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O.ll.HHaKOBbI, TaK KaK OTHOCHTeJlb-
A, A2
caeuienne
Hoe
I
I
CJlY'lalIX OnHO H TO )f(e.
I
cnenyer,
1.JTO
CTOT
H
I
I
I
I
OJI
I
,
«aaaparuoxy
H3
I
Macc
M1
H
M 2,
~=~_{M2 ~2
I
I I
Ml'
"'2 -
roaops,
cnenyer
TaK)f(e H3 nOJlY'leHHbIX
panee ssr
1.JTO,
_!!!!> -Ii M,
I
sa 6bITb
BeJlHlJHHbI. 06paTHolt OTHOUleHHIO
+-=------>~
I
nOJl)f(HO
OJ2
I
I I I
1
OTCIO.ll;a
OTHOUleHHe
KO pmo
paBHO
I
-111
'laCTHU B 060HX
1
KCTaTH
p a)f(eHHIt nJiSI 9THX 'laCTOT.
q>
Ii
r. 50.
Ii r. 51. cxas BeTBb pacr aer, ~
HH)f(HlISi BeTBb 06bIlJHO uasuBaeTClI
aKycmu'teCKo(J semesto;
1- pasaoaecaoe noaoaceaue: (k = 0; ~1 = ~~); 3 - TO »ce, 'IT
y6blBaeT, ~2
Ul3'
lKeHHlI
oTpHUaTeJIbHbIM.
nOJIOCbl
•
~
C1
xapaxrep, .llJISl
(00
TaKHM 06pa30M,
npa
00'> 003
> (03),
COCe.llHHe
COOTBeTcTBylOlUliR OnTH'leCKOR Be
zrexcaunrx BO BTOpoR no BOJIHbI pasna 6eCKOHelJHO sacrauu KOJle6JIlOTCfI B npornaorp .nJIHHa
x aaryxauas SlBJllIeTCSI B03paCTalOlUeR epy Ha qmr. 52 npaaenensr xpusste 3aBHCHMO
'laCTIi BOJIHOBOrO 'lHCJIa
noxysaex
nOCTOllHHa
a
H nOCTOSlHHOR 3a
§ 16. Ilepexon OT pemer cocroamea 113 .lJ,ByxaToMHbIX M K pemerxe H3 O.lJ,HOaTOMHbIX M
a
k
aaryxa aarex uasuaaer
nOCTOSlHHall
v=w/21t'.
H .lleRCTBHTeJIbHall qaCTb pasua ayJIIO. • DOCKOJIbKY , B KalK.lloR H3 nOJIOC HenponycKaHHH Jl.eRCTBH."", 'laCTb
52.
'laCTOT,
k=21t'ad=l~,
TeJIbHaH
~
CTH 'laCTOT .llBHlKeHHe HOCHT OJl.HHaKOBblR
C1
2=0.
orxyna
sin -2 ch 2- 0•
'f
K HyJIIO no Mepe npH6JIHlKeHHH 00 K 0)2, a J
Hall
e.
uenponycxanaa,
MaKCHMaJIbHorO 3HaqeHHSl H
T. e. Jl.JIlI lJaCTOT,
nonoce nenponycxanus, xaanpar aupa (15.12) .nOJIlKeH 6blTb Jl.et!CTBHTeJIbHbIM, HO MeHbwe ayJIll, ronops. aupaxenae (15.12) Jl.OJIlKHO 6blTb 'lHCTO
MHHMbIM, T.
<jl II r.
supaacenae,
cOOTBeTcTBylOlUHX BepXHeR HHalJe
II
..
Ul2 .lleRCTBlI
,/ /
1/
\
CJIe.nOBaTeJIbHO, .lleRCTBH
CTaHOBlICb
qaCTOT
n sacror
COOTBeTcTBylOlUHR aKycTH'leCKOR BeTBH, a B03paCTaeT;
a
MeHbwe e.nHHHUbl,
DPH YBeJIHqe
.llBHlKeHHe HOCH
WI'
(15.13)
TeJIbHall lJaCTb panna' 1/2d. Korna 3HalJeHHe lJaCTOTbl JIelKHT MelK.ny 0>2 H Ul3'
crosuiee
lJaCT-HUbl;
Heno.nBHlKHbl,
I
1t
06JIacTH qaCTOT Ul1
TOJIbKO
rsxexue
002
(15.12)
~ orxyna 2" = 2"' 2=0. B
KOJIe6JIlOTClI
CTOTe
.neRCTBHTeJIbHbIM,
C1
sacrs k = 2'rwd paana 1t',
TeJIbHall
6blTb
a
H
MeHlIeTClI
sacrs. TO .llJIH c06TBeTcTBylOlUHX 9THM
TOJIbKO ero MHHMall
nOJIOCaM
lIMeTb MeCTO 3aTyxaHlIe BOJIH. ¥TO'lHH/d xapaKTep 9TOfO aaryxanua.
'laCTOT Jl.OJIlKHO
Peiuerxa H3
H3
nnyxarosnux
O.llHOaTOMHbIX
peusr
B r JI.
3;
MOJIeKyJI,
MOJIeKyJI aecsu
caoncraa KOTO
.lleRCTBHTeJIbHO, B 060HX cny
rana, 'ITO lJaCTHUbl pacnoncosenu na pasau OT .llpyra
H
orpanaxaaaxaci,
B3l\HMo.neRcTB
B
4TO B OllHOM CJIy4ae MbI HMeeM ana pa3JIH4HbiX copra
TOM,
4aCTHU M 1 H M 2 , TOr na . KaK B npyroa CJIy4ae TOJIbKO OllHH. npelleJIbHbIti nepexon OT peurerxn H3 llByxaToMHbiX MOJIe KyJI
K
peurerxe H3 OllHoaTOMHbiX MOJIeKyJI
0)1 ~
CTOTa
a 0)3 ~ 0)2.
0,
Torna
TaKHM
KaK
4aCTOTa
06pa30M.
B
TpeMH pa3JIH4HbIMH cnocooaxa. a HMeHHO npennonaras. 4TO: 1) Macca M CTpeMHTcH K HyJlIO; 2) Macca M 1 cTpeMHTcH
0)2=0)3=
2
KaK
H
y
peurerxn
H3
OllHoaTOMHbiX
d).
(paBHbIM
d, OKa3bIBaeTCH anaoe MeHbwe. T. e.
B TpeTbeM CJIy4ae nepaon
d1 = d/2.
MOJIeKyJI
Mbi censac paccMoTpHM see TpH CJIy4aH.
l , Mz
CTpeMHTCJI
Ilpn
K HyJIlO.
9TOM rpaHH4HaH 4aCTOTa
V
0)2=
V
M2
-~/-2U"( 1 J! 1 M
H
0)3 -
llBH)I{eHHe
Ka)l{llaH
H3
He
HMeeT BOJIHOBOrO
M2
4aCTHU
OT apyrnx 4aCTHU,
xapaxrep
KOJIe6JIeTCH C xacro
T. e. peurerxa CBOllHTCH
HH4ecKHx
OCUHJIJIHTOpOB,
C npyroa.
lfaCTHUbl
M1
COBepweHHO
He
npHBOllHT K nOSlBJI
lleticTBYIOW,HX na 4aCTHUbi M 2 , HO He n B llBH)I{eHHH; aMnJlHTYlla KOJIe6aHHIt, CJIellOBa
rrpeauurars 3Ha4eHHH. pasaoro d, d/2, n raex, 4TO 4aCTHUbi M 2 1l0JI)I{Hbi acer.aa OC
M1
HbIMH Me)l{llY nByMH coceanaxn 4acTHuaMH ECJIH He orpaHH4HBaTbCH BaaH~.lOlleltcT
OCTaeTCH HeH3MeHHolt.' Torna KaK 4aCTOTbi
.. (2U~
H
.. (2U" V M~'
=
_ .. (2U~ 0)1-
OCT
OllHO 3Ha4eHHe 4aCTOTW. paBHoe
MO)l{HO COBepwHTb
K 6eCKOHe4HOCTH; 3) Macca M 1 cTpeMHTcH K Macce M 2•
B nepsux llBYX CJIY4aSlx nepnon OKa3bIBaeTCH TaKHM )I{e.
0)2
npene.ne Mb
1
+ M1 )-V~+2 2
-
0)1
0)2
CMe)l{HblMH
4aCTHuaMH,
a
pacnpocrpanari,
paCCTOHHHe Me)l{llY 4aCTHuaMH, paanoe d, TO CJIy4ae Mbf nOJIy4HM uen04KY CBH3aHHblX rapM JlHTOpOB, IWTopaH oyner npencraaaan, c0601t
CTpeMHTCH K
6eCKOHe4HOCTH. TaK 4TO WHpHHa nOJIOCbI npony
CKaHHH (0)2' 0)3)
0)3-
cTpeMHTCH
K HyJIlO . .lleltcTBHTeJIbHO. pa3HOCTb
0)2=VO)i+0)~-0)2=0)2[Yl +(~y -1]
(16.1)
Il pn
9TOM cuoryr pacnpccrpaaarsca CHrHaJ
HOlt c06CTBeHHolt 4aCTOTe H30JIHpOBaHHoro OC Jle)l{HT B nonoce npcnycxanas). HH)I{Helt
CK aHHH, KOTopaH HMeJlaCb B CJIy4ae peurerx MOJleKY JI,
npu 1l0CTaT04HO MaJIbIX 3Ha4eHHHX
1 2
0)2-
nOCKOJIbKY
)I{e
"'2 -
2
3. M 1 xon
"'2 .
3Ha4eHHe 0)1 ocraercs KOHe4HbiM.
B
y6blBaHHH MaCCbI
He
0)3 -
0)2
TaK)I{e
MeHHeTCH.
TaKHM
M2
noaoca nponycxauas
B
CTOpOHy BbICOKHX 4aCTOT. H Ilonoca nponycKaHHH HH)I{HHX
0)3)
4aCTOT
cTpeMHTCJI K peuretxu
H3
M 2• B
9TOM CJlY4a
nayxarounux
MOJlel
npOHCXonHT H3MeHeHHe nepHOlla CTpyKTypW: C
M1
> M 2 HMeeT
nepnon
d,
ror.na KaK nepn
4aeMolt B peay.n.rare npenem.noro nepexona
cyacaercs. cnsaraacs npeneae HC4e3aeT COBceM.
(0)2'
OT
OllHoaTOMHbIX MOJleKYJl HeCKOJIbKO 60Jlee CJl
a 0)2 CTpeMHTCH K 6eCKOHe4HOCTH, TO pa3HOCTb CTpeMHTCH K HyJIIO. I1TaK, no Mepe
reneps He 6YlleT.
CTaHOBHTCH nopsnxa
(W- 1)2- 1 wi
M2~ 0
npa
M2
06paaOM, Mbi nOJIy4aeM peurerxy
H3 OllHHaKOBblx 4aCTHU, HMCIOW,YIO nepaon c0601t .P! \_~/ /~-" x"
'_v'k,-n ........
KaK
-,
'
,
\:Y
'I!
I
-
e·- nOJlOlKl1TeJIbHaSi
rne
e),
BeJIl1ql1Ha, MHoro MeH
-L ,
-- ~
)
M 2 = M1 (1
x
1 "
_""/ \~
",
.....
'
,_/
A,
a
11;
! I 1
1
1/1
_1!i
'If
a
-2 1
;I. k, I
rr
a
21
k
-1f,
,I
2 H r. 55. a
aKYCTHlJeCKall BeTBb; (j - ormorecxaa BeTBb. CTpenKaMH YKa3aHO HanpaBneHHe pacnpOCTpaHeHHH BonHM.
C
Ta6JI.
2)
OT
COOTBeTcTByeT
-7t
BOJIHe
no C
+7t;
I1HTepBaJI
aMnJII1TY.uo"
C,
a
(-7t/2,
'1r
-"2
0
D.
3TOT
Ilpa
'1r
! ... _-:!+. 0
_!!
2
H
+21 2
yCJIOBl111
aupaacenne
(15.3)
1- M - 2+ M1
{
1
+ (MM 2)2+ 2 MM 2cos 2k 1
1
BOJIH
_ £ =+= V2+2(1-£)cos 2k1 2 cos k 1
r. 56. I1JII1, TaK KaK 2
OTHOWeHI1S1 AdA 2 .uJISI aupaxenae (15.3) 11 qmr. 50]; 11306palKeHbi aa qmr. 57, a B npen
noaeneuae
[CM.
HaMI1 xpaaue
1
2 cos k1
cos- k 1 = 1 +
cos
2k 1 ,
qmr. 53.
I1CCJIe.uOBaJII1
Tl1nOB
9TOM
+'/[
3TO n03BOJISIeT 061>S1CHI1Tb COOTBeTCTBl1e MelK.uy pa3JII1qHbIMI1 BeTBSlMI1 KpI1BO" aa
noayseauue
Al A2
I
---o........
-st
pa3JII1qHblX
M, (1+KHAKOCTH, n panna p. TaK KaK nOTOK paaen
KH.n.KOCTH. IToAo6HbiM ureaae KYlUettcSI
3HaKOM .n.ett
qaCTHUY·
In. n+1 = V" (Yn+1 - Yn)' Yn = Ae i (Olt-kn).
r
JlHqHH 3aTyxaHHSI. CMblCJI 3Tott CKOpOCTH
SlqettKe. 3HaK MHHyC CBSl3aH
(n
r
.. / -
cpe.n.HeD.
Ha .n.ettcTBHTeJlbHyID qaCTb CKO
n-tt
A~
V" k £TA2sln22
CHJly.
C TeM lPaKTOM, qTO MOIUHOCTb norJlOlUaeTCSI
TaK KaK
paaen .n.aHHott
B3S1TOtt C 06paTHblM
pOCTH qaCTHUbl, Haxo.n.SllUettcSI B a CHJla
OT
BblqHCJleHHH
.n.ettCTByIDllI,yID na He l)-tt S1qettKH. TO nOTOK 9 p r HH
0603HaqHTb
npOH3Be.n.eHHID
CTBHTeJlbHott qaCTH CHJlW
SllJettKH.
npn nepexo.ae
+
BpeMeHH OT
3HepfHH K nJlOTHOCTH 3HeprHH. 0603HaqaSl lJaeM
(20.7)
'2 .
nOTOK
ennnnuy
panna
2A2V".
n o T•
-
""
n-ID
supaxenne
ITO,llCTaBJlSISI CID.n.a
(20.5)
xaanpar na
sin "2 = En o T ••
pe3YJlbTaTe cpe.n.HSISI nJlOTHOCTb 3HeprHH
9TO
. + e-tk)(iwt =
V" A20)
M0)2A2
= ~.
"2 Re (lYn) (lYn)
V" A2
=-
..
:t
f
1
w2 C k, nonysaex
CTOTbl
B
--------;.;-
yHKUHH BeTcTByIOII.I,Ht\ et\ cnexrp lJaCTOT
nOJlyqeHHble AJlll Hero pe3YJlbTaTbl, Bo06II.1,e rosopa, BO MHorOM He aaaacar HH OT q>OpMbl BOJlHOBOrO naxera. HH 01' xapaicrepa OrH6aIOllJ,e1L 91'11 pesy Jlb1'a1'bl CBOAllTCll K CJI~AYIOllJ,HM nonoxe HHSlM: KpHBall, onHCblBalOllJ,aSl pacnpeAeJleHHe aMnJlH1'yA, nepe MeII.l,ae1'Cll oea HCKameHHi'l C BnOJlHe OnpeAeJleHHo" CKOpOCTblO;
BpeMeH
C(t)=C(-t).
Torna ooparnue
npe06pa30BaHHH opMaul1" orH6alOllJ,e" csasano. O'leBHAHO. C 01'CyTCTBHeM aaTyxaHHll BOJIH. C APyro" CTOpOHbl, oTcyTcTBl1e HCKameHH" HMeeT MeCTO TOJlbKO B TOM cayxae, «orna BOJIHOBO" naxer npeACTaBJlSleT
+00
C (t)
=
f
B ('I) cos 2rr:'1t a-
-00
c060" cynepnoaHUHIO MOHOxpOMaTH'leCKI1X BOJIH. '1aCTOTbI KOTO pblX aaKJlIOtJ.eHbl B oueus Y3KOM nurepaaae. TaK, llanpHMep.
00
f
B ('I) =
C (t)
cos
2rr:'1t dt,
-00
Oeu60WUjQR
-, -
Ug
f·"·'; ."
.\
H
r. 60:
B npe,nbl,nyIUeM caysae Mbl I1MeJlH TOJlbKO ABe BOJIHbl C '1aCTO ti.'1 H '10 - S», 3aKJlIO'leHHbIMH B KOHe'lHOM HHTep TaMI1 '10
+
aane 2ti.'1. AHaJlOrHtJ.Hbie >Ke pe3YJlbTaTbi MO>KHO nOJlY'II1Tb. pac CMaTpHBaSl BOilHOBoe 06paaOBaHI1e, cocroauiee 113 60JlbWerO '1HCJla BOJlH, npH yCJlOBHH, 'lTO HX xacroru TaK>Ke JIeIKaT B HeKOTOpOM MaJlOM aarepsaae ti.'1. llpyrHMI1 CJlOBaMI1. cnexrp opMl1pyeTcll, npl1'leM 9Ta Aeq>opMaUHll C Te'leHl1eM BpeMeHI1 CTaHOBI1TCll ace 60Jlee 11 60JIee 3aMeTHo".
cyurecrnyer
C (t) 11 B (z), KOpOTKO MOlKeT 6blTb Bbl pax Cal-Hall C (t) u ueem cnetcm.p «acmom B (' 6yoem u uems cnetcm.p «acmom. C ('I). I1cnoJlb3YH q>yHKUHIO C (t) AJIH onacana
-00
+
raio
B('1)=B(-'1). B3aHMHaH
B ('I)
orxyna, B 4aCTHOCTH. cnenyer, 'lTO q>yHKUHe"
8
3aK. 201.
CI1TeJIbHO
xepea
BepTI1KaJIbHotl OCI1. rrpoxo.nHllletl
O.
1t'llt
HaqaJIO KOOp
cnenyer 113 (21.9). cnexrp 8 1 ('I) MO)J{HO orrpe.neJII1Tb. naxoaa noaycyaxy opnauar xpasux. nonyuaeuu x 113 Kpl1BbIX 8 ('I) rrapaJIJIeJIbHblM nepenocoa OTHOCI1TeJIbHO OCI1 qaCTOT ua BeJIl1ql1HY -+- '10'
.nI1HaT
TO.
KaK
Cnexrp
C (t)KoHeqeH. HO BO BpeM co .no co. XOT5I BpKO Bblpa)J{eHHbltl MaKcI1MyM. Ooparnue )J caaaeuue QlYHKUI1HMH 8 (t). 6y.nyT KOHeqHbl yrOJIbHbltl cnraaa H rpeyr-o.n.aun Cl1rHaJI). oyner neor paaasen. DOKa)J{eM. qTO CHrHaJ KOHeqeH, pacnpocrpanserca TaKHM )J{e 06p JlHpOBaHHaH BOJIHa. onpenenaeu as COOTHOllJe
c
l±L :L1\ -~
a
CTOTOtl '10
V,
a
,~ a
_v,
a
61. a _ cnexrp; 6 - om6alOII\all caruaaa
~'I v
a
onpene.raerca
.nBH)J{eHHe
TOqKH
npn
B
ormcuaarouiea
QlopMyJIe.
o
npa
t
V,
(MOn.y JlHp0BaHHali
-
2 'II
(21. 7) 11 BbI O. noaysaea, rro.nCTaBJIH aupaxcenae sepea QlYHKUI1I
=
x
8 1 ('I)
ee +00
11 ero TpaHCQlOpMaHTbl
C 1 (t)
f
=
+00
8 1 ('I) COS 211:'1 t d» = 2
< 'II'
TpeyrOJIbHbltl
I'll> 'II
f [8 ('I
'10)]
cos 21
reneps npasenennue
asnu
+
8 ('I -
'10) +
o (21.10)
PaCCMOTpHM
Il pennonoxax,
qTO
> 'II'
cnexrp
1
8('1)={
1'1 I -;
npa
npn
I'll < 'II' I'll> 'II
81
00
=
'10
If.
f 0
-00
sin 21t'llt 21t'l1t •
TOqK
COOTHOllJeHI1H
rrpl1
CHfHaJIa
rrOJIyqaeTCH .nJIH Cl1rHaJIa
C (t)
.nBI1)J{eHl1e
'It aa ['1t-a('1)x]. rne BOJIHOBO cooreercrayioiuee paccaarpaaaeuoa cpene, sacroru 'I.
BMeCTO
1'1 I
I1M
peWeTKI1 C a6cUI1ccotl
3aMeHI1Tb
BOJlHa).
• 8('I)={ 1
C (r),
QlYHKUl1etl
[C
ne
)J{eHl1e IlP0l13BOJIbHOtl TO'lKI1, I1MelOllletl
Yql1TblBaH
Ilpaaene» naa npaaepa cnr-aana C (t) ¢ypbe 8('1) (QlI1r. 61). I. DpHMoyrOJIbHbltl cnexrp
cneaano auure
C 1 (t). Haaneu supaacenae, aocu
QlYHKUl1etl BI1.na
i;;
-1-""'=
6
KaK 9TO 6bIJIO
qTO 3aBI1CHMOCTb OT BpeMeHI1 aMrrJIHTy.nbl
I
v, v
+
OT -
npennoaarart.,
_~
H f.
Cl1fHaJIOB
rrpOCTHpalOTCH
H BBe.1leM HOBylO nepexeuayio p.
'I 8*
= '10+p..
+v ,
f
C 1 (t) =
B (p.) COS 2" (\10
+ p.) t dp..
(21.13)
-v. 9TO Bbl
pazcenne
Bepxnsa KpHBaSl, rtpencraansnouraa c060t\ rpaipux
'I
V g = a(i =(i =V, rpynnosaa H ljJa30BaSi CKOpOCTH paauu npyr npyry. aomeuae )l{e (22.13). onpenenmouiee xapaxrepacmvecxua .naHC, coanaaaer C 06blllHblM Bblpa)l{eHHeM .nJISI Z. T. e.
§ 23.
noreps
HMne·
TaK KaK WI
=
HaM
B
CMe)l{HblMH
2
BOJIH B MeCTe coe.nHHeHHSI
nayx peurerox,
COCTOHT
llaCTHU,
H3
o.nHHaKOBbIX
noaeneune
Ka)l{.naSi H3 KOTOpblX
pacnOJIO)l{eHHblX
z
-~ -T2 . Z
na
paBHbIX
r,
.uJISI
peurerox,
3JIeMeHTapHaSi SllleUKa KOTOpb
llaCTHQbl,
OTHOllleHHe
T
MO)l{HO
aanasascs
KaKHM-JIH60 KOHKpeTHblM
xapaxr
aMnJIHTy.n
paCCTOSlHHSlX npyr OT npyra. TIyCTb Ka)l{.nall H3 3THX
nsyx peuierox xapaxrepaayer cs W l H W 2 , rpynnoBblMH nOTOKaMH 3HeprHH O
TaKHM 06paaoM, OKOH4aTeJIbHO HMeeM
~~[ +~ ~ (Yn+p-Yn)Up1 +12(Yn+p-Yn) 2"J Up .
U=U o
-
E""II,
n 1»0
-
=
1-
~
A2
"
E n oT, ="2E=2d ~ Up{ p>O
nJIOTHOCTb nOTeHUHaJIbHOIi 9HeprHH paBHa nOTeHUHaJIbHOIi 9Hep rHH 3JIeMeHTapHoli SI4eIiKH. OTHeceHHoli K ee .n.JIHHe .n.JISI
MOHoxpOMaTH4eCKoli
SIBJISIeTCSI
BOJIHbI
.n.BH)I(eHHe
rapMOHH4eCKHM. TO B cpe.n.HeM aa
d.
Ka)l(.n.oli
TaK KaK
rne E -
4aCTHUbI
nepHO.n. KOJIe6aHHIi
HMeeM
nOTOKa CJIeBa
H. CJIe.n.OBaTeJIbHO • .n.JISI
cpenuea
nJIOTHOCTH nOTeHUHaJIbHOIi
snep
nupancenae
OT co
~ .1'-
1
2
(24.1)
E n oT, = 2d ~ Up(Yn- Yn+p)'
(n > 0).
4aCTHUbI
p>O
B
CJIy4ae MOHoxpOMaTH4eCKoli BOJIHbI,
snpaao,
Yn = Ae i
CTOpOHbI
4aCTHU
C
In.n+p' .n.eIiCTBYIOw,aSI na n-IO 4aCTHUy, panna
npHHSITbI
npeaome
060aHa4eHHH. a HH
onyuien,
HOMep 4aCTHUbI B H4e1iKe,
(Yn - Yn+p)2
= TaK 4TO
2A2 Re (I -
= ~
aroanon peurerxa
Re (Yn - Yn+p)(Yn - Yn+p)* =
e :?'k p)(1 - e~'k P) = A2 (I - cos kp).
B peayJIbTaTe BWpa)l(eHHe .n.JISI cpe.n.Heli
Ha
2 (4.2)
pac co
In. ntp = U; (Yn+p - Yn) 3.n.eCb
H
oTpHUaTeJIbH qaCTHUbI,
CHJIa
pacnpOCTpaHSIIOw,elicSi
(wt-kn)
pacnpocrp
.n.JISI
Y4HTbIBaSi BJIHSIHHe
BaaHMo.n.elicTBylOw,He C HHMH
-
no.
scex S aexoropoa .n.aHHolt 4acTHUbI. xo OTC4era (n = 0). MbI MO)l(eM Halt
9HeprHH,
sa Ha4aJIO
utne
rHH nOJIy4aeM BWpa)l(eHHe
nJIOTHOCTb
saanaoneacraae pacnpoc
WHe paccToSIHHSI, YCJIO)l(HSIeT BbI4HCJIeHHe nO nbITaeMCSI HaliTH
(Yn+p - Yp) = 0
sa nepaon
cpe.n.HSISI
TOT <paKT. 4TO
onuoa 4aCTHUbI. Cpenuaa aa nepnon
CHJIe.
onpeneaaercs -
nJIOTHOCTH no
nOCKOJI
Ka)l(.n.aH 9JIeMeHTapHaH SI4e .
nOTOK 9HeprHH, COO
Bblpa)l(eHHeM
.
1
In,n+p' Yn =-"2 Re (/n • f1 +
TeHUHaJIbHOIi 3HeprHH npHHHMaeT BH.n.
ElIOT,
=
A2
~
"
2d ~ Up (1- COS kp). P>O
(24.3)
3HaK MHHYC BJiHeT Ha
CHJIy.
n-IO 9
CBHaaH
4acTHUy,
3aK. 201.
C TeM,
.n.eIiCTBYIOW,YIO a He
co
4TO
BeJIH4HH
CTOpOHbI
Ha060pOT.
(
TeJIbHO,
Y'lHTblllaSl
HSBeCTBble
Bblpa)l{eHHSI ./lJISI
Yn
Yn
H
Ug
21 R e (In, n\ p
A 2U" *)• Yn - -2-PR e (-ikp e .
Hcnom.ays
1)('E U )* J 1
2
II
TaKHM oopasoa.
.", I n, n+p . f"n =
DOCKOJIbKY
_ p
+1
(24.6)
epopMyJIa
Ka)l{./laSl
HS
-2
n = O.
-
1. -,- 2•... ,
a
1
..
¢ H r.
iJoo 2
ok
II
2 ~ II M ~ Uppsinkp
=
p
OTKY./la OKOH'IaTeJlbHO
2
3
HMee'M
,
66).
'1aCTHUbl (epHr.
Buure MbI onpenemom no MOlll,H COOTHOllleHHSI
aupaxenae
TO B
./lJISI nOTOKa
P
BXO./l5lT
'1JIeHOB
aana
(24.7). P
(24.7)
In -
1
II
2
•
(24.8)
cD = ~ "2 U pA UJp SIn kp ; pacnpocrpaueuan
U e = -= =
p
E
3HeprHH
U e'
cor JIaCHO COOTHO
./lOBaTeJIbHO, HMne./laHCa
nanaoe
(24.11)
suure
onpeneaenne
CTaHOBHTCSI HeBepHbIM.
1
II
d
~
'1T06bl
nae
He06xO./lHMO
L
2d
CKOM6HHHpOBaTb
II
,UetlCTBHTeJIbHO, no onpenexenmo,
= araJ =
ih 21td iJk '
B
HMne./l
peu
•
1
cD = 2" ZA2UJ2,
p
ih
orpaxeunaa
'1aCTHuaM OT rpaunuu
=M ~Uppslnkp. (24.9) 00
M _A9 oo 2
Ug
peurerxy,
TOpOMy npn 3TOM aano CJIe./lOBaTb, yxasuaae B COOTBeTCTBHH C HHM MbI nOJIO)l{HM, 'ITO
2
2 U pA Olp sin kp
U a:
ropua Hy)l{HO 6bIJIO 6bI aar pyaare 060pBa
ume nepsux
y6e./lHTbCSI, 'ITO nonyueaaoe anaueaae connaaaer C rpyn
. nosoa CKOpOCTblO
l:!e06XO./lHMO Y'lHTbIB
'uJISI nOJIy'leHHSI xapaKTepHCTH'IeCKOrO HMne
6Y./leT panna
~ ~
Teneps
HS '1aCTHU ./letlcTBYIOT CHJIbI CO CTOpOHbI M
'1aCTHUe
p
(20.11),
BeJIH'IHHa CHJIbI, ./letlcTBYlOllI,etl ua
ee CKOpOCTb.
npaBHJIbHotl BeJIH'IHHbI xapaKTepHCTH'IeCKO ~
CKOpOCTb
xapaKTepHCTH'I
In=-ZYn.
66.
D03TOMY no nuua nOTOK 3HeprHH 6Y./leT onpenenarsca CyMMOtl '1JIeHOB BH./la
p
'ITO H Tpe60BaJIOCb' noxaaars.
3HeprHH, npoxonsuiero B./lOJIb peurerxa sepea yseJI. B KOTOpOM HaXO./lHTCSI HyJIeBaSl '1aCTHua,
d~lI.
p
4
rne OT HyJIeBotl
Jlerxo
~
U g = 200 M ~ U pp SIn kp = Moo ~ U pp
000 0
0
-1
2
=
nOJIy'laeM
( 2 4. 7 )
1 UpA-UJ II? -2 sin kp.
'1aCTHLL
000
-3
ok
M ~ U p(l-cos
d2~1I.
o
llIeHHIO
2
=
0'0
d
(l 0.1)
COOTHOllleHHe (J)2
=
p npHHHMaeT BH./l
BSaHMO./letlcTByeT C '1aCTHLLaMH, HaXO./lSlIJJ,HMHC5I cnpasa
-4
a
=
•
=-2 UpA UJsmkp,
H,
iJ"oo 1 21td ok 2n
p' no
JIy'laeM
r ae
Z-
xapaKTepHCTH'IeCKHtl
snaers , scerna neacranrenen.
cocoa
TOT epaKT,
KOM6HHauHIO HMne./laHCOB, cooTBeTcT
rpaHH'IHbIM
g*
asmenaac.
'1aCTHuaM.
XOTSI
H
HeSlBHO.
nencraas MelK.uy '1aCTHLI.aMH. 3aMeTHM, 'ITO COOTHoweHHe (24.12) 9KBIlBaJIeHTHO COOTHO weHHIO 1 -.
KHO nOJlYtIHTb
1
HCKaTb peurenne ,lI,JlSI nOJlHHOMO
nOJlH
+1)
+ Cn =
1
1 2x
I·
(25.6)
OTMeTHM,
X=
e
tITO,
nonaraa
B
BbIpa>KeHHH
(2
I I>
cos B 3aBHCHMOCTH OT roro X JlytIaeM JlHOO <jlOpMyny (25.3), JTHOO- (25.4)
lIeHYIYI nYlCKpeTHbix crpyxryp.
HeKOTopbIe
Yl3 paCCMOTpeHHblx
aanas 6ynyT Yl3J10}KeHbi HYime. 1
§ 26.
npHJlOmemte nOJlHHOMOB fereHoay9pa C" (x) K BMqHCJleHHW CoocTBeHHMx qaCTOT KOJleOaHHH orpaHHqeHHOH oAHoaToMHoH pemeTKH
I1cTOpYilleCKYI HaYl60Jlee paHHYlM npYlMepOM, B KOTOpOM BBe
CTYlU, pacnoxoxeunux ua paauux pa npyra; nepsaa YI nocnenasa xacrauu y na KOHUOB CTpyHW Ha TaKoe me paccToSlHli 0603HallYiM xepea Y1' Y2' ... , Yn pa3J1Y1l1HbiX rose« pa36Y1eHYISl OT csoer CHSl; KaK YI auure, 3TH cxeuieuas npe Torna BblpameHYISl nJlSl nOTeHUYlaJlbHol:t YI CYlCTeMbi MomHO aanscari, B anne
en 1
neHYIe nOJlYlHOMOB fereH6ay3pa (x) n03BOJlliJlO aecsua 6bICTpO npOBeCTYI He06xonYiMbie pacxeru. SlBJlSleTCSl 3analla 06 onpe.ae JleHIiYl c06cTBeHHbiX lIaCTOT nonepe'lftbtx KOJle6aHlil:t HaTSlHYTol:t
A- : • I . 1
I
• 3
: 1 : 2
0
0
0
---------
\
0
I
11 r.
YpaBHeHYISl Jlarpauxa npasonar K CHCTeM ypasueuaa, onpeaensnoiuax nBYlmelme CH
67.
..
.. [-1Yp
(n DpennOJlaraeTCSl,
T. e.
cocpenOTOlleHbi
p = 1, 2
3TO BbIpaJKeHHe
B ypaanen
(2 ~ -
aa
KOHuax
Ka)l{nol:t 3J1eMeHTapHol:t SllleI:tKYI.
1) CM. npl1MetIaHl1e na CTp. 135. 2) CM. D. E. Rut her for d, Proc. Roy. soc. Edinburgh, 42, 229
(1943-1949).
nOnCTaBJlSlSl CHcTeMy
+ 1) [-1 = pl.
liTO 3TH MaCCbI
= Ape i (wt-'!'J,
Yp
DpennOJlO}\(YlM reneps, uro B n ro-ncax pa36lieHliSl cocpe.aOTO lIeHbi paauue MaCCbI, BeJllillliHbi KOTOpbIX [-1 pasuu Macce OTpe3Ka
a (qmr. 67),
T
+ a (- Yn-1 + 2Yn
Peurenae ee 6yneM IiCKaTb B anne r apu
+ 1) a.
1= (n
+ Ii"T (- Yp_1 + 2y p- Y .. [-1Yn
+
TaK liTO
T
[-1Y1+-(2Y1-Y2)= a
B TeOpYiIO KOJle6aHYlI:t CTpyH. DyCTb 1- nJlIiHa CTpyHbI, P- Macca, npHXOnSlw.aSlcSl ua enaHHUy .llJIYlHbI, a T - HaTSl}\(eHYIe. Pa306beM crpyny ua (n 1)
a,
'2 [-1 "'" ~ Yk. k
08
n-l'. n :
CTpyHbI, npuxex npennOJlaraJlOCb, liTO crpyua COCTOYIT Yl3 60Jlb uroro lIY1CJla OnYlHaKOBbiX lIaCTYlU KOHellHol'l MaCCbI, pacnono meHHbiX na paBHbiX paCCTOSlHYlSlX npyr OT npyra, 9Ta aanaxa 6bIJIa nOCTaBJleHa JIarpaHmeM (CM. § 1) B KalleCTBe BBeneHYISl
lIaCTel:t paauoa .aJlIiHbi
Y
k
\
l-a-l
CTpyHbi nJlIiHol:t
= ~ ~ (Yk -
2EKH R• =
I-G-l I I I
2En oT.
-
~ A2 ........... ~ A p_1 +(2 ~ -[-1w 2 ) Ap ............
_!.a
[-1W 2) A 1 -
+ (2..!....a -
A n- 1
ILw2)
r
YCJlOBHe paapeuHlMocTH KOTOPO~ rrpHBo~HT K
2x 1 1 2x
C~(X)=
o
0 I
0
0 0
BepHeMClI K supaaceaaio 3aMe'lall, lJTO [
a
0 ... 1 2x
KaK
xerxo
'ITO
aaoe,
BH.n.eTb, 9TOT cy~eCTByeT
x H
=
k'1t
n
ry "k'=-;-Y
k'«
a!J.COS
"k= 1t1
= 1,
2(n +
YP
'1k'
eJle
x
CHCTeMhI,
k7t Sin 2(n+ 1)'
k
=
'1HCnO paaJlH'IHbIX C06CTBeHHbiX '1aCTOT CH
CTerreHe~
HMelOT, KaK
nerxo
cB060.n.bI, T. e. lJHcny
non
COOTBeTCTBYIO~He C06CTBeH y6 e.ll.HTbCSI,
BH~
t
-
]
,
p= I, 2, ." n, H ~k --:- sexoropue nocTOllHHble, 3Ha4eHHSI KOTOphIX aa k BHClIT OT HOMepa B 60Jlee 06~eM sane ~BH)\(eHHe cncrexu MO)\(HO onacare
rne A
k.
cf!OpMyJIOft
YP_
KOTOpylO He
lJTO
~ Aksm n+l COS [2 .
pk 'It
a- sm 2{n+1) t-~kJ.
(T.
k Tt
pOBaHHbIMH
nQnblTaeMcH reneps caeaars nepexon K
HenpepblBHo~
crpyxrype.
Y"~
paccxarpaaaa
Mbl
nnoe,
KaK
aaaavy
0
TO'IKaMH,
a
Ka)f(.n.all
H3
ECJlH 6bl Mbl He HMeJlH B BH.lI.Y ocy
nepexoaa HOCTH
rrenpepueaoa crpyne,
K
saaasy,
k Me)\(~y CHno~
H pa3HOCTblO C
CMemeHHlt
HHIO 0 TOM,
cnnu,
'ITO
TO
BBO.n.lI TOJIbKO K09q>e
B CJly'lae MaJlblX
scerna
9K
.n.e~CTBYlOmHe
nopuaouansnu
H3MeHeHHIO paCCTOllHHlI
~BH)I(eHHlI
9TOM 6y.n.yT HMeTb BH.n.
npa
HHu,elt, 'ITO Ha BenH'IHHy
exoasuree
B HHX OTHomeH
k. 'ITO
noayseaasre
TaKH
cnpaBe.n.JIHBhI rrpa rex )\(e npennonoace Mo.n.eAcTBHlI '1aCTHu,.
k-l
n pH
TOnbKO CO CMe)\(lIbIMH '1aCTHu,aMH.
O'leBH.lI.HO,
n
k=
uenosxa, COCTOll~e~ H3 o.n.HHaKOBbIX na paBHbIX paccrosnasx .apyr OT .n.p noes, 'ITO KpaAHHe '1aCTHu,bl yrrpyro
.n.HTb 9TY
9k
p'
'1i[ 3a.n.alJa,
(26.2)
~T
crpynsr.
pasnux
1, 2 •.•. n.
1)
xoponro H3BeCTHoe COOTHomeHHe, ypaaaeuna B nacrnux npoaaaon
)\(eHHe KOJIe6J1IO~e~ClI OCHOBHOA rapMoHHKH
2, .. , n
T. krc = A k Sin. npk« + 1 COS (2 -Yr Ii sm 2 tn + Il
_
3TO 1f3
k =1, 2, ... n
I) '
BH)\(HbIX lJaCTHU CHCTeMbl. HopManbHble THnbI KOJle6aHH~, lIaCTOTe
n.
»r:» -V
He
Tt
oecsoueunocra.
il
c0601l
CTeMhI CHOBa paBHO lIHCJlY
HO~
rropH,UKa
K
k
p-sm 2(n+
,
(T. ap.
Mbl nOJly'lHJIH, 'ITO
C;(x)
c06CTBeHHhIX· '1aCTOT
1 ~
HJlH
k'
c060~
rrpe,UcTaBJllIeT
paaJlH'IHbiX aHa'leHH~
n + l'
cos
cOOTBeTCTBeHHo
n
+ 1 _ rr .
reneps n
YCTpeMHM
a ~eTepMI1HaHT
sane
B
'1k=-;- - t - V
-2x.
rereH6ay9pa
KaK nOJIHHOM
~OBaTeJlbHO,
T
samrcars
ero MO)\(HO
1 n
=
p.=
a=n+l'
r~e
2T -!J. w2
(26.2).n
cncrea u,
0 0 l = 0,
.n.OJlbHhIe
Ilpa
H
.n.llll
MaJlhIX
9TOM BeJIH'IHHbl
cMemeHHlI
npooo.uoc Yp
npenc
'1aCTHU OT csoero
KOTopoe oHa HMeJIa B CJIy'lae rrOrrepe'lH~X KOJIe6aHHlt. STOT H CJleAYIOllI,He napar parpu Mbl rrOCBSlTHM OrrpeAeJIeHHIO co6CTBeHHbiX 'laCTOT KOJle6aHHlt pa3JIH'lHbIX THrrOB
uenosex, k
nyio
SI'leltKa 6yAeM
conepxorr
KOTOpblX
OOITbITaeMCSI HaltTH
TOJlbKO
rrpH'leM
csumams cBo600HblMU cnysaex, xorna
or paHH'lHM CSI
2x+ 1
paaJlH'lHolt B aaBHCHMOCTH OT roro, nponom.nue HJIH nonepesnue KOJle6aHHl1. coficrueuuue nacroru uerrO'lKH, COCTOSl H pacnonoaceanux H3 paaaux paccroa
aa n O)l.HHaKOBblX HHSlX npyr OT )l.pyra 'laCTHU,
llI,elt
6yoeM
orpannsenaux
1
0
1
2x
1
o
o
0
OAHy 'laCTHUy. OOCTOSlH
C'lHTaTb
paCCMaTpHBaeM JlH Mbl
co CMe)l{HbIMH
(!fJHf.
oee tcpatiuue «acmuusi 68). KaK H npexne, Mbl
Y'lHTbIBal1
Heooxonnaue
/l
2
j
11
/l
4 -----n~,
HnbTp
3JIeKTpI14eCKHe,
HaMH CHCTeMhI,
HMeJlH
KaK MexaHH4e
n creneuea csooonu
HHTepBaJIe, npH4eM BeJIH4HHa
OT
':!HCJIa
n
3JIeMeHTapHbIX
sroro
R4eeK,
PaCCMOTpHM nJIR npOCTOTbI ueno-ncy 4aCTHu"
63,
coaepuiaer 4aCTOToiL
-
2' Yn+ k(- Yn-1 +Yn)=O.
H ti
HHTepBaJIa
06pa3ylOLU,HX
nanayio cacreay. tPHr.
H npennOJIOlKHM,
r-apaoaaseckae T.
e.
aanauo
4aCTOTa KOJIe6aHH5I
4TO
Kpa~H5I5I
KOJIe6aHH5I ee
CMeLU,eHHe
cosnanaer C
onHo~
C
H306palKeHHylO JIeBaR
Craunonapnue peWeHH5I 3THX ypaaneaan Yp = Ape i wt • Ilonaras 2x = - 2 (prlk CTeMe
H3
3JIeMeHTapHbIX
BaeMa5I CHCTeMa,
+
onpeneJI5I1OL
4aCTHu,a
xA 1+A 2=-A,
nexoropon sanaaaon
Y
JIHHe~HbIX ypasnenna,
=
Y cos 2n'it. ECJIH coocraenasrx 4aCTOT
A p_1
+ 2xAp + A P+ 1=
An
+xA" = 0,
CHCTeMbI, TO Ha6JIIOnaeTCR pe30HaHC. npennOJIOlKHM reneps, 4TO 4HCJIO
Yp+1
4aCTOT, 3Ha4eHHR KOTOpbIX 3aKJII04eHbi B HeKOTO
KOHe4HOM
HeSaBHCI1T
aa
~
+ k (- YP-1 + 2yp -
R4eeK, 1;\3 KOTOpbIX COCTOHT paCCMaTpH
HeorpaHH4eHHO B03paCTaeT. Torna 4HCJIO co6
10*
-1
0,
n-It
BblpalKeHHe .nJISI aM IlJIHTyJlbl xoneoauaa
4aCTHUbl
•
orpaaaxeaaof pemerxa, co
nayxaroaaux
H3
A=(-1i'A
(x~ -
n
ECJIH
\X
I < 1.
1
1) C~_2 (x)
TO MOIKHO IlOJIOIKHTb X
(_On+l A
A (! -
IX I >
1,
= (- 1t
A
ECJIH IKe. Ha060poT.
An
sin
(j
= COS
sh
(j
Hccnenoaauae
1_ sin (n - 1) (j
sh (In_
1)~
TaKIKe.
eH
TO nonaraex X
n-It
ch
CTBeHHO
eH
KOJIe6aHHIt
n
4aCTHUbl cytuecrseuaua oopa
IX
I·
An n-It 4aCTHUbl CTpeMHTCSI K HyJIlO BO BTOpOM CJIy
+
2 cos e= - 2 (plk) 0)2 3aBI1CHT OT 4aCTOTbl. TO yCJIOBHSI I X I > 1 HJIH I X I < 1 OIlpe.neJISIIOT 06JIacTH H3MeHeHHSI 4aCTOTbl, IlpH4eM o aaa H3 HI1X (I X I < 1. HJIH 0 < 0) < 2 VkIp.) coorsercrayer 'laCTOTaM. npn KOTOpblX
=
KOJIe6aHHSI. B03HHKWHe ua OJlHOM KOHue CHCTeMbl. pacnpocrpa HSiIOTCSI HJIH
0)
H
.nOCTHraIOT
> 2 VkIp.) -
HHKWHe
na
O.nHOM
ee npyroro KOHua. a npyras
'laCTOTaM. KOHue
He
peurericy
M1
npyr
n
aa
4aCTHU
M 2•
H
paCIlOJIO
OT
npyra,
8JIeMeHT
(I X I > 1.
Mbl
6y.neM
npennoaaran.,
.nOCTHraIOT
'ITO
B3aHMo.n
TOJIbKO MeIK.ny CMeIKHblMH 4aCTHuaMH, a rHMH, rrpOIlopUHOHaJIbHblMH CMemeHHIO. PaCCMOTpHM CHa4aJIa CJIy4alt. xorna
o
npuuaosexcatuue I{ iJ8Y.M. pa3/lU'tHbl.M. cop c iJBy.M.fl fjJUI{CUp08aHHbl.M.U mowca.uu A
B H Mbl
9TOM cnysae 'lHCJIO 4aCTHU. oripasyi MOIKeM rrOJIOIKHTb n 2m.
=
Ilpennoaar-aa, H 0603Ha'laSi JIOIKeHHSI B BH.ne
'ITO nepaas CJIeBa
yp
uepea
cxeurenue
paBHOBeCHSI. aanaureu
sacr
p-It
4a
ypaBHeHH
M 1Yl +k(2Yl- Y2)=O
npn KOTOpblX KOJIe6aHHSI. B03
CHCTeMbl.
Bh
CHCTeMbl aHaJIOrH4Ha Sl4eltKe peurenca, pa
HeorpaHH4eHHO B03pa~TaeT. aMIlJIHTYlla
2x
I'erenriay
y6e.nHMcSI.
COcToSimelt H3 KOHe4
MaccaMH
C
paCCTOSlHHSlX
•
xae, rorna KaK B IlepBOM CJIy4ae OHa OCTaeTCSI KOHe4Holt. nOCKOJIbKY BeJIH4HHa
celt'lac
paBHOBeCHSI rrorrepeMeHHO B.nOJIb HeKoTop
30M 3aBHCHT OT TOrO. 60JIbWe HJIH MeHbwe ennuauu BeJIH4HHa neltcTBHTeJIbHO, xorna
IlOJIHHOMOB
M hi
Paccxorpax
•
=
KaK
4aCTOThI peweTKH, HhlX MOJIeKyJI.
MOIKHo 3aMeTHTb, 'ITO npa .nOCTaT04HO 60JIbWOM 4HCJIe 'laCTHU xapaxrep JlBHIKeHHSI nocnenaen
Mon
.
M 2Y2
ee .npy
+ k (-Yl + 2Y2-
roro KOHua, eCJIH. KOHe'lHO, 4HCJIO 4aCTHU .nOCTaT04HO BeJIHKO. TaKHM
06pa30M.
paCCMOTpeHHaSi CHCTeMa Ilpe.nCTaBJISIeT c060lt
M 2Y2m
MexaHH4ecKHIt
HaATH
6bIJIH
)2
)2
4
-~SI
4 - M 1M2
aHaJIOmqHbI Bbl.pa)/(eHlfHM
pacnonoxenu aa
aa
2
1
aupaxenna
xoropue
Xl
BeJll1ql1H
H
X2'
1, 2, .,. Zm,
qaCTOTbl paCCMaTpl1BaeMOA CH CJIe.uyIOIllI1M COOTHOWeHl1eM:
M 2 2) -_ TOJ
1
n03BOJIHIOT I
2
cayxae CTOHlllHA no nncxpernue SHaqeH
BepXHeA H HH)/(HeA BeTBH
49. npeaunyume aupascenas
KHO
aanacars
OTCIOD.a.
aaaexas. fIoli
TaK
'ITO
Xi+l -
'ITO
+ k [2Ii -
CHCTeMa
Xi
(li-I
= Ii'
cos
OKOH'IaTeJlbHO
TeOpeTH'leCKHX
Bappaons
aI = O,
onacusaioinax
ypasueuaa,
Cnenyer yxasari.,
C
KOJle6aHHlI
pac
yxerox
2 cos
al =
+ k [2Ii - (I i_ + Ii+!) fIoi~_1 + k [2I n _ - In _
fIol,
1
2 cos
1
Ilonaras reneps Ii = Liei(wt-KeHHlI
Ilpenn
paBHOBe
xacroru lO. H oooauasaa 'lepea nonyxaea cncrexy n JlHHe"Hb
KOJle6aHH", CHTeJlbHO i,
fIolO2) L n _ 1 = O.
(29.2)
a
ncnsrrsmaior
O.
panmcaau
ace
Coxpauss npenno.roxenae,
rrOTeHUHaJlbHO" CHCTeMe JlH
MOlKHO
''laCTOT UHKJlH'leCKHX
pacrrOJlO>KeHHbIMH B
xepes
3
KaK UHKJlOrreHTaH, UHKJlOreKCaH
aaaeaars
MO>KHO
rrOTeHUHaJlbHO" MeTOD. Bbl'lHCJleHHlI
uenosex
c06CTBeHHbiX
DOCKOJlbKY
OTHOCHTeJlbHO HeHaBeCTHblX
k
onacanuun
uanpaaep,
0,
a
Ln _ 1 (2k -
cos
(l /21t) V
pesyasraros. YKa>KeM TaK>K Lllaneans t), B KOTOpO" 3Ta
aaBHCHMOCTH
TOJlbKO 'ITO D.eJleHHlI
+ k [211 -1
H
aJlH<paTH'leCKHX
CMaTpHBaeMO" CHCTeMW, 6YD.eT HMeTb BHD.
fIo!'1
'ITO BeJlH'lHHa
flo. 3Ty sacrory, TaK >Ke KaK H BeJlH sxcnepnxeuraasno. TaKHM 06paaOM MO>KHbIM. paayaeercs hpn YCJlOBHH crrpaBeD npennonoxenatt, onpenenars nee C06CTBe CMaTpHBaeMO" yr JlepOD.HO" uenoxxa, B yn 60Te DapOD.H MO>KHO Ha"TH cpasnenne sx
al = o.
noxyxaea
+ Ii+!)
, 1.=1,
Ha"TH
cos
1
An
1 +cosOcos n
MaCCbl
cos
1 cos
HJlH
V fL V
c060" c06cTBeHHylO 'laCToTy KOJle6aHH" D.B
+ k [Ii-l a-IiI = O. fIoXi+1 + k [Ii - li + a1= O, flo [Xi+1 - xiI + k [2I i - (I i_ + Ii+!) fIoXi
v)..= 21t
HMeeT SHD.
-
-
kL n
cos
a+ L
1
-
kL I
cos
0
2
kL n _ 1 cosf
(2k -
mlO 2) -
kL
+ L (2k - mlO2) - kL + L n (2k - mlO 2) - k
r.ne
41
2x = -
2k - fLw 2
k \;05 0
I) CM. Bar rio I, C hap e II e, Journ. de
i
('lk - m(2 ) ('lk - M( 2 ) k 2 cos2 6
x l 0 •.• 0 0 l lxl
•.. 0 0 0
OOO
lxl
100
01x
~
=0.
r.ue
x=
). = 0, 1, 2, .. . (n -
§ 30.
I:
'lk-mw 2 k cos 6 •
KaK
6bIJIO
peay JIbTaTbl nopsnxa oopauiaerca
3TOT .ueTepMHHaHT n-ro
rrOKa3aHO
PaCCMOTpHM,
2
,
=
V m'
0 VI-cosB,
'I p =
'10
VI-cosBcos
KOJIe6aHHtt c06CTBeH
'l~P
(29.5)
npn p= 1, 2, ... ( n - I ) . OOCKO.'IbKY, KaK y)J{e 6blJIO yxaaaao. BeJIHlJHHa '10 H3BeCTHa, TO MO)J{HO onpenenars 1'1 ace c06CTBeHHbIe lJaCTOTbl paccaarpasae 3aMeTHM,
HaKOHeu.
'ITO
H3JIO)J{eHHbItt
auure
MeTO.u MO)J{eT
6bITb HCrrOJIb30BaH TaK)J{e .uJIH BbllJHCJIeHHlI c06CTBeHHblX lJaCTOT
nsyx THrrOB lJaCTHU COOT BeTCTBeHHO C MaccaMH m 1'1 M, pacnoaoxenaux nonepeuenno B Be pumnax rrpaBHJIbHOrO MHoroyrOJIbHHKa. Ilpennonaraa, 'ITO UHKJIHlJeCKHX
uenosex,
rrOJIHOe
lJHCJIO
lJeHHlI,
MO)J{HO
COCTOlIlUHX 1'13
n lJeTHO, noxaaars 1), 'ITO
lJaCTHU
, B
1'1
coxpanss
B
paxxax
npeaorae 0603Ha c.ueJIaHHbIX
1) CM. M. Parodi, Cornpt. Rend., 231, 609 (1950)
auure
2),
C
lI'
Az ---------
I I
82 1 I
T
I I I
I
Bn
A' A; I
I,1
KaK
aro
npyr
rrOKa3aHO
na
KaSi npaayio H nesyio aa d2• nOJIy4aeM
7ll = (a . d.) = a l•
(31.18)
4aCTH CHa4aJIa aa d l • a nOTOM
712 = (a . d 2) = a2'
Bexrop a nepnenauxyxspen npsxoa. onpenensexon ypaane
HHeM (31 .17). a OTHOllleHHe c /1 a I COOTBeTcTByeT paccrosamo 0 OT 3TOR npSlMoR .no Ha4aJIa Koop.nHHaT. - 1{00p.nHHaTbl ~l H ~2 JII060ro Y3JIa peurenca H3 T04e4HblX 4aCTHU npHHHMalOT JIHlllb uenue 3Ha4eHHSI 11 H 12 H n03TOMY npOH3BOJIbHblR Y3eJI peurerxa onpeneaaercs aanaaaex sexropa
r
=
Ild l + 12d2.
h = hlb l rne h l
H
h2 -
+ h2b2•
(31.20)
uenue ..
ECJIH B ypaauenue (31.17) BMeCTO sexropa a noncraaars netcrop h. TO OHO 6y.neT onpeneasri, cexencrao napaJIJIeJIbHbIX npsaux (nepneH.nHKYJISlpHbIX aexropy h). aaBHCSlllI.HX OT napa MeTpa c. npaxeu nexoropue na HHX. KaK MbI ceaxac y6e.nHMcSI. npoxo.nSlT 4epea 6eCKoHe4Hoe 4HCJIO yaJIOB npsxon pemerxa.
O.
•
12 =
-
hl •
Icl= 1;
3TO ero HaHMeHblllee anaxenue. CJIe.noBaTeJIbHo. paccrosnae Me>K.ny sepea yaJIbi peurerxa npSlMblMH cexeac napon ueJIbIX 4HCeJI (h l • h 2 ) . pasao
.~.
(31.19)
BeKTOp h. KOTOpbIR onpenenser Y3JIbl 06paTHoR peurerxa, nupaacaerca CJIe.nylOllI.HM 06pa30M:
=
Ilpyrne npaxue sroro cexeacraa npn COO 'napaxerpa c 6y.nyT. 04eBH.nHO. npoxonar yaJIOB peurerxa. Haaaea reneps paCCTOSlHHe OT Ha4aJI >KaRlllerO psna ceuencraa HJIH. HHa4e rOB MeHblllee OTJIH4HOe OT HyJISI 3Ha4eHHe napa BeJIH4HHbl h l H h 2• TaK >Ke KaK ~l H ~2' >KHTeJIbHblM HJIH OTpHuaTeJIbHblM 4HCJIaM. BeTcTBylOllI.HR KaKoMy-JIH60 pSl.ny peurerxn. 4HCJIy. ECJIH BeJIH4HHbl h l H h 2 He conep JIeR H BeKTOp h npHHHMaeT csoe nanxens 4eHHe. TO napaxerp
IleRcTBHTeJIbHo. nOJIaraSi
a
2/ 2
np
3TO COOTHOllleHHe y.noBJIeTBOpSleTcSI. B 4aC
B 4eM JIerKO uenocpencraesno y6e.nHTbCSI. no.nCTaBJISISI B nepsoe
a
+h
hl~
1
0= ThT' \
IlpYrHMH CJIOBaMH. yaen (h l • h 2 ) oop ) .neJISIeT ceaeacrso PSl.nOB yaJIOB npsxojt p CTBa PSl.nOB nepnennaxynspnu aexropy panaux paccroaunax 1/1 h I .npyr OT np aHa4HTb CHMBOJIOM (hi' h 2 ) . TaKHM oop K 060aHa4eHHl'IM 5paB3 .nJISI KpHCTaJIJIOrpa4 TaK. aanpaaep, na 4>Hr. 75 ceMeRcTBO THKaJIbHblX PSl.nOB. ceaencrso [01]- as HblX nexropy d l • ceMeRcTBO [11] - na P
l noaareasuo, npOH3BeAeHHe 3THX tPyHK J AaJlbHeftllleM AJl5I nOJlYlJeHH5I HaH60Jlee nar JISlAHblX pesy Jlb " nepaonasecsoa hz 1Jh2 anrerpaau, CTOSl
hz = h z OTJlHqeH OT HyJlSl H, KaK aenco a OTcJOna rroayxaex 1
Ch,h, =
penxax cnyuaax. ----fl(}KameM reneps. qTO rrepnonnvecrcyto KHO rrpencraaart, B sane CyMMbl nnyx nepaonns
HbIX rro x H y
flpUM. pea.
I
Paccuorpaa nnocxyio aonay, pacnpocrpaamouryiocs
B ~BY
k~ = k i
MepHOM npOCTpaHCTBe. OHa HMeeT BH~
= Aei (",t-21u,x-21ta,y).
~
uenue 4Hcna, a t = I, 2), HH anaseaae 4aCTOTbl He HaMeHllIOTCll. I1Ta
HH
npH4eM I
r ne BeKTOp
a
2
1
=
2
I
a1 1
+
2
I
a2 1
1
= 12 '
t
aaexca C neonpenenennocrsio B Bw60p nonoouo TOMy KaK 9TO, HMeno MeCTO onaoaepaux crpyxryp. BenH4HHaM k~ COOTHoweHHeM (33.6), Mbl MOIKeM con HOBbllt sexrop a', paaaua at
a onpenenser nanpaaneaae pacnpocrpaueuaa,
aero
BenH4HHa 06paTHO nponopUHOHanbHa ~nHHe BonHw.
nOCKonbKY, KaK
k~ =
k~
§l
30HQ
~ 30HQ
3
IIlIIID
30HQ
5
It
~ 30HQ4
Z
H
r. 78.
B cnysae nacxpernon crpyxrypu ~YHKUHll ~ onpeneneua ronsxo B rex T04Kax r. rne aaxonsrca 4aCTHUbl, T. e. ronsxo B yanax rI,l. npaxoa peurerxa, nOCKonbKY II H l2 - uenue
+ 21t (a . l2d2) =
llkl
+ l2k2'
(33.3)
r~e
k 1 = 21t (a . d 1) , TaKHM oopaaoa.
k 2 = 21t (a . d2) .
supaaceuae (33.2) ~
(33.4)
npHHHMaeT BH~
= Aei (",.t-k1l1-k.I,) ,
+ m + m2b2 1b 1
caenyer aa onpenenenas
+ 21tm (b, . + 21tm2 (b d = = 21t (a . d z) + 21tml (b, . + 21tmz (b d =
= 21t (at.
d 2)
1
2 •
1)
2 •
2)
Hanpaanenae pacnpocrpaneaas onpen at, xoropua, KaK H ~m1Ha Bon UHelt m, H m2 [CM. (33.7)]. I1TaK, paanasaue aeicropu at cooraerc HOMy sexropy a; uexoropue aexropu at rne aaoopaxena ooparnaa peurerxa. Ilpaaeneuaue paccyxneaas nOKaaWBaIO B ~BYMepHolt peiuerxe aanaerca nepno
aexropa a B ooparnoa peurerxe C 6aaHCHblM Ilonuraexca onpeneaare reneps OCHO TOpOlt ~onIKeH 6WTb aa~aH BeKTOp
a.
BblXO~HT
na nax 6b1no 6bl or pana-urrs o6nacTb ero HaMeH rapnca 114eltKolt ooparaoa peuierxa, T. e. C4HTall, 'ITO BeKTOp
lJHCna, TO
21t (a . rI,l.) = 21t (a . lldl)
a
TOpOM
a
30HQ 1
=
21t (at. d 1) = 21t (a . d.)
u
c::::J
j
(anecs m, -
(33.1)
Oooanasaa sepea a aexrop C KOMnOHeHTaMH (al' a2) H xepea r sexrop C KOMnOHeHTaMH (x, y), MOIKeM aanncars 7 = Ae i [",t-2 1t(a.r)], (33.2)
+ 21tm
a
aepunmu xoroporo, KaK 9TO noxasaao na ~H [DO], [01], [10], [II]. O~HaKo npH aarpynnenus. Bo-nepaux, npa TaKOM onpe nanpaanenas pacnpocrpaneans, xoropue pOM b1 yr on, MeHbwHlt 4eM 1t H, B T04Kax
e,
(33.5)
BOilHe
He
0611aaTenbHO
COOTBeTCTByeT HaH
xoropoe
MbZ
4T06bl
onpenemrn,
npa
rrpHHSlJlH
060ltTH
3TH
no,U06HO
paCCMOTpeHHH O,UHOMepHbIX
aaTpy,UHeHHH.
TOMy.
KaK
OCHOBHylO
3TO 6blJlO
crpyxryp. Ilpeacne ncero noaosceaae aouu
aOHY
cnenano
sanas. MomHO
B
cnysae
O,UHOMepHbIX
Bbl6HpalOT TaKHM 06paaOM.
ooparnon peurerxu. I'pa npoxonsurae sepea cepennnu
'lT06bl ee UeHTp COBna,UaJl C Ha'laJlOM HHUaMH
souu
HBJlSlIOTCH rrpSiMble.
I
I I r'
(
peurenca, B 'leM nerxo y6e,UHTbCH. paacaaa na 'laCTIf, xoropus orxeveau na ar. ulfpaMH, If cpaasaaas roxcaecraeunue 3J1e Ilocrpoenae BTOpOtt aOHbI, a TaKme aOH 60Jlee CJlOmHO. QHa ,UOJlmHbl 6bITb orp npoXO,UHlUHMH 'lepea cepezmnu orpesxoa, co
"
If
r. 80.
II r. 79. OTpeaKOB. HbIMIf
COellHHSllOlUHX
yaaaxa,
npaxue
If
oopasyior
kOTOPblX
COBrra,UaIOT
'or-panaxeaaas
H
TaKIfM QHa
06paTHott
COBOKyrrHocTb C
rro
oopaaox,
C
6YlleT
pasnas
orpeaxax.
MHoroyrOJlbHHKOB, nJlOlUa,UH
If
peurerxn
3TlfM
Ha'laJlOM 06paTHolt
HaHMeHbWIfM
nocrpoeanux nepayio aOHY. BeKTopOB
UeHTp
nepneunaxyaapaue
peurerxa.
3m
uenrpu Q6J1aCTb.
MHoroyrOJlbHHKOM,
npencrasnars cocoa
He aaBHCHT OT BbI60pa CIfCTeMbl 6aaHCHbIX
rrOaBOJlHeT
paCCMaTpHBaTb
nponsaonsnue nanpas
~eHHSI pacnpOCTpaHeHHSI; DOJlee Toro, TaKoe onpelleJleHHe npH BO,UHT
K
paCCMOTpeHHIO
,UJlH
aaaasuoa BOJlHbl
HaHMeHbwero
aexropa a (HJlH. 'lTO TO me, HaH60Jlbwelt llJlHHbl BOJlHbl A). Haxoneu, rrOJlHbltt nepaoa pacnpocrpauenns, COOT· sercrayiounot uexoropoay HanpaBJleHHIO, ueJlIfKOM YKJla,UbIBaeTCH BHyTpH aonu, Q,UHaRO HeOnpelleJleHHOCTb aa rpaanuax aonu,
no BeJlIf'lHHe
l1MelOlUaH MeCTO
B O,UHOMepHOM caysae, anecs raioxe OCTaeTCSI~
ooparnoa peurerxu C ee paaJlH'lHbIMH y3J orpesxax, npasex Hif O,U He ,UOJlmHa npoxonars BHyTpH aonu. Ba>KH lleJleHHH 6Y'ueT HCHa na ,UaJlbHel'lwero. Dpe, HOCTIf, 'ITO BOJlHa pacnpocrpaasercs B KyJlHpHbIMH 9TlfM
xapaxrepuayexoa
MaJlblM nepHO,UIf'leCKIfM Bo
06paaJM, 'lTO KOHeu ee
aa
auure
YKaaaHHbIX
BOJlHOBOro BeKTopa
npHMblX,rrepneHlllfKyJ
coeanrenounra aaxaao ooparnoa peurerxn yaJlaMIf. B 3TOM caysae, KaK MbI yBH,UIfM, JlHlOlUaH cocoa aaBIfCIfMOCTb sacroru OT
YHKUHH
3TlfX aOH. 3rOT nOJly'leHHoMy (CM. § 17).
12
aaK. 201.
'I
BO
acea
pesym.rar
H3MH
rrpa
06J1aCTIf,
rrOJlHOCTblO Hay'leHHIf
sa
IfC
auaa
O,UHO
aOHbI. TaK, nanpnxep. sropas sona npencraanaer c060R 06JIaCTb, aaKJII04eHHYlO Mem,ny Hal1MeHbWI1M 11 nenocpencraenno c.tenyio . W.I1M aa HI1M no BeJII14liHe MHoroyrOJIbHliKOM, npasex croponaan
•
•
•
• •
Kam,nylO sony na 4aCTI1, naxonsumecs B CpaBHI1BaH TO)l(,neCTBeHHbIe 3JIeMeHTbi nacr H4eeK Mem,ny c060R. BOJIHOBbIe setcropu pacno.roacenu BHYTpl1 cooTBeTcTBYIOW.I1X no,n aOH, npuaonsr K onnon 11 ToR me yHKUHtl (M) U k p ( ECJlH P 51BJl5IeTC5I eJIHHCTBeHHbIM Y3J10 meHHbIM na oxpyxaocrn (M), TO
l»
kp(M) nJl51 scex T04eK KJlaCCa HMeeM
jp(M)=j;
0
(PM) =jp (M).
kp(M)=k,
'ITO 3KBHBaJleHTHO YCJlOBH5IM (33.12). B 3TOM cnysae Mbl rOBopHM, '-ITO M - oc06a51 TOl(Ka l-ro nOp5lJIKa, a l(HCJlO nepeHocoB, npana.anexaumx K rpyrme 0 H nepeBOJI5Iw,HX TOl(KY M na rpanauy 30HbI Zi' pasuo k 1. Paccxorpaa reneps 3J1eMeHTapHylO 51l(etlKy D peurerxn R, nanpaxep 51l(etlKy, npeJICTaBJl5IIOiUYIO c060tl napanaeaor-paax, nocrpoeuaua na sexropax b, H b z' BbIXOJI5IW,HX H3 TOl(KH O. B 3TY 514etlKy BKJlIOl(aeTC5I, Ol(eBHJIHO, KaK 06J1aCTb, aatono-iea Ha51 BHyTpH napaaaeaorpaxxa, TaK Hero rpanauu. JII06ylO TOl(KY nJlOCKOCTH MomHO nepesecru C noxouisto onepaLlHU nepenoca 0 BHYTPb 51l(etlKH D. BTO nOJlOlKeHUe CnpaBeJIJlHBO JIJl5I scex TOl(eK 30HbI Zj; 06paTHO, cor JlaCHO TOJlbKO 'ITO JIOKa3aHHotl reopexe, JlI06ylO TOl(KY 51l(etlKu D npu yCJlOBHU, 'ITO OHa 51B,151eTC5I perynapaoa, MomHO nepesecru B 30Hy Zi' npHMeHHB onepaumo nepenoca O. TaKHM 06pa30M, MemJIY TOl(KaMU 30Hbl Zi H perynspnuxa TOl(KaMH i-ro nOp5l,LI.Ka 51l(etlKu D cyuiecrayer B3aUMHO OJIH03Hal(HOe COOTBeTCTBue. PaCCMOTpHM reneps ocoriue TOl(KH i-ro nOp5lJIKa 51l(etlKu D. OHH pacnoxoacenu aa KOHel(HOM l(UCJ1e OTpe3KOB, "npnnanae »caurax np5lMbIM, npOXOJI5IW,HM sepea CepeJIIiHbI BeKTopOB, COeJIUH5IIOw,UX JIBa pa3J1Hl(HbIX Y3J1a peUIeTKH R, Ii nepneHJIHKY Jl5IPHbIM 3TUM aexropaa. BTU oTpe3KH JIeJl5lT 51l(etlKy D na KOHel(HOe l(HCJlO MHoroyrOJlbHHKOB, conepncaumx per ynapuue TOl(KH. nOCKOJlbKY TOl(Ka i-ro nopsnka Pi' COOTBeTCTBYIOw,a51 TOl(Ke M, MomeT MeH5ITbC5I, TOJlbKO ecnn TOl(Ka M CTaHOBHTC5I oc060tl, TO KamJIbItl H3 3TUX peryJIHpHbIX MHoroyroJlbH\tKOB
+-
=
H MbI nonaraev
Paanocrs (PM) -
1 panna l(lfCJlY Y3J10B p BHyTpH oxpyacaocra (M). npeJInOJlOmHM reneps, 'ITO ua oxp TOl(KH P, HaXOJI5ITC5I e.ue k p (M) Y3J10B p nae Mbl 6YJIeM rosopart., 'ITO paCCT05lHU roaaaunon Q>YHKIJ,lfetl kp(M)
=F 0,
(PM)
KHO nocrpoars MeTo.nOM. YKa3aHHbIM B § 31. Ilo.auepxnea, 'ITO B 3TOit nJIOCKOCTH b., b 2 aanaercn BOJIHOBOlt BeKTOp. onpe .neJIHlOmHIt nanpasnenae pacnpocrpaneuas BOJIHbI B npaxon peure rke; '1aCTOTa 3TOit BOJIHbI HBJIHeTCH KeHHSI K 4aCTOT
nonpasxa me
OTRH4Ha OT HyRSI H pasna at.
CAyttafi. a 3TO
C",,1n,
3Ta nonpasxa npHBo,UHT K cRa60My HCKame
(D)
HRH
w2 -
OT 00 0 H. CRe,UOBaTeRbHO. K03ljlljlHUHeHTbl B B 3aKRlO4eHHe 3aMeTHM. 4TO B paCCM
(34.18).
B BH,Ue
2 V2 0 Wm,lII,
Awo2
Ia'm1m'J [4'Ia I. TO 3Ha4eHHSI 00
nOCKORbKY
rtlOmeT 6bITb
OTCIO,Ua cnenyer.
aBTOMaTH4eCKH
,
(8 m,m
2
wml1n~
w O-
m2' T. e.
OT BeKTopOB a lll , 11I, npH RlO6blX 3Ha4eHHSlX m t H
=
npHpaBHHBaSl 'upyr 'upyry K03~ljlHUHeH 3KcnOHeHUHaRbHWX MHOlliHTeRSlX. nORy4aeM
OTRH4Hbl OT HyRSI, ratose paaen HyRIO. PaCCMOTpHM Tenepb yCRoBHe opTorOHaRbHOCTH ,URSI OCTaRb HblX
B
0
= -
nOCKORbKY BwpameHHe
a -.a~ = mtbt
(tl2
(Umtt/l']
--2 V
o
'
(34.20)
ananorunnu peurerxa npH
peureaasa,
a=
nORY4eHHblM
HaM
-+- 1/2d. H nepexonsr B
1
BeTcTByeT O.llHOMY H3 4aCTHbiX peuienan
Ce-2~i (a-r) + C'e -21ti (a~,'II ..r).
U = o
(34.25)
=
f Je21ti
1n1
_
z {~ OOO...::::.J
+ ~ C 'C ~
"J1H
6.u 1
-
k1
-
vg
+
m 1m';je
-21ti{(a~n -m,b1-m1b.).r]} 1
~
1
0
0
I1TaK, COOTHoweHHe Me>K.IlY
t
) e-21ti (a-r] -'11,-'11.
I
k~~' + OO~CCn,'II')
:J
e-21ti ( a~,n,·r)
~ CC m,m. e -2Iti (a:n,m.·
2 OOO...::::.J
m,*n,
ffl,
)
*
I
-) }
nOJlO>KHM reneps
~\ ~. (34.26)
nBe CYMMbI, CT051IUHe B npason 4aCTH npensrnyuiero COOT HoweHHSI. He npHBO.ll51T HH K KaKHM 3aTpY.llHeHH51M npn HCCJle .Il0BaHHH YCJlOBH51 opTOrOHaJlbHOCTH. nOCKOJlbKY menan, nasae Mbllt HMH B anrerpaa, pasen HyJlIO. nettcTBHTeJlbHO, nocne yMHO>KeHH51 aa e21ti (a' ·r) B 3THX cyxxax He 6Y.lleT HH O.llHOIt
e-21ti (a.r)
a" =
a'
nlnJ
H
• fIOJlY4HM
0
(D)
HJlH aHaJlOfH4HO
,0
'11,'11,
npensrayiueay
( k~'o + OO~CCnl'll') f f do = (D)
OTClO.Ila nOJlY4aeM BTOpOe COOTHOWeHHe Me
k1C V z' o
HyJlIO. OCTaeTC51 paCCMO
.r) e -21ti (a'n,'II,.
C'
v2 + oo2C'C -n,-n ff /1t i (a ~,n" .r){, ( klC + (k1C' + oo2CC ) e-Z1ti (a~,n ..r V oz .
n"
3KcnoHeHTbI C noxaaarenex, paanua TpeTb POJlb 4J1eHOB C 3KcnOHeHTaMH
H
o
+
.
k 1, C
k~~ +oo~C'C_n,_n,,=O.
+
0
+
-7I,-n"
nOCKOJlbKY B CHJlY pa3J1H4H51 nanpaaneun HHTerpaJl OT BToporo 4J1eHa paBeH HyJlIO.
2 ~ C'C m1mKeHHe (34.14) .IlJl51 4JYHKUHH nOJlY4aeM
A
OTJ1114eH OT BeKTopOB
3THX 4J1eHOB B mrrerpaa (34.17) pasea HyJ 4alt. KOr.lla BeKTOp a" paBeH O.llHOMY H3 3 Ilpn a" a YCJlOBHe opTOrOHaJlbHCCTH
rne C H C' - nna npOH3BOJlbHblX K034J4JHUHeHTa, K onpenene HHIO KOTOpblX Mbl BepHeMC51 nosnnee. J{aK 6Y.IleT nOKa3aHO HH>Ke, 4J1eH C a'n cooTBeTcTBYeT orpaaceunoa BOJlHe npa OTpa>KeHHH, ~
uUl
a"
ECJlH BeKTOp
+ OO02CCn,n. = O.
COOTHOWeHH51 (34.27a) H (34.276) 06pa HeltHblx O.llHOP0.llHbIX ypaaaenaa OTHOCHTeJl H C'. ECJlH 3Ta CHCTeMa nonycxaer cyuiecrn
13
3aK. 201.
C
c\JIIUl1eHTOB
H
k1
C'.
sroro
LlJIH
kl
Ul6C-n,-n,
V~
2V
o
= C~,n,. Paspetuan npenunyuiee
kl =
-t-
C'IC.
JII1Tb K09!p!pHUl1eHTbI
(34.18)
JIO)l{eHl1l1
3HaH
Bm,m"
1=0.
(34.28)
TO
C-n,-n, =
COOTHOWeHl1e OTHOCI1TeJlbHO
kl ,
CIIMOCTb
axoasuure
B
ljJYHKUl111 UI' lei
!
!7oAoca nponyCKaHuR
TO BpeMSI KaK AJIH HeB
JlI1HeltHolt
rrpSiMolt
lJaCTOTbJ
OT
3aBI1CI1MOCTI1
aaoopaxeaa aa sexropa
rpl1r.
lJeHl111
a
YAaJleHI1H OT TOlJKI1
I a I cyuiecraeauua sacroru OT I a I AJIH 83 WTpI1XaMI1. OPI1
xpaaas ncnuruaaer paa paapusa IICKa)l{eHl1e ysc
--
a
JlYlJelt OT KpIICTaJlJIOrpac\Jl1qeCKI1X rrJIOCKocTe
,~/'
Bpsrr noxaaa.r,
.
.> I
orpaxan.cs
101=10;',"2'
penrreuoacxae
JIylJl1
OT KpIlCT3J1JIOrparpl1lJeCKIIX
rrJl
'ITO
secxort
peWeTKI1
Me)J{AY
KpIICTaJlJIOrparpll'leCKI1MI1
pacnpocrpaaeaas
npn
yCJIOBI1I1, eCJlII AJIIIHa
[a 1 -_
113 9TOrO o6cy)l{AeHI1H cne.ayer, .lJTO B paCCMOTpeHHOM CJlY
2
00 =
KpOMe roro, B
2
000
+
k
naer nonpasxy K qaCTOTe, 2
2
E 1 = 000 -t- EOOoV20
aupaxceuan
1
Cn,n, "I
rro-
psnu aTOMOB
(34.30)
AJIH pa3.lO)I{eHI1H BOJIH, COOTBeT
C') BMeCTO OAHOro B nepsox cayxae, ocraasuue, xoropue I1MeIOT nopHAOK E.
npeauuiaio
B 3aKJlIOlJeHI1e 3aMeTI1M, 'ITO TaK )l{e, KaK 11 B sae, aoaayuieune cJIa60, eCJIII aaasenae aexropa a
nepsoa CJlY
(C
ume see
11
6J1113KO K 3HalJeHI1IO TI1BOrrOJlO)KHOM
B03Myw,eHl1e
BOJlHbI
)
He CJlIIWKOM
B03paCTaeT II
rr pa
a'Jl
l
11 I 2
(pH.lI.bf, KOTOPhlM B cnyxae rpe
BeTCTByIOT K plICTaJlJIOrpa!plIlJeCKlle rrJlOCKOCT cooraercrnyet yCJlOBIlIO, nonysennoay B HHII cJlYlJaH a = a' .
§
n 1n2
ofiparuott
PaccMOTplIM BeKTOp
B = nib I
onnoro 113 BeKTopOB a'n,n.,• O,llHaKO B npo
cnysae
nexorop
,llJlll BeJIl1lJHH nl H n2' onpene.onouurx B cnyxae
CTBYIOW,I1X 9TI1M lJaCTOTaM, C yseroa nonpasxn U I I1MeIOTCH ABa qJleHa
nJIOCKOCTlI
yAOBJIeTBOpHIOT
MbI ce~lJac YBlIllI1M. rrOJIy'!eHHoe Ep9rrOM 9KBI1BaJIeHTHO yCJIOBIlIO
Hr. 83.
CKOJIbKY
BTo
Ilpennoxaras B .lI.aJlbHeltweM OCTaHOBI1T nenpepusnon KpllBOIt B paapunnyio B OKpeC puaa, Mbl celt 'lac CpaBHI1M yCJIOBIIH, OTHO ,C rpOpMyJIOIt Bpsrr a .IlJIH orpaxe a = a,'n
/'
6.00
!7OAoca nponyacauun
rrpH6JIH)I{eHI1e
~J1H
BOJlHW
11 op9rroBcKoe OTpa}KeHH
1n2
,," . .
sae y)l{e nepsoe
Ia
c\JYIlKUllelt
JII1HIII1, AJIH B03Myw,
§ 35. nap~HaJlbH~e
AJIH KOTOpbIX MO)l{HO
9TO OTHOWeHl1e, MO)l{HO
Ilo soca nenponyceanue
coor sercrayer
(34.29) onpene onpene aupasceaue AJIH pas
(34.29). B
HBJIHeTCH
Kpl1BaH
oo~V~ I Cn,n, \.
kl,
Mbl rroJlyql1J111 ABa 3HalJeHI1H
lJaCTOTa
f nencraarem.aa.
3aMeTHM, 'ITO TaK KaK c\JYHKUI1H
JII1Tb OTHOWeHl1e
k1 2 Vo
UlgCn,n,
HaXOAI1M
ljJOpMyJIOIt
2 * UlOCn,n.,
V~
-
k1
oo~C lI,n,
KOTOpblX COOTBeTcTRyeT OAHOMy 113 3HalJeHllI
Heo6xOAI1MO, qT06bl
IKeHHOrO (BeKTOp a~) JlY4a. 3J1e MeHTapHaH reopas OTpa>KeHHH Bpsrra 3aKJl104aeTCH
KHbl uaxonari.cs B K Tpe60BaHHlO Ilyrs AGB 20 cos cp m
om =
1
1
b
I Bm I = m \ B I = Ii
Y4HTbIBaH reneps 3TH pH.llbl nO.ll06HO T C.lleJl3HO suure, MO>KHO nOJlY4HTb oouiee Y BbJllI~H3J10>Ke:lHoe noxaauaaer, 'ITO pa cJlY'la~, a
=
,
an,~, B KOTOpOM BOJjHa xapaKT
OT
psna
aTOMOB,
4TO
C'
npl1BOnHT K n05lBJIeHI11O BOJIHbI
(a~),
U=UO+EU 1 ,
11
~=~+Ef.
I1MeIOw.etl 60JIbWyIO aMnJIHTyny.
§ 3S.
reneps
PaCCMOTpl1M
rne
B
paapsrna
TOlJKH
TaKyIO 06JIaCTb I1SMeHeHH5I
sexropa a,
COOTB::TCTBHH C Ke aTOMOB H M H3MeHSlTb xapaKTep pacnpOCTpaHeHHSI. B B03MYllJ,eHHSI nponopunonansna nJIOTHOCTH CTaJIJIe, a BUn ypaBHeHHSI, onpeneamouie CTpaHeHHSI, cxonen C BHnOM ypaBHeHHSI (34 qJIeH f nponopUHOHaJIeH 3JIeKTpOHHo'l nJIO B KpHCTaJIJIHqeCKot! crpyxrype, r ne a urerxe paCnOJIOIKNlbI ynopaaouenno, KaIKJ 9JIeKTpOHHbIM 06JIaKOM, xoropoe MO>KeT !laTbCSI C npyraxa '!JIeKTpOHflbI!>IH 06JIaKa
paBHbIM nepnony peurerxa: 3TY KeHHe 'lJ1eKTpOHoB OT KpHCTaJIJlOB, nonraepxnaio uiee BOJIHOByIO npapony snexrpona. neACTBHTeJlbHO, B MHoro qHCJIeHHbIX sxcnepuaeurax (n'lBHCCOH H n>KepMep) OTpa>KeHHe
Me>Kny
c060A
yroa
TC/3.
Ooparnas p
.'
• • • • PRoamOHlOB
II
CmpyRmypa
r. 87. I'excaroaaasuaa peurerxa.
30Hbl3
V3 ' o= O.8660 d = -2- d.
~
'0
npoacxonar OT noaepxnocra KpHCTaJIJIa, H 'lJ1eKTpOHHble BOJIHbI He MoryT npOHHKaTb B r Jly6b ero. Ceaxac Mbl npanenex HeCKOJlbKO npnxepos npaxux H 06 paraux nsyaepaux peiuerox H YTOqHHM xonqmr ypauna COOT BeTcTBYJOUJ,HX aOH. ECJIH aanaua npaxaa peurerxa, TO Jlyqrne scero nOCTpOHTb COOT BeTCTBYIOUJ,YJO eA 06paTHyJO peurerxy, ncnonsaya COOTHorneHHe " (31.22). nJili sroro B npsxoa peurerxe onpeneasior paccros \' HH~ 0 Me>Kny napannensnux« panaxa yaJIOB. Ilpoaons reneps OTpeaoK nJIHHOA 1/0 B HanpaBJleHHH, nepneHnHKyJIlipHOM COOT BeTcTByJOUJ,HM panaa, nOJ1yqaJOT sexrop 06paTHoA peurerxn. rIa qIHr, 87 npHBencf! MeTOp, nocrpoean» pell!eT~~ Il CJlyqa~
II
r. 88. Pacnoaoxesxe
30H
B
re
peurerxe,
J\
f i
:A'
>ii
-:~[1.11;
reKCarOHaJIbHoA. OHa noayxaerca as nps na yrOJI Tt/6; MonyJlb ee 6aaHCHblX BeKTo
Ibj=
2
V3
1
/lfl'
CTpYKTypa nliTH nepsux aOH 06paTH >KeHa aa
KHO nposec paaox. Ilnouraas Ka)f{noA soasr A b pann
6aaHCHbIX
sexropax b,
A =lbl12sin ~ = V 3 Ib !2= _ 2 3
b
2
1_.
V3 I d 12
I
'
naouiaas me 'OlJIeMeHTapHon 1I4enKH IlpSlMOn peinerxn panna
Ad =
aa
2.
1t
Id I S1I1:3 =
V3
-I d
-2
2 I
1
= Ali .
Jl106yIO aony MomHO CBeCTH K nepnoa aoue, paa6HBaSl ee nOJl06JIaCTH H npHMeHHSl K HHM COOTBeTcTBylOllJ,He onepauna D6pamHQH peuiemua
Itpnma» peuiemua
• • • • •
r·
•
•
A
• • • •
1
L. • • • • • • • • • c::::::::J
~ 30Ha
2
3
m:rm 30na 4
Kaanparaas crpyxrypa npaaeneaa aa
TblX H3 ony6J1HKoBaHHot1 B 1930 r. pa60TbI HJIlllOCTpHpyeT paarinenne pa3JIH4HbIX 30H u .lleHHe HX K nepBolt.
B naurex HaJIOmeHHH Mbl OCHOBbIBaJlHC • 0 nepUOJlH4HOCTH KeT OKaaJTbCH, 4TO cHMMeTplfSl MyweHHH (npe.llCTaBJIlJlOWeft, nanpaxep. nJl MomeT 6blTb !H!me cHMMeTpaH aTOMHOn KeHHolt na KHble SHalJeHHSI. TaKHM oopaaou. Mbl npaxo.nax K peay JIbTaTaM, xoropue aHaJIOrHlJHbl pesyJIbTaTaM, nOJIYlJeHHblM B KOHue § 31. Ehy reopeay MomHO raioxe CKeTClI, oriparnoa MbI
cennac
'ITO
narepnperauaa.
npyras as
TO'lKH
pannyca
3Ta
crpepa
0
Ia I C
06P3THOit
nanpa npn
TO
ABH>KeHHe
nanaiouiea
TaKHM >Ke
06paaOM,
anaseana
Ia I
BeKTopa
MbI
'ITO COOTBeT
KpHBOIt 'I
a
3
Bpsr
onnoro 1'1
nonysaex
ARII
aHa'leHHlI '1aCTOTbl.
paspuay
BORHe,
(HRH BORHe
xorna
=
'I
(a).
HMeeT MeCTO 6p3r
HaXOAHTClI B nROCKOCTH,
1'1
nepnennmcynapnojt
K
BOAlITClI
nROCKOCTH.
MeHbwHIt
peurenca
Y3Rb
nexy.
C paaRH'IHbI
nepnennasyaspnue
MHororpaHHHK,
31
nepea
COAep>Kaw.HIt
3T
BH
peurerxu, oopaayer nepayio aOHY, 06RacTb, rpaaauen nepnoa aOHbI 1'1 rpausuea cnenyt MHororpaHHHKa, OXBaTbIS31OIl.lerO
HaHMeHbw
oopaayer
BTOpylO aony. Al/aROrH'IHbIM >Ke RlIlOTClI aOHbI 60Ree BblCOKHX nopanxoe.
u,eHTpOM B TO'lKe
P.
npOXOAHT nepea BTOpylO
~YHKIJ,HlI
'I
§ 40. 061IJ,He peayns rar OTHOCHIIJ,HecH K pacnpocrpaaen B TpexMepHMx nepHO~HqeCKHX cT
DyCTb peurerxu
peuierxn HRH B OKpeCTHOCTH 3TOit TO'lKH, yBHAHM,
'ITO
0
xa
Ry'lelt 01' KpHCT3RROB. DOA06
HMeeT MeCTO 1'1 B
BeKTOp,
(a~),
'I
cynepnoaauaea ABYX
OAHa aa KOTOpbIX
coeAHHlIlOw.HX H3'13RO
THeM KpHCTaRROrpa~H'IeCKolt nROCKOCTH. a -
=
RlIeT YKa3aTb np3BHRO nOCTpoeHHlI aOH:
pasuocn
reoxerpaxecxyio nnrepnperaumo
OTpa>KeHHlI peHTreHOBCKHX
ra).
peurerxu, ABa
Ka
onncan
BORH,
roro
KOHeu,
.1'
yCROBHIO 1'1 nOKaa3RH, 'ITO
'I
paaRH'IHbIMH
npyras orpaxeanon
CTByeT
ARII
cepeanny nexropa H, coenanstourero
a~ - a= H = h1b1 +h 2b2 +h3b3 06paTII01t
1'1
AonyCTHTb,
COOTBeTcTByeT
a~ HRH a~
ARHHa KOTOpbIX nO'lTH OAHHaKOBa, a
sexrop
(a)
3aMeTHM, 'ITO,
onpenexmorca COOTaoweHHeM (39.4). MeCTO rorna, xorna MQ)KHO HaltTH
npeACTaBRlIeT c0601t
'I
onpeneasercs
(39.5)
HMeeT
a,
=
pacnpocrpaneuaa,
YKaaaHHbIX
I a;, I ~ I a I,
r ne aexropu CRy'lalt 2
'I
BReHHlIMH
ABa
BbI6paTb
DOCKORbKY BORHbI, COOTBeT
paKTepHaYIOTcH
~
onpenennrs.
HeBOaMO>KHO
a~ cnenyer
1'1
CTBylOw.He
CHOBa paaRH'IaTb ABa cRy'lall.
1. Bee aexropu a;,
a
peurerxn,
nucoxoro nopsnxa oxpyscaior nepayio sony nocrpoenu MeTOAOM, ananornnnua nOCTpoeHHIO B cay-rae nsyuepnux crpyxryp, COOTBeTCTBYIO:l(He npavepu npasenenu B AaRbHeltweM. Buoop nnocxocrea, OrpaHH'IHBalOw.HX aoau, OCHOBblBaeTClI na paccyscaenaax, nOA06abiX acnonsaoaanaua B § 34 1'1 36 B cay-rae nsyaepaax crpyxryp. B caMOM nene, reopns pac npocrpaaenas BORH B cnnotuaort cpene npH HaRH'IHH MaRblX nepHOAH'IeCKHX aoaayuienaa nORHOCTblO aHaROrH'IHa reopna, paapaoorasaoa suure B cnyxae ASYX HaMepeHHl-t. He06xoAHMO 30HbI
ynce
TeRbHO,
H = h1b1 +h2D2 +h 3 b3 •
(a) 6YAeT p3apbIBHa.
l1ay'leHHIO
pacnpocrpanenas
HaRH'IHH MaRblX nepHOAH'IeCKHX
§ 34-36,
BORH
B cn
soaayuienn
S KOTOpbIX paccuarpaaancs cRy'l
Ilonysennue
TaM
pesym.raru 6ea rpyna M
CTpaHeHbI TaK)Ke 1'1 na ananoru-raue rpexxep
nepHOJlHlIeCKHX B03MYlll,eHHtl; B CJIeJlYlOlUHX napar parpax Mbl PHJl npaxepoa. OTHOCHIlI,HXCH K 9TOMY 061l1,eMY CJIYlIalO. B CJIylIae Tpex H3MepeHHtl BOJIHOBOe ypaBHeHHe HMeeT BHJl
1
m'llJl,'l,w'J
1
2
2
npasenex
Ul~
TPH nOCJIeJlHI1X lIJIeHa B npaaon lIaC onpene.nnor aexoropua Y33JI o6paTHotl peW ITOJlCTaBJIHH reneps aupaaceuue (40.6) JlJ HOBoe ypasueane, nO.'IYlIaeM COOTHoweHHe,
(40.1)
~ljI+V~ ljI=O.
ITOCKOJIbKY Mbl XOTHM paCCMaTpHBaTb pacnpocrpaneune BOJIH B nepHOJlHlIecKotl cpene. TO 6YJleM cnnrari., liTO
1
v~ =F(r),
lletlcTBI1TeJIbHO, TaK KaK
~e
(40.2)
-21tl
, . · r) (o"',m,"'. =
4,,2 I a
-
I
mjm>Jma
12
e
TO rne K.u.ylO 30Hy aucuiero nopsnxa MO>KHO pa36HTb na no.u.06J1aCTH H cooTBeTcTBylOLUHMH nepenocaxa CBeCTH K CTpyK Type, nOJIHOCTblO coananaiourea C nepnon 30HOtt. Ka>K.u.a51 30Ha naer, TaKHM 06pa30M, nenpepuanyio BeTBb KeHHotl ua KeT 6bITb aanncauo B BIi,lle ~
= A (r) e2~i [vt-(a.I')]
+ r' + ntd l + n d + nad a•
(44.1 )
r.ae r 0 - BCKTOp. onpenensnounta nonoxenne nexoropon T04K Ii 0 nepBol'l. SI'Iel'l.KIi OTHOCIiTeJlbHO ee Ha4aJla. BCKTOp nldl-t n2d2 nad a onpenenser nonoacenne uasana Sl4el'l.KIi (Ill. n2' n a) OTHOCIiTeJlbHO aauaaa peurerxu, a BeKTOp r' coenanaer pac cxarpaaaeuyio T04KY P C T04KOl'l. 0 5I'Iel'l.KIi (n l• 1l2' na). Bexrop r ' seerna, KaK MomHO naners 1i3 eplir. 113. HaXO,llIiTCSI BHYTPIi Sl4el'l.KIi IiJlIi 3KBliBaJleHTHOro 06beMa.
+
+
+ r') =
i = 1. 2
e2~i [(a.r,,)+(a
Ii npennonoxorx. 'ITO BeJlli4liHa A" (r') B nepBol'l. H'Iel'l.Ke. nO,llCTaBJlHSI 3TO aupaxcenne ,llJlSl A (r) ~
HBJlHeTCH nepaonuxecxoa epYHKllliel'l. B nps MOl'l. peuierxe. YHKllIiID A (r) MomHO rorna onpenemrn, Jlli60 BHYTpli 3J1eMeHTapHol'l. H4el'l. Kli (d), Jlli60 BHYTPIi orpaua 4~HHOl'l. aauxnyroa noaep x HOCTbID 06J1aCTIi, 06beM KO ropoa panen 06beMy 9J1eMeH rapuon H'Iel'l.KIi (d). 3TO no noacenae HBJlHeTCH ornpaanon T04KOl'l. MeTO,lla (nl.n2,n3 ) d, Baruepa - 3el'l.Tlla. npliMe HHeMoro B Teoplili TBep,llOrO
YIlKUI1I1 ~ 11 ee nopaans HOl\ npOI1SBO.llHOl\ B T04Ke. onpenenseaoa r'. Henpepuanoc'n, ~ Ha rpaHHue 06eCne411BaeTCH YCJlOBl1eM A" (r') = A" (r) e :'
(n,k ,+n.,k,+na ka) , ,
§ 45.
(44.7)
I1SJlOlKeHHble B npenunyuiax naparparp CTaBJlHIOT 60JlbWOl\ mrrepec, nOCKOJlbKY OH nyio POJlb B 60JlbWOM 411CJle aanas TeOpeT He CTaBI1M csoea ueJlbIO paCCMOTpeHl1e sc OCTaHOBI1MCH JlI1Wb na O.llHOM Hal160Jlee paCCMOTpHM TeOpl1IO TenJlOeMKOCTI1 raepns PesIOMl1pyeM cnasana nexoropue OCHO ILJlH acex Tl1nOB BOJIH (ynpymx. 9J BOJIH ne-Bpoana), pacnpOCTpaHHIOlUl1XCH 4ecKol\ crpyxrypoa. q>yHKUI1H 'I (a), CBHSb BblM BeKTopOM a. ncnurusaer paapus H BeKTOp a OKaH4HBaeTCH aa nJlOCKocTH. np rpaHI1UY KaKol\-Jm60 SOHbI. 3TH nJlOCKOCTI1 po.n, .llJlH scex BOJlH, KaKOBa 6bl HI1 6blJla 4ae 9J1eK TpOMarHl1THblX BOJIH. aanpaxep, UI1I1 '1(a) HBJlHeTCH npHMol\ npl14l1Hol\ CeJ peHTreHOBCKI1X JlY4el\ (nHTHa Bpsrra JlaY 06lUI1M CBol\cTBaM 91'11 nJlOCKOCTI1 Bbl61lpaIO HI1U SOH. onpenexcuuux 13 npenunyurrx n 4TO B .llaHHOM xpacranne rpopaa SOU O.llI1
nO.ll06HOe nee COOTHoweHHe BBO.llI1TCH .llJlH 06ecne4cHHH uenpe PWijHOCTH HOpMaJlbHol\ npOI1SBO.llHol\. (lpRNla" pewemKQ
u r. 114. 30Hbl Bnruepa -
3eHTl-la.
B cay-rae 04eHb 60JlbWI1X .llJlI1H BOJIH BeJll1411Hbl k l • k 2 • k 3 • TaK lKe KaK 11 411CJla "I' "~, "3, MaJlbl. TaK 4TO na rpauaue MOlKBO np116J1l1lKeHHO nOJlOlKHTb A" (r') = A" (r).
nOB BOJIH.
(44.8)
MOlKHO nOJlbSOBaTbCH YlKe 9TI1M nepnux npI16J1I1lKCUHeM, HO 4aCTO 6blBaeT He06xo.llHMO nenan, .llOnOJlHI1TeJlbHO ynpouiaio uiee npennonoaceaae, nocxoasxy xuoror-paaaaa rpopaa rpaHI1U SOH Bl1rHepa·- 3el\Tua CMJlbHO sarpynnaer pacner. 3TO ripen 1I0JlOlKeHI1e COCTOI1T B aaxeue MHororpaHHol\ nOBepXHOCTI1 nep BOl\ SOHbl crpepon panuyca R. I1MeIOlUel\ TOT lKe 06beM. Tor aa cooTBeTcTByIOlUHe npyr npyr y T04KI1. onpenexsexue BeKTOpaMI1 r 11 r", saMeHHIOTCH B H4el\Ke naaxerpaasno npOTI1BOnOJlOlKHblMH T041Ke COCTOSlmaSl 1i3 '-IaCTIiU onnoro runa, HBnHeTCH, KaK MhI Blinenli, petuerx ot!. C 6a3IiCOM, H e" COOTBeTcTBylOT rpn axycrusecxae BCTBIi Ii Tpli BeTBIi C 60nec BhICOKIiMIi '-IaCTOTaMIi, xoropue MhI Ha30BeM onTIi'-leCKIiMH BCT BSlMIi. HanpOTHB. MOHoaTOMHaSl xpacranmrsecxas crpyxrypa rnna Ky61i'-IeCKO" r-paneueirrpnpoaamroa linli 06beMHoueHTplipoBallHoft peurerxa He IiMeeT 6a3lica. Ii ua xaacnyto ec Sl4e"Ky npaxonarcs onliH aTOM. Bnar onapa 3TOMy OCTalOTCH roxsxo rpa axycru
.Haace
MhI
6yneM
0603Ha'-laTh
'-IeCKlie BeTBIi. a OnTIi'-leCKlie BeTBIi HC'-Ie3aIOT.
"
nepaon 30HhI paBCH 06beMy Vb 3neMeHTapHO" 06paTHo" peurerxa. Ecnn V d 06beM 3JIeMCHTapHO" npSlMO" peurerxn, TO
H'-Ie"KIi
06beM
1
BOJIHhI
(nneaxsaan
n03TOMY
60pH
yCJIOBIi".
M.OJlelcYJl/Z-M.OJlb.
(45.2)
aaeneunoe B § 34 nOHSlTIiC peuierxn. nJISI 06W.HOCTIi peayasraroa npeanonoaora, '-ITO crpyxrypa HBnHeTCH pewCTKO" C 6a3liCOM, npnsea aaexenrapnaa SI'-Ie"Ka MhI yTO'-lHIiM, TaKIiM 06pa30M,.
UIiKJIIi'-leCKO"
npHMO" peurertcn 06pa30BaHa s aTOMaMIi onaoa MoneKyJIhI. TaKO"
orpaunuenaon crpyxrypu
3aBHCHT
He TOJIhKO OT CBO"CTB aHanOrH'-IHO" 6eCKOlle'-lHO" peurerxu, HO
rpannunsrx
yCJIOBIi".
Bo
ace
sanauu
reopua ynpyrocru
3TIi yCJIOBIiSl 06hI'-IHO BHOCSIT 60nhwlie aarpynnenas. TaK. B He npepWBHO" 1i30TponHO"
scex
runoa BOJIH:
cpene
OCTpOyMHO liM
B
UIiKnli'-
MhI ynoMIiHaml B § 34. Pac cpena, He IiMelOwaSl rpaHIiU. B KOTO pacnpocrpanarscs BOJIHhI. Ii B He" BhI6lipaeTCS IiMelOwli" rpopay napaaaeaenanena, o
KOTOphlX
He'-lHaH
0KnaeTClI
a OKaH'-IIiBaeTCSI B TO'-lKe onpenenseauxn COOTHoweHIiSlMIi TO'-leK B enliHliue 06beMa ooparuon peurer nOCKOJIhKY paccrosnas MelKny HIiMIi no paaau coo'raercrnenao 1/ L1 , 1/ L2 Ii 1/ L 3 • B 06beMe V COnep>KIiTCSI O'-leHh 60JIb OHO paBHO sN (N = 6.06· 1023) . C npyroa napaaaeaorpaxaa L l , L2 Ii L3 aasnroro 6 d 2 • d 3 3neMeHTapHO" H'-IC"KIi peuiertcn, Ta BOJIHOBO" BeKTOp
a2' a 3),
npenCTaB.~RIOw.HX
coornoiueuaaaa
coticreeunue
(45.6).
O'leHIl
Kone6aHliS
lleJIIiKO
npocrpaucraa
(aI' a2' a 3 ) ,
no,
YlJl1TbIBaH npOCTpaHCHl1H BJlCHl1H
C
yr J10M
a
r +dr,
HCKOTOpOC
l1 MaJlbIn
4l1CJlO
xouyce
MC)f{,lly
BOJlHOBbIX
nanpa
MOIKHO noaysnrs. I1
acen nepaoa 30Hc,
crpepaxa panaycos
r
paBHO
rne I1CnOJlb30BaHO COOTHOrnCHI1C
dn = L1L2L3r dr dQ = Vr'J dr dQ,
(45.7)
Hy)f{HO OTMCTl1Tb,
1
OTCIO,lla nOJlYlJaCM
=
A
1.
W-
W'
CTblC i
qJa30BaH CKOpOCTb BOJIH paCCMaTpl1BaCMOro rnna.
Cor J1aCHO POCTI1
Jlerxo
V dQ --;2-2 d (- ; ) W
U,
,llaHHoMY
§ 21
B
onpenenenmo
~::- '
r pyrmoaon CKO
npc.1lCTaBI1Tb
crpyxrypu,
YTOlJHl1Tb
iJ-;
1
d ( -; )
W=TaT' U=--;naT' TJ=(F; dn
w'
(45.9)
qJl1311lJCCKl1n
0
qJl1J1bTP
HbIX HlJCCK,
CMbICJI
UI1
UI1KJll1lJCCKYIO
B
113
n
l1,llcHTI1'1
I
KOTOpOM Ilu(,Jlt:,tJ.H>D1
.t:
HlJcltKa CBH3aHa C nepsoa. JlpYfl1M
2
npl1MCpOM
n Pl1MCT Bl1,ll _
npo
n03BOJlHIOW,l1c
KJll1lJeCKl1X YCJlOBl1n. rpauecxna
rorna BbI pa)I{CHl1C .LIJIH
eerie
p
peurerxy oopasyer, aanpnxep, 9J1CK
MO)f{HO aanacars 'I
lJTO B cnyxae pCrnCT
VIV a B npenunyinen qJOpMyJlC MOJIeKyJl N, HaXO,llHw,I1XCH B 061> exe V. rnCHI1C
(45.8)
r=lal=-x-'
(45.1).
Eho oxeas cyinecrseuuun PC3YJlbTaT T
r,llC
1 d ( -1 ) dn. = V dS_) -2
o6beM
n=f dn=V f r 2drdQ=VV o
BCKTOpOB,
2
rne
no
HanpaBJlCHI1C pac
KOHyC OKOJlO sror o
dQ.
pacrnopa
OKaHlJI1BalOw,I1XCH B 9TOM l1
HI1C (45.7) a I1MeHHO
L 1L2L 3 d« = V ds .
paCCMOTpl1M
BOJlHbi
6aHl1lt ,llaHHorO runa
paSHO
=
dti
MoryT CJlY)f{I1Tb xarue
TpOHbI: MarHcTpOH C rpaneuennans-
-;?d-;
(45.10)
dn- V d-- W?U .
HbIMl1
pe30HaTOpHbIMI1
nOJlOCTHMI1
aHaJlOfl1
C UI1KJll1lJCCKl1Ml1 YCJlOBI1HMI1 ,llJlH XJlOpl1CTO Hy)KHO
OTMCTHTb,
r pyrmosaa,
OT opnenrauan xouyca. CKOpOCTl1
LITO
B
KpI1CTaJlJlI1'1CCKOn
TaK 11 qJa30saH CKOpOCTl1 ne
3aBI1CHT
B
peuierxe
KaK
3aBl1CHT OT lJaCTOTbl 'I
l1
l1,llcaJlbHOM l130TpOnHOM TBCP,llOM TCJlC
OT
Opl1CHTaUl1I1,
l1
BbIpa)f{CHl1C
(45.10)
MO)f{HO npoaurerpnpoaars no BCCM HanpaSJlCHI1HM pacnpocrpa HCHI1H. Tor na. 3aMCHHH
dQ
na
41t, -;2
PaCCMOTpl1M reneps, KaI{ MO)f{HO ynpoc CY)f{,llCHl1H
B npl1MCHCHI1l1 K ll,llcaJlbHOMy 113
TCJly. CJlOBO .11,llcaJlbHoe" ynoTpc6J1HCTCH an HanOMHI1Tb 0
nonoxeana.
HCCKOJlbKO l1CKyCCTBcHHOM xa
To, lJTO MbI Ha3blBaCM 1130TponH
nOJlYlJaCM
B ,llcltCTBl1TCJlbHoOCTI1
d»
MaJleHbKI1X
dn=47tV W?U'
(45.11)
npencrannser
6ccnOp5I,ll0'lHO
c060n
0PI1CHTl1pOBaHHb
lJTO l1,llcaJlbHOC TDCp.llOC TCJlO MO)f{CT npC,llC KOTOpWc CpC,llHl1e CDonCTDa KpI1CTaJlJla.
OTMCTI1M,
oasuca
(s
'ITO
=
1;
l13
ypaBHCHil5I
O,lll1H aTOM
(45.7) B c/lyttae petuemtcu 6e3
aa HlJCnKY npHMOlt pCrnCTKl1) MOIKHO
06w,l1n PC3YJlbTaT: npu uiuounecuux YC/lO BUftX i)/lft «aucooeo muna ynpywx eoui no nnoe 'WC/lO c06 cmeeuuu.x IW/le6aIiUU moseo paeno N, m, e. 'IUC/lY amO.MOI,l B 06beMe V. noaysart. OlJCHb
n pn l13J10)KCHHI1 TcO pl1l1 TCnJlOCM KOCTI1 6Y.Ll.CM nOJlb30DaTbCH CJ1C.llYIOW,I1MH ynpouiai HI1HMl1. a) npC,llIlOJlO)f{I1M, 'ITO KpHBblC na qJHr. npHMbIMH, lfeCKI1C
KaK nOKa3aHO na qJHr.
BeTBl1
6y,llyT
115.
COOTBCTCTBOBaTb
TaK
TOJ
BO/lllbt
6ynyT HMeTb nOCmOflHllYIO rpaaoayro CKOpOCTb
W = U = const.
(45.13)
6) 3aMe1lUM nepeyio 301lY crjJepoii. Ilpn 9TOM He nano aa6bIBaTb 06 06w.eM pesylIbTaTe. nOllY4eHHoM B nasane sror o napar parpa, a HMeHHO, 'ITO crpyxrypa aOH He aaBHCHT OT npn ponu paccxarpanae aux BOllH. B 'IaCTHOCTH, Mbl nOlIlKHbI C4H TaTb, 'ITO 9Ta aOHa HBlIHeTCH cljlepH4ecKol\ nllH Bcex aKycTH'Ie CKH x BOllH, KaK npononsnsrx, TaK H nonepevuux. EClIH 'Iepea R 060aHa'IHTb panayc 9TOl\ cljlepbI H npenrto 1I0IKHTb, 'ITO COOTHoweiIHe (45.12) OCTaeTCH cnpaaennaauu. TO nllH ee 06beMa nOllY4aeM
- 4 7t R3 _ 1 _ N V b-3 - Va-V'
(45.14)
rne V d - 06beM H4el\KH npHMol\ peuierxu, N - 4HClIO MOlle Kyll B nOllHOM 06beMe V. nllH ror o 'IT06bl ynoaaeraopun, Tpe60BaHHHM 06w.el\ Teo PHH, Mbl nOllIKHbI C4HT3Tb, 'ITO nllH ncex THnOB BOllH aean 4HHa R onHHaKOBa, T. e. onHHaKOBa npenensuaa nllHHa BOllHbI AMR H •
R=
MaK e •
=
Am lH • = ( 3N i :
ne6al\
XOTII
(45.15)
'il. ripen.
=
'it, npsn.
= Wt
Wl
(
3N )'/, 41tV '
(i!!_)'/' 41tV •
(45.16)
H HeCKOllbKO OTlIH'IHble peaym.raru 6blllH B reopna TenJIOeMKOCTH rnepnux Tell. Koraa ony6J!HKOBlIJI CBOIO paoory, TeopHli aOH 6blJIa eiue
Ileoae»
Vo
KHO aupasnrs KaK
U = F (V) +Ul +U t, +U t. ,
f /(V)dV.
(47.4)
HX 3Ha4eHHSlMH
o:
+D,(~)[06m T O
(
TaKHM 06pa30M, MO>KHO C4HTaTb. 4TO - pannauuouuoe
dU -J;.dW
~ ~{D(6;)dT+D(6;)~
"
m
F(V)=-
H
m
/(V)=- ~Pm'
nOJlO>KHTb
dU
=
sanaurea
TO 9TO cyxxapnoe .1I.aBJleHHe TO>Ke .1I.OJl>KHO 06palU,aTbCSI B HyJlb.
Bbl3bIBaeT TeIlJlOBOe pacurapeaae. nOJlHYIO 9HeprHIO TBep.1l.0rO rena
KOJlH4eC
dQ=dU+dW•
Bbl3BaHHblX
P=/(V)+ ~Pm' na
TeJIOM
BHyTpeHHHM
pa.1l.Hau,HoHHoro
CBSl3aHHoro C BOJIHaMH rennoaoro .1I.BH>KeHHSI. T.
eCJlII
/dV
----av-'
ypaanoeeumaarsca
SlBJlSleTCSI
MOJleKyJl, /
ECJlH .1I.eltcTBYlOlU,ee
-
TaK KaK rror JlOlU,eHHOe .1I.OJl>KHO
xoropoe
B3aHMO.1l.eltcTBHeM
V
1
OV =3- W m
6m
eo
HJlH
oW
1
6~
dU= oV dV+ oT dT=-/dV+ '\1 OUm + ~ sv
41t V] = n 3N Wm
(4
CTOpOHbl. H3MeHeHHe 9HeprHH
su
nonyxaex
-.l1
C IlOMOlU,blO q>OpMyJlbl
m
U m 06 m
o In 6m _ _1_ 0611, _ ~ [I ~ W oV - 6m.OV - oV n k m
Um
dW=/dV -RT ~ [DC;)
oWmlaV•
Pm- --0,;;- oV ' TaK KaK. Y4HTblBaSi
HJlH. 3aMeHSISI
MO>KHO sarmcars B BH.1I.e
_
/ dV - ~ ~:: ~ m
c:
rrOCKOJIbKY B 9TOM CJlY4ae OTCyTcTByeT 4J1eH
(47.1)
~~;
aV'
Pm. PaBeHCTBO
dW - P dV =
HJ1'H. 3aMeHSISI
(aamlaV) dV
aa
dam'
dS=R~(D'(~mIT) _ D(6e':T»)(d m
Iloaaraa X m = amlT.
dS = R
nonyxaea OKOH4aTeJlb
m)- D~:m)
~ [D' (X m
oanasaer,
'lTO paccxarpasaeaoe HaMH pSieT npHHUHny KapHO. 113 Bblpa>KeHHH .1I,JIH dS CJIe.1l,yeT
d~m B
TeJIO y.1l,OBJIeTBO
= R [D' (Xm ) - D l~m)].
Ileoas
Xm
3
D(Xm) = - 3 Xm
f
(47.8)
aupasan.,
--.!!.£-'R dXm -
1_
E3 dE 3 -~-+8Xm'
\lJIH
dS _ dXm -
--3-f X m
3 eX.... _
e~ - 1
3R eX m
12R
-=! - x:n
4
0
Xm
f 0
E3 dE e~-I'
~ e~-1 '
(47.9)
h» hlkT
e'
+;)-; nh»
-1
=--. g
1'1
.!!- _
1
g -
e~-1
•
1
m~l,
t" t.
s;
In ; ] .
n g
h»
~=--
emiT
f
[e~ e~ -~--ln-~--e-l
o
3~
Xm
xm f [~-ln(e~-1 e -1
0
I1HTerpHpyH no 'laCTHM. MO>KHO H36aBHTb JIOrapHK.1I,blR pe30HaTOp 06JIa.1l,aeT cpenuen snepruen, onpe .1I,eJIHeMOR no <popMyJIe IlJIaHKa, TO ·U.=
OT 0 .11,0 '1m
Iloaaras CHOBa X m emiT H rpynnnpya 'l npe06pa30BaHHR nOJIy'laeM HaKOHeu
Sm=
OTMeTHTb, 'lTO supaxeaae (47.9) cosnanaer C pe3YJIbTaToM. K KOTOpOMy npHBO.1l,HT BOJIHOBaH MexaHHKa .1I,JIlI 9HTponHH HallieR CHCTeMW rapMOHH'leCKHX OCUHJIJIHTOpOB. ,UeR CTBHTeJIbHO. KBaHTOBaSi xexauaxa n03BOJIHeT npoaecrn CTaTHCTH secxoe Bbl'lHCJIeHHe 9HTponHH. Pacnpeneneamo n KBaHTOB saep rHH h» (nOJIHaH sueprns nh'l) Me>K.1I,y g pesoaaropaxa COOT BeTcTByeT 9HTponHH ;)In(1
41t~2
d» -wa
=
Hurepecuo
S=kg[(I+
V
~
S= T )3 Sm=3Nk ( -6-
Xm
~+_
0
run
.1I,JIlI BOJIH
I1HTerpHpyH no 'I B npeneaax 9HTponHIO TBep.1l,OrO TeJIa
m
xm
--9-f X~n
Torna
rne
pe3YJIbTaTe nonysaea
-
9TOMy 'lHCJIy.
BBO.1l,H KHO
UHlO
B
rnepnoe
e
3Re m
12R
~1 -
X4 m
e
-
~ = 3R [ dX , m
e
1 1 + 1] -
Xm
12R 4
f (_1+ 1) ~3 d~ ; ee -
Xm 0
-.
1
JIerKO BH./leTb, qTO OHO nOJIHOCTbIO cosnanaer C COOTHoweHHeM (47.9). )leRcTBHTeJIbHO, sropoe caaraexoe non 3HaKOM anrerpana ./laeT nOCJIe HHTerpHpOBaHHH
_
m= X:, [f.]X 4
12R
_ 3R .
0
STOT qJIeH COKpaw,aeTCH C COOTBeTcTByIOW,HM CJIaraeMblM B xaan paTHbIX cK06Kax, 6JIarO./lapH qeMy H nonysaex CHOBa (47.9). ITO./lBO./lH nror, cnenyer CKa3aTb, qTO H3JIOmeHHaH reopas BnOJIHe y./lOBJIeTBOpHTeJIbHa, rorna KaK aupaxeaae ./lJIH snyrpeanero naaneaaa no reopan )le6aH, B KOTopoR BBO./lHTCH TOJIbKO onna xapaxrepacrasecxaa rexneparypa, He cornacyercs C TOqHbIM BblpameHHeM (47.2). BOJIee nOJIHOe H3JIOlKeHHe reopaa raepnoro TeJIa MOlKHO HaRTH B paoore BpHJIJII03Ha 1), r ae npaaeneno, B qaCTHOCTH, ./lOKa3aTeJIbCTBO COOTHoweHHH (47.1). 3./leCb me MbI XOTeJIH TOJIbKO noxaaars B03MomHOCTb BBe./leHHH B 3TY TeopHIO nOHHTHH 30H H YKa3aTb aa nonpasxa K reopau Ileoas, COCTOHw,He, KaK OTMe· THJI Bopa, BO BBe./leHHH ./lBYX xapaKTepHCTHqeCKHX rexneparyp llMeCTO onaoa.
JHITEPATYPA 110 TEOPJ;fJ;f 30H 1. B rill 0 u i n L., Quantenstatlstik, Berlin, 1931. 2. B rill 0 u in L., Structure des corps solldes, Paris, '1937. 3. Mol I N. F., Jon e s H., Theory of Metals and Alloys, Oxford, New York, 1936. 4. S e i I z F., Modern Theory of Solids, N. Y., 1940. (CM. nepeson: 3 e HT I.( ., Coapeaeanaa TeopHII raepnoro rena, M. - JI., 1949.) 5. Par 0 diM., Recherches dans I'tnfrarouge loin lain par la methode des rayons restants, Paris, 1938. 6. B I 0 c h F., Zs. f. Phys., 52, 555 (1928). 7. M 0 r s e P. M., Phys. Rev., 35, 1310 (1930). 1) CM. npaxeuaaue na CTp. 257.
v
Ii' I .'
(1933); 7, 401 (1936). 10. Jon e s H., Proc. Roy. Soc., 144A, 225 (1 11. Jones H., Molt N. F., Skinner H. (1934). 12. S I a I e r J. C., Phys. Rev., 45, 794 (1934). 13. Hun d F., Zs. tech. Phys., 16,331,494 (193 (1936). 14. P e i e r I sR., Ann. Inst. Henri Poincare, 5 15. S e i I z F., Ann. Math., 37, 17 (1936). 16. B 0 u c k a e r I L. P., S mol u c how ski R of Brillouin Zones and Symmetry Proper in Crystals», Phys. Rev., 50, 58 (1936). 17. Her r in g c., Phys. Rev., 52, 361, 365 (19 18. Chodorow M. I., ManningM.F.,Phy 19. B i e b e r b a c h L., «Ueber die Inhaltsgleic Zonen» [Monalshefte f. Math, u. Phys., 48 20. H 0 u s ton W. V., Rev. Mod. Phys., 20 16 21. Sialer J. C., Phys. Rev., 84,179 (1951). 22. K a I z E., Phys. Rev., 85, 495 (1952).
MOCTH OT BpeMeHH
noaysaex cnenyiouree d'u
dx 2
+ ('Y/ + 'Y
cos
=
2~) u
r.l\e
'Y/
§ 48. 06II{lfe aaaesanaa I13y'lall
pacnpocrpanenne
BOJIH B
cnaouraoa cpene C nepao § 40 HeKoTopwe CBOtt
C
.llH'IeCKott CTpyKTypOtt. MM pa306paJlH B
ypaBHeHHlIMH
H3y'leHHH
9J1J1HnTH'IeCKolt
sanaxa H 06CY.l\HJlH psn npauepoa. sroro xpyra sanas lIBJllIeTClI cJly'latt onnoxepnoa crpyxrypu, .l\JllI KOTOpOtt ypannenae (40.1) CBO
'1aIOTClI
H3
9J1J1HnTH'IeCKolt O.
1=0
nepaonasecxae peureuas
npH YCJlOBHH p3cnpocTp3HeHHH nOJlYlJ3eM
1 V-2'
xorna npn
CJlYlJ3IO,
DOJlO)KHTeJlbH3H nOJlYOCb 7j npOXO./lHT TOJlbKO
a3rnTpHXOB3HHble 06J13CTH. nOCKOJlbKY H3 yp3BHeHHH (48.4)
n 2 (n -
B TOlJK3X H3 OCH 'Yj C 36cUHCcoR 7j =
B
noro
60Jlee
06meM
Vll =
(n -
in
xorna
CJlYlJ3e,
'Yj He
lJHCJl3. HO 1=0, yp3BHeHHe M3Tbe
./lHlJeCKHe perneHHH BH.u3
uenocpencrsenao npaneraioutejt
B 06J13CTH,
KOCH 7j. 3H3lJeHHe
I ~ 'Yj. lJTO npHBO./lHT K nOHBJleHHIO M3J1bIX BoaMymeHHR.
xpasue,
cos mt,
Ilse
Orp3HHlJHBaIOmHe a3rnTpHXOB3HHble 06J13CTH, H3lJHH3IOTCH
1.10 = tm = t
r./le
BCer./l3 B TOlJK3X OCH 7j C 36cUHCcoR
me.
sin
v1i.
A (e). B (e) CBO.uHTCH rorna npO nauie peureaae npHBo.uHTCH K oouiexy
YHKUHH
7j = n 2
(n- LleJIOe)
(48.11)
3
MombIO c!JOpMy JI 3RJlep3 H HMeIOT B 3THX TOlJK3X K3C3HHe
(n -
1)-ro nOpH./lK3, T3K lJTO
rp3HHlJHble KpHBble
npa n = npa n =
1 H3lJHH3IOTCH B
2
HMeIOT,
onuoa
KpOMe
cos
TOlJKe;
roro,
06myIO
BepTHK3J1bHYIO
aa nepaon
me =
eim~
+ e-im~. 2
•
Sin
c!JOpMyJlbI
K3caTeJlbHYIO;
npa n = 3
1
HMeIOT 06myIO BepTHK3J1bHYIO K3C3TeJlbHYIO H
H3KOBYIO KPHBH3HY H T • HCCJle./lOB3HHe
aepnexcs,
6blJl0
D t = D2 = T '
ona
n.
3
rp3HHlJHblX
KpHBbIX,
anepsue nposeneao
K
KOTOpOMy
MbI
aa
BTOpoR
euie
1
Dt = - D2 = 2 1 '
M3Tbe.
Byneu
§ 49. Oduiee
yp3BHeHHH
HCCJlelJ.OBaHHe ypaBHeHHH MaThe
3
Buuie
MbI OTMelJaJlH, lJTO nOJlYlJeHHOe JlOKe
HHe yp3BHeHHH M3Tbe MO)J{HO npe./lCT3BHTb B nOJlYlJeHHH
me =
nOJlHoR
KapTHHbI
./lOCT3TOlJHO
oouiee peure
sane
HCCJle./lOB3Tb
KOM6HH3UHH
CTBOB3Tb CTOHlJeR BOJlHe.
3TOT
3THX THn
c!J3KTOPOB peuieuns
I
=1=
0
MoryT
Cem(I'
m
H,
e).
CJle.uOB3TeJlbHO.
Sem(I'
1.1'
e).
O.uH
a3BHCHmeM
p.=im=!('Yj,I),
acnonsayerca
06bIlJHO npa lJHCJleHHOM HCCJle./lOB3HHH yp3BHeHHH M3Tbe.
1=0
paanoac
06e 9TH c!JYHKUHH COOTBeTcTByIOT lJeHHIO
6Y./leT COOTBeT H
6b1Tb
DpHMeM .uJlH 3THX q,yHKUHR 060aH3lJeHHH
JlI060R
BeJlO 6bI K P3CC'MOTpeHHIO CB060./lHbIX BOJIH THn3 (48.6). BbI60p
xoropue npa
(48.4).
npa scex
(48.5). nJlH
na 3KcnOHeHUH3J1bHblX q,3KTOPOB. BXO./lHmHX B (48.5). lJTO npa ./leRCTBHTeJlbHoR
H3abIB3Tb 0606meHHbIMH c!JYHKUH
npaxex
m
CTpeMHTCH K
V 11.
KOr.u3 1-+ O
KaK S1CHO BH,llHO H3 epHr. 121, KalK,llOMY 3H3lJeHHIO p. (HJlH m), neacauieay B aaurrpaxoaaaaoa 06J1aCTH, B nJlOCKOCTH ('lJT) COOT BeTcTByeT onpeneaaexaa COOTHoweHHeM (49.3) KpHBaSi (I' m)'
~I\~ "'" '""
t
.",
I
Ifl"'.
J. _ Co 1 -
I
I
•
Eho COOTHoweHHe onpenenser npsxyio, OTMel HH,lleKCOM Ul 2, KOTOp3S1 BhlXO,llHT H3 aasana cexaer xpaaae (I' m)' cooraercrayiouiae P33 xeuraacs B,llOJlb 3TOtt npauon. Mbl nocne
125 I I
2C 1
-4
-j
-2
I
•
I
-411
-sn
-2 tr
Hr. 121.
-I
1.• I
a -~1--~2-~ 0 tr z« • I ,
I
I
30Ha 1
Hr. 122. :J1106ott TOlJKe KOTOpOtt MOlKHO COnOCTaBHTb nne epyHKUHH Cem ,H .Sem . I1CKJllOlJeHHe npencraaamor co6ott xpasue, OrpaHHlJH B31OI.lJ.He 3aWTpHXOBaHHhle 06J1aCTH, KalK,llott H3 KOTOpbIX, KaK BH,llHO H3 epHr. 121, COOTBeTcTByeT TOJlbKO O,llH3 H3 3THX epyHK UHtt. rpaHHlJHhle KpHBhle (r m) BhlXO,llSiT nonapHO H3 TOlJeK OCH a6CUHCC 'Yj m2 L 4; 9, 16, npaxea onaon H3 HHX COOTBeT .CTByeT epyHKUHSI Cem , a npyroa - Sem . Kaacaoe H3 3THX peure HHtt y~aBHeHKSI MaTbe CBSl3aHO C ,llPYfHM HenepHO,llHlJeCKHM
=
=
nonanars B o6J1acTH, coorsercrsyioume nono HJlH nponycKaHHSI. Ha epHr. 122 CXeMaTHlJeCKH H306palKeHa pH3y101.lJ.3S1 3aBHCHMOCTb lJ3CTOTbi OT m, CooTBeTcTBHe MelK,lly ynoTpe6J1S1eMhlMH a Ii 0603H3lJeHHSlMH, HCnOJlb30BaHHblMH B np BblTeK3eT H3 COOTHoweHHtt (48.6) H (49.3):
k
lp.=--=-m. 11:
peuieaaea. C noxouiuo epHr. 121 MOlKHO cnexars BalKHhle BbIBO,llbl o pacnpocrpaaeaaa BOJIH B nepaonasecxon cpene, x3p3KTepH ayexon K03epepHUHeHTaMH Co H C1 • K3K cnenyer H3 (48.4).
O"eBH,llHO. BeJlHlJHH3 paloleTpa k.
18
3aK. 201.
+ mt: arpaer POJlb BB
sae O,llHOP0,llHOt;! neaosxyineunoa cpenu, BCIO 06J1aCTb H3MeHeHH5I k (HJlH m), B KOTOpOt;! onpeneneua HenpepbIBHa51 KpHBa51 v (k), MbI
B CJlY4ae B03MYllleHHot;!
3Ha4eHH5IMH
Ik
cpenu
I H nepenecex ace
r O. Ilpn 9TOM "1/1 H 1Jz nOJlOlKHTeJIbHbl H
>
1tV ="21 1tXI = "2 -1JI = tx-, "2 1tXz =
1tV "2 - 'Yjz =
f" = if'" = 121ta,
lx z•
T
MoryT HMeTh
1
p. = p.o· rne p.o
-< ch p.d- O. npH 1t
1
XIII
2) 1/
"2XI
•
•
L
2 ch p.d=
=2 [chY.1/IChXz/z +-2
ch p.d = ch XIII ch
,~ (~+~ X2 X
COS 1tp.' =cos XI cos Xz -
npoaaseneune KOTOpbIX
-;~)
Ilpa YCJIOBHSlX ,a· H ,6· B CHJIy epaK B:(50.6). nOJIyqaeM 9KcnoHeHLI.HaJIbHO 3aT npocTpaHSlIOUJ,l1ecSi B nepsox cnyuae nan HaJIeBO H OTJIH'IaIOUJ,HecSi TOJIbKO epa30lt repsaaa (fl' I z). CooTBeTcTBYIOUJ,I1e HM He 3aIIlTpHXOBaHbl.
~:::'; KeHHIO TOrO >Ke BHAa. tITO H COOTHOme
sin2 «ad
const.
PaCCMOTpHM Tenepb ypaBHeHHe XHJIJIa
2
K oropoe OTJIHtIaeTCli OT oouiero COOTHomeHHlI OtIeHb MaJIbIX aoaayuieana MO>KHO sanacars
Fau,
H, CJIeAOBaTeJIbHO,
. 2[( Vi1 + V1)1tvd]+(V~Vi)~ Ttvd. icvd 2 4V1V~ sin ~stn V;'
.2
sin 1tad=stn
H
+ Bv (x
uo" = ou" = -
HJIH 2
npencraaars
06m.HR HHTerpaJI MO)l(HO
-p.1t
.
Paccxorpaa censac HCnOJIbayeMbIR BbltIHCJIeHHlI MHO>KHTeJIli e (HJIH po) C noxou BaJIe (-1t/2 x 1t/2)
(50.38)
.
B 060HX 9THX cnyaasx YHKUHSI F (x) SlBJlSleTCSI nenpe PblBHOJ:t na rpauauax -+-rr/2 nnrepaana (-rr/2. rr/2). ratoxe uenpepuana H YHKUHSI J (x), XOTSI ee nponaaonnaa repnnr paspuau B TO'lKaX x = -+-rr/2. Boaspauiaacs K ypaBHeHHSlM (50.32) H (50.33), nenco BH ners. 'ITO aaaaua XHJlJla peureua. D,eJ:tCTBHTeJlbHO. B nepsou cJly'lae (a = - b)
+
sh 2 !J.1t 2 a BO BTOpOM (a = -
1t
sh2 f'-2
=
n neuenux
OOJlOlKHB B (50.44) npH
OJ
a (x)
n
n
'/2.
ebx ' /2/ (x),
= a'1 v 1 = b"'::'2 eb1t'/4/("'::') 2 •
(50.39)
r.ne
F (x) = 0 2 (x),
.
H paccuorpax YHKUHIO
y =
x
a-'/.e i S
rtpn
S=
f0( o
3b)
b:2+ brre b1t'/4/ (; ) + b2
JIerKO aaners, 'ITO OHa ynonxeraopaer ypa
1t:
eb1t'/4/(;). (50.40)
+
(Z2 -
Ilpa n neue nux y(z) = AJn(z)
n2) y = O.
+ BJ_ n (z),
rne J'l (z) - YHKUHSI Beccena nepsoro pona nopsnxa aens D ypaBHeH~U1 (50.41) 33MeHy nepexeanux
x=lnz.
z=e",
y" y
MeToAoM YHTTeKepa 9TH peaym.raru He MOrJlH 6bITb nonyuenu. B xauecrse Broporo n)HMepa paCCMOTpHM YHKUHH Bee cens, xoropue SlBJlSlIOrCSI peurenaea ypaBHeHHSI
d ( z ddzY) z dz
yff+0 2(X)y=0.
(50.41)
(50.42)
dx 2 +(e 2 "
-
n 2) y = O.
(50.44)
(0'0)2 _ ~2 Off _ 0
yHK UHSI Y (x) 6y.neT SlBJlSlTbCSI TO'lHbIM B TOM cay-rae, eCJlH 0 (x) y.nOBJleTBOpSieT
l(~)2 4 0
_!2 Off -0 0 .
T. e. eCJlH
o (x) = , ~ A,
tIaXOJl.HM d 2 y
4
orxyna cnenyer. 'ITO YHKUHSI y (x), onpe npencraaaser co60J:t npH6J1HlKeHHOe peurenae eCJlH .nByMSI nepB»IMH CJlaraeMbIMH npaaoa '1aC MOlKHO npeneopeus. T. e. eCJlH 0' Off 02,.....,e:, 0 3 ,.....,e: 2 , e:·~
n. Ilpo (50.43)
= l
rne
a H A - nOCTOSlHHble. 19
3aK. 201.
_,q,
u = (2iO) -1/,
«:",
1\
GI \
(50.52)
v'
~
II
II \\ I
A'/'
(2iO)_'/' e+ i S .
=
v
IIIrxc-a>O
\
/
I
B
I
4eTHa5I;
3HaK
(0, 7t/2), 3HaK "MHHYC« sane (- 7t/2, 0). BeJll14HHa k
U'
ypbe nepaonasecxoa epYHKlJ,HH lex), COOTBeT CTBYIOllle~ F (x), HMeeT BH.lI.
=
+00 ~ 6n ei 2na; ,
(50.75)
n=-oo
r11e
6
n
=
f
J(1+~r 21
z·
a ~-
,llJIlI
npyrne 0603Ha'leHHlI:
21
x;
X,
4 aCTO
CJIeBa
(60.3)
nponenars H ,llJIlI 138). Oooanasas
MO)l(HO
(CM.
Xl 1 71
b22Z ' - b12 -b 21Z ' +b ll •
=
Z=X; TaKHe
na HMne,llaHC
ua HMne,llaHC Z (BXO,llHO~ HMne,llaHC). npH'IeM
saauxaerca
329
cnenyer,
'ITO
b ll ) H
Z -
(59.4).
(60.3)
bl 'l = O.
HJIH
(60.4),
K
n
+
I-MY
aanpaaceuae
me KOMnJIeKcHylO BeJIH'IHHy
(60.6)
TOK
9TH
yMHomalOTClI
na
asmenaacu (CM. lj>Hr. 137).
O,llHY H
ry
E: n+l
=
EXl n,
(61.1 )
= EX 2n '
BblpameHHlI C BblpameHHlIMH
(56.5),
O,llHOP0,llHYIO JIHHe~HYIO cacreay OTHOCHTeJlbHO Xl, n
xapaKTepHCTH'IeCKHe
ooparaa yr JIOBblM K,09tPtPH UHeHTaM ocea MaTpHUbl
X 'l,
Croranuaaa
1
H
Xl, n+l
noaynaex COOTHoweHHlI
~=- S2'
r. 139.
H
EXl n
= bllxl n + b12X 2n '
EX2n
=
b'l 1X l n
+b
22X 2n •
noayvaex H X2, n
(61.2)
330
10
FAa8a
MaTpU'IHaH aAze6pa U pacnpOCTpaH. 80AH 8iJOAb 3AeKTp. AUHUU
eCJlH
YCJlOBHeM pa3peWHMOCTH KOTOpOlt SlBJlSleTCSI
I
I
ba b
bl1-E b
22-E
2i
=~2-(b11+b22H+l=O, (61.3)
1.
YpaBHeHHe (61.3), n03BOJlSlIOllJ,ee HaltTH BeJlH'lHHY ~ KaK q,yHKUHIO napaxerpoa 'leTblpeXrrOJlIOCHHKa, SlBJlSleTCSI He 'leM HHbIM, KaK ypaaaeaaex llJlSl coocraennux 3Ha'leHHlt MaTpHUhI (bi j ) , H, CJlenOBaTeJlbHO, HCKOMble BeJlH'lHHbI ~ pannu ee co6CTBeHHbIM 3Ha'leHHSlM. KOPHSlMH ypaBHeHHSI (61.3) 6ynyT t
~i'
t
~ (b
_
~2 -
2
P acnpocrpanenae
11
6e3
+b
Vr (btl +4 b
) -I- ~
22 -
22 )2 _
3aTyxaHHSI HMeeT MeCTO
1
(61.4)
B TOM
cay-rae,
eCJlH
I ~ I=
(j = V-
~ = ej~
1,
e=
e'
=
e~+j~
(1 -
rrOCTOSlHHaSi pacnpocrpaaeuas). nOCKOJlbKY npoaasenenae nsyx xopaea ~l H ~2 paBHO e.na uaue, MomHO sanaca'n,
=
~2 =
e-T
=
eT =
e-~-j~,
e~+j~,
+
2 E12 E 1 b11 - = Ch T = 2(
(61.5)
+ b22)·
3neCb Ct - rrOCTOSlHHaSi 3aTyxaHHSI 'leTblpeXrrOJlIOCHHKa, a ~ q,a30BbIlt cnaar. coanaaaexua HM B 6eCKOHe'lHolt JlHHHH. Ilpn Ct = 0 BOJlHbI pacnpocrpaasiorcs 6e3 3aTyxaHHSI, xorna, KaK yme OTMe'laJlOCb, ~l
ej mt ,
KOTOpbIlt MbI ace
BpeMSI
orry
=
e-j~,
~2
=
Il pennonoxorx, 'lTO anoas scea JlHHHH HMeeT MeCTO CTa uaoaapnoe pacnpenenenae TOKOB, nonxunmomeeca ypaane HHSlM (61.1), (61.3) H (61.6). B JlI060lt TO'lKe CO'lJleHeHHSI 3J1e MeHTOB uenoxxu OTHoweHHe uanpsacenas Xl K TOKy X2 rrOCTOSlHHO, 'lTO, cor JlaCHO (61.1), cOOTBeTCTByeT uexoropoxy HMrrenaHcy Z =
Xin X2n
=
ej~.
(61.6)
e
ECJlH a oF 0 H, cxaxea, noaoaorrensno. TO peureuae l COOT BeTcTByeT pacnpocrpaaeumo cnesa aanpaso, a peureane ~2 pacnpocrpaneumo cnpasa HaJleBO; 3TO aenocpencraeuao BHnHO,
b12
E- bu
=
E- b22
(61 .7)
•
b2i
Ilsa peureuas ~l H ~2 nJlSl pa3J1H'lHbIX uanpaaaeuaa pacrrpo cTpaHeHHSI B JlHHHH naIOT nna B03MomHblX 3Ha'leHHSI Z = Zl H Z = Z2' Jlerko nposepars. 'lTO Zl H Z2 COBrranaIOT C xapax TepHCTH'leCKHMH HMrrenaHcaMH, paCCMOTpeHHbIMH B npenuny llI.eM na par-padre,
zl=kl=Z,
1).
ECJlH I ~ 1 I, TO pacnpocrpaneuae cneaa aanpano npo HCxonHT C saryxaaaex. eCJlH me ~ 1, TO 3aTyxaIOT BOJlHbI, pacrrpOCTpaHSIIOllJ,HeCSI cnpasa HaJleBO. lITo6bI aepayrscs K npemHHM 0603Ha'leHHSlM, nOJlOmHM B 06llJ,eM cnysae
~l
MHOmHTeJlb
CKaJlH.
I bi j I =
nockonsxy nerepaanaur
ysecrs
331
z2=-k 2=-C
Il penCTaBHM reneps or-panasenuua OTpe30K JlHHHH, conep mallJ,Hlt 'leTblpeXrrOJlIOCHHKH n, n I, ... , n p, KOTOpbIlt 3aMKHyT cnpasa na HMnenaHC Z. nOCKOJlbKY ace npoueccu npoacxonsr npa 3TOM TaK me, KaK eCJlH 6bI JlHHHSI He 6bIJla orpanasena cnpana, TO axnenauc raxoro OTpe3Ka cnesa. aa axone, onan- pasen Z, npasea, eCJlH sees OTpe30K CBonHTCSI K onuoxy 'leTblpeXnOJlIOCHHKY, MbI CHOBa npaxonaa K CJlY saio. H306pameHHoMy aa q,Hr. 138, a. ECJlH me JleBbIlt KOHeu orpesxa JlHHHH 3aMKHyT na m,menaHC ~, TO OTpe30K MomHO paCCMaTpHBaTb KaK JlHHHIO, HeOrpaHH'leHHyIO cnesa H npen CTaBJlSlIOllJ,YIO co6olt nJlSl pacnpocrpaumouieacs cnpasa BOJlHbI HMrrenaHC ~. lI.JlSI onnoro nerupexnomocnaxa 3TOMy COOTBeT CTByeT cxexa na q,Hr. 138, 6. cho paccyacneune, xoropoe npasonxr K pa3J10meHHIO BOJlHbI na nse COCTaBJlSlIOllJ,He, pacnpocrpanmoumecs anons JlHHHH B npoTHBOnOJlOmHbIX nanpaanenaax, 3KBHBaJleHTHO reoaerpn 'leCKHM cxeaax, paccxorpeaaux B § 59 H npHBOnSlllJ,HM K TO'lHO TaKHM me BbI'lHCJleHHSlM. Ilpnsenex npocroa npuxep. Paccxorpas OTpe30K JlHHHH, conepacauraa n HneHTH'lHbIX 'leTblpeXrrOJlIOCHHKOB, H, npennonoxaa, 'lTO Ha JleBOM KOHue 3TOrO ()Tpe3Ka HMeeTCSI nanpaacenae X IO H TOK X20' HaltneM nanpaxenae X 1n H TOK X2n sa auxone n-ro 'leTblpeXrrOJlIOCHHKa. reOMeTpH'leCKH 3TY sanasy MomHO crpopxyaapoaars cne nYIOllJ,HM 06pa30M: sanaaascs Ha'laJlbHbIMH KOOpnHHaTaMH (XIO' X20)
+
+
332 TO'lKH
nJlOCKOCTH
BaHHR.
Tpeoyerca
Po.
MbI
n
nOJly'leHHoR nocJle
n
TO'lKH
P n'
n
nepeauoscenae
rperiyexoe
MaTpHlJ..
BeKTOp
X2
CTHM.
y,
'ITO
I1J,He
COOTBeTcTBylO
pannu
KOMnOHeHTbl
9TOM OP lO H OP 20 .
Koopnaaaru TO'leK P lO H
P 20 060aHa'lHM COOTBeT CTBeHHO (ZlYlO' YlO) H (Z2Y20' Y20)' a yr nosue K09cf>cf>H npn
... ,l.i"&,u
XlO = ZlYlO YlO= H
r. 140.
onpeneaaioume
0 Y2 •
nOJlO>KeHHSI
TO'leK
r.;
TO'lKa
Pn
SlBJlSleTCSI
paBHOrO cyxxe BeKTOpOB
Pln
H
I1cKoMaSi
P 2n.
'H
=
06e
K TO>K.u.eCTBeHHbIM ssncnanxax.
~l
'1HHa
(61. 8)
OCSIX
0 Yl Pn'
OKOH'IaTeJlbHO nonysaex
~l -
e
= enT =
-e
-n(a+j~)
e.,,(a+j~).
.
cneaa
(J.
> O.
aanpaso.
paCCMaTpl1BaeMOR JlHHHH. TO BeJlH
KaKOB
KOHue JlHHHH.
6bl
~
HH
_
6blJl
HMne.u.aHC
H3
~~
JleBOM
X10
K.u.eHHSI
TIPH
npeoopaao
xoopnauaru (X l n• X2n)
npe06paaOBaHHR.
TIoCJle.u.OBaTeJlbHOe JlO>KHTb
cosepuraea nan HHMH
on penemrn.
333
MaTpUllHaR aAee6pa U paCnpOCTpaH. BOAH BOOAb 3AeKTp. AUHUa
Fnaea /0
ypaBHeHHSI
SlBJlSlIOTCSI
KaHOHH'IeCKHMH
ypaBHeHHSlMH 9J1eK
TpH'IeCKHX JlHHHR. B HHX cf>HrypHpylOT TOJlbKO TpH
(61.11 )
naa xapaKTepHCTH'IeCKHX
(61.14)
napaaerpa
e-
HMne.u.aHca '1eTblpeXnOJlIOCHHKa H no
CTOSlHHaSi pacnpOCTpaHeHHSI
T·
334
§ 62.
MaTpurUWR aMe6pa u pacnpOCTpa/{,. BOA/{, BaOAb aAeKTp.
10
rAaBa
npHMeHeHHe reopau K CHMMeTpHlJHoMy lJeTWpeXnonwcHHKy
Bupaxenae (60.6) onpeneaser I1Mne.naHCbJ uerupexnomoc HI1Ka KaK KeHa na
Kyl1l,eMycH HMne
naucy ZI2' B 9neKTpOTexHHKe Ka>KyI1l,HMcH HMnenaHcaM '1aCTO yneJlHIOT 60JlbllIOe BHHMaHHe, XOTH OHH He COOTBeTcTBylOT OCHOBHblM caoacraax MaTpHu,bl npe06pa30BaHHH B.
336
Marpuwtan aneetipa u pacnpocrpan.
F'naea 10
§ 63. no~ocw nponycKaHHH H no~ocw HenponycKaHHH B uenax lfeTwpexnomocHHKoB Bsnne OT
HyJISI
MhI OTMeTHJIH, HJIH
'lTO B 3aBHCHMOCTH OT roro, OTJIH'lHa
HHSI
ex
(61.5)
=
0,
TO
aaryxanne
~1' ~2
=
cos
OTCyTcTByeT.
~ -+- j
sin
ECJIH rrOJIO)/{HTh
ypasne
(61.4»),
+ b22) --+- 'Vr (btl - 4 b b = 1/2 (b ll + b TO rrOJIy'lHM
22 )2 _
1 = b -+-
rV 1-
b2
(j =
cos B= b. anposex, 'lTO
V -1).
(63.2)
(63.3)
=
~= ~
(61.5);
ne pxar
oopaaosannue
aKTHBHhIX
BhlTeKaeT,
'ITO
S1BJISIIOTCSI
CJIe.llOBaTeJIhHO, 3JIeMeHTaMH peXnOJIIOCHHKa, 6Y.llYT:
B
'leThlpeXrrOJIIOCHHKaMH.
He
CO
TO see K031fJ1fJHUHeHThI C ik H K031fJ1fJHUHeHThI Xik H Zik H3 ypaa
napaaerpu
'laCTO
MHHMhIMH.
(55.8) xapaxrepacravecsoa b
KOTOpOM
H
C
H3
OTCyTCTBylOT
4HCTO
Orciona MHHMhI.
MaTpHUhI 'leThI conpOTHBJIeHHSI,
abu,
b22 , ~12
H
~21 .lleACTBHTeJIhHhl.
~,
(63.3),
KOTopoe/
,lleACTBHTeJIhHa, T. e. 'lTO
(b =
b;2)' 'lTO pacnpocrpaaenae
Ib u + BO
b221 0),
Z
Be.neT ce6S1 Ha 9TOA
TO nOJIOca nponycxaaas
CTOpOHy 66JIbWIiX qaCTOT, ecna
H3
IKe
OH
(65.11).
B 2 = (N npe.nnOJIOIKIiM,
~eJII1K
+ 1)rr:,
sro
aener
TO nOJIOca nponycxauas COOTBeT KOTopaSi
yrsepxnenae. 3HatleHHSlMH B, paBHbIMH
1I.oKalKeM 9TO
Bl=~t=Nrr:, H
TIo.ncTaBJISIlI
nonyuaea OKOH
Mbi
CTByeT 60JIee HH3KIiM qaCTOTaM no cpaBHeHHIO C ToA.
~JIytlalO
I
W -
uenoe qIiCJIO, a
(65.8)
MHHMhIMH.
TOtlKY TaKoA nOJIOChI
TIOJIoca nponycxanaa orpauasena
qaTeJIhHO
1
rne
ce6S1 KaK eMKOCTb
BhInOJIHlIeTCSI
YCJIOBl1e
nOJIOChI. ECJIH .nByxnOJIIOCHHK C HMne.naHcOM
JIHHHH
Z' oopa
KOTOpoA
(65.9) seerna Z' npa JII160 orpuuareasnue 3HaqeHHSI. onpeneasercs H3 yCJIOBI1S1 (65.9).
wl ~t=w=Nrr:,
npOCTHpaeTCSI
-< 2.
qHCTO
Z' = 0
rpauau
paccxarpaaaexoa narpyxennoa COOTHoweHHlI (63.5), orxyna 1
npa
pesonancnas qaCTOTa' nsyxno
JIytll1M. nOJIOIKHB
Tosxa,
ch I I
IB ll + B 22
JIH60 nOJIOIKI1TeJIhHble,
a urapana nOJIOChI nponycxanas
H yCJIOBHe
(65.7)
rana
ceasac ABa
no 06e CTOpOHb1 OT 9TOA qaCTOTbI
sin ~t =
ll
TIOJIOChI nponycxanaa MOIKHO nOJIyqHTh H3
HHMaeT
qaCTOTbI,
(anaxe rOBOpSl,
1I.eAcTBHTeJIhHO, npn
.nByxnOJIIOCHHK),
sh 1', - Z 'h 11- k sh 11) •
11 +, ;
Ma
onpenenaa
MhI
6) Cmpytcmypuue no socu,
1
-Ii sh I I
nOCKOJIhKY
'ITO
HMeeT BH.n
(1 - Z ).
nOJIHbIA 9JIeMeHT JIHHHH (OTpe30K
8 (8
Z,
.nByxnOJIIOCHHKa 1/2
b21 b22
OTCIO.na.
nenocpencrsenno,
BH.neTh
yCJIOBHe,
Ilosocu iJ8yxno.llIOCHUIW8. BHyTpH 9THX nOJIOC JIelKHT
y.noBJIeTBOpSleTcSI.
C apyroa CTOpOHW, JIerKO rpaua, npe.nCTaBJIlIlOUJ,aSl naa
(65.10)
nOJIOC nponycxanaa, ynoMHHaBWHeClI asnue.
nOJIIOCHHKY C MaTpHu.eA
b11 'b12 (b 21 b2 =
1;
(65.4) PaCCMaTpHBaSi
I1TaK,
pacnpocrpauenas, I'paaauu
nOCTOSlHHaSi
9JIeMeHTa narpyacennoa JIHHHH.
cos
cosB 1 =
B2 =
-:os
-+- 1 N« = -+-J.
uinpnna nOJIOCbI s MaJIa, 'ITO COOTBeTCTByeT
.nByxnOJIIOCHHKOB,
I1Mne.naHC
KOTOp::'IX
no MO.nyJIIO. Ha BTOpOn rpaHHue nOJIOCbI
~t=
N« +e;
Z
.nOCTaTOqm;>
344
rAa8a
MaTpWlHall aAi!e6pa U pacnpOCTpaH. 80AH 8iJOAI;, 3AeKTp. AUHUU
10
H llOJDKHO BbInOJlHSlTbCSI COOTHOIlleHHe
cos(N
z' sin (N1t +e)= + 1)1t=-cOsN1t=cos(Nrc+e)-2k
=
COS
Crpyxrypnue nOJlOCbJ pacnonoaceuu B 06J1aCTH '1aCTOT H JleIKaT MeIKllY aHa'leHHSlMH qaCTOTbl NnW
w=-t-
Z'
Nrc - 2k cos Nrc . e.
W
w=(Nrc+e)-t-~
H
60JlbIIIHX
(65.13)
rlle
4k
B peayasrare
4k
Iz I
3Ta BeJlH'IHHa MaJia. eCJlH MOllYJlb BeJlHK no cpaBHeHHIO C xapaKTepHCTH'IeCKHM HMnellaHCOM J1HHHH k. 3HaKH e H Z'
~r t.,
1
k=V C;' 5
(65.14)
a sepea L 8 H C, oooauaxeau COOTBeTCTBeHHO HHllYKTHBHOCTb H eMKOCTb orpesxa JlHHHH llJlHHbl 1; lleOCTBHTeJlbHO,
r
;/
4 Ls
4kt
e= Low = LoNnW =Nn ~'
(65.12)
e=Z'.
ep=pl=Z7T
345
w= VLses
(65.15)
'
no Mepe nOBblIlleHHSI qaCTOTbl nonocu nponycxauns CTaHOBSlTCSI ace 60Jlee yaKHMH. 3TH peaynsraru CXeMaTH'IeCKH npencraaaenu
//
/'1-4
ep
//
/
3
8 -I/C
IT
2~
3rr
4~
5'Jt
61r
~ ~
H
r. 145.
OllHHaKOBbl, 'ITO nosaauaaer cnenannoe auure yrnepacaenne o nOJlOIKeHHH nonocu nponycxanaa OTHOCHTeJlbHO ee rpannuu,
onpenenaeaoa ypaaneanea (65.11). npHMeHHM npHMepaM.
reneps
HaJloIKeHHYIO TeopHIO K nsya npocrua
1) JI UHUfl. HazPY:JICeHHafl uHiJytcmusuocmn.uu. 060aHa'lHM Lo K03~~Hu'HeHT caMoHHllyKU,HH J1HHHH; Torlla Z' = Low. EllHHcTBeHHbJA HyJlb nonysaerca npn w = 0; 3TO xopoiuo ana
'1epea
KOMaSi TOTax,
noaoca
nponycKaHHSI neyxnomocnnxa
na
HH3KHX 'lac
-'l!
a
'lC
2rr
Bn
4n
511
8 6rr
--ll-----------ll-
--ll--------ll-
C
C
ct>Hr. 146. na ~Hr. 145, r ne no OCH opnanar OTJlOlKeHa BeJlHqHHa ~1, nponopuaonansnaa qaCTOTe, a no OCH a6Cu,HCC - yroa B. 3lleCb HMeeTCSI YIKe ncrpevaaurascn HaM neonpeneneunocrs ; MOIKHO nOJlOIKHTb, 'ITO yron B 6eCKOHeqHO YBeJlHqHBaeTCSI, HO OO/lee yll06HQ scerna BblqHTaTb H3 aero aeJHi'HlHy. xparuyio 21t,
346
F'naea
10
MaTpUClHa!! aAze6pa U pacnpocTpaH. 80AH 8aOAb 3AeKTp. AUHUt1
ttTo6bl nonyxenuoe anasenae acerna JlelKaJlO B anrepaane ( - 'It, 'It). TaKHM 06paaOM, cymecrsyer nsa 9KBHBaJleHTHbiX npe.n.CTaBJleHHH peayJlbTaTOB, HO nepBOe nOaBOJlHeT npOCJle.n.HTb, KaK acxeaaer neacrsae nar pyaosnux xaryurex, xorna HX K09ep KeHHSI MoryT
paccxarpasarsca
3TH xarpnuu B ,llHarOHaJlbHolt epopMe
(II o
o
0
KaK
3pMHTOBbl
MaTpHU, KOTO
oopaaoaaanue H
MoryT
6bITb
o 0 o 0 (0 0 aanpaanenan
cooTBeTCTByeT
raxas
L. B rill
0
npn pac ue
I).
TOKH
BXO,lle
H nanps H
TBOPSIIOT YCJlOBHSlM, sepea
(Vi' II)
MaTeMaTH4eCKoro
BWXone
npH HCCJle,llOBaHHH nepnonnse
cenuac
nexoropue peayasra'ru,
it =r' 1~
H
crpopaynapoaannux
(V 2 , 12)
aanpsarenae H
13 0 0 0 C
!I
r. 151.
B r a.
TOK
axone H asixone 4eTblpeXnOJllOCHHKa (epHr. [CM. epOpMyJly
(56.2)].
crauaouapnou
B
10.
0603Ha4HM
COOTBeTCTBeHHO na
151).
KaK H3BeCTHO
pe)KHMe 3TH BeJlH4HHbl
CBSl3aHbI COOTHOIlleHHSlMH
......
•
,
I I
(71. 8) -'II
0
0 0
-"(2
0 0
0
-IS
J
pacnpocrpanamueacs B He " pacnpocrpanenns Ti' pacnpocrpanmoiuaaca B 06paT
nOCTOSlHHOI'I
>Ke BOJlHa,
HOM nanpaaaenaa C nOCTOSlHHOI'I 1)
MaTpHU
aanacanu
0
'upyrHMH CJlOBaMH, Ka)K,llolt BOJlHe, KOTOpOM
H3
Ha
4TO
O,llHOrO HJlH He
sror o
4eTblpeXnOJllOCHHKOB ynosae
12 0
0
y,l106HOrO
H3
CHCTeMax
nououn.io
B ,llaJlbHel'lilleM Mbl 6Y,lleM C4HTaTb,
BBO,llHTCSI coaoxynnocn, 4eTblpex He3aBHCHMblX pue
C
nOJlY4HM
uen04KH
Tbl peXnOJlIOCHHKOB
-
Ilayxa,
Mbl
acrpevaaumecs
cMoTpeHHH
MexaHH4eCKHX
H3MepeHHI'I.
CKHX
/I
H
CKOJlbKHX
+
I
VI = II
=
+ V +
a U V2
a 12'2'
a21
a22 /2'
2
(72.1 )
r,lle aUa22 -
PaCCMCTpHM
ueno-ncy.
peXnOJlIOCHHKaMH, H
al2a21
(72.2)
= 1.
o6pa30BaHHylO
O,llHHaKOBbIMH neru
nonsrraexca ortpenexrrrs
nanpsneenae H TOK
Til).
u i n, Journ. de Phys., 8, 401 (1936),
I) M. Par 0 d i. Memorial des Paris, 1944.
sciences
physiques,
Iasc.,
47.
368 B
fAaBa
npoaasoasnoa
nOCTOSlHHblX
Va
12
TOqKe J1HHHH (q>Hr.
xerupexno.nocnaxa aiJ
it
npUJI. TlOAuHOMOB feeeH6ay3pa C~(x)
152)
B
aaBHCHMOCTH
H nOJlHHOMOB
V, ~ ;p---~"q; ~_.:'I!..-1
,
2
Ii
n
OT
l'erenoayspa
Jl
rpaaaexoa SlqeltKH K ee BbIXO.uy. BeJlHqHHY ornoureaaa (73.2): _
all -
Vn llByM HHlI
r. 152.
KOPHSIM
1 H
a
nOHSITHe
raioxe nonuraexca noxaaan., KaK MOlKHO onpenemrn, xapaxrepacrnsecxoro HMne.uaHca npH noxouia nHX
1/2=aU
Bblpa>KeHHe .lI.JUI TOKa H HanpJl>KeHHJI
npOH3BOJlbHOA
TOIJKe
u,enOIJKH
H3
= - -atll Xt
X2a21=a 21 -
-
Ilpeoopasyea cuasana ypasneaae xerupexnomocnaxa (72.1) oonee Y.u06HOlt .uJlSl paccaarpasaexoro cnysas q>opMe. BbISlCHHM, CYlUeCTByeT J1H raxoe anaseuae BeJlHqHHbl x, npn KOTOpOM au paxceaae V o - xlo nponopuaouansno BeJlHqHHe V t - xII' npH qeM 6y.ueM npennonarars, qTO TOKH H nanpsxeaaa csaaanst COOTHOlIleHHSlMH (72.1). liMeeM = (au -
Xa2t) V t +(a I2 - xa22) It, xa'J1 _
1
-
atl! -
xa211
-
3aMeTHM,
qTO,
eCJlH
B
STOM
ypaBHeHHH
+ a22) + 1 =
xerupexnomocnaxa (72.1)
MoryT
6blTb
X2 /0 =
1/2 (V t
X/I)'
- X2 /2)'
(73.7)
(i= [,2)
npH nepexo.ue qepea qeTWpeXnoJilOcHHK.
Ha suxone In
n-ro
OTHOCHTeJlbHO
Vo
H
10
Vo-x/o= 1/'HVn - x /n ).
V o-
x
ai.J sepea cooraercrayioume HM, cor J1aCHO (56.6), BeJlHqHHbl bi j , TO OHO TOqHO cosnanaer
nanpaxenne V n
serupexnomocnaxa
6y.uyT CBSIaaHbI C BeJlHqHHaMH
(73.3)
ypasueanex .uJlSl xapaKTepHCTHqeCKOrO HMne.uaHca, noaysen HblM B npenunyutea raaae. Haxce MbI euie BepHeMCSI K STOMy sonpocy. OYCTb XI H X2- KOpHH sroro ypasaenna, KOTOpbIM COOT BeTcTBylOT nse q>yHK1.I,HH: V o - x/o H V o - X2/0' yMHomalO umecs ua nOCTOSlHHoe uncao npH nepexone OT axona paccaa C
(73.6)
O.
1/
Vo-xi/O
aaMeHHTb BeJlHqHHbl COOTHOlIleHHIO
x2
(73.5) ;
(73.7) MomHO paccxa'rpaaars KaK pesyppenrnue ormcuaaioume npeoopaaoaauae cosoxynuocra
(73.1)
TOK
(au - a22) x - a t2 = O.
atll
HMeeT BH.l1
nOJlyqaeM
a21x2 -
a2t.
K q>opMe
Vo -
(73.2)
-x·'
ypaBHeHHSI
1/
Vo- x/o = ""It (Vt -
q>OpMyJlbl,
oTKy.ua, yqHTWBaSi YCJloBHe npOnOpu,HoHaJlbHOCTH all -
Hrax, nsa npasenenu
YpaBHeHHSI
Vo-xlo=auVt +a I2 /t- x(a2IVt +a22/t) =
(au
1/2 -
lJeTbipeXnOJIIOCHHKOB
K
(73.4)
.
X2 coorsercrayior nna ana qe
KaK J1erKO BH.l1eTb, ypaBHeHHe .l1J1SI
B
oouryto
xallll
-x
q>yHKU,Hlt.
§ 73.
xepea 1/
1/: 1/t = au -Xt a21
c~ (x),
an -
-
ypaBHeHHSI Xt
uay'lf!HUIO sne«. ¢UJlbTPOB 369
060aHaqHM
xallt _
1/-
Ie
3aMeqaSl, qTO 1/1""12
x/o=
= I,
1/~ (V n -
x/n ) ·
HenOCpe.l1CTBeHHbIM
H
COOTHoweHHSIMH
(73.8) nO.l1CqeTOM
no
J1yqaeM
Xl-XII
n
'h
In = 24
aSK.
201.
on
-1j2
Xt-XII
(73.9)
V
0
370
Ilpu». nOAUHOMOB tezeH6ayspa
fAaBa 12
Ilpyrnaa CJlOBaMH, aHCaM6J1b tJeTblpeXnOJIIOCHHKOB 1. 2•... , n xapaKTepH3yeTcH MaTpHue~ n
n
X1"11 -
X 2"12
_
Xl-X~
T=
X
Xl-X~
X l - X 2 = - "11-"12 . a21 •
x~
C npyron CTOpOHbJ. H3
'ITO MOmHO
= X1 nepenacars B aane
a12 -
T=
T=
(au -
"12) "1~
"1~-"1~
(au -
• "11 -112
"1f-"1~
all---
"11 - "12 (
a12
"1f-"1~
-
"12) "11 - (au "11-"12
-
n
"11) "12
'
all---
"11 - "12
"11 - "12
B sanacannyio TaKHM 06pa30M MaTpHUy HOMhI I'erenoayspa c~ (x). BeJlHtJHHhI 111 H 112 HBJlHIOTCH nOJlO>KHB
aU
+
a22
"1~-"1~)
a12 - -
"1f - "1~
a21---
sh nO - a 12 S1i!l
an sh nO - sh
a~l S1iT
C; (x)
2x = 2 ch 6. )
=
(n -1) 0
(t i j ) ·
sh 0 MaTpHue~
tJeTblpeXnOJIIOC
C~_2) ( - aUC~_l + C ; ) -
C ytJeTOM H
roro
pexyppenruux
a12a21
(C~_IY;
COOTHoweHH~ .nJIH 10 .
V nIn- p
p
=
Zn_p
(74.10)
374
npUA. nOAUHOMOB FezeH6ayapa C~ (X) I' uay'leHUIO aMI'. t/JUAbTpOB
FAa8a 12
n03BOJISIeT
Vn -
aanacars
COOTHoweHHSI
(74.10)
Torna,
B BH.ne
HCXO.nSl
1'13
yCTaHOBJIeHHylO p
=Zo Orciona
AIC~_I (x) - C~_2 (x)
A2C~_1 (x) - C;_2 (x)
I AICn _ 1 (x) -
I A2Cp _ 1 (x) -
I '
Cn _ 2 (x )
BH.nHO, 'ITO eCJIH HMne.naHC
p
Mbl yme
(73.6),
T.
BbISlCHHJIH. 'ITO'l/
e.
(74.11)
I '
Cp _ 2 (x )
_
an
l/1'l/2-
na KOHue lPHJIbTpa TaKOB,
TO .nJISI npOH3BOJIbHOrO
p
=
npaaenea M3TPHUY
BWnOJIHSleTCSI yCJIOBHe
asmenanc
u uneoancou
noxa aars , 'ITO
aupaacenaa (74.4)
(74.6)
H
xapaxrepacrnsecxaa
,U.JlSl AI'
A2• JIerKO SlBJISleTCSI peure
HMne.naHC
KOTOpOe TO'lHO
§ 75. (an - a22) P- al2 = O.
coanaaaer
C ypaauenaex
(73.3)
B OTHOCHTeJIbHO
TaK 'ITO aaenenaue ssnue BeJIH'IHHbl XI H X2 SlBJISIIOTCSI
x,
xapaxre
pHCTH'IeCKHMH HMne.naHCaMH lPHJIbTpa. 3aMeTHM. nsmenanca
rparpe.
anpouea,
MOmHO
'ITO
6blJIO
6bl
nOHSITHe BBeCTH
lleltcTBHTeJIbHO, H3 ypaBHeHHlt
Zn Ilpa Zo = Zo = X2 -
XI
H
Z"
=
X2'
'ITO
npn JII060M ueJIOM nOJIOmHTeJIbHOM 'IHCJIe
Zo
n.
n
HMeeT Ty me BeJIH'IHHy. a
Boaapaiuaacs
K 0603Ha'leHHSlM
=Z,
x 2 = - C.
- 2ZC sh nl ) Ze"T
~e-nT
2 sh nT
KaHOHH'IeCKolt
•
+
·lPopMoR.
= 1)
(74.15)
•
nOJIy'leHHoR
auure.
CJlyqaA cHMMeTpHqHblX qeTblpeXnOJllOcHHKOB
all = a22'
H.
ceasac .nOKa)J(eM. MaTpHua T Tome CHMMeTpH'IHa. 113
aannca T noxasars
B lPOpMe
(73.12) cnenyer,
'ITO ,U.JlSl
sroro
.nOCT3TO'lHO
COOTHoweHHe I
- all Cn-1 (x)
+ Cl (x) =
l all Cn-1. (x) - C nI - 2(x) (7 5.1)
n
XI' a npn
BJIeTBOpSiIOT nOJIHHOMbl
C~ (x),
cnpaaennnsu a
au
Zn
+ C~-2 = O.
(75.2)
9TO .nOKa3bIBaeT. 'ITO
XapaKTepHCTH'IeCKHe panuu
npyry,
HMne.naHCbl TaK
KaK
B
paccaarpaaaexox
ypasneaae
K ypaBHeHHIO
npeo6pa30BaHHSI
npenuayuiero napa
npyr
nOCKOJIbKY
+ a22 = 2au.
a2lx2 -
al2 = O.
X= -+-
.r»: V a21'
OTKy.n3
rpatpa, aanaurex xapaKTepHCTH'IeCKHe HMne.n3HCbl B sane
XI
1
_
coananaiouiee C pexyppenruua COOTHoweHHeM. KOTOpOMy yno
Z" =
OTMeTHM euie CBSl3b. cymecrsyiouryio Mem.ny xapaxrepacrn HHBepCHH.
+4 a,~)2
K lPopMe (noCKOJIbKy l/1l/2
+ ~enr
l1TaK, eCJIH HMne
TO'lKaMH
(all
ypaBHeHHSI
(74.14)
XI H X2 SlBJISIIOTCSI xapaKTepHCTH'IeCKHMH HMne.naHcaMH. 'IeCKHMH HMne.naHcaMH H B3aHMHbI MH
peureauea
.
c~ - 2aIlC~_1
paaen O.nIlOMY H3 3Ha'leHHlt XI HJIH X2' TO HMne.naHC
npa JII060M
nOJIy'lHTb
12 = e- r
HJIH
(73.8) cneayer
OCTaeTCSI
C
xapaxrepacrnse
'Iepe3 lPHJIbTp.
(ze- nT _
BHOBb
cay-rae CHMMeTpH'IHbIX 'IeTblpeXnOJIIOCHHKOB
Mbl
XapaKTepHCTH'IeCKOrO
scerna BbWOJIHSleTCSI paBeHcTBO
paaencrso
K3K
B npenunyurea napa
~O-Xl (~)" Zo- X2 \ lJl •
Z l I - Xl _ X2 -
nanc
coananaiouiert
ypaBHeHHSI
a21p2 -
r
= er•
(73.9)
Z+ '0
xa paicmepucmuvecxu.u
Ha3bIBaeTCSI
JIHHHH.
Y'IHTblBaSi HHeM
p
1
(74.13)
Zn_p=Zo· TaKoR
l/I
(74.12)
A2•
JIerKO lPOpMy
SlBJISleTCSI
+2 a~~ -V +- ~ r
nOJIOmHB
'ITO AI
K3HoHH'IecKYIO
pacnpocrpauenas
cKoR MaTpHlJ,bl
-,--=Zn_p 'n-p
(73.9).
lPOpMyJIbl
nsnue
375
.nJISI
X
cnysae CBO.nHTCSI
376
C
npyrott
all = a2 2 = npencrasan,
CTOpOHbI, nonaras
(73.13»)
[CM. yp3BHeHHe
MO)l{HO
ch
TIe
-
nB
-
e,
M3TpHl.l,y
.
ch nO
all = ch 0
nOCKOJIbKY MbI nOJIO)l{HJIH
ch
T
B <jlopMe
a 12a21 =
(75.3)
nOCJIe.uOB3TeJIbHbIX
C HHTeHCHBHOCTblO
nanaouiea
OTp3)1{eHH3S1 BOJIH3
rozce
T3KHM
06p330M, B
npHHHM3eT
TOT )I{e
cnenaea
CHJIy
)I(I1Tb
B
a21
B CBOIO BHHOM
P3CCM3TpHB3eMoM BH.u,
x sh e.
AHaJlOrHH Mem.n.y
cnysae
M3TpH1.I,a
(74.15),
(75.3) nOJIO
§ 63
MbI
H3YlJeHHH
06ll.1,HMH
H
OTMelJ3JIH,
MoryT
npOCTp3HeHHH
BOJIH
K OTOp3S1
sepea
TOJIbKO
cpenu,
6PHJIJII03H
HMelOll.I,He
YK333JI 1) H3 3H3
pacnpocrpaaenuex
npoxoxnenaea
onncannua.
B
3JIeK
peHTreHOBCKHX
lJ3CTHOCTH, B p360Te
lJTO
paCCMOTpeHHoe
onncanne
3JIeKTpHlJeCKHX
nOSlCHSIeT
31'0
cpaanenae.
H3
lJTO BCSI
BOJIH,
pacnpocrpanaoumx
CSI Me)l{.uy .uByMSI
(p-
CTSlMH,
nJIOCKO
l)-A
H
p-A,
lp_'
rp-, n-t
1
r. 154.
q-'tp
1'1i r. 155.
npencraaaena
KOTOpbIX
1
p-
T3KOBbI:
pOHbI,
L. B rill 0 u in, Journ. de Phys., 7, 401 (1936),
H
A-
,llJISI
,UJlHH3
P
H
't -
oT.ueJIbHott
BOJIHbI
H
B
nsyx
oKpecTHOCTH
q- 1t p '
COCT3B JISilOll.I,HX, H3np3BJIeHHIO Hanp3BJIeHHH.
q-
nJIOCKOCTH
pacnpOCTp3HeHHH
e-i'P,
HJlbTpOB
Cp3BHHMOA
nOCKOJIbKY 31'3
6eCKOHelJHOCTH. nap
------t~~~~'---
peHTreHoBcKHX J1YQeH B KpHCTaJlJlaX
B
ro
------~"""=---------"
H 3aKoHoMepHocTHMH pacnpocTpaHeHHH
npn
no
sroro
eCJIH B no
lJeTbIpeXnOJIIOCHHKOB
reopaea
YlJeT3
t2~12
Z = C(= x).
§ 76.
cnocoo
O.uH3KO
OTp3)1{eHHbIe BOJIHbl.
OTp3)1{alOTCSI, H T3K
6blJI YK33aH
CTaTb
MO)l{eT CB060.uHO pacnpOCTp3HSlTbCSI
':
1
lJTO H MaTpH1.I,a
cHMMeTpHlJHOCTH
osepens,
H
1,
=
He
MO)l{eT
BOJIHbI.
B KpHCT3JIJIe, B03HHKalOT HOBbIe
nOJIylJ3eM
a12"":'--xshO,
nJIOCKOCTeA,
377
orpaxenaux
1.I,HH HHTeHCHBHOCTb CJI36bIX no OT.ueJIbHOCTH BOJ1H, 01'
sh nB) S1l"6
a12
sh ( - a• ' lsh- B a~l
npUA. nOAUHOMOB Feeeutiaqspa C ~/x) ICU3Y'leHUIO aMIC. !jJUAbTPOB
12
FAaBa
Flyers,
OTp3)1{eHHSI
01' onaoa
C
npyr oa
H
npOXO)l{.u eHHH
CTO
Tor na, paccMaTpHB3S1 BKJIa.u 01'
p-A (r, t, p H 't - KOMnJIeKCHbIe
H
TO.l\bl\O
a(icoJllOTHYIO
BeJIHlJHHY
378
r/laoa
npUll. nO/lUItOMOO rezelt6ayapa C~(x) K uSYlleHulO 3AeK. cPUIlbTPOB
12
'ITO aaer
axnaarynu, HO H <pa3y). nOJIyIJaeM COOTHoweHHSI:
+ tqrp' pqrp + ttp_I' ptp + sqr pqr + ttp'
q-Itp = rp =
+ ,q! _
p2q = ,
(76.1)
H
(76.2)
BbI'IHCJIHM
(76.3)
+ tqrp; pq2rp + tqtp_
tp =
(76.4)
p+1
reneps
aMnJIHTY.Ilbl
npourenuren
(76.1)
nnocaocrea fC01UltHO. OpMyJIbl
pexyppenruux
H
orpaxenuon
H
(76.4)
COOTHoweHHSlM
JIerKO npRBO
MeiK,1J.y aMnJIRTy
,1J.aMH npOWe,1J.WHX H orpascennux BOJIH. a HMeHHO
1
p
rp=-~ t p_ 1 +~rp_I' l ;
I< 1.
p2q ,
BOJIH B npe,1J.nOJIOiKeHHH, 'ITO '1HCJIO napaJIJIeJIbH ~IX OTpaiKalOJ.JJ.HX
CJIe.IlyeT
r p_ 1 =ptp_I
+ J._ ,q
j tq
(76.7)
,1J.SlT K TeM iKe 113
1 2
.l\eilCTBHTeilbHoe '1HCJlO,
(76.2)
p+l.
q-Itp+I =
tq
(76.1 )
r p-I = ptP_I
379
t p = pq2 [ - ~ t p_ 1
+ ,~ r P-I] +tqtp_I
r p-I -
2),rp
tp _ l
2A.tp
p+1
p
= O.
=
I
(76.8)
O.
r.lle
2A=tq+_1 _ ,q
HJIH
p'q ) pq t p= ( tq--,- t p _ l +-,-rp_I'
p2q . "
3Ta BeJIHIJHHa S1BJISleTCH He '1eM HHbIM. KaK weHHSI (76.8) H (76.9) ,1J.alOT B03MOiKHOCTb
(76.5)
p 1 rp=-~ tp_ 1 +~ rp_I'
-
+r +t +
rynu
npOWe,1J.WHX H
orpaxeuaux
BOJIH.
(76.9)
+
b22· COOTHO BblIJHCJIHTb aMnJIH
bll
nYCTb
t n +1 -
aMnJIH
Ty,1J.a BOJIHbI, npoureaurea IJepe3 BCIO KpHCTaJIJIHIJecKylO nJIaCTHHKy. TaKHM
06pa30M.
no KpaltHelt
Mbl
noaysaex ypaBHeHHSI.
xepe no <pop Me.
C ypaBHeHHSlMH
Mor3H
cosnanasnuae, '1eTblpeXnOJIIOC
p2q " P
b21=-~'
rp = b21tp _ 1
1
npHMeHHM
-:cq'
b22 =
+ b22r
pexyppearnue
p2q
+ -.,.q =
tn _ 1 =
1. Ii T.
napanaeasnsrx naocxocrea 6eCKOHeIJHO. TO yCJIOBHe
noayxaerca TaK iKe. KaK H paasure, T. e.
1
.I\e!icTDHTeilbHoe '1IfCilO,
COOTHoweHHSI
2"1 au + a221< I,
!!I
\
,1J.. B
tn +l
[_,lq 2A -
0).
n
TaKOBbl:
(76.10) C Ha'laJIbllblMH
r n = ptn
1] =
yCJIO
(76.11)
r.ae
tn+1
[,~ C~ (A) - C~ (A)J
06ll1,eM CJIyIJae
tn_p=tn+I[,lq
fI
Heo6xo.IlHMO
=
OT
S1BJISleTCSI
1 I
tn=tn+I-Co(A). ,q
pacnpocrpaaenna
~2
He
H nOJIyIJHM
(76.6)
p-I'
1 ( p'q ) bub 22- bl 2b21 = -,q xq - - ,
orcroauiea
OHa
BHSlMH
t n+ l •
r.Ile
+
(noCKOJIbKy
rn=pt w
tp=blltp_ 1 +b I2rp_I'
au
1
q-1tn+ 1 = ttn·
b - pq 12--,-'
nOJIyIJaeM
ECJIH '1RCJIO
3TY BOJIHy B nJIOCKOCTH.
paccrosnae
=
.lleItCTBRTeJIbHO, nonaras
ll
n ua
aToMHolt nJIOCKOCTblO, Mbl nOJIOiKHM rn+l YCJIOBHSI npOXOiK.lleHHSI xepes nJIOCKOCTb
HHKa.
b = tq -
paccxarpusax
nJIOCKOCTH
C~(A)-C~_I_(A)].
(76.12)
C~ S1I;lJllleTCSI nQJH\HOMOM I'erenoayspa k-t« nopanxa.
380
fAaBa
Ilpun.
12
Mbl nORmHbl nORY4HTb
(76.13)
i
I
axnnaryny nanaiouiert
BORHbl
1 C 1 - (A) f 1 = t n +! [ :cq n 1
Cn1 - 2 (A) ] ,
C; (x)
-1
H
K
usuwenuto 3AeK.
+1,
cPUAbTPOB
coorsercrnytor
381
pacnpo
saryxanas. Iloxyseaaae TORbKO "ITO pesyas raru nonuepxnaaior aHaROrHlO Memny nByMH YK3aaHHblMH auure aonpocaxu. 3 a a e 4 a H He. rlOTTOH H OpTlOaH 1) yxaaana na B03Mom HOCTb oCyLl.l,eCTBReHHH 3ReKTpOMafHHTHbiX epHRbTPOB nyrex nacsperuoro paaouenas BORHOBOna na paamrsaue YlJaCTKH. cTpaHeHHIO
rn~ p = f n + 1 't~ c1 (1-).. 1
I'eeentiaqspa
3aKJII04eHHbie B HHTepBaJIe
AHaRorH4HO nRH oTpameHHoR BORHW nORy4aeM
Ilpa P = n -
tlOAUflOAfOB
6ea
Kamnwtt paapwB npH nepexone OT onHoro Y4acTKa K npyroMy TaK "ITO f n _ p a HMeHHO
H
rn-p
1
t noxasan, naa cayxaa:
r 1
H
1
rne
npotuemuett f n + 1
xoacer
1
2.
1
(A)
(76.15)
npH
oopauiarsca
P.I < 1.
1
TaKHM HHHX
5lCHO,
06pa30M,
1
(cos
C;_1 (cos 0)
B= Bk
= ~ n
"ITO aHa4eHHH
I"
B HyRb.
sin /16
oopauraercs B HyRb
k = 1, 2, ... (n -
npH 3Ha4e
1).
COOTBeTCTBYIOLl.I,He pacnpocrpaaemno 6ea
nnocxocrea
nHaepparMHpoBaHHoM
paccaorpenaa
sanase
BORHOBone
nByxaToMHott
0
peurerxn
50pHa, noaBORHIOT
MomHO nORYlJHTb 60Rbwee 4HCRO
1) 0 u t ton, 0 r t u s i, Compt. Rend., 216, 67 (1943).
0) = -. -6-' Sin
4HCRa
TaKOM
nosone paannsuux paapuaos
3aTyxaHHH, nORmHbl scerna RemaTb B HHTepBaRe -
B cayuae oecxoueuuoro
B
6bITb TOlJHO CBeneHO K
nOROC nponycKaHHH.
Torna, noaaras 1,=cosO, naxonux
Cn -
asoopaaceunaa na epHr. 154, onpenenserca K03epepHuHeHToM npony K03epepHUHeHToM orpaxcenas p. [l03TOMY asyuenae
cneRaTb BWBon, "ITO npH nepHOnHlJeCKOM pacnpeneReHHH B BOR
"ITO neRH4HHa ). neRCTBHTeRbHa. 3neCb B03
He MomeT
pas
npoxoxoieaan peurre HOBCKHX RylJeR 4epea KpHCTaJw. [lOROCbl nponycxanus onars onpenenmorca ypaaneaaea (76.7). Peaynsraru, noxysennue
1.= ch 0, ncnysaea
1 C n-I (h C 0) = sh n6
sh f
Cn -
H paBHoOTCTOHLl.I,HMH
nROCKOCTb
pacnpocTpaHeHHH , tj
'tq C1 (A) _ 't C 1 n-l q n-2 (A)
KamnaH
CKaHHH 't: H
6ynyT
1. I AI> 1. Ilonaras
H
BORHOBOn C OnHHaKOBblMH
ce6H TaK me, KaK CROHCTaH CHCTeMa,
l '
BORHbl
-t
MOf3H
oripasox:
pWBaMH HenpepwBHocTH B cRy4ae pacnpocTpaHeHHH BORH BeneT
Cn _ 1 (A) - 'tqC n _ 2 (A)
C;_1 (A) t P----,--'-'-- 1 1 C1 (A)_ 't C n-l q n-2 (A)'
n+l -
MomHbl
1
orpaxenaoa
1-
xapaxrepaayerca KOMnReKCHblMH K03epepHUHeHTaMH orpaacenus H nponycxanna. Pacnpocrpanenne BORH B TaKott CHCTeMe MomHO paccxo rpers no HaRomeHHoR auure cxexe, paccyxnaa cnenyioum«
fl'
(76.14)
pc1 (A)
rn _p = f1
r -
OT
1_d
n-p
A MnRHTynbl Tor na panau
B aaBHCHMOCTH
A) c~ (A) - 'tqC C;_1 (A) - 'tqC;_2 (A) ,
= t
f
aupasun,
MomHO
ace
1 < A
C,
nO,ll,06HbIX
BonpOCOB H HJlJlIOCTpHpyeM ero HeKOTopbIMH 4aCTHbIMH npnxe
HeHHe. KOMnOHeHTbI
3HTb KaK ~YHKU,HH
§ 78. 06w.He ypaBHeHHSI pacnpocrpansicrca B,lI,OJlb OCH z, OT t H Z MO>KHO npencraanri, BbI
Ez
KaIOllI,HX H3 ypaBHeHHR
E _ k2 x - -
pascenneu BH,lI,a
z,
t) =f(xy)
ei UJ t - rz,
(78.1 )
k 2E v _
r:
1) 3aMe.L\JllIlOmHe BOJIHOBO.L\Hble CHCTeMbl HMelOT urapoxoe rrpaxe HeHHe TaKlKe H B aHTeHHOH TeXHHKe. HaH60Jlee crapoii cacrexoa raxoro poaa lIBJllIeTCli BOJIHOBO.L\HbIH xaaa» rana Y.L\e - 51rH. B conpe MeHHblX cacrexax, 3aMe.L\JllIlOmHX BOJlHbl cauraxerpoaoro .L\Hana30Ha, '1aCTO npaaeuaercs rpetieauaraa crpyxrypa. TaKHe CHCTeMbI xapaxre PH3YlOTCli H3J1y'leHHeM BOJlH, .creaaunrxcs- B.L\OJlb BOJIHOBO.L\HOH CHCTe MbI, Teopna nnaaxeuaux BOJIHOBO.L\CB Me.L\JleHHbIX BOJIH paspaooraaa 51. E. aHH6eprOM [CM., nanp., 51. E. a HH 6 e p r, Arowaaa sneprus,
6, aurr. 4, 431, 447 (1959»).- Ilpu u. peo,
k 2H k2
y -
-
,
H"
iJEz
-.::1-,
uy
. oe, 7iX + /(JJsoay' iJHz
rne k 2 = (JJ2S0P-O
. /UlSO
+ ,2 =
supa
oE
z ox
iJHz ----ay -
MO>KHO
COOTHoweHHR. BbITe
z , ----ax'
+./(JJP-O-.::I--' iJH .
H
noxoun.io
. oHz /(JJP,o 7iY -
x=-,
H _
n;
Ex. E".
Hz C (78.2).
H
Dpe,ll,nOJlO>KHM, 4TO BOJlHbI
p (x, y,
div H = O.
rne So H P-o - COOTBeTCTBeHHO ,lI,H3JleKTpH4ecKaSi H MarHHTHaSi nponauaexocru cpenu, B KOTOpoR npOHCXO,ll,HT pacnpocrpa
Ilonepeunue
paxa.
TaK 4TO 3aBHCHMOCTb nOJleR
iJHa: • E ~ = /(JJso z', iJy
06bI4HOMY
B BOJlHOBO,ll,aX, xorna
MeTO,ll,
(78.2)
iJE{J! = _ i(JJP-oH z, iJy
div E = O.
BOJlHbI 6bIJla
npOTHBOnOJlO>KHOCTb
BOJlH
cenuac
W
H
iJEy iJx u
~
CKOpOCTb BOJlHbI ,lI,OJl>KHa 6bITb 6JlH3Ka K CKOpOCTH 3JleKTpOHOB, a ,lI,JlSI
.
/(JJP-o y'
-
B BOJl
CKOpOCTH CBeTa, Ba>KHOCTb
TeMH
nocnennee TM)
HCCJle,ll,OBaHHIO
pacnpocrpansroumxcs
CBoRcTB 3J1eKTpOMarHHTHbIX BOJlH,
raxoro
,lI,eTaJlbHOMY
oe, ox =
npH
H3Y4eHHH TeopHH 3J1eKTpH4eCKHX ~HJlbTPOB, 3Ta
- , Ea: -
lI)~
(78.3)
iJEz ----ax '
C2
+ ,2.
. (78.4)
384
f'AaBa
13
3aMeaAHIOll{ue
ecau
KOM6HHHpySl 9TH COOTHOllleHHSI C YCJIOBHSlMH paBeHCTBa HyJIIO .nHBepreHUHH nOJle{;\ (78.2), nOJlY4aeM ypaaueuaa iJ~Ez
iJ2Ez
+ iJy~ + k E 0, iJ~Hz + iJ~Hz+k2H =0 iJx2 iJy2 z' iJx2
xoropsre
S1B.~S1IOTCSI
(78.5)
Z -
OCHOBHblMH
-
2_
ypaBHeHHSlMH
W
1=
> Ole.,
i~
=I
BOAliOB03lible
TO
. OJ
W H~BOilHbl pacnpocrpanaiorcn B BOilHOBO,Il,e 6e3 aary (78.106)
X3HHSI.
(786)
Bo BTOpOM
.
cnynae
B CHJIy (78.4)
k 2 = w2
reopnn BOJIHO HJIIf
YpaBHeHHSI (78.5) H (78.6) xoryr 6blTb paspeureau, eCJIH sananu rpanusnue YCJlOBHSI. nJiSI 06bl4HbiX BOJIHOBO.nOB, ripen CT3BJlSlIOW,HX cocoa MeTaJlJIH4eCKHe rpyou, peurenaeu S1BJISIIOTCSI: Bo suu mana TE (nonepennue 9J1eKTpH4ecKHe), .li.JISi KO
W>C. 3TH peaym.ra'ru npHMeHHMbl K BOJIHaM, pacnpOCTpaHRIOw,HMCH anyrpa MeTaJlJIH4eCKllx Tpy6 npOHSBOJlbHOrO ce4eHHSI, 1'1, cne noaareasuo, .nJlSl acex TaKHX CHCTeM YCJlOBlle W C y.lWBJIe
a Hz onpenenserca yp3BHeHHeM (78.6) 1'1 rpaHH4HbiM YCJlOBHeM 113 CTellKax BOJIHOBo.na
Boneu mana
TM (nonepenaue MafHHTHble),
Hz = a
E,
0,
WCo
Me.n.JleHHble BOJlHbI.
Ilpennoxoaoor, 4TO MO)/(HO nOJIY4HTb BOJIHbl, rpaaoaas CKO
(78.86)
Ha CTeHKax BOJlHOBo.n.a. B 9TIIX nsyx cnysaax Ez 1'1 Hz S1BJlSlIOTCSI co6cTBeHHblMH q>yHK UHSlMH ypaaueuan (78.5) HJlH (78.6) 1'1 cooTBeTcTByIOT HeKOTO pblM CQ6CTBeHHbiM 3Ha4eHHSlM k~ 0 I). Cor JIaCHo COOTHOllleHHIO (78.4). nOJlY4aeM 4acToTy BOJIHOBo.n.a
1
< C2"
II, CJle.nOBaTeJlbHO.
Ez=O,
l(
W~
1 1 k2 W 2 =C2"- -;2
ropux
rne d/dn 060sHa4aeT npowsnonnyio no HOpMaJlH CTH BOJlHOBo.na.
(_1 __1_) c 2
BO.nOB.
iJHz = O. iJn
385
CUCreAtbl
~
1) Boansr, aonaoasre '1HCila KOTOpblX onpeneamorcs xepea c06 CTBeHHble 3Ha'leHHR aexoroporo ilHHeHHoro oneparopa, Ha3b1BaIOTCSI
HopMailLHLIMH BOilHaMH. - Ilpu u. peo,
1(2=
k2 =
w2 (-~2 -
-
1(2,
c1~) > O.
k=IK, (79.2)
KaK cnenyet HS ypasneuan (78.3), (78.5) 1'1 (78.6). Mbl MO)/(eM KJlaCcHq>HuHpoBaTb BOJlHbl, OTHOCSI HX K onnouy H3 .llBYX THnOB: TE (Ez = 0) HJlH TM (Hz == 0), xoropue MO)/(HO p3CCM3TpHBaTb OT.lleJlbHO.
25
_,.,L
W '
E z = SIn -a- sh
-+- --+- --+- -+- -
-b
Peurenae (79.56) ucnonsayerca npa JlByX rpana-mux yCJIOBHSlX, sanaunux na nJIOCKOCTSlX y=-+-b. HH>Ke Mbl yKa>KeM, xorna TaKHe r paanunue yCJIOBHSI HMeJOT MeCTO (KHO HCnOJlb aosan, peureuns tuna z
cHMMeTpHlO, ynoe B nJIOCKOCTH xy,
L.LJ__LJ.J t
-0
KHTeJlb npH 3TOM pasen eiwt-r'y. BbIOOJlHSleTCSI YCJlOBHe
d~A=21'C Wro -I
rne
A ~ .uJlI1Ha
(80.5a)
•
BOJlHbl B CTpyKType; 113 (80.5a) nonysaex
d~A=21'C: • rne
A-
.uJlliHa BOJIHbl B cBo6o.uHOM npocrpaacrae
(y""
0). HJlI1
2n
w~cd'
KHO Bbl6HpaTb TOJlbKO $YHKUHIO Jw '1T06b1 H36e>KaTb paapusa He npepusuocra aa OCH. B cnysae KOaKCl1aJlbHOrO Ka6eJlSl HJlH crpyxrypu. conepxcamea saexrpouuun nysox. B 06J1aCTH. npa MblKalOIUelt K nysxy HJlH uearpa.n.noay nposonnaxy, B xase CTBe peiueana Hy>KHO HCnOJlb30BaTb KOM6HHaUHlo
Zn=AJn+BYn·
389
CUCTeMbl
(78.3) - (78.6), nOMeHSIB B HHX MeCTaMH H3MeHHJlOCb HanpaBJleHHe pacnpOCTpaHeHHSI:
peuieaaa rnna Ez
80AH0800Hbte
nocxom.xy W C. 9TO npennoaoxeaae npl1HSlTO HaMH aa nepsoa srane 113-3a ynpourenna, K KOTOpblM OHO npnaonar: onuaxo OHO He SlBJISleTCSI He06xo.uHMb1M, 11 H3JIO>KeHHbllt HI1>Ke MeTO.u peWeHI1S1 MO>KeT 6blTb pacnpocrpaneu aa cJly'lalt, xor.aa d 11 A npltHHMalOT npoasaom.nue 3Ha'leHHSI. B YCJlOBI1I1 (80.5a) co.uep>KI1TCSI npennonoaceaae, 'ITO
d
«:t.
(80.56)
TaK KaK Mbl 6y.ueM paccaarpasars cnyuaa, xorna .uJlI1Ha BOJlHbl B cB060.uHOM npocrpaacrse nopsnxa l, Ha OTKPblTOIt rpanaue Ka>K.uolt IUeJlH nona, onpenenennue Bblpa>KeHHSlMI1 (80.3), MO>KHO C'II1TaTb He 3aBI1CSlIUI1MI1 OT Z H HMelOIUHMH TO anaseaae. xoropoe OHH npHHHMalOT npn HeKOTopOM
390
rAaBa
cpennea CMOTpHM,
Z.
3Ha4eHlIH
COOTBeTcTBylOmeM
.naHHo'l
meJlH.
Pac
B
=
KOTOpO'l
Mbl
113
ypaBHeHH~
rana
6y.neM
C4HraTb
paBHblMH HX 3Ha4eHHSlM
TE
(80.4)
nOJlY4aeM
H
TM (80.7)
HHlI
I'm
KaK
cnenyer
H3
CTBOBaTb OCH
ranu
pa3J1H'lHble
y:
(80.5a).
r ana TEM rana TE
BOJlHbl
runa TM
meJlH
Ey = H y E y = 0, H y = O.
B.nOJlb
saase
xoropue,
(l)m'
.
BOJlHaM
me
(I):
m1t
aurexaer,
OTCIO.na
(80.9)
'ITO K03Ke OHH
l,
= ( _ W2)lj,' 1
(80.10a)
1
KOMnOHeHTa
(80.7)
HO
w2
C
Ey = I WK
Dm SIn -----a- e TmY, (m1t)2
KTI
w
-TmY de.
1
,2 _
MOlKHO
cos m1tZ
r.ne
,
wf
(80.7) B036ym.naeT TOJlbKO
m>l
km=(f'
. E"
-lTj-=tg-=~ =K - = W 2 Hx C w'o ro
ua
MaJlOM
paccroauan,
.nJlSl 06ecne4eHII1I HenpepbIB
paccaorpaa
HX B caeayiouie»
392
13
rAaBa
§
3aMeOAJlIOU{ue
YC~OBHH. BTOpHqHWe BO~HW.
81. rpaHHqHWe
10 = l.'" W
>
~ o; sin
m;z e-T~Y
k m --
m«
'( I -
d'
(
2. 00 )1/,
k2
m-
1m -
c2
'-J '-J
E sz H H sx ' onpenenaexuan
ih = 10
se
:-, 1m oEm. 8, Y = - } : -2k OZ m m
H 8,,, = -
.
~-
km
m
d
. -lWE
d ~D -----!!! COS m1tZ -T I m
Y
H
demo
h
ah~d'
KI1M
nHIOllJ,YIO
F o = 0,
OCHOBHOIt
Oouiee
21ti hz
-.
~
F h l I e-
w'-od ~ Ph H fx = - i -21t-
>KHTClI B Bblpa>KeHI1H
-
~
lJT06bl He YlJl1TblBaTb euie paa COCTaB
BonHbl,
KOTopaH
B
HeHBHOM
sane conep
(81.7).
peaym.rapyiouiee
none
BonH B llJ,enH H CB06o,ll,HbIX
(80.11),
.llnll xero
.i;K' £sz+£fZ=O, Hsx+Hfx=O.
(81.11)
BOJIH .llOn>KHO paBHHTbcH OCTaTOlJHOMY no.no He06xo.llHMO nOnO>KHTb
2h7tZ
(2p+ I)T= ~ cphsm~, h;;'.1
."
h""O
HMeeT BH.ll
X'I
~
m
h?!'O
(81.7)
Mbl 6Y.lleM paccxarpaaars cnyxan aexeruux
= 7tZ
-2Tti~
~ D 2p + 1 ' 2h ~ e
y= 0
EfZ=
11;;>1
cos
= w'-od ~ 21t ~
Ef y = i
He CTOHT nl1IlIb ,ll,nH
nona (80.11)
(KHbl
(81.2))
2h·
4
+ 1= - - ; (2p + 1)2 -4h
BOJIH B llJ,eJIH B nnOCKOCTI1
P +--=.!! I f yHz he. 21th (
HeM
h>O
Ilpeacne lJeM aanacart, nOJIe B
2p+ 1
Hcnom.ays paanoacenae (81.1) (81.6)
•
395
CUCTeMbl
+ 1)2- 4h2 '
2h c ph = - SPh 2p
21th ( .)} If Y+1Z
. ) P 21th -i ..!!..- ~ ~ IT (Y-1Z H fX Ul!O 21t ~ { h e
.
Sp_h = Sph = -; (2p
,
+F _he
aOAflOBOaHble
a
B6nHaH
')}
21th (
{
Fhe
none
3TH nons HMelOT BI1.ll
~ Fhe IT (y-.z -F._heIT Y+1: h>O
}:{
Ef z =
3aMeaAJlIOU{ue
13
rAaaa
ESII+Ef ll
396
13
FAaBa
3aMeOAJlIOU{ue
:$1
397
cucresiu
BOAflOBOOlible
:~f
nOCJIellHHM H3 3THX cOOTHOWeHHtl MO)/{HO
H sx
MarHHTHble nOJIll
(80.5)
COOTHoweHHll
h =1= O.
l
Hf x
H
TaK KaK
d
HMelOTnoplllloK BeJIH'IHHbl
(80.9)}.
H
npeaeopeus,
ypaanenns
TaK 'ITO OCTalOTCll
h 1 ~ ThT Ph + 2" ~
D2p+1Sph
=
[CM.
MbI
YBHllHM
nansure, B nexoropux cnysasx <pa30Ball CKOpOCTb
Mo)/{eT CTaHOBHTbCll H 60JlbWe c. B paCCMaTpHBaeMOM cnysae Mbl
(80. lOa)} 0,
t
p>o Ph
~
+ 2"i
2h
~ D2p+1Sph2p+ 1 = O.
(81.12)
h=O. h
CJIy'laH
>0
H
h
0).
8TO
31t
He
HMeeT
3aMe'laHHe
co!
< c < 2'IT. 3 2" '1/ < < 2'1/, 2"
'I
c '1/='51'
~H3HqeCKHe CBOHCTBa BOJIH
onpeneneuuaa
cB06011
Y (y
peurenne
'ITO peurenas He cyiuecrnyer, eCJlH
"2
eCJlH
cymecraenuoe
oauasaer.
1t
CMbICJla.
e.
3KCnOHeHLI,HaJlbHO aapacraioume
O p>O
(80.10).
oI
1. k
..k
IJ.-
aue BOJlHbl
p>o
ECJIH K03eptPHLI,HeHTbi
(82.2)
•
Sph=
1 ~
'ITO
I
<Jl
4 1 D 2p + 1 -;2p+ 1--2h =0,
~
I
co! c
COs -
fl8.1lflemCfl nO.llo:JICUme.llbHOU ee.uixuuoti, -
TaK
,l
npnsoxar K ypannenaax,
p>O
h=O.
I
w
(81.13) D:
')n-1-1
+ tgll ~c!)'I• = 1
1
113.1laZaeMafl meopust npuzoiuta, ecnu mo astco
c
D2P+lSph=-P-h'
2p+ 1 +2h
BOJIHY H ee <pa30BYIO CKOpOCTb
h>- 0;
p>O TaK KaK
(82.1)
OTMeTHTb OrpaHH'IeHHOCTb 06J1aCTH npHMeHHMocTH 3TOtl
o 1 ~ D . co "2 ~ 2p+1 SpO= t WK' p>O
nonysaex
OllHaKO,
KaK
8TO BeTBH
06CTOllTeJlbCTBO
OTMe'leHO
na
epHr.
160
H
161.
r ne
KpHBOU, He HMelOllI,He 41,
l
H x=-TchKyelw -rz. rparpa,
TOJlbKO 'ITO
pacnpeneneuae noaert
E y-- KI ch Kyeiwt-rz . ,
nycKaHHSI.
Peaynsratu.
tS
~ /'.,,----
/./'
/~'/' 1
~,~1
t
.//'
/./ I /'
/
-irr --- --~--,-~._~'-~.=.~ /
/ / ./.".... V, = vh'V'/
o
et>
Hr.
!
-----
1/2d
168. 1·- peuieaue .uml aHTHcHMMeTpH'lHblX 2 - peureuue .uJlR cHMMeTpHllHblX BOJIH.
A=~ BOJlH;
eM. noanacs K
> =
(83.22)
/
>
=
+ l) = 2h,
/8
.::..
poA 1, a xpnaue nJlll cHMMeTpHlIHblX
1
H BO
TaK,
scex
ynoanernopaer
06J1aCTliX
non
na
CHCTeMoA.
cj>Hr.
160
H
CBoAcTBa
nJlOCKaH c rpyxrypa C OTKpblTblMH lIJ,eJlHMH KOTOpOMy
neu
nJlll
nJlll
HCCJlenOBaHHlI
peurenaa
H
npyrux aanas.
Mbl nplfMeHHM ero ceAlIac
CBoAcTB nJlOCKoA CTpyKTypbI C OTKpblTblMH
llI,eJlllMH, cxeMaTHlIeCKH H306pameHHoA na cj>Hr.
Hccnenosanae
MeTOA rrpur-o
pacnpocrpaaeuas
BOJIH
170.
B TaKoA
(84.2) Han CHCTeMoA llI,eJleA
I .~,
noaen,
8
162. nocrpoari, peurenne
H3 BOJIH C cHMMeTpHlIHoA
e, (y) = n, (- y), tuesn:x
COOTBeTcTByeT
CBOAHTCli K paccxorpenmo Tpex pa3J1I1l1HbIX THnOB BOJlH;
(84.3)
cnenyiouiee pacnpeneneaae rnna TM:
OTHOCllllI,HXCli K OCHOBHOA BOJlHe
Ez = A cos k'y,
Ey=O. cTpyKType
/11, pacnoaoxennon BOJIH noscnenu
aHTHCHMMeTpHlIHblX
MomHo TaKme
npenunyumx naparparpax
06llI,eMY YCJlOBHIO
npOTHBOnOJlOmHO no 3HaKy nOJlIO B 06J1aCTH
2.
YlcnoJlb30BaHHblA B
k'=~ c
Ez (y) = - Ez ( - y), xro, nanpnuep, none E z B 06J1aCTH /
crpyxrypoa
§ 84.
r. 170.
Ez = A sin k'y,
H B
u,Hcj>poA
H
HHK. pacnonoxennun B nJlOCKOCTH y = O. 8ToT THn BOJIH ana aorasea BOJlHaM, paCCMOTpeHHblM B § 80-82. BHyTpl1 llI,eJleA peurenae HMeeT BHn
cf> H r. 169. 1- peureane )l11lI aHTHCHMMe. TpHlIHbIX soan: 2 - peurenae )lilH CHMMe TpHlIHblX BOilH.
BOJIH -
11/
H
,J;
= -
(84,4)
. A . k' y. lOlE O Ii' Sin
412
TAaBa
3aMea/lJlIOU{ue
13
Eho pacnpene.reuae aaxeaser TO. xoropoe 6bIII0 aanauo COOT nourenaaxu (80.66). Bnaronapa YCIlOBIiIO (84.3) .lI.OCTaTO'lHO paccaarpaaari, none TOJIbKO B O.ll.HO~ ns 06JIaCTett I Ii III.
COCTaBJIlIIOw.lie HHe (npn
~: = _
w
6Y.lI.YT HenpCpbl13Hnl. eCJIH oruoure
lW;0
+ lWEo ~~ k'l
=
(84.5)
npHHHMaeT O.ll.HO H TO me SHalJeHHe B ooeax 06JIaCTlIx. B yCJIO BHH (84.5) nepsoe oraourenne cooTBeTcTByeT-BOJIHaM B 06JIaCTH III ypaBHeHlie (80.3)]. a sropoe - BOJIHaM B uieaa (84.4). Cor JIaCHO
\2 Ilz'i
Hx
H
y = -l)
~-
C
\
Ez
413
cucreuu
BO/lHOaOaHble
-W
\"'----_-----"L-_~
Pz I'
3/Z Ji -...L
ZVl
1-21
1
V
",-zr
yCJIOBHIO (84.5). nrpaiouiexy .lI.JIlI clfMMeTpHlJHblX BOJIH r'y me pOJIb, lJTO H YCIlOBHe (80.10a) .lI.JIlI aHTIiCHMMeTpHlJHbIX BOIIH.
K=
5/2'1
'1 l/zl1
........ I 312Ji u- c
..8
3'1
312l'z
'1
0,
(84.6)
I I
K2
c2
c2
k'2 -K2 -;;;2-
v-=
k' = ~.
BeJIlilJHHa K AOllmHa 6bITb nOJlOmHTeJIbHO~, lJT06bI BOIIHbI 9KCTIOHeH U,HallbHO saryxana, xor na y npHHHMaeT 60JIbWHe auaueuus. BOSBpaw.alIcb K cootlioweHHIO (79.2). MomHO aanncan
/-'2-- 1'1=zl
I
orxyna I
B
paccMaTplIBaeMoM
w2 -
cnyuae
1• .lI.lllI
aHTlicliMMeTpHlJHblX
BOIIH c nenyer
1Jzd
c2
_= 1
euie nanacars YCIlOBlilI. anaaornunue YCJlOBlilIM (80.10). xoropue .lI.OllmHbI ooecnesaaan, aenpepusuocn; COCTaBJllIIOw.IiX E z H H x B nJIOCKOCTIi Y = -t. 0 aenpepuanocra E y MbI noxa He aaoo TUMClI. nOCKOJIbKY, Clle.ll.YlI HSJIOmeHHoMy B § 81 NeTo.ll.y. ~TO YCIlOBHe MomHO yAOBJIeTBOplfTb noanaee.
nYCIWftUfl:
I
7t
< c < 1t; 37t 'u! 2< C < 21t; "2
OJ!
1
< v < v" v,< v < 2v,.
"2 v, 3
2
rnaea
414 Il pn
9THX
HeHbI q>Hf.
YCJlOBHlIX
na
171-173,
q>Hf.
3TH
160-162 llOJl>KHbl 6blTb saxe
KOTOpbIX
runaa
CTBylOlUHe paSJlHqHbIM
3a,lIedIlJlIOU{ue
J!)
BOJIH.
naaecenu xpnaue, COOTBeT UHq>pa 1 Ha KpHBblX CHOBa
peureuae llJlll aHTHCHMMeTpHqHbIX BOJlH, a uHq>pa 2 - peureaae llJlll cHMMeTpHqHbIX BOJIH. 3TH peurenas Mbl nOJlY qHJlH B npennoaoaceaun, qTO ll:IHHa BOJlHbI BeJlHKa no cpasne HHIO C WHPHHO~ J..UeJle~ d; B npOTHBHOM cnysae xpnaue He
~
II
060SHaqaeT
CKOJlbKO
HCKaSlITClI,
nOll06HO
TOMy
KaK
noxasano nynx
9TO
I
II. B III. B
Crpyxrypa, npencraaaeunas na crpyxrype,
sa HCKJllOqeHHeM
q>Hr. 174,
paCCMOTpeHHO~ B
roro,
qTO
reneps
§
cosepurenno ana
83 (CM. q>Hf.
157),
J..UeJlH OTKpbITbI.
y
.
I
t
b':lJI~I~~~=~~~~~
I
I
I
pacnpocrpaneaae BOJIH rnna
TM,
HH>KHHX
YCJlOBH~ cHMMeTpHH (85.1).
Hasaea C HsyqeHHlI cHMMeTpHqHbIX BOJIH. B 06J1aCTH III 6y nyr Ha6J1IO.lI.aTbClI BOJlHbI rana a B
B
uienax - crossae
Bblpa>KeHHlI
q>asa
Ez(y)=- E z(- y).
HHX
reneps
llOJl>KHa
I
(85.1)
HO
BXOllHTb HeH3BeCTHall
=
A cos (k'y
+ r.p);
Hx
=-
k -
Bynea npennonarars,
qTO Hall
i7JoA sin (k'y+ r.p),
w
_
Woo _
7Jo - IT - eoC.
C'
CTPYKTYPO~, T.
(85.2)
e. B 06J1aCTH
I,
Ez = Be- K y
"
" t
,I [r
(83.1),
(80.66),
nons HMelOT 9KcnOHeHlI,HaJlbHbI~ xapaxrep:
"
,
runa
r.p:
~~
cHMMeTpHqHble BOJlHbI:
aHl'HCHMMeTpHqHble BOJlHbl:
llJlll
oouiero
BOJlHbl
Ey=O,
' .'
Ez(Y) = Ez(- y),
uienax:
' _
MbI 6y.neM pas
llByMlI CHCTeMaMH
KllY
V. B cB060llHOM npocrpancrae non CHCTeMO~: y b-l. Ilse nocnenune 06J1aCTH MO>KHO He paCCMaTpHBaTb, nOCKOJlbKY csenenas 0 noaenenaa pacnpOCTpaHlIlOJ..UHXClI TaM BOJIH MO>KHO
nOJlyqHTb HS noseneans BOJIH B 06J1aCTlIX I H II C yseroa
H a; = Ey= WK Is» B -K e v.
Hr. 174.
Hccnenys
npOCTpaHCTBe
- b - l
7 ___
Ii'
i
I1rTI
?
,..,.-,0
KHO
aanacars
B BHAe
i '(
oqn
(
ypa~HeHH!\
o
2
,,
HenOCpeACTBeHHO CJIeAycT
1
-
2V n) =
_
02qn _
B
-
+ Vn+1-2Vn_LC
(j[2 -
Flpennonaraa, AJIHHO!\
3aMeHHTb 'ITO
'ITO
BOJIHbI,
nepaua
AJIHHa
(fl2 (CoVn - Pndo)
02Vn = - L 02Pn.
ot2
AJilI
JIHHHH
nepearn
roro
§ 89.
o(pv)
HMeeT
(88.6)
•
CMbICJI
CKOpOCTH He
MaJia
aapsna
nyxxa
no cpaBHeHHIO
MO>KHO
op
(88.4)
BOJIHOBO" MexaHHKH
ypasneuae ABH>KeHHH ypannenaea liJ penanrepa 2m 00/ + til (f[ -
'1aCTHu,bI AOJI>KHO
2m
fi2 eV (x, t) ~ = O.
(89.1)
xepea 1i 0603Ha'leHa BeJIH'IHHa h/2rr. (h - nOCTOHHHaH V - nOTeHI.I.HaJi B AaHHo!\ TO'lKe B MOMeHT t, y'lHTbI aaiouiaa B03MymeHHe, sneceanoe 3JIeKTpOMarHHTHbIMH BOJIHaMH. YpaBHeHHe (88.6) npOaOJI>KaeT OCTaBaTbCH cnpaaenanaux: B '1aCTHOCTH. 6e3 H3MeHeHHtt OCTaeTCH ero npasaa '1aCTb, y'lH DJIaHKa),
,.1
CB5I3b
Me>Kay
9JIeKTpOMarHHTHott
BOJIHOtt
H
3JIeK
TpoHaMH.
L npeACTaB,1HeT HHaYKTHBHOCTb eaHHHI.I.bI aJIHHbI urpaer POJIb K03qJKeHHeM
oanasaer
KOMnJIeKCHO
COnpH>KeHHyIO
BeJIH'IHHY)· YpaBHeHHe ABH>KeHHH '1aCTHI.I.bI
npa
3TOM
ECJIH
HMeeT BHA
aJiH
BeJIH'IHHa
do
e oV
dT=-m ox' rne e
H
m
(88.5)
0603Ha'laIOT COOTBeTCTBeHHO 3apHA H Maccy '1aCTHI.I.bI,
POCTbIO
L
W.
naxaaa
MaJIa), TO MaJIO
V
npeanOJIO>KHTb. BOJIHa 6YAeT
OT,1H'IaIOme!\C5I
= V o +2U
'ITO
OT
O COS(Ul' t
W O'
29*
caaoa
W~Wo.
(T. e.
CO CKO
H MO>KHO nOJIO>KHTb
-k'x).
rae
Ul'=k'W,
CBH3b
pacnpocrpauarsca
(89.2)
Flpa (
!ZeU:JICeHUe 3,UICTpOHa 8 6ecyU{eAI nepuoiJU'teCICOM nOAe
fAa sa U
4:28
9THX yCJIOBHHX ypaBHeHHe
02 2
ox
(89.1)
npeapaiuaerca B ypannenae
2m 0 + hi at t-ro+ltoel'l'. , +ltOe-l'l'. ,) ~(x. t)=O.
r.n.e
1"'0=-
If'=o/t-k'x.
2~e v; lto=_2~e ii;
(89.3)
COOTHOrneHHH (90.2) H (90.3) MOlKHO paccMaTpHBaTb KaK 0t506ll.leHHe TeopeMbl JIOKe - Bnoxa, OTHOCHll.lef,\cH K ypaane HHIO Marse. I{09$$HUHeHTbI An' BXO,D.HlUHe B pa3JIOJI<eHile (90.1). .n.OJIlKHbI y.n.OBJIeTBOpHTb 6cCKOHeIJHof,\ CHCTeMe JIHHef,\Hblx ypaBHeHHf,\
(89.3a) (-
C
9THM XpaBHeHHeM Mhl 6y.n.eM pafiora'rs; OHO aHaJlOl'H'IHO ypaaneumo MaTbe H CBO.n.HTCH K uexy, eCJIH nOJIOJl' t k' x) u>'2 (~2 me o. W
aa
(89.2)
HHTepBaJIa.
qaCTblO
ypaa
9JIeKTpOMarH
B 06J1aCTH
> 0.
C HaHMeHbilleH
C
C
06pa30M.
W3aXB•
k=k oo-
k=k ooKaK
(92.46)
onpeneneu,
JIerKO
nplIHIIMaeT 60JIbWlle OTplIIJ.aTeJIbHble 3HaQeHIIH.
(92.36)
-< k o -< k oo + ~ k'
(92.4a)
CB060.LlHbIMII
OTBeTcTByeT IIHTepBaJIy
k oo -
< Woo 'TaKIIM
[0]
onpenenaiouryio C, BXO.llHT .LlBe BeJIII'IIIHbI: [+1] II [-1] [CM. (90.4a)]. Hanpnxep, aupaxenae [+1] nOJIOlKlI TeJIbHO nan napaooaoa, 0603Ha'leHHolt aa epHr. 181 HH.LleKCOM [+1]. II OTplIIJ.aTeJIbHO non Helt. 3HaK C 3aBllCHT OT nOJIO lKeHIIH peayasrapyioiuen KpHBOlt no OTHoweHH~ K napaooaax [+1] II [-I]. HlIlKHHH '1aCTb peaym.mpyiouiea KpllBOlt co
6bITb
CSH311
ycxopaercs
B TO'lKaX
B epOpMyJIy,
MOlKeT
cJIa60lt
npaKTII'IeCKII CBO.LlIITCH K
2. CB060.LlHble 9JIeKTpoHbI:
II
9JIeKTpOHa K09epepllIJ.lleHT
Bupaxeaaa (92.3a)
CJIe
3aBllCIIT OT 3HaKa
W
(91.14).
I1cCJIe.LlyeM reneps pesxe AlBl• rne
saepraea,
xpasoa
BeTBHX
113
sneumpon
nJlOTHOCTb tj;tj;~ MaKCHMaJlbHa B ,TO'lKaX
-< k o -< k oo +- ~ k',
3aXBa'leHHOrO
1. 3aXBa'leHHble 9JIeKTpoHbI:
C
k'
CJIe.LlyeT
C8060ilHbJU
Hr. 184.
nonysaex
BO scex OCTaJlbHblX HHTepBaJlaX H3MeHeHHH k o.
{[;1] + [ \] }
BblpalKeHlIll
II V"'OKC,
~
k oo -
C ua scex npyrax
'uJIll
IPIMUH
H MbI
2) CB060.LlHble 9JIeKTpoHbI:
i
x
nOJIOlKIITeJIbHa.
1) 3aXBa'leHHble 9JIeKTpoHbI:
3HaK
m
sroa xpasoa
B.LlOJIb'
j
MOlKHO
C.
rI :"
C
nyiouiae YCJIOBlIll:
Pa3JIII'IlIe
9JIeKTpOHaMII
3HaK K09lj>epllIJ.lleHTa
I
431
6ezyU{eM nepUOaU'leCKOM nOM
8
°
[] =
2m
[+1] =-+- TE.
(92.9)
438
118uJlCeHue aAeKTpOHa a 6e2yU{eM nepuosusecxo«
14
F/la8a
Ii
e = 2m lt o
nO.uCTaBJlSlSI
noxysaex
TaKHM 06pa30M. Y'lHTbIBaSi (;) -2
"2
(x, t) Mbl MomeM rorna
H Tor.n.a
, mW kOO=T' 113
i("'+~
~(x. t)=e
~
441
F p (cp') = P (cp') e-iP'f'
r.n.e '1 ~ = 2'CP' =
tJeeyU{eM nepuoduseceou nOAe
Oouiee peurenne .n.JllI epyHKUHH sanacars B cnenytouiea epopMe:
(93.1)
e(iOJt-kuoX)1I> (;).
8
o
-1Ke cnenyer na 06W.HX CBoRcTB ypaBHeHHSI MaTbe, KaK BH.llHO Ha cnenyiouiero: I. lj - i . 06J1acTeR nponycxanas HeT:
• I
< lj < r-
Hble 3J1eKTpoHbI):
(93.12a)
v=
Vo
+ ~ VpeiP'f'. p
(93.13)
rae rpl = O)'t k'», P - ueaoe qHCJlO. OpMbl, pacnpocrpaaaiouryiocs C nOCT051HHolt CKO POCTblO:
ur.
V
Torna
ypasneane
o"V ( 1
(89.3)
= V (x -
urt).
npHHHMaeT CJIellYlOlll,Hlt BHll:
1 )
at" W2- w~
=-
02p
L@
=
02
•
Lie I ot2 (0/0/ ).
lia .u.eH-n aY9J1 J1 (Baden-powell) 14-17, 23, 24, 31, 47
Bappaoni, (Barriol) 157
5epHy.~JlII LI.. (Bernouilli D.) 12
5epHyJlJlII 11. (Bernouilli J.) 12
5H6ep6ax (Bleberbach M. L.) 265
5J1ox KepMep (Germer) 204
LI.>KOHC (Jones H.) 264, 265
Ll.llpaK (Dirac P. A. M.) 363
Ll.9BIICCOH (Davisson) 204
3eHTl.-l (Seitz F.) 244, 264, 265
KaJlbMaH (Kallman H. E.) 66
KaY9p 308
Kau (Katz E.) 265
KeJlbBIIH (Kelvin) 14, 15, 21-25,
28, .50, 145. 307, 406
Knepo (Clairaut) 13
Koran C. C. 308
Kourn (Couchy) 14, 21, 24
Kpauepc (Krarners) 289
KpacHywKIIH n. E. 25, 27, 28, 52,
65, 364
Kpouar (Kronig R. de L.) 282, 306
K9Mn6eJlJl (Campbell) 27, 307, 308
K9HeK (Canac) 68
Jlarpaios (Lagrange) 13, 14, 138
Jlannac (Laplace) 11,185
JIeHHap.u.-CIIMOHC (Lennard-Sim mons) 154
MaH.u.eJlbWTaM JI. 11. 299
MaHHIIHr (Manning M. F.) 265
Mar se (Mathieu P.) 267, 270, 306,
425
Mor9H (Mauguin) 376, 379, 380
Mopc KeHHe nonHHOMOB
§ 72. Bsenenae . . . . . . . . . . . . . . § 73. Bupaaceaae ,llJIli TOKa It aanpsaceaaa
Fuasa 10.
§ § § § §
457
3aMeJ\,nHIOUI,He BonHOBOJ\,Hble CHCTeMbl
§ § § § § § §
77. BBeAeHlte • . . . . . . . . . . . . . . 78. Ooutne ypaBHeHltli . . . . . . . . . . . 79. MeAlleHHble BOJIHbI. Hexoropue npnxepu . 80. I'panauuue YCIIOBltli. OCHoBHali BOJIHa . 81. Fpaxasmae YCJIOBltli. Bropauaue BOJIHbI 82. lt3ltqeCKite CBOHCTBa BOJIH . . . 83. BOIIHbI Me}J{AY AByMlI CItCTeMaMIt nnocxnx napaaaem-asrx
.
.
.
.
.
.
.
.
401
OIlOCKali CTpyKTypa C OTKpbITbIMIt IlleJIliMIt .
.
.
.
.
.
.
.
410
IllelleH .
§ 84. § 85.
BOIIHbI
.
.
.
Me}J{AY
.
.
.
.
,llBYMlI
OTKpbITbIX IllelleH .
§ 86.
382
382
385
388
392
396
UItJIItHApltqeCKali
•
.
.
.
.
.
CItCTeMaMIt •
.
•
.
.
.
.
nnocxnx napaanexsuux
414
421
.
crpyxrypa . . .
I'sasa 14. F'naea 11.
Henpepsmasie § 67. Ilepexon IIItHItIt
•
OT •
.
.
.
.
.
§ 68. Flpuaepu aenpepusaux nOJIIOCHItKaMIt.
§ 69. Henpepsmaue XItJIJIa .
.
.
§ 70. HOpMltpOBKa § 71. Tpexrpaanue
.
.
aneKTpHqeCKHe nHHHH
serupexnomocansoa
IIHHItIt .
,lI,BH>KeHHe
.
.
.
•
.
.
II liHlt H, .
.
.
.
.
K .
.
nenpepuaaoa
.
.
.
.
.
.
.
It
.
.
.
.
.
351
354
.
.
.
.
.
.
.
.
.
Marse .
Mar pnuu Ilay na . . . MHorocjJa3Hble aneicrpaaecxae IIHHHH
MaTp1t1l (£ij)'
.
..
nepuoaaxecxae II It H1i1t. YpaBHeHHlI .
.
npeacraanseaux serupex It
358
360
363
§ § § § § § §
87. 88. 89. 90. 91. 92. 93.
aaexrpona
B 6eryUl,eM nepHoJ\,HQeCKOM none
Baeneaae . . . . . . . . . . . . . . . . . . . .
Ypasaeaaa 3aAaqlt B paMKax KliaCClIqeCKOH MeXaHItKIt Ypaaneaaa sanaua B paMKax BOJIHOBOH MeXaHItKIt .
Hccnenoaaaae
ypaBHeHltli B3aItMO,lleHCTBItR
CJIyqaH CJIa60H CB1I31i.
.
.
anrepnperauna Peuieaae aanaxn caenenaea
lt3ltqeCKali
).1MeHHoH YKu3aTellb . . .
Il p e n a e r u u f y x a a a r e a s
.
.
.
.
.
.
.
.
.
.
•
.
.
.
.
CllyqalI clla60H CB1I31t .
K
ypasaenmo
MaThe
.
424
425
427
428
430
435
440
445
447
H3nATEflbCTBO HHOCTPAHHO~ flHTEPATYP~
KHUZU no rjiu3uICe
>T·'l
'~
"~i
JI. B p
H n n 10 9 H H
M. nap 0 JI H
PACnpOCTPAHEHI1E BOJIH
B nEPI10.lUIt{ECKI1X CTPYKTYPAX
~r, "
PeJlaKTOp A. 1