sin /α < 1 R , (2.68) .
. αm,
= m. ,, (2.66) α αm m:sin2
22α µm
mB mv= .
, , 0 ,. (2.66) :sin2
02
2α µB mv
= .
, ,sin sin2 2
0
α αm
mB B= ,
sin sin sin2
0
2 2α α αmmB
BR= ≡ .
, « »R sin2 1α ≤ , . . ,
(2.68). , (2.67), , .
, ,. ,
, , . -
- .
, (2.67) -− , ,
, ( ) . ,
:, ,
.
( , σc ∼1
2E, . § 6),
, “ ” ., . −
, ., ,
,v⊥ − m/ 0,
. ,, (2.65),
v||.
.
sin2 αsin2 α
, ,
, , .
. , , R(t), ,
E R dc
Bt
rdrzR t
ϕ
π
ϕ∂∂
π= − ññ1
200
2 ( )
, (2.69)
ERc
dBdt
zϕ = −
2. (2.70)
R(t).
v cEB
edz
r= ϕ, (2.71)
dRR
dBB
z
z= −
12
, (2.72)
R B constz2 = . (2.73)
, ( ),
( . .2.13, , ) - ,, , ,, .
(2.59) (2.73), . (2.59) , , ,
, (2.73) , , , ,. ,
.N=nΩ ( Ω - ) , ,
Ω ~ R2l,N n R l nR l= =0 0
20
2 ,
nl n lrr
n lBB
= åçæõ÷ö =0 0
02
0 00
. (2.74)
,( )
.
“ B” ( .2.13)
, ∇B . −,
( . .2.13, ), , , , ,. « »
. ,.
- ., ,
.
. , ( . .2.13, ) – « ».
« » ,,
..
, . 2.13, .,
, E B×,
.:
.,
, . .. ,
, , , .,
, . , ,« ». ( . [11])
., , l S,
., .
.−
. ,Ω = ñ dSdl . (2.75)
,
Ω = ñ ( )BdSdlB
. (2.76)
Φ = ñ BdS (2.77) , ( !) ,
Ω Φ= ñdlB
. (2.78)
.,
. , « »
δdlB
<ñ 0 (2.79)
. ( )« ».
,: “ ” ( .2.14)
( ,
), “ ” ( .2.15) . “ ” ,
.
E B (B E )
v cEBE = . ,
. F mv rE= 2 / ,, .
( ,), vE.
,Fm , ,
., - .
, ,, , .
, ,. .
( ) .
.2.15. “ ”.2.14.“ B”
§ 20.
, , :.
,.
,, , .
, ,« » « » . ,
, , , ,. , , -
. « », .
« » .
[12], ,, . ,
. , ()
,, .
, ,, :
D D vTmTe
e
eei| | = = =0
13
λ τ . (2.80)
ρ, -, τei, ,
D x~ ( ) /< >∆ 2 τ ( . §10), :
Dei
⊥ ~ρτ
2
. (2.81)
, (2.81) ,
- .vTe ω :
ρωe
Te
Be
v~ ,
, (2.81),
DD
ei⊥ ≅ | |
( )ωτ 2 . (2.82)
ω τBe ei (), . .
. ,. (2.82)
, .D⊥/D||~1, ,
, D D Be ei⊥−
| | ~ ( )ω τ 2 . [13]:
D DBe ei
⊥ +~
( )||1
1 2ω τ. (2.83)
, [12]:
DT T
TDe i
e Be ei⊥ ≅
++
021 ( )ω τ
.
, ,. ,
. (1.54), D0∼ 5/2/n,
ω ∼ , τei∼T3/2/n,
D⊥ ∼n
B T2 . (2.84)
,, , ,
!,
,
. , ,, ,
., . ,
,, .
. ,,
:elE∼T. (2.85)
, , ( ) ,. , l,
× . ,−
v cEBE = , (2.86)
. ,τ ~ l/vE.
, . () , l
(2.85) vE (2.86),
Dx l
v lcTeBE~
( )~ ~ ~
< >∆ 2 2
τ τ (2.87)
, :
Dce
TBB =
116
. (2.88)
, ,. DB D⊥
. (2.82) (2.88),
D /D⊥∼ω τ i∼ T3/2/n, (2.89) . . ,
,. ,
D⊥ , , D , − . , « »
.
, ,, ( Q- )
, ., ,
., ,
,.
, , ,, .
( . 2.16), ..
., ,
,:
D q D= + ⊥( )1 2 . (2.90) q>1 - ,
..
, h. , v,t ~h/v,
∆ dv t~ .
vd - , , v||~v⊥~v , , (2.36),
vv v
Rv
RdB
T
B=
+⊥2 2 222
|| ~ω ω
, (2.91)
R – , ω – , v – .
∆ dB
Bvhv
vR
hv
q~ ~ ~2
ωρ , (2.92)
ρ - , q=h/2πR. – . q
, - :D q D~ 2
⊥ (2.93)q2 1+q2,
, q→0, (2.90).
,- ,
. -,
.. ,
. , ~ν −1 ~h/v,
, - :
ν ν>v
qR~ . (2.94)
,
.
, [14],
., « »,
( . 2.16, ).:
.- ,
, - .
, ,,
. , « » , - ,
:.
. , ,, . ,
,ε=r/R<<1, « »
~ε. , ,( (2.67)),
v v|| < <<ε 1 . () , , ,
n n < <<ε 1 . ,, .
.2.16. ( );
, ( )
,. .
, « », . ,
thv
hv
t~ ~
|| ε ε= , (2.95)
∆∆
∆dv t~ ~ε
>> . (2.96)
, , , (2.96),
.
« » .,
( .2.16, , ) − «». [11]
:DD
Rr
= = åçæõ÷ö >>−ε 3 2
3 2
1//
. (2.97)
« » , ,
ν ν ε ν ε< = =v
qR3 2 3 2/ / . (2.98)
, ν ν> , ,ν ν< , , ,
( . . 2.17), .
, « », ,
,−
.,
-ν ν= .
,, ,, , , .
.
. 2.17. « »
, (1.61): κ = Dn.
, κ||,κ⊥. , ,
, , ,. ,
κ||, (2.80) (1.61),
( , ~(mi/me)1/2, ) , , (2.82),
( , ~(mi/me)1/2, )., ,
,− .
,.
. ,
.
• .
, , ,, :
j E|| || ||= σ , σ σ| |
/
~= 0
3 2Te
Λ. (2.99)
σ0 - .• .
, , ,,
v cEBE = . ,
, − :,
∆e, ∆i. , , ( . . 2.2), ∆e,i
,.
Z , - Y ( . .2.18).
X.<∆e> <∆i> :
< >= = < >= =∆ ∆e Bee E
i Bii Em v c
eBm v c
eBρ ρ; , (2.100)
v cEBE = , .
“ ”
∆ ∆ ∆=< > + < >=+ ⊥
e ie im m c E
eB( ) 2
2 . (2.101)
,
P nen m m c E
Bc
EB
e im= =
+=∆
( ) 2
22
2ρ , (2.102)
ε π πρ⊥= + = +1 4 1 42
2PE
cBm . (2.103)
ρm e in m m= +( ) - ., ε⊥ ,
. , n=1010 -3, =103 ε⊥≈102.,
, ε⊥ . , (2.103) .
, ε|| ε⊥ ,. ε|| ,
., ε⊥ −
.• .
, , ., , -
. Z, Y − , X ϕ
. , -
ε⊥ . − − .
- ,
,.
() F= −∇p/n.
,
− . ,
.. 2.18
Z, - YX. -
, ; ∇pF. F Y, -
Y. j, Y, - « ».
.2.18.
, ,.
, [13], ,, σ⊥≠σ||, . . .
σ⊥/σ|| . Z=1, ,σ σ⊥ ≈ 0 5, || (2.104)
, [13]:
′ = + +⊥
⊥
Ej j
encjB||
||
[ ]σ σ
1, (2.105)
, ,, ,
′ = + + ∇ −E Ec
VBen
p Re T1 1
[ ] ( ) . (2.106)
− , ,,
, - , -, :
R n b Tn
b T bBBT e e
e
e eie= − ∇ − ∇ =0 71
32
. ( ) [ ],ω τ
. (2.107)
(2.105) , ,. ,
,
′ =Ej
||||
||σ
. (2.108 )
, , :.
, :
′ =⊥⊥
⊥
Ej
j σ. (2.108 )
,, - :
′ = =⊥
Eenc
jB jBBe ei1[ ] [ ]
ω τσ
. (2.109)
, , ,(2.108, ) (2.108, ). , .
.• .
, , , ( ), F p ne i e i, , /= −∇ .
vce
F BB
cen
B pBd =
×=
× ∇2 2 , (2.110)
., . . ,
j nev cB p
Bde i
= =× ∇ä
,2 . (2.111)
−, , -
. [15] , I
jµ
j c rot Iµ = . (2.112)
: I n= < >µ ,, (2.9),
I nmv
BB
e i= − < >⊥ä
2
22,. (2.113)
, , , ,, p⊥=nT⊥,
j c rotp BB
= − ⊥2 . (2.114)
,. , (2.114)
(2.111).
3
§ 21. .
..
,, , ,
. ,− .
( ),, ;
.
..
,. , ,. ,
“ ” − .,
, ,, .
[15], τs ( )δ
τπσ
δδ
s c D= =
42
22
, (3.1)
σ - ;
Dc
=2
4πσ - (3.2)
. t<<τs.
,, t
τs.,
. , , , L. ,
.t<<τs,
L>>ρi (3.3) Zni-ne=0.
( , ),,
.
, −. ,
,, .
, ,,
.
§ 22.
−, . ,
, ,, .
, ,.
.:
•ρ α α
α= =ää n m n mi i
i( ),
α - ( , )., mi>>me
ρ ;•
v n m v vi= ≈ä1ρ α α α
α( ),
;•
ρ α αα
q i en q e zn n= = −ä( )
| |( ) ,
|e| - ;•
j n q v=ä α α αα( )
.
, vevi,
j e n ve e= −| | ., :
• ( ):∂ρ∂
ρt
div v+ =( ) 0 . (3.4)
• ( ):
ρdvdt c
j B p F= × − ∇ +1
, (3.5)
p=pe+pi - ,
, F - ( , ),dvdt
vt
v v= + ∇∂∂
( ) −
( ). (3.5)
( . (2.1)) .
(3.5) F .
• :div E e zn nq i e= = −4 4πρ π| |( ) ;div B = 0 ; (3.6)
rot Ec
Bt
= −1 ∂
∂;
rot Bc
j=4π
.
,, (3.6).
• ( [13]):
′ = + + ×⊥
⊥
Ej j
cn ej B
eσ σ||
|| | |1
, (3.7)
′ - ,, ,
; σ||,⊥ - ( )
;1
cn ej B
e| |× - .
. ,
( .[13]).:
j Ec
v Bn e c
j Bn e
pe= + × − × + ∇ëìí
ûüý
σ1 1 1
| | | |. (3.8)
• - :p p T= ( , )ρ . (3.9)
,. , ,
= nT. n – ( ) .
« » − « » ne « »ni
p = neTe+niTi.,
Te=Ti= , ne=ni=n,
p = 2nT. (3.10)
− , ,. ,
, ,,
. ,, , [13].
. , , , ,, ρ=const , ,
(3.4), divv = 0 ,
. ,, p ~ ργ .
T ~ ργ −1 . γ −, , γ=5/3.
, , γ = +1 2 N , N=1,2,3… - .,
.
. , ,. , , ,
, (3.3) .
§ 23.
(3.4) (3.5). F ,
( )
ρdvdt c
j B p= × − ∇1
. (3.11)
,a b ,
∇ = ∇ + ∇ + × + ×( ) ( ) ( )ab a b b a a rotb b rota , (3.6), :
1 14 8
14
2
cj B rotB B
BB B p× = × = − ∇ + ∇ ≡ − ∇ ⋅
π π π( ) . (3.12)
( ),^
pB
= −2
82
πδ ττ (3.13)
; δ - , τ =BB
−
, .z, B ,
:
p
B
B
B
=
−
å
ç
ææææææ
õ
÷
öööööö
2
2
2
80 0
08
0
0 08
π
π
π
. (3.14)
:,
,−
.
pB
m =2
8π (3.15)
. (3.12),
. ,,( ) ( )( ) ( ) ( )B B B B B B B∇ = ∇ = ∇ + ∇τ τ τ τ τ τ2 .
, ,
( )τ τ∇ =nR
,
n - , R - ,
18 4
2 2
cj B
B BR
n× = −∇ +⊥ π π, (3.16)
∇ = ∇ − ∇⊥ τ τ( ) . (3.16) “ ”, ,
,, “ ”. , ,
, j ≡ 0 , (3.16)
∇=⊥ B
BnR
,
. (3.12) (3.16) ,
.,
,. R→∞,
.
βπ
= =p
pp
Bm
82 , (3.17)
., β>1, , β<1.
,, βmax<0.1.
β ~1., , ,
(3.15), ,pm / 3, .
§ 24.
, ,, . . ,
(3.4) − (3.10) .
, . ,v = 0 dv dt = 0 , , ,
. , (3.5), :
1c
j B p× = ∇ . (3.18)
. , , , j B ,,
p=const. ,, , , ,
. ,
.,
, , (3.16)
18
2
cj B
B× = −∇⊥ π
. (3.19)
z ,Bz
divB = 0
∂∂ z
Bz = 0 ,
. (3.19) , (3.18)
∇ + ∇ =⊥ ⊥B
pz2
80
π,
∂∂ z
p = 0 ,
, ,:
Bp constz
2
8π+ = .
, ,, Bz=B0. ,
Bp
Bz2
02
8 8π π+ = . (3.20)
, , , . , « » , .
(3.20) “ − ”,
,. ,
- ,, , .
(3.20) .,
.
-
Z- ∗. - r ( .3.1). :
− =∂∂ ϕpr c
j Bz1
.
, jz=const
Bc
rjzϕ π=2
,
B dl B rd B rc
J rc
r jzϕ ϕ ϕ
π
ϕϕ ππ π
π≡ ≡ ⋅ = ≡ññ0
222
4 4( ) ,
J r j rdr r jz z
r
( ) = =ñ2 2
0
π π
− r. ,, :
B dl B rd B rc
Ic
a jzϕ ϕ ϕ
π
ϕϕ ππ π
π≡ ≡ ⋅ = ≡ññ0
222
4 4,
∗ Z- , ,
, z.
.3.1. Z- : −, −
, − ( ) (pm)
I - . , ( . 3.1, ):
B r B a
ra
r aar
r aϕ ϕ( ) ( )
,
,=
≤
>
ë
ìî
íî
,
B aI
caϕ ( ) =2
.:
B aI
aA
ϕ ( ).
, [ ][ ]
[ ]=
0 2.
∂∂
πpr c
j rz= −2
22 .
, , ( r=a) ( =0),
pac
jra
Ia c
ra
B a ra
= −åçæ
õ÷ö = −
åçæ
õ÷ö ≡ −
åçæ
õ÷ö
ππ π
ϕ2
22
2
2
2
2 2
2
2
2 2
21 14
1( )
.
,
pB r B a r
aconst+ = −
åçæ
õ÷ö ≠ϕ ϕ
π π
2 2 2
28 82
( ) ( )
( (3.20)). ,,
.
pB r B r
pB r B a
const+ − − ≡ + = =ϕ ϕ ϕ ϕ
π π π π
2 2 2 2
8 8 4 4( )
(( )
)( ) ( )
.
,.
( . 3.1, ).p=2nT, ,
N ( ),
N nr drIc T
a
= =ñ240
2
2π ,
42
2NTIc
= ,
.,
. ,.
, , ,. ,
, .
,,
, .Z- ,
.− .
,, . ,
∆1J∆2J ( .3.2), , , .
,z, z,
( .3.2). , F1∼J2,, , F2∼J z. [10] ,
,
BJ
cRR
az = −åçæ
õ÷öln
8 12
,
J - .
( ),.
,z.
a,R, ∆r
.
. ( )
.
,.
.3.2.
§ 25.
(3.18) .. , ,
..
− , ..
, ,“ ”
, . ( snow-plow − )
[16], - .
« »,, , -
..
, , , , ,, , ,
, - , . ,, ,
,, . ,
( !) ,Z- ,
, z .,
, ,, − [16], :
ddt
mdrdt c
j B rB
rråçæ
õ÷ö = × ⋅ ≡ − ⋅
12
82
2
( ) ππ
πϕ , (3.21)
m -
m mra
m a= − =0
2
2 021( ), πρ ,
- , m0 -;
BI tcrϕ =
2 ( )
., , .
(3.21), ,,
! (3.21) k,.
I(t) - . , ,
. ,,
I t I t( ) = 0 ,I0 - ( ),
. (3.21) [12].
,t t ac I m= = 1 5 0
1 201 4. ( / ) / / , (3.22)
, . ( , ,
), (3.22) ( . [12], t
2 - 10 ). , t , ,.
Z- ,. 3.3. :
,,
« »nτ - .
.
. 3.3. ( , ) ( , )Z- : − Z- , − , −
, − . − , − ,−
§ 26.
, ,t<τs. , t>τs,
“ ”. ,
, ,− ,
, , ,, .
σ, ,,
, ,( ) :
∂∂t
B D B= ∆ , (3.23)
D , (3.2). , - .
lm
l D tm ~ .,
,.
, . . .
∂∂t
n div D n= ∇⊥( ) , (3.24)
D e
ei⊥ =
< >ρτ
2
. (3.25)
lp t:
l D tp ~ ⊥ ., [17]
D D⊥ =12
β , (3.26)
, ,β<1 β / 2
., ,
, . . ,. ,
,. ,
. , , (3.11) , ,
10
cj B p× − ∇ ≈ . (3.27)
, (3.8)
Ej
cv B
n ep= − × + ∇
σ1 1
2 | |. (3.28)
,=const
, z. , .
E (3.28)
rot Ec
Bt
= −1 ∂
∂,
(3.4),
1n
dndt
div v= − , (3.29)
ddt
Bn
DBn
z zåçæõ÷ö =
∆. (3.30)
, (3.27) 1
80
2
cj B p
Bpz× − ∇ = −∇ + ≈⊥ ( )
π.
,:
Bp
Bz2
02
8 8π π+ = , (3.31)
0 . , (3.29) − (3.31) β,
. , ,,
βπ
00
28
1= <<p
B,
(3.30)
B Bz ≅ −0 0112
( )β . (3.32)
β0 , ,, (
),(3.29), (3.30)
∂∂
βt
n div D n= ∇( )12 0 , (3.33)
(3.24). , , (3.32),
, , «»
. (3.33) , - .
.,
, x t~ , , ,,
ξβ
=x
D t12 max
, (3.34)
βmax - ,β ξ β( , ) ( ) maxx t f= . (3.35) f(ξ), (3.33), dd
fdfd
dfdξ ξ
ξξ
( ) + =2
0 ,
. 3.4. ( .
3.4, ). , ,.
( , ,. 3.4, ), .
,, (3.33),
. , (3.33), , .
. 3.4. :− ( -
|ξ|1/2); −
) )
4
.
, .,
( ).. ,
( ). , ,
[18,19]. .
−. , ,
, - , ,
.
( ) .,
.,
.- - .
, ,. ,
. ,, ,
, .− − :
, .
,.
§ 27.
., ,
, :ω ω= ( )k . (4.1)
vk
kkk
= ⋅ω( )
, (4.2)
vk
k( )
=∂ω
∂. (4.3)
v c< .− −
, ., , ,
. ,, ,
. :, .
:ω ω= ( , )k a . (4.4)
, , [17]. . ,
, .~, ~E B :
rot Ec
Bt
~~
= −1 ∂
∂,
rot Bc
jc
Et
~ ~~
= +4 1π ∂
∂, (4.5)
divE q~ ~= 4πρ ,
divB~ = 0 .~ ,~ρq j −
. , , (4.5),
∂ρ∂
~ ~q
tdiv j= − , (4.6)
. , (4.5).
(4.5), (4.6) ,~ exp( )− +i t ikrω .
( )−iω ( )ik ,∂∂
ω ω ωt
i t ikr i i t ikrexp( ) ( )exp( )− + = − − + ,
∂∂
ω ωr
i t ikr ik i t ikrexp( ) ( )exp( )− + = − + ,
(4.5) :
k Ec
B× =~ ~ω,
ik Bc
jic
E× = −~ ~ ~4π ω, (4.7)
ik E q~ ~= 4πρ ,
k B~ = 0 .
~ ~Bc
k E= ×ω
,
, , (4.7) . , (4.7) :
ic
k k Ec
jic
Eω
π ω× × = −( ~ ) ~ ~4
, (4.8)
ik E q~ ~= 4πρ .
( ) . ,:
j k E= ( , )σ ω , (4.9)
σ , , , ,.
( , )ε ω k :
( , ) ( , )ε ω δπω
σ ωki
k= +4
, (4.10)
− . , (4.6),
ωρ~ ~q k j= ,
(4.8) ., ,
:
( )ε
δijij
i jjN
k kk
E2 2 0− + = , (4.11)
Nkc2
2
= åçæõ÷öω
(4.12)
. (4.11), , ,
. , (4.11) :
Detk
Nk kk
ijij
i j(( , )
)ε ω
δ2 2 0− + = . (4.13)
, (4.1) , .
,.
, , ,
,− εl εtr
( l tr – longitudinal – transversal - ).
:
ε ε δ εij tr iji j
li jk k
kk kk
= − +( )2 2 , (4.14)
, , :
Detk
Nk kk N N
ijij
i j l tr(( , )
)ε ω
δε ε
2 2 2 2
2
1 0− + = −åçæ
õ÷ö = .
, , :εl = 0 , (4.15) εtr N= 2 . (4.16)
, − . , ( . .
) ,
ε ε εl tr= ≡ ,ε - .
ε = 0 , (4.17)
ε = N 2 . (4.18) , , , , , (4.10)
ε = 1 ,, (4.17), ,
, (4.18) (4.12), ω = kc .
,.
, , ,, ,
, (4.14) , ( , ,
« » , . [20]) -(4.15), « »
ε εli j
ij
k kk
≡ 2 .
§ 28. .
, ,, ,
. −.
,. - ,
( . § 11):
( )ε ω ω= −12
p / , ω ω ωp pe pi2 2 2= + (4.19)
ω − , ω p − ( ) ., ? , ,
. (!) − ( ), ,
,:
vk
v vT Te i≡ >>
ω, . (4.20)
,, , , (4.19)
. - , , ( . §3 §11)
∆ϕ =ϕrD
2 , (4.21)
rD - ., (4.21)
. (4.21) ϕ ~ eikr ,
:
ε = +112 2k rD
,1 1 1
2 2 2r r rD De Di≡ + , (4.22)
,, (4.19):
vk
v vT Ti e≡ <<
ω, . (4.23)
, ε :ε δε δε= + +1 e i .
− − ,.
.
, . ,, .
- ,,
, .
.
,, . ,
, , ,,
.: .
, ,.
., noe=Znoi=no=const,
Eo=0 v voe oi= = 0 ., ,
:
m nvt
e n Ee ee
e0 0∂∂
~ ~= − , (4.24)
m nvt
Z e n Ei ii
i0 0∂∂
~ ~= .
(4.24) , ,.
, . ,,
,~,~ ~,E v ee i
i t ikr− +ω ., (4.24) ,
, . :~ ( ~ ~ ) ( )~j e Z n v n v
i e nm
Z e nm
Ei i e ee
e
i
i≡ − = +2
0 0
20
2 20
ω.
,
σω
= +i e n
mZ e n
me
e
i
i( )
20
2 20 . (4.25)
, , , ..
(4.10), , ,, (4.19).
,.
, − :
m nvt
e n E n mv v
e ee
e e ee i
ei0 0 0
∂∂ τ
~ ~ ~ ~= − −
−, (4.26)
m nvt
Z e n E n mv v
i ii
i e ee i
ei0 0 0
∂∂ τ
~ ~ ~ ~= +
−.
, ,. ,
.. ,
! , ,:
m n v m n ve e e i i i0 0 0~ ~+ = .,
:
~ ~ ~vm nm n
vZmm
vie e
i ie
e
ie= − = −0
0.
(4.26) , ,:
m nvt
e n E n mZm m
ve ee
e e ee i
eie0 0 0
1∂∂ τ
~ ~ / ~= − −+
,
.:
~ ~,( / )
( / )j E
i Zm mi Zm m
e nm
e i
ei e i e= =
++ +
σ σω ν
11
20 , (4.27)
. ν τei ei= −1 − - , σ −. :
εω
ω ω ν= −
+ +1
1
2p
ei e ii Zm m[ ( / )]. (4.28)
, . ,, . , (4.28)
- -. . (4.26)
f n mv v
ei e ee i
ei= −
−0
~ ~
τ,
f n mv v
ie e ee i
ei=
−0
~ ~
τ,
− - , ,, « » , ., . , ,
§ 9, :
τµ
µeie ei
e i iei
e i
e i
me e n
m mm m
~ ,2 20
=+
.
, ,.
(4.24), (4.26) ,, ,
(4.20). ,,
, ,.
. , . ,- :
νei eT~ /−3 2 ., (4.24)
.,
:∇ = ∇~ ~
, , ,p T ne i e i e i ,
, ,. ,
.:
∂∂~
( ~ ),, ,
nt
div n ve ie i e i= − 0 .
, ,
:p n T ne i e i e i e i
e i, , , ,~ ,= γ ,
γ e i const, = .
∇ = ∇~ ~, , , ,p T ne i e i e i e iγ .
, «»:
m nvt
e n E T nα αα
α α α α α∂∂
γ0 0 0
~ ~ ~= − ∇ , (4.29)
∂∂
αα α
~( ~ )
nt
div n v= − 0 .
α = e i, « » − . ,, e ee = − − , e Z ei = − .
, :
~~ ~
n nkv ie
mkE
k csα α
α α
α αω ω= =
−0 2 2 2 ,
~~ ( ~ ( ~ ))
;viem
E k c Ekk
kk
E
k cc
Tm
s
ssα
α
α
α
αα α
α
αω
ω
ωγ=
− −
−≡
2 2 2
2 2 22 0 ,
, , csα « »., ,
:
~ ~~ ( ~ ( ~ ))
, ,j e n v
i n em
E k c Ekk
kk
E
k ce i e i
s
s
= =− −
−= =ä äα α
αα
α α
αα
α
αω
ω
ω00
22 2 2
2 2 2 .
, , (4.9), (4.10) :
σ ωω
δω
ωα α
αα αp q
e ip q
p q
s
p qki n e
mk kk k c
k kk,
,,( , ) ( )= − +
−=ä 0
2
2
2
2 2 2 2 , (4.30)
:
ε ω δ πω
σ ωωω
δω
ωα
ααp q p q p q
pp q
p q p
se i
p qki
kk kk k c
k kk, , , ,
,( , ) ( , ) ( )( ) ( )≡ + = − − + −
−=ä4
1 12
2 2
2
2 2 2 2 . (4.31)
(4.30), (4.31) p, q .−
, (4.14).
εω
ωα
ααl
p
se i k c= −
−=ä1
2
2 2 2,
(4.32)
εωωtr
p= −12
2 (4.33)
. ,
. (4.32), (4.33) (4.19) ,, ,
! ,, ,
, ., ω → 0 ,
εω π
γ γ γωα
αα
α α
α ααl
p
se i e i e De i Dik cn e
k T k r r→= =
= + ≡ + ≡ + +åçæ
õ÷öä ä0
2
2 20
2
2 2 2 21 14
11 1 1
, ,, (4.34)
(4.22), γ γe i= = 1 ,., , ,
. ,. ,
γ γe i, « », . , ,
, , (4.32) :
εωω ω
γα
α
α α
αl
p
e i
k Tm
≈ − +åçæ
õ÷ö
=ä1 1
2
2
2
2,
, (4.35)
[18] γ α = 3 .
§ 29.
,. ( . § 27)
εl = 0 ,
εtr N= 2
, :. .
.
(4.33), :
12
22
2 2
2− = ≡ωω ω
p Nk c
.
, ,ω ω= +p c k2 2 2 . (4.36)
( . 4.1). ,, k→∞, ω→kc,
(4.36) . , k→0,
ω ωω
≈ +å
çææ
õ
÷ööp
p
c k1
2
2 2
2 .
, (4.36)
v ck c
c= + >1 02
2 2ω
,
.. ,
, . ,, ,
vk
c
k c
c= =+
<∂ω∂ ω
1 02
2 2
,
, , .
. 4.1. .
- ω = kc
§ 30.
(4.36), , ,
, . ,.
, ( - cut off). − ,
ω ωπ
pe
e
i
n em
Zmm
= ≡ +åçæ
õ÷ö
410
2
, (4.37)
. , ,.
.
, .
- , :
δωp
c= . (4.38)
, , - ..
., , . 4.2.
:
ω ω2 2 22 0
0 0− =><
ëìí
k c xx
p , ,, .
.,
k i x→ − ∂ ,
∂
ω ω
ωx
p
f cf x
cf x
2
2 2
2
2
2
0
0=
−>
− <
ë
ìîî
íîî
, ,
, , (4.39)
f : ,.
:∂ ∂x x x x
x x
f ff f| | ,| | .
=+ =−
=+ =−
==
0 0
0 0 (4.40)
(4.39), (4.40), ., ,
, :
( )f f i
cx i
cx x
x x=
åçæ
õ÷ö + −åçæ
õ÷ö <
− >
ëìî
íî0
0
0
exp exp , ,
exp , ,
ωα
ω
β κ (4.41)
κω ω
=−
>p
c
2 2
2 0
. 4.2.
- , f0 −. α β ,
(4.40), :
αω κω κ
βω
ω κ=
−+
=+
i ci c i c
,2
.
, - .
| |α = 1 ,, , .
., ω → 0 ,
αωω
βωω
κ δ≈ − − → − = − → ≈ −1 2 1 2 0 1i ip p
, , ,
- .
§ 31. .
.,
., ,
εl = 0 ,.
, (4.19),
1 02
2− =ωω
p .
, .ω ω= p (4.42)
,.
vk k
p≡ =ω ω
(4.43)
,:
vk k
p≡ = ≡∂ω∂
∂ω∂
0 . (4.44)
, :,
., .
(4.32) :
εω
ωα
ααl
p
se i k c= −
−=
=ä1 0
2
2 2 2,
,
1 02
2 2 2
2
2 2 22−
−−
−= =
ωω
ωω
γpe
se
pi
sise i e i
e i
e ik c k cc
Tm
, , ,,
,. (4.45)
.,
,. mi→∞, (4.45)
:
1 02
2 2 22−
−= =
ωω
γpe
sese e
e
ek cc
Tm
, .
, , ,:
ω ω= +pe sek c2 2 2 . (4.46)
.
,,
:
( )ε = +mc p c2 2 2 2 . (4.46) « - »,
« »,.
, (4.46) :ω ω γ= +pe e Dek r1 2 2 . (4.47)
,.
,,
v vTe~ .
,. (4.47)
:
ω ωγ
≈ +åçæõ÷ö <<pe
eDe Dek r k r1
212 2 2 2, .
,, :
ε ≈ +mcpm
22
2.
, (4.47),
γ e = 3 . (4.47),
ω ω≈ +åçæõ÷ö <<pe De Dek r k r1
32
12 2 2 2, . (4.48)
., , ,
k rDe2 2 1<< .
(4.48) ,
. (4.48)
. ,
,, ( . .4.3):
vk
kr v v krpe De Te De,= = =∂ω∂
ω3 32 , (4.49)
vk k
vkr
pe Te
De
= ≈ =ω ω
3. (4.50)
.4.3
, .
§ 32. . -
(4.45).
v v vTe< < .,
:v vTe<< .
, (4.45) ω 2
1 02
2 2
2
2 2 22+ −
−= =
ω ωω
γpe
se
pi
sise i e i
e i
e ik c k cc
Tm
, , ,,
,.
:
ωω
ω2 2 2
2
2
2 21= +
+k c
k c
sipi
pe
se
.
,:
c T me n m
rse
pe
e e e
ei ee De
2
2 22
4ωγπ
γ≡ ≡ .
ωω
γ
2 2 22
2 211= +
+k c
k r
sipi
e De
. (4.51)
,,
, :
ω ω ωγ2 2 2 2 2 2 2 2 21 1≅ + ≡ +
åçæ
õ÷ö <<k c
TZT
k r k rsi pi pii i
eDe De, . (4.52)
. (4.46), .
, , ,.
,,
(4.51) , , , :ω γ ω2 2 2 2 2 2 2 2 2 2 1≅ + ≡ <<k c k r k c k rsi e De pi s De, . (4.53)
c c rZ T T
ms si e De pie e i i
i
2 2 2 2≡ + =+
γ ωγ γ
. (4.54)
(4.54) [18], ,,
,
γ γe i= =1 3, , (4.54) :
cZT T
mse i
i
2 3=
+. (4.55)
(4.53)
ω = kcs , (4.56)
, ( ,
!). ,
ω∂∂ρ
γ= = =kc c
p TMs s, ,
- , - . (4.55), (4.56) - .
, ( . .4.4).
, -:
v v cZT T
mse i
i= = =
+ 3. (4.57)
.T Ti e≥ ,
-. ,
. -,
« » . (4.57) ,
-:
cZTm
T Tse
ie i≅ >>, . (4.58)
,,
, «». ,
,- :
vT
mv c
ZTm
vT
mTii
is
e
iTe
e
e= << ≅ << =
3 3~ . (4.59)
,. ,
,
. 4.4. -
,, , , −
, , −. ,
~E = −∇ϕ ., ,
:| |e n T ne e e∇ − ∇ =ϕ 0 , (4.60)
, ,. : « »
, , , :
n neTe e
e=
åçæ
õ÷ö0 exp
| |ϕ.
ne0 - (). , ,
, , ..,
, ,, . Z-
:
mdvdt
Z eii = − ∇| | ϕ . (4.61)
,,
( )∂∂nt
div n vii i+ = 0 . (4.62)
, (4.60) − (4.61),
n Zne i= , (4.63) , , ,
. ,, (4.63)
( )∆ϕ = −4π| |e n Zne i . (4.60) − (4.63) ,
, - ..
ϕ ϕ= = + =~, ~ , ~, , ,n n n v ve i e i e i i i0 ,
.(4.60) − (4.63), , ,
| | ~ ~e n T ne e e0 0∇ − ∇ =ϕ ,
mvt
Z eii∂
∂ϕ
~| | ~= − ∇ , (4.64)
( )∂∂~nt
div n vii i+ =0 0 ,
~ ~ ,n Zn n Zne i e i= =0 0 ., , ,
∂∂
ϕ ϕ2
22 20
tc c
ZTms s
e
i
~ ~ ,− = =∆ , (4.65)
, , (4.56), (4.58).
. , 4.1:
4.1
ωLe ωLe k 0 krDe → 0
« »ωLe Dek r1
32
2 2+åçæõ÷ö
≈ ωLe k ( )v kvTe Te Leω krDe << 1
- kcs cs csc
ZTm
T Tse
ie i≅ >>,
krDe << 1
ωLe k c2 2 2+ c k cLe1 2 2 2+ ω c k cLe1 2 2 2+ ω
§ 33.
, ,, ,. ,
( . [20]), ,
., , ,
ω − =kv 0 . (4.66) , ,
. ,, , .
, ,.
( ).,
. 4.5, [21].
, ,. ,
, , , , ,. , . 3.5, ,
∆v ( (1)) ( (2)) , .
, , , ( ).
,, ,
« » ,. « » ,
, , « » ., ,
« », ., , ,
! ,.
. 4.5. .: « »
; (1) (2) . :
.
[19]. ,
. , , ,, ,
, . , , . .,
.,
.
§ 34.
:.
:
, « » ,
..
., , . ,
, , ., ( ,
,).
,. , ,
..
• ( ) .,
, , ,.
,.
, ( ),
:
ε ε ωω
ωπ
ωα α
ααα
α||
, ,, ,= = − å
çæõ÷ö
= ≡= =ä ä0
02
02
221
4 n eme i
Le i
ωLe i, .
, ,
,
( )B0 ( .4.6).
«0»,.
,.
,, .
.
• .
.4.6.
B0 B|| 0
, ,. ,
,:
j E E Ec
v B= ≡ + ×∗ ∗σ , ,1
(4.67)
σ - . – , σ→∞,, , .
:
E Ec
v B∗ = + × =1
0. (4.68)
− =1c t
B rotE∂∂
,
:
( )∂∂t
B rot v B= × . (4.69)
.,
( . 4.7), .
,:
Φ = =ñ ñBdS B dSS
nS
, (4.70)
n –. (4.69)
, , , ,:
ddtΦ
= 0. (4.71)
.,
, , « ».
.
• .,
, . ,,
,.
ε ε ε ωωω ω⊥ >> ≈ = = − åçæõ÷öBe i, || .0
02
1 (4.72)
. 4.7.
,,
( .4.8),−
,.
, , ,§ 17. ,
, ( . 4.9)
u u cEBe i= = .
,
, , § 20.
(2.103). , ,
εωωω ω
α
αα⊥ <<
== + äBe i
L
Be i,
,.1
2
2 (4.73)
, ( ,
). ,. , ,§ 20,
, ,.
(4.72) (4.73), ,,
:
εω
ω ωα
αα⊥
=== −
−ä12
2 2L
Be i,. (4.74)
[18]. ( ):
,||
εε
εε
= −å
ç
ææ
õ
÷
öö
⊥
⊥
igig
00
0 0 (4.75)
,« » :
.4.8.B0 B⊥ 0
. 4.9.
g B L
Be i== −
−=ä ω ω
ω ω ωα α
αα
2
2 2( ).
, (4.76)
, , z.
§ 35.
- ( « » – « »),
. − , (, ),
, , , « » ,, - -
, ,( ) « » .
,, . .
k , B E .
,. 4.10. .
,
• ( E k B|| || 0 , . 4.10, ), ,
, « »-
( ) ( . §§ 31,32).
• ( E k B⊥ || 0 , . 4.10, )
, ,. ( . § 27)
εω⊥ = ≡ ≡N N
k cv2 2
2 2
22, .Φ
, (4.73) . , (4.73)
:
. 4.10. :
− ;− ( ) ;− ( );− ( ).
ε⊥ = +12
2ccA
, (4.77)
( )c
B
n m mA
i e
=+4π
(4.78)
, ( . , 1942).
. ,
v vc c
c cA
AΦ = =
+2 2. (4.79)
, >>c,, .
, cA<<c,:
v v cAΦ = = . (4.80) –
.,
, « ».
«» − ( . 4.11),
, .,
:
ωπ
= = =kc cB
nmv cA A
iA, ,0
4. (4.81)
,,
.,
,,
.
. ,.
. ( . 4.12).
. , – « -
»., ,
. 4.11. , . . « » –
.
. 4.12.
: 1 – ( , ),
2 – ( ,), 3 – , 4 – -
,− . ( . . 4.12).
,
• ( k E B⊥ || 0 , . 4.10, ),
, :ε ω ω|| , .= = +N k cLe
2 2 2 2 2 (4.82)
• ( k E B⊥ ⊥ 0 , . 4.10, )
,:
v v cAΦ = ≅ , (4.83) :
( . . 4.13) -
, . .,
. ,, ,
, ,, , . . (
)., ,
., (4.83) :
vBnm
pnm
Bnmi i i
22 2
4 41
12
= + = +åçæõ÷öπ
γπ
γβ . (4.84)
. 4.13.
:
. 4.14. :
1 – , 2 –, 3 –
, 4 − -
( . ., β→0),; , β→0,
., ,
, , . ( . 4.14).
:. ,
,. ,
ω ω ωBe Bi≅ | |, (4.85) ,
, ( . [18]) ω ω ωLe Be≅ +2 2 . (4.86)
ω ω< ,ω ω> ,
ω ω ω< << .
§ 36.
, ,, ,
.,
, , , ,.
: , ,, ,
,, § 10.
, ,,
.:
,,, . , ,
,,
, , ,,
, .,
,
. , ,,
., ,
« », -. , , « »,
,,
.: , ( ),
« » .
., « »
,. ,
,,
. [12] ( ), ,
–
, , . .,, ,
. ,,
. ,
, ,,
.
• - ,
. ( . § 19),
[22].
.
( . 4.15). ,
,.
,
.
,. ,
« » , ,.
. « » , ,, ,
. 4.16.
β π= 8 2p B ,
( . § 23). β<1,
, β>1, – .
β~1, –
. 4.15. – : ρ -
. 4.16. – .
« », ,, .
, >B2/8π,. ,
, .,
( - ). , ,β<<1.
, « – »:
δ dl B <ñ 0 , § 19.
( ) « »
. ,,
.
• – .,
.
,
– ( . 4.17, ).
, , ,. 4.17, ,
, .
,« »
.
, - ( . 4.17, ) « » ,
.
• .
– ( . 4.18). .
.
.
. 4.17.
.
,
., , ,
, – .
., - ( : tearing instability).
,, « »,
( . § 23). « ». 4.18, ,
« » , ( . 4.19). , .
.
• (Z – )Z – – ( z) ,
.
, § 24. ,§ 25.
, , -
. , ( .
. 4.20):
p p pB
BI
ca= = =, ,ϕ
ϕπ
2
82
. (4.87)
– , I – , ,.
. 4.18. . ( ).
, ( )
. 4.19.
•,
(4.87). ( , ,
. 4.20 ), ,
.
,« »
( . 4.20, ).
Φ = = − −πa B const B a p az z z2 2 4, ~ , ~ , (4.88)
,.
, (4.87) (4.88), .
. , ,:
BB
z > ϕ
2,
ϕ - ..
• « »« » – ( . 4.20, ).
: , . 4.20, ,, – .
, ,.
Z- .« » ,
( ,).
-, « »
( . . 4.20, .). ,
. 4.20. : − ; −
; −:
,; − -
« »; −
,, , .
, , , .
, .
• . - ,
. ,.
, ( . 4.21, ).
« », . 4.21, ,
. . ,, ,
. ,. .
2π ,h, ,
,
h aBB
z= 2πϕ
. (4.89)
, « », . 4.21, , L
, , ,,
h aBB
Lz= >2πϕ
. (4.90)
. ,
.
, (4.90) .
,
, ( .§ 17), .
R ( . . 2.6), , (4.90) L2πR,
– [23]:
. 4.21.
qhR
aR
BB
z= = >2
1π ϕ
. (4.91)
q – , Bz
, Bϕ − .– ,
.,
BI
caϕ =2
,
,:
I I caR
Bz< =max
2
. (4.92)
( )., –
, . ., . ,
( )8
114
022 2π ′
− + > ′ = =p rB
q s p rdpdr
sddr
q( )
( ) , ( ) , ln . (4.93)
– ( ).
p(r) , ,, q>1 –
. ,q(a)>1, .
q(0)<1, , « » « »
,.
§ 37. .
• .,
. – ,
. ()
,
ω − =kv 0 , , ,
, .
, (§ 33). ( . . 4.22, « »
, ),.
., ,
– . , -, , ,
. , ,
( ),:
, . .,
, .. ,
(§ 33) « » ( . 4.23).
. 4.22. . 4.23.
« » ( , – – [12]),
.,
, .
(§ 17). , ,,
v v LT B~ ρ ,L – .
. , , B||z, ∇n||x, y:
ω = k vy .•
, .,
(Te→0, Ti→0) noi=noe=no, (vno≠0, nno≠0) .
:
εω ω
ωω
ω= −
+−
−=1 0
2 2
2
2
2Le Li n
nokv( ). (4.94)
( . § 29), . ωLe, ωLi ω –
,, v – .
4- , .. .
, ( ,) .
. ,
. . 4.22 , ,v 2. , : e→0,
, ,. e→0 ,
( ), , ,, . , .
ω ωLi Le2 2<< , ω ω→ → nokv . ,
, , nn<<no,ω ω δω δω δω ω≅ + ≅ + <<Le no Lekv , | | . (4.95)
(4.94) , 1 2 12
2− + =δωω
ωδωLe
n ,
. .2 03 2δω ω ω− =n Le . (4.96)
, ( ) :
δω γ γ ω ω ω= ≅ åçæõ÷ö =
åçæõ÷öi
nnn Le Le
no
o,
/ /12
21 3 1 3
. (4.97)
( n v n v Ee e n n, , , , ) e i t ikr− +ω ,
e tγ . γ. ,
( ),. ,
. , (4.94) , ,| |k vLe no>> ω , . ,
, ,
( )[ ]| | ( ) ( )k v n n vLe no n Le no= + ≅2 1 202 3 3 2
ω ω . (4.98)
, , ,, ,
..
, ( ), (4.94)
εω
ωωω
= −−
− =1 02
2
2
2Le
eo
Li
kv( ), (4.99)
, ,
γ ω≈åçæõ÷öLe
e
i
mm
1 3/
. (4.100)
, , ,.
, , ,.
. 5.1.
5
§38.
( - ).
: ñ =1
0
0t
t
Ldtδ , t0 t1
, L(q, q ,t) – , .
,,
, ( )
( . 5.1).
L = T – U, ,
L = mv2/2 – (-eϕ). B
L = mv2/2 – (-eϕ) + (-e vA /c), A -: B = rot A .
lAcepP −= , l - ,
, δS = 0, (
ñ=1
0
t
t
LdtS , )
: 0=ñB
A
ldPδ .
: ñ =B
A
ndl 0δ ,
. ñB
A
ndl
– , n – , dl –. n = const,
.,
α β (
):1
2
nn
sinsin =
βα , n1 n2 – [27].
,, , U ,
U – ,. U, :
1
2
UU
sinsin =
βα , (5.1)
, U1 U2.
,U1 U2 ( . 5.2).
,
:
][25.12][
UmvhA ≈=λ
, . ..
, :1. –
.2. , ,
, .3. ,
n ≤ 2.5. 4. , ,
.5. , ,
– ( )
( ).
§39.
,,
., , -
. 5.2. (a)
( )
U(z,r) θ.U(z,-r) = U(z,r), U r :
U(z,r) = b0(z) + b2(z)r2 + b4(z)r4 + … + b2k(z)r2k + … (5.2)
( ):
012
2
2
2
=∂∂+
∂∂+
∂∂
zU
rU
rrU (5.3)
( 02
2
=∂∂
θU ),
, :
...2)!(
)()1(...)()2
(21)()
2()(),( 22
)2(4
2''2 +−+++−= k
kkIV
kzUzUrzUrzUrzU , (5.4)
U(z) = U(0,z). . (r2/L2 << r/L , L – ),
, .r,
:
m z = -eEz = eU´(z) m r = -eEr = -erU´´(z). (5.5)
,
. ,, z,
z: 22
2
)(21
dtdz
dzd
dtzd = . ,
z = 0 U(z) = 0 dz/dt0 = 0 ( ), dz/dt = mzeU /)(2 .
z: )(2
2
dtdz
dzdr
dzd
dtdz
dtrd = , r(z)
:
0)(4)(
)(2)( '''
2
2
=++ rzUzU
dzdr
zUzU
dzrd (5.6),
.
2-U(z) r(z)
,, . .. 5.3. .
k (), ,
.,
e/m,.
z = a,z = b ( . 5.3),
:
)()(
arbrM = , r(a) r(b) .
,
)()(
'
'
1
2
arbr
tgtgG ==
γγ ( . 5.4).
[28]:
)()(
bUaU
GM =⋅ , (5.7)
-
:2
1
nn
GM =⋅ .
f1 f2h1
h2
,r1
r2 ( . 5.5):
)()(
'1
11 ar
brf = ,
)()(
'2
22 br
arf = .
, z = az = b.
(b - a) << f1, f2 , . .,
:
ñ=b
a
dzzUzU
aUf )()(
)(411 ''
1ñ=b
a
dzzUzU
bUf )()(
)(411 ''
2
. (5.8)
. 5.4. ,
. 5.5.
:)()(
2
1
bUaU
ff
−= .
:
ñ+−==b
a
dzzU
zUbUaU
aUbUbU
bUfD
)())((
)(81)
)()(
)()((
)(411
2/3
2'''
2
. (5.9)
, D > 0, . ..
:
dd UEE
fD
41 21 −
== , (5.10)
E1 E2 – ,Ud – .
– f1 f2l :
2121
111ff
lfff
++= . (5.11)
-:
îí
îì
ë
≈−=
−≈−=
⋅⋅
⋅⋅
)(
)(2
'
''
zeUeEzm
zUereErm
z
r, (5.12)
. .. U′′(z) > 0, ,
U′′(z) < 0, .
§40.
-
,
vz >> vr.
BvceF ×−= .
: Fr = -(e/c)vϕBz ( .5.6).
. 5.6. -
: Fϕ = -(e/c)(vzBr + vrBz) ≈ -(e/c)vzBr , vz >> vr.vz ,
Br ,vϕ ( ),
, Fr . ,
., BzBr -
B (Bz,Br) Br = -(r/2)(dBz/dz),:
22
22
2
2
2
4 dzdB
Brcm
edt
zd zz−= . (5.13)
( ) :
mceB
dtd z
2=ϕ (5.14)
( ), . ., ,
. :
ñb
az dzzB )( = 0, . ,
vz>>vr ( 0
2
2Umv ≈ ),
:
rUmc
eBdz
rd z
02
2
2
2
8−= , (5.15)
U0 – ., .
,Bz. , ,
.r, , -
. (
) :
ñ=b
az dzB
Umce
f2
028
1ñ=b
az dzB
Uf][
][022.0]1[1 2
0
. (5.16)
ñ=b
az dzB
mUe
cz
081)(ϕ ñ=
b
az dzB
U][
][15.0][
0
ϕ . (5.17)
I R Bz=2/3
2
2
)1(RzBm
+,
Bm – ( - ). (5.16), :
][][][
8.96][ 20
AIRU
f ≈ . (5.18 )
N :
20
])[(][][
8.96][ANIRU
f ≈ . (5.18 )
:
][][7.10][
0UANI≈ϕ . (5.19)
f = kf, k – , k =0.5÷0.7.
§41. -
,.
.
.
( . 5.7 ), α U0:
α(U0) = ∆UlE/(2U0d), (5.20)
∆U - ,, lE -
, d - .
,, .
( . 5.8).
. 5.7.
E(r) = a/r, dU(r)/dr = a/r,a = (U2 – U1)/ln(R2/R1), E(r) == (U2 –
U1)/(rln(R2/R1)), U1, U2, R1, R2 – .
« » ,, : mv2/r = qE (), . . , :
U0[ ]= q(U2 – U1)/(2ln(R2/R1)). (5.21)
:mrqarr −=− 2ϕ ,
constr =ϕ2 ,u : r =
r0+u (u << r), r0 - ,.
: 02 20 =+ uu ω , ω0
2 = (qa)/(mr02),
2π/ 2 ,π/ 2 =127.3
().
-
r =
eBvmc ( )
v m , ( )
,,
.
( . 5.9).
2r . ,α ,
,, ,
2α, .
. 5.8.
. 5.9.
δ,,
2α: δ =2r (1-cosα).
- ( . 5.10)
,,
, .
ϕ
.
( ), ,
, . . ε1 + ϕ + ε2 = 180 ( . 5.10).
,
( . 5.11).
, . α,
l = τ vcosα ≈ 2πmvc/(eB), τ = 2πmc/(eB) –
. ,
,
- -.
( ,. .) ( . 5.12).
. 5.11.
. 5.12. : 1 – , 2 – , , 3 –
, 4 – .
. 5.10. -
( ) . ,
. 500 .
,- .
. ( )., ,
, .,
, .
( . 5.13),
.
.,
,
.
.
,
.. , ,
, .-
. ( ,) ,
. – . ,
,, «3/2» ( .
§ 42). , .
.
. 5. 13. : 1 – , 2 –
, 3 – ,4 – , 5 - , 6 –
, 7 – , 8 – , 9 – , 10 –
, 11 –
.
-
.
λ = h/(mv)
(
).. 5.14.
.,
.
( 10-4÷10-5 . .),,
.
.-
, .
.,
.
. 5.14. ( ), ( ) ( ) :
– , – , D – , L1, L2, L3 – , I1, I2 – , S – .
)) )
6
§42.
d
: U(x)=U(a)dx (
∆U = 0). ρ(x) ,
-« »,
( . 6.1).
∆U= -4πρ(x),
j = - ρv. ,
(,
),,
« »,
: 00 == =xdxdUE .
Umej
dxUd 1
/24
2
2 π= (6.1)
( , mv2/2 = eU) :
3/4))(()(dxaUxU = . (6.2)
,, Ua
d:
. 6.1. (I),
(II) (III)
][][
1033.29
2]/[ 2
2/36
2
2/32
2/3 dU
dU
mej aa
e
−⋅==π
. (6.3)
- ,«3/2». :
][.]..[][
46.59
2]/[2
2/3
2
2/3
dMU
dU
Mej
i
aa
ii ==
π. (6.4)
v0,:
)/(21
4200
2
2
mveUv
jdx
Ud
+= π , (6.5)
U(x) = (mv02/2e)((±(x/xm-1))4/3-1), (6.6)
( “+” x > xm, “-“ x < xm). «» ( ) eUm=mv0
2/2
ejmv
xm π18
30= ( .6.1).
, «3/2», (
)
)(k
a
rrβ , ra rk –
:
)(
292
2
2/3
2/3
k
aa
a
e
rr
r
Um
eJβπ
= . (6.7)
, , I3/2=J3/2Sa (Sa=2πrala – .):
)(][
][][1033.2
)(
291][
22
22/36
22
2/3
2/3
k
aa
aa
k
aa
aa
e
rr
r
SU
rr
r
SUm
eJββπ
−⋅== (6.8)
- - .ra/rk [29].
:
3/22
22/3 )
)/()/(
()()(ka
k
aa rr
rrrrUrU
ββ
= . (6.9)
Ia:
)(
][103.29
)(
294][
2
2/36
2
2/3
a
k
a
a
k
a
ea
rr
U
rr
Um
eIαα
−⋅== , (6.10)
α(ra/rk) – [30]. :
2))/()/(
)(()(ak
k
rrrr
aUrUαα
= . (6.11)
§43.
-, , .
( ) dM, U0 ( . 6.2).
:
UUeMj
dxUd
−−=
02
2 2/4π . (6.12)
U(0) = U(d) = 0
[31]:
Ε0 = dψ/dξ0 < 2 , (6.13)
ψ = U/U0, ξ = x/rd,)4/( 22
0 neMvrd π= - .
:
d < (4 2 /3)rd = dm (6.14).
dm:
Um = (3/4)U0. (6.15)
. 6.2.
Um, « » Um = U0,,
4.5 . ,:
2/32
2/30 82
98 j
dU
Mej ==
π. (6.16)
,
. ,.
.,
( ),-
,. ,
, – , ,
.,
( .§37).
d < πrd, () :
2/3
2
2
2/30
3/1 492
))/(1(4j
dU
me
Mmj ππ ≈
+= . (6.17)
. (
):
),(42
2
2
2
yxdy
Uddx
Ud πρ−=+ , (6.18)
. ( .6.3),
2X,
, ,
.:
Ex = J/(2ε0v)=J(/2ε0 meU /2 0 ), (6.19)
J – (
), U0 – ,
.xeExm = ,
,x(z):
x = x0 + tgγ⋅z + pz2/2 , (6.20)
2/300
24 Ume
Jpε
= , γ -
, . .
z.
, « » x ,
:
dx/dz = 0, . . z = tgγ/p. (6.21)
,
zr0, r(z)
:
ññ ==RR d
IUd
IU
me
rz
12/1
4/30
12/1
4/304
0 ln][][
3.32ln2 ς
ςς
ς , (6.22)
I – , U0, R=r/r0 (). ,
γ z ( . 6.4):
ñ⋅+
=R
tgUmeI
mUe
dUme
rz
1 20
0
00 2ln8
2
γς
ς . (6.23)
. 6.3.
. 6.4.
( )
: γγ 22/3
0322/3
0
min
0
][][
1004.12
ln tgI
Utg
me
IU
rr
⋅≈= , (6.24)
.,
.
.
,.
,
,
( ).,
,
: 02
2
2
2
=+dy
Uddx
Ud ,
dU/dy = 0. 3/4))(()(
dxaUxU =
,:
Uxyarctgyx =+ ))/(34cos()( 3/222 . (6.25)
(U = 0) arctg(y/x) = 3π/8 = 67.5 .
, , , 67.5 ( .6.5).
. 6.5. ,
7
§44.
. 1883 ..
. ,:
ψk( r ) = )exp(12/3 rki
L, (7.1)
L3 = V – , k = 2π/L,- , . .
f(E) = )exp(1
1
TkEE
B
F−+
. (7.2)
= 0 . ,« » 1/h3,
, , « »kF: N = 2(1/h3)(4/3)πpF
3V. ,: pF = h(3n/8π)1/3, n = N/V – 1 3.
EF = pF2/(2m) = 3/22
2
)3(2
nm
π . (7.3)
E:
n(E) = 2/322 )2(
31 mEπ
, (7.4)
( ,E E + dE)
ρ(E) = dn = dEEm 2/12/322 )2(
21π
.
- :
. 7.1.
ρ(E) = )exp(1
)2(2
1 2/12/3
22
TkEE
dEEm
B
F−+π
. (7.5)
, , EF ( . 7.1). > 0
, « »EF, ,
,
. .
Wx
Wa ∞ , Wa - ., ,
Wx - EF >> kBT , - :
)exp(2
kTe
ATj at
ϕ−= , (7.6)
ϕa = Wa - EF – ,, , kB
– . A = A0D , D = (1 - r ), r –
, ,« »,
4.1204
][ 3
2
220 ==h
mekA Bπ (7.7)
- . , ( ),
:
ϕa = ϕ0 + α(T-T0), (7.8)
α = dϕ/dT|T=To = 10-5 ÷ 10-4 / – ,, .
15 350 /( 2⋅ 2).ϕa « »
« », ln(jT/T2) 1/T.
ϕa, ln(A) .
(7.6) ,
:
j = enev /4, (7.9)
)exp()2
(2 2/3
TkeTkm
nB
aBee
ϕπ
−= (7.10)
- ,e
e
mkT
vπ8
= - .
( )
,. ,
j3/2,j j ∼ Ua
3/2.Ua , ,
, ,Ua. , ,
Ua .
()
ϕ = ϕa - ∆ϕ (7.11)
( . 7.2). x
E :
U(x) = EF + ϕa - e2/4x – eEx. (7.12)
,
: eE = e2/4x2m,
xm= Ee 4/ , (7.13)
Um = EF + ϕa - e3/2E1/2. (7.14)
:
e∆ϕ [ ] = e3/2E1/2 = 3.79⋅E1/2 [ / ]. (7.15)
:
j = jTexp(e3/2E1/2/kBT) = jTexp(4.39E1/2[ / ]/T[K]). (7.16)
§45.
E(106÷107 / ),
( ), -
– « ».
,
,
,.
( )Wx,
Wa,h
:
)/)(24exp()( hWWmWD xaex −−= π . (7.17)
dx,.
:
)]()(3
28exp[)( 2/3 ζθ
π⋅
−⋅−=
EWW
hem
WD xaex , (7.18)
θ(ζ) – , ,
xaxaxa WWEe
WWW −=
−∆
=∆
=2/12/3
)(ϕ
ϕϕζ .
7.1.
7.1. ∆ϕ / ϕa 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
θ 1 0.98 0.94 0.87 0.79 0.69 0.58 0.45 0.31 0.16 0
0 < ζ < 1 θ(ζ) ≈ 0.955 – 1.03ζ 2.- :
. 7.2 –
),]/[
)/()(1085.6exp(
]/[/102.6
)/
)/()(exp(][
2/3726
0
2/32
02
Ee
eEEeE
EEe
eEBj
aa
aF
aF
aa
a
ϕϕ∆θϕϕ
ϕ
ϕϕ∆θϕϕ
⋅⋅−⋅
+⋅⋅=
=−⋅⋅=
−
(7.19)
EF – , B0=e2/(8πh), E0=8π em2 /(3he). E2,T2 - ,
.eϕa.
. ( )
, 106 / .
§46. -
,,
, ,
v u :
îí
îì
ë
=
=−=
zz
yy
axx
vuvu
Wmvmu 22 2/, (7.20)
x .ux
ux + dux :
xxB
xaBFB duu
TmkummW
TkETkhmdN )
22
exp()/exp(4 22
3
2 +−= π =
= xxB
xBaB duu
Tkmu
TkeTkhm )
2exp()/exp(4 2
3
2
−− ϕπ = xxB
x
B
duuTk
muTk
mN )2
exp(2
− =
= xB
x
B
dWTk
WTk
N )exp(− , (7.21)
N = jT/e – .:
TkdTkN
dNWW BB
u
uxx
x
x
ññ∞∞=
=
=−==00
)exp( εεε , (7.22)
ε = Wx/kBT.
: TkW By 21= TkW Bz 2
1= .
s:SBzyx TkWWWW 2=++= . (7.23)
,: mev2/2 ≥ -eUa,
Ua < 0. :
ñ∞
−
=−=
e
a
meU B
ax
B
x
B
eexaa Tk
eUIdu
Tkmu
Tkm
NuSI2
2
)exp()2
exp( , (7.24)
I – . Ua,. ,
,
. (, 1 ≈ 11600 ) ,
.,
,. w =
(jT/e)(2kBTS+eϕa). « »
, , : E < EF.W = - ϕa – (EF-E),
: w = (jT/e)( EF-E)., ,
– ( ).
§47.
( ).,
, ,.
1887 . ,,
:1.
( I[ / 2])
j ∼ I ( – 1889 .). (7.25)
2. (1899 .) –
– , ( ):
aehmv ϕν −=
2
2max ( ). (7.26)
, n :
aenhmv ϕν −=
2
2max (7.27)
3.( ) λ ,
:
λ < λ , ν > ν = c/λ = eϕa/h, (7.28)
, :λ =12300/( eϕa[ ]) = 0 . ≠ 0
, EF, .. ,
, , ( ).
4. - , 10-9 .
Y – (Y = 10-3÷10-1), j .
, –
., , ,
,.
.
,.
-U -(U + ∆U) ,eU e(U + ∆U).
. ,, - ν
ν : U0 = h(ν - ν )/e. U0,ν, ,
ν .
,
,
( ) ( hν)
( .7.3).
,,
Wa Wa - hν,
,
., ,
,
- . ,
.
-.
, ,, ν
1.5ν .:
îîí
îîì
ë
>+−
=≤−
=a
BTkhh
TB
hekT
hhTB
jνν
νν
ϕνννν
),)(
(
/),exp(
322
22
2
21
, (7.29)
. 7.3. « » ,
. 7.4. .
B1, B2, B3 – , A0.
j (kThγ ) [32]. ,
≈0 j → 0 ν . ≠ 0 ,
ν < ν , ν > νj ∼ ν 2. ν
> 0. x = hν/kT y = ln(j /T2). :
ln( j /T2) = B + F((hν- hν )/kT) = B + F(x- hν /kT). (7.30)
F =F(hν/kT) y B x hν /kT ( . 7.4).
xν .
§48.
-
,,
-.
.7.5.
.
, ,
Is.
: γe = Ns/Np.
. 7.5.
. 7.6.
. 7.7.
W, Mo, C, Be
Is/Ip.
, 5.
( . 7.6)
. (< 50 ),
. ,, ,
.
, .
, , - , ,, , - ,
, ,. ,
,..
.,
Wmax,-
, 5 - 100 Α ,
,.
-
δe=Ns/Np , Ns –
, Ns - ,
.
. 7.8.
ηe =(Ne+Nu)/Np, Ne Nu - .
γe = δe + ηe. δeWp (Kollath):
)2exp()72.2()(
maxmax
2
max WW
WWW pp
e
pe −=δ
δ, (7.31)
δemax= aeϕ35.0 , Wmax – .
7.2. δe ( . 7.7),
,.
Wmax ( , )
- ( ,
, ). δe
α ( ) α < 60:
αδαδ βcos/)0()( ee = , (7.32)
β = 1.3 ÷1.5. α, ,.
ηeWp ( .7.8) z:
))(exp(),( )( zCWzW zmppe =η , (7.33)
m(z)=0.1382–0.9211z-0.5, C(z)=0.1904–0.2236lnz+0.1292ln2z–0.01491ln3z(Hunger). ηe α:
αηαη cos)891.0
)0((891.0)( e
e = (Darlington). (7.34)
7.2.
Al Be C ( )
( ) Cu Fe Mo Ni Ta Ti W
δemax 1.0 0.5 2.8 1.0 1.3 1.3 1.25 1.3 1.3 0.9 1.4 Wmax[ ] 300 200 750 300 600 400 375 550 600 280 650
,
. 7.9.
.
-
, -: γi = ne/ni, ne -
, ni - ,
.γi = ne/ni =
je/ji,Z γi = Zje/ji.
: ,
– γ ;
–
γ .,
γi = γ + γ .-
1928 .
,
, ,.
,.
, -, Vi
ϕa:
Vi >2ϕa. (7.35)
- .,
, . ,E1, ( . 7.10).
. 7.9.
. 7.10. -
Vi – E1 ,E2. ,
: mv2/2 = Vi – E1 – E2 > 0. , E1 E2 ϕa, Vi >2ϕa.
, γ Vi -2ϕa – .
:
γ ≈ 0.016(Vi -2ϕa)[ ]. (7.36)
γ , ( ):γ (A+) < γ (A++) < γ (A+++). , ,γ Ep: dγ /dEp ≈ 0.
Vi -2ϕa >> kBT -.
: Vi -2ϕa ≈ kBTγ
, Vi < 2ϕa.-
. , (Ep) ∼1.5 ,
γ ≈ 0. (Ep < 10 ) : γ
= (Ep - (Ep) ), = const. ≤ 0.2⋅10-2 -1.γ ∼ Ep
1/2 . γ,
:.
-( , , )
. « » ( ,
, ).
-, . ,
-.
. ( , ),
, .,
. (1923 .), ,
, .
α = ni/na, ni - , , na - ,
.β = ni/n = ni/(ni+na) (β=α/(1+α)).
.
α,- :
))(
exp(kT
Uegg ia
a
i −=
ϕα , (7.37)
Ui – , eϕa – , gi/ga –
½ 2 - . ( ,
Cs, K, Na W) ,
.,
.,
eS..
,- :
))(
exp(kT
Segg a
a
ϕα −= −
− . (7.38)
8
− ..
( , ) -.
:,
, . ., ,
, ,, .
( -" "!),
,, « » «3/2»,
,.
,. ,
" ",
.,
., ,
, , .
§49.
.,
. ( )
.. ,
Ui.,
..
, ,,
, -.
U :
fi = a(U-Ui)exp(-(U-Ui)/b), (8.1)
a b − ., ,
τi = 1/νi.n, v
σi :νi = nvσi. (8.2)
λi ( , )
λi = vτi = v/νi = 1/(nσi) = 1/Si, (8.3)
Si = nσi . (8.1)
U:
Si = a (U - Ui) exp(- b(U - Ui) ) ( ), (8.4)
a b – .:
)1exp(maxmax
maxr
r
r
rr UU
UUUU
UUSS
−−
−−
−= ( ), (8.5)
Ur – , Umax Smax – ,.
N, dt : Ndt = wnadt, w – na -
. : Ndt = -dna, dna = - wnadt. ,
:
na(t) = na0exp(-wt), (8.6)
na0 – .t0 = 1/w « » .
.t0 ∼ 10-8 ÷ 10-7 ,
,,
, ..
, 10-4
. . 4.7 ,
.,
,.
. ,, Ei
, – E0. A = E0 – Ei .
( He, Ne, Ar, Xe, Kr,..)
. (F, Cl, K, Na…), ,
. :−FA = 3.4 ÷ 3.6 , −ClA = 3.82 .
Ek, A+ Ek. : e + a → a- + hγ,
X + Y + e → X+ + Y- + e X + Y → X+ + Y- .
.,
, . . .
.
,. ,
( )., ,
ud, . ( )
:
b[ 2/( ⋅ )] = ud/E. (8.7)
,
. τ
emeES τ
2= , ud = S/τ , :
e mve
b2
λ= , (8.8)
λ - , v - .
( ):
.]..[]/[64.064.0
22
]/[ 1
pE
vme
Evm
emeE
kTm
ue
ed ⋅===
λλλπ
, (8.9)
pλλ =1 - 1 . . .:
pE
mm
vme
au i
ii
iid ⋅+=
µ
λ11 , (8.10)
ai – , 0.5 ÷1, mµ - .
, « »
α, ,, .
αn: dn(x) = αndx. x:
ne(x)=n0exp(αx), (8.11)
:
α = (1/n)(dn/dx). (8.12)
Yi – ,:
Yi = (1/n)(dn/dt). (8.13)
:
Yi/α = ud
α, Yi, ud ., α(E), Yi(E), ud(E) ,
, Yi(E) α(E)( , ).
§50.
80- . ., U pd
( p – , d – )
γ, (. 8.1).
.
,
20- .
,
1) .
, ,Yi ,
: ) Ya, )Yd, )
β. ,:
Yi(E) > Yd + Ya (8.14)
:
Yi(E) = Yd + Ya (8.15)
" ". , Yd = 1/τd, τd D
λd : τd = λd2/D.
1/λd2 = (2.4/R)2 + (π/L)2 (R L − );
: 1/λd2 = (π/L1)2 + (π/L2)2 + (π/L3)2 (L1, L2, L3 −
). (8.13) (8.14)
:
dne/dt = ne(Yi(E) - Yd - Ya), (8.16)
ne = ne0exp((Yi(E) - Yd - Ya)t) = ne0exp(t/θ), (8.17)
8.1.
θ - . ,, (8.14), t
. , t (),
, . . (!). (8.17):
θ -1(E(t)) = Yi(E) - Yd - Ya = ln(n(t)/n0)/t (8.18)
, t → ∞.. ,
, .
,
. .
2) .
, .,
( ) ,
().
, -
- , ,
. .-,
( . 8.2). x ( αx -1)
., – .
.-
γ, ( )
, ., γ = 10-4 ÷ 10-2., , γ( αx -1)
. ,, :
γ( αx -1) ≥ 1. (8.19)
( αx -1). , n0,
. 8.2. ( ) ( )
,,
d,γ( αx -1) < 1, , ,
:
)1)(exp(1)exp(
0 −−⋅=
ddnn
αγα . (8.20)
µ = γ(exp(αd)-1) (8.21)
. µ < 1 , µ = 1
( ) ., ( ,
, ),, α = const E = const
d,. ,
, :
α/ = exp(-Bp/E), (8.22 )
, - , - .
, λi,,
exp(-λi/ λ ). α = Nexp(-λi/ λ ), N = 1/ λ -
1 : N = N0p, N0 – 1 , .
, λi = Ui/E , (8.22 ):
α/ =N0exp(-N0Uip/E), (8.22 )
.α γ , α/p
γE p, : α /p=f1(E/p) γ =f2(E/p).
, ,U :
1)1))()(exp(( 21 =−pdU
fpdU
f . (8.23)
(8.23) , Upd, pd = const .
. ( . . 8.1),
U pd, .α (µ = 1) (8.21)
(8.22 ), :
E/p =B/(C + ln(pd)),
C = ln( /(ln(1/γ+1))). U = Ed,pd:
U =Bpd/(C + ln(pd)),
. , p, d, E" ", " " pd ( . . p = ngTg, ng Tg -
, Tg = const, pdd), , , / , . . "
". U (pd)min :
(pd)min = ( e /A)ln(1/γ + 1), (8.24)
e ≈ 2.72 - , .U min = B(1-C)
, ( / )min = . , ,
., , ,
, . . .,
E/p.,
,( / )min .
( . .8.1). : (pd) / ( .8.1), ,
Yi U , U min.pd ( ) ( )
/ , . . U .
p d. d., . .
, α.p α .
d α,. d
- . (8.22)
( .8.1).
( )
( ) – ( ) ( 10-5
)., ,
. , 1900 .,
(8.19) . .
., ,
.
§51.
– ,,
..
( ). -.
,
. .
. 8.3, 1- ,
1-3 , 1 ; 2 - ;
3 - ; 4 -
( );– -, -,
.1
.: ) , Uk,
200 ÷ 700 ; ) - :, ;
- : ,,
, , ; ), " ",
. , " ", ,.
. 8.3.
Ua (10 ÷ 20 ) ( ).
: .dk,
. dk,p, , pdk
Uk ( !).
Uk. – .,
( ), Uk. " " :
, - 180°,
. " " ( . 8.4). " ", ,
, ,.
, ., ,
« » ,, .
,,
. « », .
, ,, ,
" ". (ρ ≈0) . E ≈ 0
.. .
– ,.
. ," " ( )
. : SS = I/j , : I -
, j – « »,.
. I ( ,R ε) S
, ( ).
Uk .I Uk - " "
,
. j = const - .
.8.3 ,
-,
,,
.,
, U = ε - IR – Uk - Ua.
,.
,
,
,
.
,, . .
.
.
, .
. 8.4. J, U, ,
je, j+,ne, n+ ρ = e(ne - n+)
., " "
– . , ( ) , , ,
. e = 1 ÷ 2 , (
), .
– ., ,
. ,, –
.,
(20 ÷ 30 ),,
.,
., ,
, , .
., , , .
" " – , ,
., . –
" " " ".( )
, ( arcing, " "), ,
Ti - ,Ti . ,
δne↑,δj↑ (j = neev), δσ↑ (σ = nee2/τ)
δw↑ (w = j2/σ). δ g↑,δng↓ ( pg = ngTg
), δE/ng↑,δYi↑, δne↑ – :
δne↑ → δj↑ → δw↑ → δ g↑ → δng↓ → δE/ng↑ → δYi↑ → δne↑ → … (8.25)
j g, . " -" (
δTg↑).
, .
. Ia = Sven ee41 (Ia
> I), , (Ia < I) . ,
( p j), ,
.
,
- . (
, , )
(
( . 8.5), ),
.
~ 450 ÷ 500
, ,, . .
. , 300 ,.
., ,
,. ,
( . 8.6). 30% , 70% ,
. ( ), CO2 10
. 8.5. 02 -: 1 – , 2 –
, 3 –, 4 – , 5 – , 6 –
, 7 – , 8 –, 9 –
" " 80 – 100 / . , (
): " ", .
– : " "
Ti ,.
, ,, ,
, .R ( . . 8.3), -
, : . ,,
( ~ 100 ) , ,.
, ,
.
§52.
() ,
(jk ≥ 10 ÷ 102 / 2). 1802 . ( . .),
., .
, : ) ; ) ; ) ; ); : ) (p ≤ 10-3 ÷ 1 ); ) ( ∼
1 ÷ 5 ); ) ( > 10 ). :
.8.6. 2; - (
); ,; - ,
,
1) – , ( ) ; 2) ,
;( Ti ); )
, .
,, -
( ) ( ) ( ).
,. " "
, (" 3/2" ) .
10-20 , .
. ( ) (
).,
.- .
, 2 ÷ 9 .
( ) , - ( ),
. ( ) ( Al, Mg).
,,
. , j ∼ 103 / 2
∼ 3 (Ek ∼ 106 / ). 108 / 2
,
.
−: ,
( ), .
. ( 108 / 2!),, .
, . 1903 .
, ,j, (!) Hj × ... .
: (j ∼ 106 / 2), ∼ 107 / (
) - , ,j
. ( ),
( ). - .
( " " ),. -
" ", .
" " , .[33] , , " " −
. " "-
( ) ( ).
,, . (p ≤ 0.1 )
(I ∼ 1 )(Te > Ti)
., ,
p ≥ 1 ,
()
. 8.7.
,T ~ (10 ÷ 12)⋅103
.
−ne ~ exp(-r/r0) (
(σ ~ ne)),. , r ( . 8.7),
. I,r0 R, w ,
"min w". I R
. 8.7. σ .
- σ (r)
r0 , (, " min w " , ).
, ( . .8.7) ( )
,. (p ≥ 10 ) ( ≥
12000 ) , 90% .
.
- , , ,, . -
j ≤102 / 2 , (1 ÷ 3 ) ( ). :( . .), ( )
( ) j = 102 / 2. (!), ,
( , ,...). .
, (, ), .
( )
, -.
, .−
,( ). (
ε),,
, ..
1802 ., , ,, . ,
, , .: .
, ,. 19-
:
. , . " " (− I, − I)
( , ,, ) 20- . XX .
, .,
( 12000 ~ 5000 ). 10 (!), 10 ,
( ) ., ,
(« »), . , ( ≈ 3500 ) ( ≈ 4200 ).
§53. , - -
,: (" "),
.XVIII . . . , . .
: −" " ( " "), − (+), −
(-). , (+) (-)... , (+) (-)!..
, , ., () .
− ,.
, ,, ,
, E.0, − ,
(). ( 108 / )
,. ,: 1) (E ∼
E0); 2) . ,
, ,− . ,
− ( − ). (, ,
...) ,
, - ( . 8.8),
,
,,
.
,
" ",.
. [34].
.
−,
( − ,− , , . .).
" "( , ). −
( " "), , ,
α ( ) () ( ),
ñ −+=−1
0
1 )1ln())()((x
n dxxax γα , (8.26)
x1 − , ,: E ≈ 0. ,
, x1. " " - ( " "),: ,
( . 8.9). , ,. ,
− , ., . . .
, ( );
- ., , ( )
. 8.8. ,,
, , ,
.
− 104
, − 106 −
,.
, ,,
,
,, .
" " .−
, ,, .
.
( )
- (10-1 ÷ 102 ) E~ H~ −.
E~ ( ) , ,- (
,
−).
(
),E~
−.
( H~ - )40- .
, , , - . ,
. 8.9. 2 150 125
; - , , - ,
; -
. 8.10. R,; r0 - ,
–
, H~ ., ,
. ,−.
:( , , − )
. − (!), ( . 8.10).
H~ , E~
. E~ ,. ,
, ( ),, .
:
<S> = ( /4π)< > , (8.27)
( ).
( ),
(- /δ), δ −,
:
δ2 = 2/(2πσω) , (8.28)
σ −, ω − .
,
R δ.δ < R,
, δ,.
σ. 8.11,
,,
"".
,p
( !) < >,
. 8.11. ( ), ( )
( ) ; - σ (r)
, J – , S0 - , δ - -
: < > ~ IN ( I − , N −).
-
( ). ( )
, . -, , -
90% [35]. - . 8.12.
, ( .8.12)
( ):
− ( - ).
, ( !)
(,
(8.19), E = f(p) - ,
/p), [33].
− , . 1963 .
30 3⋅10-4 , ≈6⋅106 / . ,
. (8.17), Ya Yd
− ,, . " "
( ). ,−
, .,
1976 . " " 8 ,.
. 8.12. ,H01 : )
, ;)
.1. . . . – :
. , 1984. 2. Zhdanov S.K., Kurnaev V.A., Pisarev A.A. Lectures on Plasma Physics. M: MEPhI, 1998. 3. . ., . . .
: . , 1999. 4. . ., . . . : , 1994. 5. \ . . . \, .2. :
, 1990. 6. . \ . . ., . .\. :
, 1991. 7. . . . .: , 1978, .132, 2.23. 8. . ., . . . .:
, 1988. 9. - . . . - .: , 1964. 10. . . . - .: , 1961. 11. . . . - .: , 1996. 12. . ., . . . -
.: , 1997. 13. . . . - .: , 1963, .1, .208-209.14. . ., . . . - .: , 1973. 15. . . . - .: , 1946, .432.16. . ., . . - , 1956, 3.17. . . . - : , 1976. 18. . ., . ., . . .
- : , 1978. 19. . . . - .: , 1977, .11-23.20. . ., . . ( : «
», ). - : , 1979. 21. . . . - : . , 1996. 22. . . . -
.: . , 1958. 23. . . . - .:
. , 1958, .224. . . , , ,
198025. . ., . .,
, , 1989 26. . ., . .
, . 1972. 27. , , , 1957, .6428. . ., . ., , , 1959, .12529. . ., , , 1956, .13830. . . , , 1971.31. . ., . ., , , 1966. 32. . ., . ., ,
, 1992, .35233. . . . : , 1992.34. . ., . . . : , 1975. 35. ., . . . : 1984.