rnABA 6

HECTAQHOHAPHbiE IIPOQECCbl IlPH UAPAMETPWIECKOM B031iY)I(JlEHHH BOJDI

HecraUHoHapHbie npouecchi npH napaMeTpH'IeCKOM B036Y)I(lleHHH BonH

Moryr HoCHTh caMbiH pa3rthm xapaKrep. Hx ycnoBHo MO)I(HO pa36HTh Ha rpH

rpynnbi: K nepBOH . OTHOCJITCJI nepexOOHble np01-4eCCbl yCTaHOBneHHJI CTaUHO·

HapHOrO COCTOJIHHJI, B03HHKaiOUlHe npH BKniO'IeHHH HaKa'IKH HnH npH llPY·

rnx cKa'IKoo6pa3HbiX H3MeireHHHX ycnoBHH 3KcnepHMeHra: nepeKniO'IeHHH

cpa3bi, 'laCTOTbi, aMnnHTYllbl HaKa'IKH, BHeWHero MarHHTHOrO nonH H T.ll. qaCTb H3 HHX onHcaHa B § 28. BropaH rpynna - 3TO BblHYJICOeHHble H-eCTQijUOHapHble

npoijeCCbl orCJie)I(HBaHHH HenpepbiBHhiX H3MeHeHHii ycnoBHH 3KCnepHMeHra :

nepHOllH'IeCKHX, MOTOHOTOHHbiX HnH wyMOBbiX. 0HH OnHcaHbl B § § 25-27 npellblllYUleH rnaBhi H § § 29, 30. TpeTbro rpynny coCTaBnHIOT cnoHTaHHble

HeCTQijUOHOpHble npOijeCCbl, B03HHKaiOWHe H3·3a HeyCTOH'IHBOCTH OCHOBHOrO

cocroHHHH IICB H npHBOllJIWHe K aBTOKone6aHHHM HaMarHH'leHHOCTH. 3TH

npouecchi cywecTBeHHo pa3nH'IaiOTCJI ,IJ;1IH CHTyaUHH, Korna Bo36y)l(lleH WH·

poKHH naKer IICB ( § 31) H Korna nopor napaMerpHlleCKoro B036y)l(lleHHH

MHHHManeH ,IJ;1IJI OllHOH HnH HeCKOnhKY.X nap BOnH, TaK 'ITO npH He ManbiX Hall­

KpHTH'IHOCTJIX B036y)l(lleH Y3KHH naKeT IICB ( § 32) .

§ 28. llepexo.llllhle npoQeCCbi B CHCTeMe napaMeTplfCieCKHX BOJIH

DpH BKJII()qeHHH H8K8'1KH

3ror Bonpoc paccMorpeH B.E. 3axapoBhiM, aaropoM H C.M. MywepoM B

I972 r. H nanee aarop cn~nyer 3:roe pa6ore [102]. 28.1. OWiaCTL MBJihiX ~KpHTH'IHocreA. B noM nYJiKTe H3yqeH npouecc

ycraHOBneHHJI CTaUHOHapHOro COCTOJIHHJI S -TeOpHH (I9 .1) OT ypOBHJI TeDnO·

Boro wyMa Tllf: =no npH ManbiX npeBhlllieHH.IIX h v - 'Y < 'Y· OlleBHllHO, 'ITO

no rex nop, noKa N = ~ nk < N0 , aMnnHryllbi nap 6ynyr paCTH raK, KaK npen-k

CKa3biBaeT nHHeHHaJI TeOpHH, T .e. C HHI<peMeHTOM h V - 'Y. IlpH 3TOM B k-npoCT­

paHCTBe 1W3HHKaer Y3KHH naKer CB WHpHHoii !::. w ~ (h V - r) . .IlanhHeewee

ero noBeneHHe OllHCbiBaeTCJI ypaBHeHHHMH {18 .9) :. J.h HHX BHllHO, 'ITO BpeMH penaKCaUHH aMnnHTYll H 4>a3 nopHUKa (h V - 'Y) -t H 'Y-l COOTBeTCTBeH­

HO, no3TOMY MO)I(HO C'IHTaTh, 'ITO cpa3bi Wk aUHa6aTH'leCKH cnellJIT 3a aMMHTY·

~MH, r:e . B (I8.9) MO)I(HO npeHe6pe'lb a wk jat . To, 'ITO naKeT Y3KHH: t:.w< r. n03BOnHeT pa3nO)I(HTb B ypaBHeHHHX (18.9) tpHr0HOMeTpH11eCKHe cl>YHKUHH B

pHll H npellCTaBHTb 3TH ypaBHeHHJI B BHlle

I at , , --- =x2 +x(2a-I)(Jfdx -l)+affdx -a+

f ar + [ffdx' -I J:l a 2 + fx'fdx',

f= nk(r)/No, x=(wk - Wk0)/(2SN0 ), (28.1)

1- = rt(2SNofr):l, · a==: Cl T + S)/S.

OrMeTHM, 'ITO B noM ypaaHeHHH 3aBHCHMOCTH or HallKpHTH'IHOCTH p = (hfhth)2

HC'Ie3na, a napaMeTpbi CHCTeMbi IICB BXOllHT B Hero nHWh 11epe3 a. YpaBHeHHe 9. 131

(28.1) HMeeT aBTOMOJJ;eJibHOe perneHHe

f(x, r) =A (r)(r/1T) 112 exp { - [(x - x 0 (t))/d(r)J 1 } , (28 .2)

rne A(r) =Jf(x, r)dr - cyMMapna.ll HHTeHCHBHOCTb naKeTa 11CB,x0 (r) = =A -I f fxdx - norro:>KeHHe ero ueHTpa T.ll)l(ecm, d (r) - IDHpHHa naKeTa.

OHH ynoBrremopmoT ypaBHeHH.IIM

1 dA 1 d(rx~) · 1 - +- - = - Ax0 -o:A(A -1)[o:(A - 1) + 1].

A dr 2r dr (28 .3)

d(rx0 ) ( 1 ) = - (A - 1) o: - -2

,' dr

d(r)=t- 1 12 .

3TH ypaBHeHH.II HMeiOT CJie;::J;yiOlliYIO aCHMnTOTHKY npH T "* 00 :

x 0 =-1/2T, A-1=-[2(4o:-1)r2) - 1 , d(r)=r - 112 . (28 .4)

TaKHM o6pa3oM, npH ue6orrblliHX npeBbillieHH.IIX nan noporoM npoH3BOJibHa.ll

<l>YHKUH.II pacnpenerreHH.II nap nk perraKCHpyeT K HMeiOllieMy BHJJ; o-<I>YHKUHH

CTai.J;HOHapHOMY COCTOHHHIO (18.9) no CTeneHHOMy 3aKOHY (28.4) :

No - N-No (hV- rt 1 r 2,

(28.5)

w2 - (w )2 "'r[(h v T r )/tP'2

. k k ~.

JlnH )J;eTaJibHOrO H3ytleHHH HeCTai.J;HOHapHOrO noBe)J;eHHH CHCTeMbi npH npoH3-

BOJlbHLIX h V, S, T ypasHeHHH (18.9) pernarrHcb tiHcrreHHo . Ha pHc. 28.1 .H3o-

6pa:>KeHa <l>Y"KUH.II pacnpenerreHH.II nk(t) nrr» JJ;Byx noCJie;::J;oBaTeJibHbiX Mo­

MeHTOB BpeMeHH H 3Ha'leHHii napaMeTPa o: = 1/6 H o: = 1. Ha'laJibHoe pacnpene­

rreHHe Bbi6pauo onuopoJJ;HbiM, h V = 1 ,4-y. Buaqarre BOJIHbi paCTYT JKcnoHeH­

UHaJibHO, MaKCHMYM nk(t) HaxO)J;HTCH Ha noBepXHOCTH Wk = Wp/2, KaK 3TO

crrenyeT H3 JlHHeHHOH TeOpHH. EcTeCTBeHHO, 'ITO BH)J; <l>Y"KUHH nk npH 3TOM He

3aBHCHT OT napaMeTpa o: . B JJ;aJibHeiirneM, Korna 'Lnk He Marra no cpasHeHHIO k

C N 0, noBe;::J;eHHe nk (t) cymecTBeHHO 3aBHCHT OT OTHOllieHHH KOJ<l><l>HI.J;HeHTOB S H T, onHcbmaiOIUHX B3aHMoneiicTBHe BOJIH, a HMeHHo, npH T = 0 naKeT nk. npOJJ;OJI:>KaeT paCTH H Cy:>KaTbCH, OCTaBa.IICb Ha noBepXHOCTH Wk = Wp/2. llpH

T * 0 naKeT pacTeT H nepe;::J;BHraeTCH KaK uerroe, uecKOJibKO ne<l>opMHpy.IICb, K

I I ' noBepXHOCTH wk + 2T0 N0 = wp 2 (T .e. B CTOpoHy OOJiblliHX HllH MeHblliHX k B 3aBHCHMOCTH OT 3HaKa T). Korna MaKCHMYM naKeTa nk (t) 6rrH30K K noli

noBepXHOCTH , HatiHHaeTCH aCHMnTOTH'IeCKOe Cy:>KeHHe naKeTa. 0TMeTHM, 'ITO BH)J;

KpliBoii In nk(t) (eM. pHc. 28.1) npH 6orrblliHX t 6rrH30K K napa6orre, 'ITO

no;::J;TBep:>KnaeT BbiXOJJ; Ha aBToMonerrbHoe perneHHe (28 .2) - rayccoB naKeT .

lh pHC. 28:2 BH)J;HO, 'ITO npouecc ycTaHOBJleHHH CTaUHOHapuoro COCTOHHHH OT

TellJIOBOrO ypOBHH llpH BKJliO'IeHHH napaMeTpH'IeCKOH HaKalJKH llpOKCXO)J;HT B

TpH CTa)J;HH : B nepBOH - JlHHeHHOH - CTa)J;HH llpOHCXO)J;HT JKCllOHeHI.J;HaJibHbiH pOCT

aMnJIHcy;::J;bi, a JaBHCHMOCTb OT HeJIHHeHHbiX xapaKTepHCTHK T, CHCTeMbl BOJIH OT­

CYTCTByeT. EcTeCTBeHHO noJToMy, 'ITO JJ;Be KpHBbie N(t) (nrr.11 o: = 1/6; 1) Ha JTOH CTaJJ;HH COBna;::J;aiOT. Ha BTOpOH - HeJIHHeHHOH - CTaJJ;HH CTaHOB.IlTC.II

CYIUeCTBeHHblMH B3aHMHLIH HeJIHHeHHblH CJJ;BHr 'laCTOTbl H napaMeTpH'IeCKOe

132

N

Na -----

1,2

0 10 20 Zyt

PH C. 28.1. Cl)yHKUHJI pacnpe.n;eneHWI llCB n (k) DpH p = 2. JlHHeltHax CTa.n;Hll : 1 -a= 1/6; 2- a= 1; aCHMDTOTHlleCKaJI CTanHJI: 3- a= 1/6; 4 - a= 1-

p H C. 28.2. 3aBHCHMOCTL cyMMapHolt aMDnHTYIU>l mip N(t) OT BpeMeHH Mll p = 2: 1 -a = 1/6; 2- a= 1

B3aHMO,!l;eHeTBHe nap BOTIH. JlpH 3TOM eme OTeyTCTByeT xapaKTepHaff ,IJ,J"UI CTaUHO· Hapuoro eOCTOffHHff B S~TeOpHH KOMneueaUHff 3aTyxaHHff BOTIH 'Yk nonHOH Ha­Ka'IKOH p k . Ilose,o;eHHe eHeTeMbi Ha 3TOH CTa,!l;HH HaH6onee CTIO)I(}JO H ,!l;OeTyrrHO nHIIIb 'IHcneHHoMy Mo,o;enHposaHHIO. 0Ka3bmaeTen , 'ITO npH T < 0 HHTerpamuan aMrrnHTy,o;a npoxo.o;HT qepe3 MaKeHMyM, a npH T > 0 pacTeT MOHOTOHHo. Ha TpeTbeii - aeHMITTOTH'IeeKoii - era.o;HH I P k I .:... -'Yk <t. r~ H npoHexo.o;HT Me,!l;]leH­Hoe npH6nH)f(eHHe K CTal(HOHapHOMY COCTOffHHIO ( 19.1) , OnHCbiBaeMOMY S -TeO­pHeH. 0TMeTHM, 'ITO JHaK pa3HoeTH N- N0 B ~mcneHHOM JKenepHMeHTe (eM. pHC. 28.2) eosna,o;aeT e TeM, KOTOpbiH cne.o;yeT H3 aHanHTH'IeCKOH aCHMllTO­THKH (28 .4).

282. lionLIIIHe npeBLillleiDUI ua.o; noporoM ueycroA'IHBoC'I1f. 4HcneHHhiH JKenepHMeHT BblffBHn eymeeTBeHHbie Ka'leeTBeHHbie OTTIH'IHff B HeCTal(HOHap­HOM noae,o;eHHH CHeTeMbl TICB npH ManbiX h v - r <t. r H 6onbiiiHX h v - r_'Pr npeBblllleHHnx Ha,o; noporoM (pile. 28.3 H 28.4 ,IJ.Jiff h V = 4r) Ha BTopoii -HenHneiiHoii - eTa.o;HH nepexo,o;Horo npoueeea. IlaKeT nk (t) He ae.o;eT ce6H KaK uenoe, Kor.o;a aMrrnHTy,o;a N (t) ~e He Mana no epaaHeHHIO c N0 , "npopacTaeT" BTOpOH MaKeHMYM tPYHKl(HH nk (t) B TOM MeCTe, r,o;e z;k = 0 B ,o;aHHbiH MOMeHT apeMeHH. 3aTeM aMrrnHTy,o;a BToporo MaKeHMyMa pacreT ,a nepaoro YMeHbiiiaeT­cn . Ha pHe. 28.3 iipH rt = 6 BTopoii MaKCHMYM y)f(e 6onbiiie nepsoro. KpoMe TO­ro, KaK BH,!l;HO H3 pHC. 28 .4, CYMMapHaH aMITnHTy,o;a OCl(HnnHpyeT, npH6TIH)f(aJICb K N0 . B npe.o;emHoM cnyqae He.o;HccHnaTHBHoii cpe,!l;bi, Kor.o;a . ( r I L v) -+ 0, cHcTeMa TICB soo6me He BbiXOAHT Ha cTaUHoHap, ocuHnnHpyn tiecKOHe'IHo ,o;onro. 3'fo CBH3aHo C TeM, 'ITO npH· 'Y = 0 cymecTByeT HHTerpan ,!l;BH)f(eHHff j{s

(18.13). B Ha'lanbHbiH MOMeHT BpeMeHH j{s = 0 H He COBna,o;aeT CO 3Ha'leHHeM JC9 = - (2T + S) N~ B OeHOBHOM COeTOffHHH S -TeOpHH, KOTopoe nOJTOMY n}m 'Y = 0 He,o;oeTH)f(HMO .

Ha nocne,o;HeH - aeHMnTOTH'IecKOH - CTa,!l;HH, KaK BH,!l;HO H3 pH e. 28.3, naKeT nk rayccoa. 3Toro H cne.o;oaano O)f(H,!l;aTb, TaK kaK na cra.o;Hn onHchrnaeTcn ypaa­HeHHnMH (28 .1) , HMeiOII.IHMH aaToMo,o;enbnoe peiiieHHe (28 .2) - rayccoa naKer. JleHCTBHTeTibHO, KpHTepHH cnpaae,!l;]IHBOCTH ypaBHeHHH (28 .1) eCTb ManOCTb OT· HoiiieHHn S (N- N0 ) !r<t. 1. B cooTBeTeTBHH c acHMITTOTHKOH (28.4) N(t) npH-

133

~nn

N

5

ko If PH c. 28.3. ~lQJ,HJI pacnpeAeneHHJI n (k) . npu p = 16; 1 - nHHeltHaH CTaAHJ[ (01 = 1/6; 1); 2, 3- HenHHeltHaH CTaAHJ[ (01 = 1/6; 1 COoTBeTCTBeHHo); 4, 5 - aCHMirrOTH'IeCKlUI CTa­AHJI (01 = 1/6; 1 COOTBeTCTBeHHo)

pH C. 28.4. 3aBHCHMOCTL CyMMapHolt aMJIJIHTYP.bl llCB N(t) OT BpeMeHH AJIJI p = 16: l - 01=1/6; 2-01=]

6JIH)I(lleTCH KN0 CBepxy HJIH CHH3Y, K3K 3TO BHAJIO H3 pHC. 28 .4, B 3aBHCHMOCTH OT BenH'IHHbi T/S. IIpJ:!,.IHa ociJ,HJinJmHH N(t) B nepexo,u;HoM pe)I(HMe O'leBHA­Ha H 33Knl0'13eTCH B YA3pHOM B036y)I(AeHHH KOnneKTHBHbiX KOne6aHHii:. l1x AeKpeMeHT 33TYXaHHH paBeH 'Y' a '13CTOTa no = [2S (2T + S)] 1'

2 - h v npH

h V ;p r. Ilo3TOMY 'IHcno Kone6aHHii nopHAKa h V /r = p 112•

§ 29. llapaMeT})H'IecKoe Bo36y)I(AeHHe BOJIH B yCJIOBHJIX ApeAcjla 'laCTOTI>I

AHanH3 3KCnepHMeHTanbHbiX AaHHbiX B.C. )IU{THIOKa H r .A. MenKOBa [ 80] noKa3biBaeT, 'ITO npu napaMeTpH'IeCKOM B036Y)I(AeHHH CB B )JG1T B onpeAeneH­Hoii o6naCTH 3Ha'leHHH ITOCTOHHHOrO nonH H B03HHKaeT Apeii:cjl C06CTBeHHOH 'laC· TOTbi CB ~ B k-npocTpaHCTBe, T .e. rioHBnHeTcH MOHOTOHHaH BpeMeHHaH 3aBH· CHMOCTb Wk (t) . IlpH'IHHOH TaKOrO Apeii:cjla HBnHeTCH HaKonneHHe JTepaBHOBeC­HbiX CB/ (HCB), B03HHKaiOIUHX B pe3ynbTaTe ,u;HCCHnauHH IICB. H aKonneHHe 6omumro 'IRena HCB B o6nacTH ManbiX k Ha6niOAanOCb AaBHo X. Jie.fannoM H AP· [ 143], B R. BeHHuKHM HAP· [ 62 ). EcTeCTBeHHo AYMaTb, 'ITO B Te'leHHH HeKo­Toporo BpeMeHH 'IHcno HCB nHHeiiHo HapacneT co BpeMeHeM, a 3aTeM BbiXO,u;HT Ha crau;HoHapHbiH pe)I(HM. B pa6oTe E.B. Ilo,u;HBHnoBa H B.E. qepenaHoBa [198], ITO pe3ynbTaTaM KOTOpoii: HanHCaH 3TOT naparpacjl, ITOKa3aHO, 'ITO B Te'leHHH Bpe­MeHH nopHAKa 102r-k MO)I(eT npoucxo,u;HTb HaKonneHHe HCB c nocTOHHHoii CKOpOCTbiO , npHBOAHUiee K ITOCTOHHHOMY Apefupy 'laCTOTbl ~ C 6e3p33MepHOH

CKOpOCTbiO -

Tlk=_1_ awk~ 21TI (p - 1)1/2. (29.1) 2rk at s

3Aecb T- xapaKrepHoe 3HalleHHe semi'IHHbi Tkk', onucbrBawureii HenuHeHHbiH

134

cromr qacroTbi ITCB rro.n; .n;eiicraueM HCB. OrMeTHM, qro, rrocKon&Ky aenu­'lHHbi T H S, aoo6we roaopn, o.n;uoro rropn.D;Ka, oueuKa (29 .1) .n;eMoHcrpupyer B03MO)!(HOCTb B03HHKHOBeHH.II 3Ha'lHrenbHOrO ,n;peii<Pa qacTOTbl, 11pH KOTO­pOM wk H3MeH.IIercn Ha aenHqHHY 'Yk Ja apeM.II 1 I'Yk. PaJyMeercn,. yqer raKoro .r(peii<Pa .n;on)l(eH rrpHBO.n;HTh K cywecraeHHOMY H3MeHeHHIO peJyn&raTOa S­reopuu, a qacrHOCTH K B03HHKHoaeHHIO KoHequoii: IIIHpHHbiiiaKera ITCB. Mo.n;!i­<PuKaUHH S-reopHH c yqeroM .n;peii<Pa H rrocanweH uacrO.IIIIIHH rraparpa<P.

29.1. KaqeCTBeiDibiA aHMH3 cl>opMLI rraKent ITCB H ero aMDJIIfTY.D;hi npu 11101eA:noM .n;peAcl>e qacron.I. YpaaueHH.II S -reopuu rrpe.n;cTaBHM a BH,n;e

a~ o - + 2-yk nk + 2ImPk ak = 2-yk nk , at (29 .2)

aak + [i(2wk - Wp) + 2-yk 1 ak + 2iPtnk = 0 . at

OHH omuqaroTC.II OT (18.6) .n;orronHHTeJlbHbiM cnaraeMbiM 2-yk nz, OIIHCbiBaiO­lllHM reiiJloaoii rnyM. Crporoe o6ocuoaauue ypaaHeHHH (29 .2) 6y.n;er .n;auo B rn. 10. 3,n;eCh Mbl TOJlbKO OTMeTHM, 'lTO B OTCYTCTBHe HaKa'lKH OHH OIIHCbl· aaror penaKCal(HIO nk K repMO.D;HHaMH'leCKH paBHOBeCHOMY 3HaqeHHIO nZ. TaK )l(e, KaK H B § 28, rrepeii:,n;eM K HOBbiM rrepeMeHHbiM €k = wZ - Wp/2 -paccrpoii:Ka qacrOTbl H Q = (6, r.p) - II011.11pHbiH H a3HMYTMbHbiH Yf1lbl BOJl· noaoro aeKropa k . B 3THX rrepeMeHHbiX ypaaHeHH.II (29.2) rrepexo.n;nr a

anen 0 -- + 2'Yn nen + 2ImPf1 a en= 2-y0 n ,

at aaen . -- + 2 [ig(t) + ie + 'Yn 1 a en + 2iPn nen = 0 ,

at

g(t)=wk - wZ =at+c.

(29.3)

(29.4)

3.n;ecb g (t) - crauuouapHbiH .n;peii:<P co6craeuuoii: qacroTbi. B orcyrcraue /lpeii<Pa (IIpH 0! = 0) ypaBHeHH.II (29 .3) HMeiOT e.n;HHCTBeHHOe peiiieHHe

1Pnl 2n° __ nen - ~0 = _2 2 2 , e = e +c. (29.5)

e +-y0 - 1Pnl

B rrpe.n;ene n° -+ 0 3TO perneHHe rrepexo.n;ur a ro H3 perneHHii S -reopHH, KO­ropoe y.n;oaneraopner ycnoauro BHeWHeii: ycroii:quaocru (18 .18). lleii:crau­renbHO, nen MaKCHMaJlbHO Ha pe30HaHCHOH IIOBepXHOCTH € = 0 H B rrpe.n;ene MaJibiX n° MO:lKer omuqaJbcn or uynn ron&KO a rex ee ro'lKax, r.n;e I P nl = 'Yn · B ocranbHoii: qacru rroaepxuocrH o6n3aHo BbiiiOnH.IITbC.II yCJloaue I P nl < 'Y, HHaqe B03MO)I(HO rrporuaope'lHe: n0 < 0. Ilpu IPnl >-y perneuue cucreMbi (29.3) Hapacraer 3KCIIOHeHIJ,HaJlbHO C HHKpeMeHTOM

vn = - 'Yn + (!Pn 12 - €2)1/2 . (29.6)

06nacTb ueycroii:'lHBOCTH rro qacroraM wk orrpe.n;ennercn uepaaeucraoM

v n > 0. 3ro .n;aer

I €1 < (IPn 12 - 'Yh)112. (29.7)

llpH JlHHeHHOM .n;peii:<Pe qacTOTbl B03MO)I(H0 cymeCTBOBaHHe TaKHX IIaKeTOB IlCB, KOTOpbie C IIOCTO.IIHHOH CKOp0CTbiO IIO qacrore /lBH~TC.II B k-rrpocrpaH-

135

CTBe 6e3 H3MeHeHH11 Q>opMbi, JJ;pyrHMH CJIOBaMH, ypaBHeHHH (29.3), (29.4) JJ;OJDKHbi HMeTh CTaUHOHapHbie perneHHH BHJJ;a

n(n,e,t)=nn(e+at+c). (29 .8)

IlpHBeJJ;eM KaqecTBeHHhiH aHaJIH3 JTHX perneHHH. IlpH JJ;peJiQ>e B Ka)f()J;biH MOMeHT BpeMeHH OJJ;HH BOJIHbi IIOIIaJJ;aiOT B 06JiaCTb HeyCTQHqHBOCTH y pe30-HaHCHOH IIOBepXHOCTH, a JJ;pyrHe H3 Hee BbiXO,!J;HT. IlpHXOJJ;HIUHe BOJIHbi Ha­pacraroT JKCIIOHeHUHaJibHO C HHKpeMeHTOM V n (t) (29 .6), HaqHHaH C ypoBHH TeiiJIOBbiX Q>nyKTyaUHH. 0HH IIpOXOJJ;HT 06JiaCTb HeyCTOHqHBOCTH (29.7) 3a BpeMH

To= 2(1Pn 12 -'Yh) 1/2 /lal ~ 2(2-y(IPI - 'Y))1/2/Ial, (29.9)

a HX ypoBeHb HapaCTaeT IIpH JTOM JJ;O

"To

nmax=nn(r0)=n°exp[2f v0 (t)dt] ~ 0

(29.10)

llanee, 3TH BOJIHbi BbiXOJJ;HT H3 06JiaCTH HeyCTQHqHBOCTH H 3aTyXaiOT 3a BpeMH r 1 co cpeJJ;HHM JJ;eKpeMeHTOM v ( r 1 ) ~ ( ar 1 )

2 /2 'Y· OrcroJJ;a H H3 (29 .1 0) cne­JJ;yer oueHKa JJ;JIH T 1 :

(29.11)

8JJ;aJIH OT 06JiaCTH HeyCTQHqHBOCTH JJ;peHQ>OM MO)I(HO rrpeHe6peqb, H B COOT­BeTCTBHH c (29.5) (nn - n°) - e- 2

• MaKcHMaJihHan HHTeHCHBHOCTh B rraKere orrpeJJ;ennercn Bhrpa)l(eHHeM (29 .1 0) H ne)I(HT Ha JaJJ;HeH rpaHHUe o6nacrH HeycTOHqHBO.;TH (29.7):

(29.12)

illHpHHa rraKera l5 orrpeJJ;ennercn BpeMeHeM, Ja Koropoe CB y6hrBaror B e pa3, IIpOHJJ;H rpaHHUY o6nacrH HeycroJiqHBOCTH :

(29.13)

IlpH BhiiiOJIHeHHH ycnoBHH aJJ;Ha6arHqecKoH MeJJ;JieHHOCTH JJ;peliQ>a:

(29.14)

rrpH KoropoM crrpaBeJJ;JIHBhi orrHcaHHbie Bhrrne coo6pa)l(eHHH, rnHpHHa rraKera l5 (29.13) MHoro MeHhrne, qeM ero cJJ;BHr t::.wk (29 .12). l1HrerpanbHaH HHTeH­CHBHocrh Nn oueHHBaercn nerKo:

(29 .15)

IlpeJJ;JKCIIOHeHUHaJibHbiH MHO)I(HTeJlb B (29.15) Ha HarneM ypoBHe CTpOrOCTH He orrpeJJ;ennercn, JJ;JIH Hac cywecrBeHHO TOJlhKO, qro B orcyrcTBHe rnyMa n° = 0 aMIIJIHTyJJ;a IICB Nn = 0 . OrMeTHM raK)I(e, qro (29.15) He HBnnercn perneHHeM 136

Jap;aiiH, raK KaK )].11.11 orrpep;eneHH.II IIHCJia fiCB Heo6xo,ll;HMO caMocornacoBaH­HhiM o6pa30M HaiirH rrepeHopMHpoBaHHyro HaKaiiKy Pn .

,llrr.11 KalleCTBeHHOfO aHaiTH3a aMIIITHTY.r:lbl H <Pa3bl fiCB 3a rroporOM yp;o6HO BOCIIOITb30BaTbC.II YCJIOBHeM paBeHCTBa BKaiiHBaeMOfO H ,[IHCCHIIHpyeMOfO IIOTOKOB :meprHH :

h Vsin'll = "f , (29.16)

rp;e 'I! - pa3Hocri. <Pa3 aHOManhHoro Koppen.~~:ropa fiCB H HaKaiiKH. TaKHM 06pa30M, Ha BeKIOpHOH ,[IHarpaMMe (pH C. 29.1) BeKrop Sa HaKITOHeH IIO,[I yrnoM 'I!= arcsin ('Y/h V) K HaKaiiKe P (Mbi CIIHTaeM )].11.11 orrpep;eneHHOCTH, 11ro S > 0) . CyMMa h V +Sa ecTh rrepeHopMHpOBaHHaH HaKaiiKa, On;eHHM rerreph yron Me)l(,!l;y P H Sa . 113 ypaBHeHHH (29.3) cnep;yer, liTO B cran;HoHap­HOM CJiyllae

ak =.- , 'Yk + i(wk - wp/2)

1Pkl 2 (wk - wp/2)nk RePZ ak = - 'Yl + (wk - wp/2)2

(29 .17)

flpH He CITHIIIKOM 6biCipOM p;peif<Pe llaCTOTbl IIaKeT HBITHeTCH Y3KHM ITO llaCTO­TaM, rro:noMy B n;eHrpe naKera I Pk 12 = 'Yk + ~w, H

RePZ ak = -~wnk . (29 .18)

flocKOIThKY rrpH rronmtmrenhHOM p;peii<Pe ~w > 0, yron Me)l(p;y Sa H P ryrroii .

PH c. 29.1. KoMnneKCHaJI nnocKOCTh ueKTo­poB HaKa'IKH. h V - BHeiiDIIDI HaKai!Ka, W -<f>aJa nap, 1' - JaTyXaHHe, Sa = ~ S kk' a - no-

k ne peaKUHH CHCTeMbi IlCB; OA - caMocorna-coBaHHaJI HaKa'IKa 6eJ Apeit<f>a, OB - c nono­lKHTellbHbiM Apeit<f>oM, 0C, 0C' - C OTpHUa­TellbHbiM Apeit<f>oM. ITHHIDI ] COOTBeTCTByeT Henpepb!BHOit MOAH<f>HKaUHH p eiUeHHH S-TeopHH; nHHHH 2 - AononHHTellhHoe p ewe­HHe llpH CHllbHOM OTpHUaTellbHOM Apeit<f>e

~ ~ 0 hV

flo:noMy BenHIIHHa !Sal MeH&IIIe, 11eM ee JHalleHH.II B orcyrcrBHe p;peii<Pa,

r .e. I Sal < ( I h VI 2 - "f2) 1/ 2. B cny11ae orpHuaremHoro p;peii<Pa ~ w < 0, yron Me)l(,!l;y P H Sa ocrphiH H, cnep;oBaremHo, Sa> ( lh Vl 2 - "f2

) 112 rrpH

nro6oii cKopocrH p;peii<Pa. Crporo roBOpH, rrpH orpHn;aremHOM p;peii<Pe ycno­BHIO (29.16) MO)I(HO Y.r:IOBITeTBOpHTb H B TOM cny11ae, Korp;a BeKIOp Sa Ha­~OHeH K h V rrop; yrnoM (1r - arctg"//h V) (rrpHMaH 2) . ,llrrH roro 11ro6bi TaKOe peiiieHHe cyiUeCTBOBaiTO , He06XO,[IHMO, IIT06bi p;peif<P 6bm ,[IOCTaTOIIHO BeiTHK . 0,[1HaKO JTO ,!J;OIIOITHHTeiTbHOe peiUeHHe, KaK IIOKa3aHO HH)I(e, Heyc­TOHIIHBO .

29.2. OcHOBHbie ypaBHeHHJI S-reopHH c p;peA<PoM. ,llrrH Hccnep;oBaHHH rraKe­roB CB, BOJHHKarowHx rrpH p;peii<Pe c rrocroHHHOH cKopocThro, yp;o6uo rrepeiirH or rrepeMeHHOH f K rrepeMeHHOH w - reKyiiieMy JHalleHHIO paccrpOHKH llacro­Tbi w· = f + (Xf. 3ro aHaiTOfHIIHO rrepexop;y B ,[IBH)I(YlUYIOCH CO CKOpoCTbiQ (X

137

CHCTeMy OTcqeTa npH npeo6pa30BaHHH fanHJieHo flpH 3TOM ypaBHeHHH nepexonnT s cnenyroi.UHe:

(29o3)

(~a~ +rn)n(w, .Q)+ImPna(w, n)=rnn°,

(~ (~

~ + iw + rn)u(w, .Q) + iPn n(w, n) = 0, aw

~ -iw+rn)a"(w,n)-iPnn(w,n)=Oo aw

.(29019)

CTaUHOHapHbiH npeii<P naKeTa onHCbJBaeTcn peweHHHMH (29 019) , He 3aBHCH· I.UHMH HBHO OT BpeMeHHo 0HH AOJI)I(Hbl, oqeBHAHO, YAOBJieTBOpHTb rpaHHqHbJM YCJIOBHHM

(n(w,n) - n°], a(w,n), a"(w,n)~o npH w~±ooo (29020)

CTaUHoHapHbJe peweHHH OAHOpOAHOH CHCTeMbi (29 .19) npH n° = 0, I w I ~oo HMelOT CJie)l;yJOI.UHe aCHMllTOTHKH :

ImP n Own"' exp [(rw ± iw2 )/o:] w-3 exp( -2rw/o:) 0 (29021)

HH OAHO H3 3THX peweHHH He CTpeMHTcn K Hynro npH w/o: ~-oo, noJTOMY HHKaKaH JIHHeHHaH KOM6HHaUHH perneHHH OAHOp0AHOH CHCTeMbl He YAOBJieT­BOpHeT rpaHHqHbJM YCJIOBHHM (29020) o <l>H3HqeCKHH CMbJCJI 3TOrO B TOM, qTo B OTCYTCTBHe TellJIOBOrO rnyMa napaMeTpHqecKOe B036y)K)l;eHHe BOJIH npH ApeH<t>e HeB03MO)KHO o Il03TOMY AJIH Ollpe)l;eJieHHH <t>opMbl naKeTa 'fiCa llpH ApeH<!>e HaM He06XOAHMO HaRTH perynnpHoe CTaUHOHapHOe pellieHHe HeOAHO· pOAHOH cHcTeMbi (29 .19), ynosneTsopnromee rpaHHqHbJM ycJIOBHHM (29 020) 0 Tpe6osaHHe perynnpHOCTH peweHHH 3KBHBaneHTHO ycnosHro BHernHeH ycTOH· qHBOCTH 6e3 Apeii<Pao

YpasHeHHH (29 .19), s crauHoHapHOM npeH:<t>e, MO)KHO npHsecTH K 6e3pa3: MepHOMY BH)l;y, BBOAH HOBbie nepeMeHHbJe:

Xn = wfrn, 71n = o:/2r~, fin= Pn/rn,

u(x)=ReiT"a(x), V(x)=-ImiT*a(x),

m (x) = n (x) - n° 0

(29022)

3aMeTHM, qTo yroJI .Q B ypasHeHHHX (29.19) HBJIHeTCH napaMeTpOM, ll03TOMY npH HCCJie)l;OBaHHH <t>opMbl naKeTa llO C06CTBeHHbJM qacTOTaM llpH <!>HKCHpoBaH· HOM n MO)KHO onycKaTb 3aBHCHMOCTb oT noro yrnao Yno6Ho TaK)Ke npoH3· BeCTH npeo6pa30BaHHe <l>ypbe, ll03BOJIHJOIUee llOHH3HTb llOpHAOK AH<t><t>epeH· uHanbHoro onepaTopa, H cnenaTb eme OAHY 3aMeHy nepeMeHHoii::

f(x) = ffkexp(ikx) dk,

z = (I + i77k2 )/i71o

B pe3yJILTaTe H3 (29.19) B cTaUHoHapHOM cnyqae nonyqHM

a2 v -

2 - (1/4 - i iiii 2 / 477z) V= i I Ili 2 n°b (z + i/2)/2TJZ o az

(29023)

(29 024)

PeweHHH noro ypasHeHHH Bbipa)KaJOTCH qepe3 <PYHKUHH YHTTeKepa c HHAeKCOM 138

(29.25)

3To peweuue ypaBHeHH.II (29.24) 6bmo nonyqeuo E.B. fiOAHBHJIOBbiM H B.E. qepenauoBbiM [198).

29.3. PeweHHJI yp8BueHHA S-TeOpHH c npeA~oM. 0TMeTHM, 'ITO no nopora napaMeTpH'IeCKOH HeyCTOH'IHBOCTH H DpH MaJIOH CKOpoCTH p;peii~a HMeeT MeCTO o6bi'IHbUf .u;onoporOBbiH nonorpeB cucTeMbi BOJIH. B cnyqae a.u;ua6aiH'IecKoro )J;peii<Pa, KOr,IJ;a BblllOJIH.IIeTC.II ycJIOBHe (29.14),

n,o=[7Tifln l2 n°/(1-lfln l2)

112 exp[2(1fln 12 -1)312 /317]. (29.26)

l1MeHHO JTOT CJiy'laH Mbl Ka'leCTBeHHO o6cy )J;HJIH B ll. 29 J, H DpHBe)J;eHHbiH J,u;ecb OTBeT (29 .26) KaqeciBeuuo coBnanaeT c nonyqeuuoii pauee oueuKoii (29.15). flpH ueanHa6aTHtiecKOM npeii<Pe, Korna 117 n,1 > (I n 12

- 1} 312,

'IHCJIO fiCB orpaHH'IHBaeiC.II ua ypoBue, 6JIH3KOM K TeUJIOBOMY :

(29.27)

flpH 310M C p0CIOM CKOpOCIH npeii<Pa HHTerpaJibHa.ll BeJIH'IHHa Re fl •a nanaeT: -1111 -I. l1MeeTC.II eme O,[UIH Ba)l(}fbiH CJiytiaH, KOrp;a DpH 6biCTpoM Apeii<Pe ( 17 ~ 1, I n I ) ~HCJIO fiCB BeJIHKO. 3IO cnytiaH OOJiblliOH ua,u;KpHTH'IHOCTH lfll 2 ~17>l, lfll.flpH JTOM

nn=4khrn(l171 112 /un lfln l 2 )exp(7Tifln 12 /217), (29.28)

flepeiineM Ienepb K Bonpocy o <PopMe naKeia fiCB. B n. 29.1 ua KatiecTBeu­HOM ypoBHe Mbl llOKaJaJIH, 'ITO npH BblllOJIHeHHH yCJIOBH.II a,[Uia6aiH'IHOCTH p;peii<Pa (29 .14) lliHpHHa naKeia fiCB (29 .13) MHoro MeHbllie, 'leM ero c,IJ;BHr OJ pe30HaHca flwk (29.12) . TO'IHbie Bbipa)l(eHH.II AJI.II 8 H flwk COBUa)J;aJOI c JTHMH oueuKaMH c TO'IHOCTbJO p;o nocTO.IIHHOH. B cnyqae 6oJihlliOH cKopocTH npeii<Pa H 6omwoii uanKpHTH'IHOCTH MO)I(HO nony'IHib

llwk =-(I P 12 /2 a)In [(a/r)2( I P 1

2 + 2are - c)- 1],

82 = 1 P 12

{ 1·tlj4 In [(a/r)2( I P l2 +2are - cr1

]}. (29.29)

3necb C = 0,577 ... - nocTO.IIHHa.ll 3iinepa. BHAHO, 'ITO IaK )l(e, KaK H npu a,[Uia-6aTH'IeCKOM .u;peii<Pe, BblllOJIH.IIeTC.II YCJIOBHe I /l Wk I ~ r k . CoficTBeHHO fOBO· p.11, ua JTOM JaKautiHBaeTc.ll cneuH<PH'IecKa.ll 'laclb S-TeopHH, CB.IIJauua.ll c p;peii­<PoM. ,Uanee cnenyeT o6bi'IHa.ll npouep;ypa caMocornacoBaHH.II tiHCJia fiCB c aMUJIHIYAOH nepeuopMHpoBaHHOH uaKa'IKH P.

29.4. OnpeneneHHe 38BHCHMOCTH 'IHCJia liCB oT aMUJIHTynhl uaKaqKH -npoue.u;ypa caMocornacoBaHHJI. flpome Bcero no cp;enalb B c<PepH'!eCKH·CHM· MeTpH'IHOM cnyqae (uanpHMep, AJI.II A<l>M nrr) , KOrp;a AJI.II Bbi'IHCJieHH.II llOJI· uoro tiHcna TICB N p;ocTaiO'IHO yMHO)I(HTh (29.26) H (29 .28) ua 47T . 8 aK-

139

CHaJibHO-CHMMeTplftiHOM CJiy'llle, xapaKTepHOM ):VUI Ky6HqecKOfO <l>M )I{J1f, Ha KOTOpoM 6&m 1IpOJJ;enaH JKCIIepHMeHT C JJ;peH<J>OM, TaK)I(e ,ll;OCTaTOqHO npOCTO B&monHReTcH HHTerpHpoBaHHe (29.26) H (29.28) no yrnaM. B peJyn&TaTe npH l11o I ~ (I ll 0 12

- 1) 312 nonyqHM Bbipa)l(eHHe

N = Nr(Tr11o/{3) 112 [I llo 12 I ( lllo 12 -1)3 14 exp [ 3( I llo 12 ...:.1)3 12 I 111o I],

P. = (I llo 12 - 1 )112 sign 11 , (29.30)

3JJ;ec& Nr - qHCno TennoB&IX CB B cnoe lliHpHHOH t::.wk = Tr/'k B6nHJH peJo:

HaHCHOH noBepXHOCTH. EcnH I ll 0 12 ~ l11o I ~ 1, I ll 0 I, TO

N=2Nr111o IIllo l(rr{3r 1'

2 exp(rrlllo l2 /2111o I) ,

p. = (2 I llo I I 11o) ln ( l11o I I I llo I ). (29 .31)

8 JTHX Bbipa)l(eHWIX ll o H 11o - 3HaqeHHH ll n H 11 n Ha pe30HaHCHOH IIOBepx­HOCT.H , a KOJ<J><J>mXHeHT {3 ""2 xapaKTepHJyeT yrnosyJO JaBHCHMOCTb ll n :

I lln 12 = I llo 12 - {3 cos2 fJ . (29.32)

BHJJ;HO, qT0 B cnyqa.R.X pa3nH~OH CHMMeTpHH - c<J>epHqecKOH H aKCHaJibHOH -COOTBeTCTBY!Oll.IHe <J>opMyn&I OmHqaJOTC.R. JJ;pyr OT JJ;pyra TOnbKO npe,ll;JKCIIOHeH­IlHaJlbHbiM <J>aKTOpoM. llg H ,ll;aJibHeHilleM IIpeo6pa30BaHHH JTO IIpHBeJJ;eT TOnbKO K HecymecTBeHHOMY omHqHJO norapH<l>MHqecKHx MHO)I(HTeneii . llonoMy B ,ll;aJlbHeHilleM Mbl orpaHHqHMC.R. cnyqaeM aKCHaJlbHOH CHMMeTpHH, ,ll;n.R. KOTOpo­fO B.B .3ayTKHHbiM H AP· [90] nocTasneH JKcnepHMeHT.

CneJJ;yJOll.IHM nanoM caMocornacoBaHHH HBnHeTcH Hcnon&JOBaHHe CBHJH Me)I(JJ;y PH h V. 113 <J>opMyn&r (18.3) MO)I(HO rronyqHTb

I hV 12 =I P l2 +2p.'Y IS IN+ (SN)2 'Y 2 (1 +p.2 )1 IP 12,

(29.33)

PellleHHe noro ypaBHeHHH HMeeT BHJJ.

(29.34)

EcnH B&monmreTc.R. yCJioBHe anHa6aTHqHOCTH npeH:<J>a (29.14), TO qHcno llCB N BenHKO no cpaBHeHHJO c TennoB&IM lliYMOM, a p.=a(IP I2 -'Y2 )JI2IIa 1/' . Torna H3 (29.34) cneJJ;yeT

(29 .35)

HaH6onee npocT !VIH aHaJIHJa cnyqai1: Sa > 0. llpH JTOM B (29.35) HMeeT CM&Icn nHillb pellleHHe N+. llpH MaJIOM ypoBHe Tennosoro lliYMa

e=ISINrh~l.

140

(29 .36)

PH c. 29.2. 3aBHCHMocn. 'IHcrra TICB N OT aMlllJHTYIJ.hl HaKa'IKH npH pa3JIH'IHbiX cKopocnix co6CTBeHHoro ,D,peli<J>a: 1 · -1111 = o; 2- 1111 = 10-• ;3- 1111 = 3 ·Io-•; 4 - 0,1. Sawktat > 0 (a); Sawktat < 0 (6) .

113 (29.35) H (29.30) crreiJ:YeT, 'ITO

ISIN+ = /'(p-1)112-

- (a/4'}' 2 ){1n [(p - 1)/p]} 11 3•

(29.37)

3gecb nepsoe crraraervioe cooTBeT­

cTsyeT H3BeCTHOM)' pe3yJI&TaTy S -TeO­

pHH. C p0CTOM CKOp0CTH j:(peii<Pa 'IHC­

JIO IIB nagaeT. IlpH O'leHb Marroii

Ha)J;KpHTHl!HOCTH, KOr)J;a p - 1 < rt, <PopM)'rra (29.37) HenpHMeHHMa, TaK

KaK HapyrnaeTCH ycrrosHe (29.14). B 3TOM crryqae 'llicrro napaMeTpHl!ec-

SNjy J

0 5

z

KHx sorrH HeserrHKO no cpasHeHmo c 'IHCJIOM TenrroBhiX. IlpH 6orr&rnoii HaAKpH­

THl!HOCTH npHMeHHMOCTb YCJIOBHH (29.14) TaK)!(e HapyrnaeTCH, H He06XO)J;HMO

Hcnom30BaTb <PopMyrrhr (29 .31) . II pH noM KaqecnieHHOe nosegeHHe 3aBHCH­

MOCJH NOT cKopocTH gpeii<Pa T/ coxpaHHeTCH (pHc. 29.2). IJpH OTpHIJ.aTeJlbHOH CK0p0CTH gpeii<Pa CHTyauM cymecTBeHHO CJIO)!(Hee,

llOCKOJlbKY MOryT HMeTb CMblCJl o6a perneHHH (29.35). IJpH He CJlHlliKOM

Marr&rx ypoBHHX wyMa H He CJIHlliKOM 6orr&lliHX cKopocTHX gpeii<Pa cymecT­

syeT e)J;HHCTBeHHOe perneHHe, COOTBeTCTBYIOil.lee KOpHIO N+ B (29.35) H

onHc&rsaeMoe <PopMyrroii (29.37) . IlpH Sa -+ 0 3TO Bbrpa)l(eHHe, eCTecTBeHHO,

cosna.o;aeT c pe3yJI&TaTOM S-TeopHH. EcrrH yposeHb rnyMa Marr, a cKopocTb

gpeii<Pa AOCTaTOl!HO BeJIHKa, TO HMeiOTCH TpH perneHM }J)lH 'IHCJia BOJIH N H

nepeHOpMHpOBaHHOH HaKa'IKH I rr I. HaHMeH&Wee H3 HHX COOTBeTCTByeT

CHTyaUHH, KOrga ypoBeHb Wy_Ma HaCTOJlbKO MaJI, a CKOp0CTb Apeii<Pa BeJIHKa,

'ITO aMnrrHTyAa TICB 3a speMH npoxo)I(IJ:eHHH o6rracTH HeycTOH'IHBOCTH He

ycnesaeT BbrpaCTH AO 3Hal!HTeiTbHOH BeJIH'IHHbl. ,llpyrHe )J;Ba perneHHH COOTBeT­

CTBYIOT 6orr&WHM N, npoMe)!(yT<)l!Hoe 3Hal!eHHe HeycTOH'IHBO. Ha pHc . 29.2, 6 npHBe)J;eH Ha6op 3aBHCHMOCTeH 'IHCJia BOJIH NOT MOil.IHOCTH HaKal!KH p, Bbl'IHC­

JieHHbiX }J)lH pa3J1Hl!HbiX 3Hal!eHHH <;KOpOCTH Apeii<Pa Ci. H aMnJIHTYAbl TenJIOBOrO

wyMa E = 10-3 . BHAHO, 'ITO HMeeTcH rHcTepe3HC 3aBHCHMOCTH 'IHCJia IICB OT

MOil.IHOCTH HaKa'IKH npH <l>HKCHpOBaHHOH CK0p0CTH gpeii<Pa. IJpH MOil.IHOCTH

HaKa'IKH, MeHbweii HeKoTopoii, KpHTH'IecKHii ypoBeHb IICB 6JIH30K K Tenrro­

BOMy. IlpH p > p 1 , rge

P 1 ~ 1 + [ 3/4 ( I a I h 2) ln ( I a I I 41' 2

E )]2'

3 , (29 .38)

MOryT cymecTBOBaTb ABa ycTOH.'IHBbiX cocTOHHHH IICB. C pocTOM MOil.IHOCTH

HaKal!KH npH p > p2 , rge

(29.39)

141

ocTaercH eromcrBeHHOe cocrOHHHe, Koropoe npH a --+- 0 nepexoro~r B craHA<~pr­HOe perneHHe S-reopHH. AMIUIHTYA<l Jroro perneHHH OrrHJKa K HJBecrHOMY B bl pa)f(eHHIO

iSiN = "f(p-1) 112. (29.40)

Cne;:zyer non11epKHyTb, 11ro 11ucno IIB Ha perneHHH (29.37), coorBercrnywmeM S-reopHH B orcyrcrBHe npeii<Pa, npH YBeflHIIeHHH cKopocrH npeii<Pa H npH <i>HKCHpOB3HHOH MOIUHOCTH H3K3'1KH HeorpaHHIIeHHO pacrer.

B JaKnJOIIeHHe JToro naparpa<Pa npHBeneM Bbipa)f(eHHH MH 11acror&I OnHO· p<>nHbiX KOnneKTHBHbiX KOne6aHHH B cny11ae a,JJ,Ha6aTHIIeCKOf0 npeii<Pa:

n± = -h ± [ 4(SN + ~wk)(2 T + S) N- 'Y 2 ]

112. (29.41)

3nec& ~ wk - CnBHr naKera IICB, JanaBaeM&IH (29 .12) . Hy)f(Ho CKaJar&, liTO Bbipa)f(eHHH (29.41) nonyqeH&I E.B.IlonHBHnOBbiM, B.E.qepenaHOBbiM [198] B pe3yn&rare rpOM03nKOfO aKKypaiHOfO pac11era, npH 3TOM OHH COBn3Al!IOT C Bbipa)f(eHH.SIMH, KOIOpbie MO)f(HO nony'IHTb H3 npOCTbiX ypaBH~HHH S ·TeOpHH 6e3 npeii<Pa nfl.R yJKoro naKera IICB, noKanH30BaHHoro Ha noBepXHocrH

wk = ~wk + wp/2. KcrarH, ro )f(e caMoe MO)f(HO CKaJaTb orHocurenbHO Bbipa­

)f(eHHH nfl.R o6mero IIHcna IICB N H l: . TaKHM o6paJOM, B rex cnyqaHx, Korna IIHcno IICB BenHKo B cpasHeHHH c rennoB&IM IIIyMOM, pom npeii<Pa B JHaiiH· renbHOH Mepe CBO,JJ,HTC.SI K MOrof<i>HKaUHH ycnOBH.SI BHelliHeH ycrOHIIHBOCTH:

BMecro 2 wk = Wp nony11aerc.R 2 wk =. 2 ~wk + Wp, rne noJio)f(eHHe JieHrpa

naKera ~wk 3anaerc.R Bbipa)f(eHHHMH (29 .12) HnH (29 .29) .

;: 6 § 30. IIapaMeTpH11ecKoe B03 )')KJJ.eHHe CDHROBidX B011H HeKorepeH1110A: HaKaqKoA: npH UJYMOBOA MO,!zy'JISIIOIH HX 118CfOT

Xoporno H3BeCTHO, liTO K napaMerpHIIecKoH neycroHIIHBOCTH &onH npHBo,JJ,Hr He TOflbKO MOHOXpoMaTHIIeCKa.R, HO H IIIyMOBa.R H3K3'1Ka. IJopor 3TOH HeyCTOH· IIHBOCTH 3aBHCHT OT V - 113CTOTHOH lliHpHHbl H3K311KH, IJo nopHnKY BenHIIHHbl

(30.1)

CymecrByer OlliHOOIIHOe MHeHHe, 11ro npH HeKorepeHrHoH HaKa'IKe <PaJOBbiH MexaHH3M orp'aHHIIeHHH aMIUIHryn&I IICB He pa6oraer. OnHaKo nerKo YBH· neTb, 11ro aMnnHTYAll IICB H <PaJOB&Ie coorHorneHHH, Bbi'IHcneHH&Ie no S -reopHH, He 3aBHCHT or lliHpHHbi cneKrpa HaKaiiKH v npH v ~ 'Y. IloJTOMY cne;:zyer O)f(H· naTb, liTO npH V - 'Y 3TH BenHIIHHbl KallecrBeHHO He H3MeH.SITC.SI, I.e . aHOManb· Hbie KoppenHropbi ak B napax IIB He Man&I no cpasHeHHJO c HOpManbHbi·

MH nk. 3Ha'IHT, H npH V > "f aHOManbHbie KOppen.RrOpbi Be,rzyr ce6.R KaK HeKO· ropa.R <i>Y"KIIHH OT V /'Y, KOIOpa.SI crpeMHTC.SI K nOCTO.jiHHOH, KOrna V /'Y --+- 0, H ecnH 3Ta <PY"KUHH y6&IBaer nocrarOIIHO MenneHHO c pocroM v /'Y, ro cymecr­Byer raKa.R o6rracrb v > 'Y, B Koropoi% caMocornacoBaHHOe B3aHMoneiicrBHe HBfl.SieTC.SI fnaBHbiM.

Oco6eHHOCTH HenHHeHHoro noBeneHHH CB npH IIIyMOBOH HaKaiiKe H3ylla­nHCb reoperHIIeCKH B.E.qepenaHOB&IM [227] H JKcnepHMeHranbHO B.B.3ayrKH· HbiM H np. (90] . flo HX pe3yn&raraM H HaTIHCaH 3TOT naparpa<P.

30.1. YpaBHeHHH S-TeopHH npH UJYMOBoA: HaK&qKe. EyneM ciiHrarb, 11ro·

HaKaiiKa h(t) ecrb h(t) = h(t)exp(-iwpt), rne h(t) - cnyllaHHa.R <PY"KIIHH 142

c UIHpHHOH cneKTpa 11 TaKOH, 'ITO wP ~ 11 ~"'f . YpaBHeHHH S-TeopHH ,[J,]lH MeA­

rreHHbiX aMnJIHTYA llCB B 3TOM crryqae rrerKO norrylJHTb H3 ( 18.11) , (18.12) 3aMeHOH h ~ h ( t) . J.h 3THX ypaBHeHHH, B lJaCTHOCTH, CJieAYeT

cat +2n) lak\ 2 +2Im[Pt(t)ak(t)a_k(t)] = 0,

\ak(t)\ 2 =

= -2 Im f exp ( -2 'Y kT )Pic(t - r )ak(t - r )a_ k(t - r)dr . 0

(30.2)

(30.3)

fiOCKOJlbKY lJaCTOTa lPJIYKTyauHH nOAbiHTerpaJibHOJ'O Bbipa)l(eHH.ll BeJIHKa no

cpaBHeHHIO C 'Yk. TOnpHHHTerpHpOBaHHHnO T 3aBpeM.H (2'Yk)-l npoHCXOAHT

ycpeAHeHHe . 11MeHHO:

(30.4)

TaKHM 06pa30M, lJHCJIO BOJIH nk =I ak 12 ('lepTa 03HalJaeT ycpeAHeHHe no CJIY·

lJaHHbiM <Pa3aM nap llCB) , a BMecTe c HHM H nepeHopMHpoBaHHa.H qacTOTa wk He <PrryKTyHpyiOT npH UIYMOBOH HaKalJKe. ll,rr.H BeJIHlJHHbi ak = ak ak aHarrorHlJ·

HO MO)I(HO n011YlJHTb

ak(t) = -2ink f Pk(t - r)exp [- 2('Yk + iwk)r]dr , 0

Im<Picak > = -2rrnkP2 (k , 2wk),

+oo

Pk(t) = f P(k, w) exp ( - iwt)dw ,

• , 2 , <P(k,w),P (k,w )> = P (k,w)o(w - w ).

(30.5)

(30.6)

(30.7)

(30.8)

llOACTaBJI.H.H (30.6) B (30.4), norryqHM nnk = 2rrnkP 2 (k, 2 Wk) . CTaUHO·

HapHoe peUieHHe nk * 0 cymecTByeT npH

(30.9) .

EcnH ,[J,]l.H KaKoro-nH6o k BbmonH.HeTC.H HepaBeHCTBO

TO ,[J,]l.ll 3TOrO k npOHCXOAHT HapaCTaHHe aMnJIHTYAbi fiB , 'ITO npoTHBOpe'IHT

npeA"OJIO)I(eHHIO o cymecTBOBaHHH CTaUHOHapHoro cocTO.HHH.H. TaKHM o6pa30M,

,[J,]l.H Bcex k AOJI)I(HO 6biTb BblnonHeHO ycnoBHe BHeUIHeii ycTOH'IHBOCTH:

(30.10) 143

npH'IeM Ilk OTJlH'IHO <;lT HYJI.R TOJlbKO B crryqae 3H8K8 paBeHCTBa B (3.2). 3TO yc­

JIOBHe aHarrorH'IHO ycrroBHJO BHeUIHeH ycTOH'IHBOCTH, BOJHHK8JOllleMy npH

MOHoxpoMarH'IeCKOH HaKa'IKe I P k I ,;;;;; "f k. 113 o6bi'IHOro Bblpa)l(eHH.R WI.R nepe­

HOpMHpoBKH H8K8'1KH (HarrpHMep, (18.3)) H ypaBHeHH.R (30.5) 110JIY'IHM

[ S(k k 1)P(k k' w)n(k )

h~ vk v;, =P2 (k, k', w)-2 J dk1 ' ' ' 1

+ w-2w(kt)+2i"!(k1)

S(k', kt)P(k,k1, w)n(k1)] + + w-2w(kt)-2i"f(kd

(30.11)

S(k, kt)S.(k' , kt)P 2 (k1, k2, w) + J d k

1 dkz n (k t)n ( k2) __ :...__....:.:.__:..._____:..:.....___:__ _ __:___.:.....__

(w-2w (k1 )+2i"f(k1))(w -2w(k1) - 2i'Y(kz)

Pewa.11 JTO ypaBHeHHe COBMeCTHO c ycrroBHeM BHewHelf ycTOH'IHBOCTH (30.10) , MO)I(HO UOJiy'!HTb pacnpe.o:erreHHe riB Ilk .

KaK HJBecTHO, npH napaMeTpH'IeCKOM BOJ6y)l(lleHHH BOJIH KorepeHTHOH

H8K8'1KOH Ilk OTJIH'IHO OT HyJI.R B MaJIOH o6JiacTH k-npocTp8HCTBa. B /laJibHeH­

weM Mbl yBH/lHM, 'ITO npH I.J ~ kv (v - rpynrroBa.R CKOpoCTb riB) OfiJiaCTb,

38H.RT8.R riB, TO)I(e MaJia. 3TO 1103BOJI.ReT H8M C'IHT8Tb Bee KOJ<l><l>HUHeHTbl

B (30.1-1) He 38BHC.RlllHMH OT k . B p.Rile B8)1(Hb!X crryqaeB, H811pHMep npH c<t>epH­

'IeCKOH H aKCHaJibHOH CHMMeTpHH pacrrpe.o:erreHH.R riB, MO)I(HO H368BHTbC.R

B ypaBHeHHH OT yrJIOBOH 38BHCHMOCTH M8TpH'1HbiX JJieMeHTOB. 3TO 1103BOJI.ReT

ynpocTHTb ypaBHeHHe (30.11), rrepeXO/l.R oT rrepeMeHHOH k K nepeMeHHOH Wk : /~

n(wk)dwk - sf - +

wk- w

(30.12)

30.2. Pacnpe.o:erreaue UCB Ja noporoM npu wyMoooA aaKa'IKe. B.o. qepena­

HOB npoaHaJIH3HpoBarr (22 7) norry'leHHbie HM ypaBHeHH.R (30.12) Wl.R cne­

UHaJibHOrO BH/l3 CITe-KTpaJibHOH UJIOTHOCTH WyMOBOH H8K8'1KH

B o5rracTH MaJibiX HallKpHTH'IHOcTeH (p - 1 < 1) OH norry'IHJI

37TSnw = (p - l)(v'3w+p)3 12 (3p- v'3w) 112 fv 2,

J.l = (p- 1)112 v signS.

(30.13)

(30.14}

Crre.o:yeT OTMeTHTb, 'ITO :no peweHHe OTJIH'IHO oT Hyrrn: B HeCHMMeTpH'IHOM

HHTepBarre ( - J.l I v'3, v'3 J.l) H He HBJI.IleTC.Il cHMMeTpH'IHbiM; HecMOTp.ll - Ha

cHMMeTpH'IHblli npo<l>HJib HaKa'IKH (30.13). B rrpe.o:errbHOM crryqae 60JibWHX

144

HaJlKpHTHIJHOCTeH H3 (30.12) H (30.1 0) CJienyeT

37TSnw = (p - I)112 (3 - 4w 2 1v2)

3 12. (30.15)

3To perneHHe OTJIHIJHO OT Hynn B CHMMeTpHttHOH o6nacTH I w I< w 1 = v'3 v 12, 6onee YJKOH, lJeM o6nacTb HeycTOHIJHBOCTH I w I< v, B KOTopoii h~w > 0. TaKoe cyll<eHHe xapaKTepHo JlTIH S-TeOpHH. EnaronapH 3TOMy noseneHHe h~w

npH I w I> w 1 HecymecTBeHHO JlTIH Harneii Jana'IH H perneHHe (30.12) KalJecT­

seHHO cnpaBeJlTIHBO npH Jl!OOOM BHJle cneKTpaJJbHOH llJJOTHOCTH HaKalJKH.

11HTerpHpyH (30.14) H (30.15) no W , JJerKO noJJylJHTb JaBHCHMOCTb llOJJHO­ro lJHcJJa TIC8 OT HaJlKpHrnttHOCTH:

N=4v(p - I)312 I91TyJ lSI, p-I< 1, (30.16)

p ~ I. (30.17)

TipH v - 'Y Bbipall<eHHe (30.17) cornacyeTCH c <J>opMynoii (19.1) JlTIH CJJyqaH

KOrepeHTHOH HaKalJKH. liTO KacaeTC.II COOTBeTCTBH.II <J>opMYJJ (30.15) . H ( 19 .J) Jl)lJI MaJJbiX HaJlKpHTHIJHOCTeH, TO npH V - 'Y cnpaBeJlTIHBa nepBaH H3 HHX, TaK KaK B 3TOM CJJylJae HeJJbJJI rOBOpHTb 0 ll<eCTKOH KOppeJJHUHH <J>a3 BOJJH

B napax. TipH JlaJJ&HeiirneM yMeH&illeHHH v tUieH Re ( Pic ak } B (30.11) 6yneT

MaJJ no napaMeTpy (vI 'Y) 2 , H npH MaJJbiX v CJJenyeT y'IHTbiBaTb KsanpaTHIJHbrH no a tUieH .

OnpenenHM Terrep& npenen npHMeHHMOCTH npH6JJHll<eHHH caMocornacosaH-

Horo llOJJH npH napaMeTpHl!eCKOM B036Yli<JleHHH BOJJH lliYMOBOH HaKalJKOH.

JlnH JTOrO CpaBHHM tUieH lm(Pkak) B ypaBHeHHH JlJJH nk (30.2) C onymeH­

HblMH B HeM tUieHaMH KHHeTHIJeCKOrO ypaBHeHHH, HMeiOUlHMH nopHJlOK BeJJHIJH­

Hbl ( SN) 2nk I (kv). 8 peJyJlbTaTe nonyl!HM, 'ITO Ham nonxon cnpaBeJlTIHB npH

v 2 (p- 1) < 'Ykv. (30.18)

8 TBepJlbiX Tenax kvh ::::::: 104 + 106, 'ITO o6ecnelJHBaeT nocraTOIJHO lliHpoKHe

paMKH npHMeHHMOCTH JlJl.ll npHMHll<eHHH S -TeOpHH. 0TMeTHM TaKll<e, IJTO npoTHBononOli<HbrH npenen&HbiH CJJylJaH, Korna pacceHHHe TI8 npesocxOAHT HX CaMOCOrnaCOBaHHOe BJaHMOi:leHCTBHe, H3YIJa1lC.II 11.E.DeBHHCOHOM [141].

TipHseneM Tenep& oueHKY JlTIH yrnosoro paJMepa BOJ6yll<naeM&IX naKeTOB. KaK H B CJJyqae MOHoxpoMaTHIJeCKOH HaKalJKH B npHMHll<eHHH S-TeopHH, npH rnyMOBOH HaKalJKe yrnoB&Ie paJMep&I naKeTOB TI8 paBHbi Hynw. 3To nerKo ycraHOBHTb Ha npHMepe npocTeiirneii MOJleJJH

Vk = Vf(8), S(k,k') = Sf(8)[*(81), 'Yk = 'Y,

KalJecTB'eHHO cnpaBeJlTIHBOH B <l>M. TipH JTOM P 2 (k, k1

' w) = P~f(8)[*(8 1 ), H B (30 .11) ynaeTcH HCKJJIOIJHTb JaBHCHMOCTb OT yrnos, TI8 cocpenoTOIJeH&I

B yrne 8 0 , rne I f ( 8 0 ) I MaKCHMaJJ&HO . Moli<Ho noKaJaTb, IJTO cna6oe pac­

ceHHHe TI8 npHBOAHT K ylliHpeHHIO naKeTa Ha yron A8 :

A8 "'=' (SN/kv)::::::: v(p-I)112 Ikv. {30.19)

8 JaKJJIOIJeHHe JTOrO naparpa<J>a OTMeTHM, IJTO paJBHTaJI B HeM TeOpHH HaXO­

Af!TC.II B XOporneM COOTBeTCTBHH CO cneUHaJJbHO nocTaBJJeHHbiM JKcnepHMeHTOM

8.8 .3ayTKHHa, E.l1.0pna H 8 .E . lJepenaHosa [90]. 3To noJBOJJHeT ClJHTaT&, IJTO

aMllJJHTyJla TIC8 B yCJJOBH;HX rnyMosoro BOJ6y)I(JleHHH orpaHHIJHBaeTCH, KaK H npH KorepeHTHOM BOJ6yli<JleHHH, <J>aJOBbiM MexaHHJMOM S-TeopHH. 10. B.c. n~,sos 145

§ 31. HeycroA'IHBble KOJUieK'DIBHI>Ie Kone68HHJI­

llBTOKOne68HHJI H3MarJIKqeHHOC11t

Xopowo H3BeCTHo, 'ITO npH napaMeTpH'IeCKoif reHepaUHH CB cTaUHoHapH&rif pe)!(HM '!aCTO He ycraHaBnHBaeTCll H HaMarHH'IeHHOCTb COBepwaeT CJlO)f(Hbie aBTOKone6aHHH - AK - BOKpyr HeKoToporo cpe,llHero 3Ha'leHHH. OcHOBH&Ie

JKcnepHMeHTan&H&Ie <l>aKT&I, oTHocHWHecH K AK H nony'leHH&Ie Will B&ICOKo­Ka'leCT:BeHH&IX KpHCTannoB )J(J1f npH napannen&HOH HaKa'IKe, 3aKniO'IaiOTCll

B cnenyN>weM [ I7] : I. qaCTOTbi AK HaMarHH'IeHHOCTH ne)!QlT B HHTepBane OT I 04 no - I 06 ru

(B 3aBHCHMOCTH OT MOWHOCTH HaKa'IKH H BenH'IHHbl llOCTOliHHOrO MarHHTHoro llOnll). IlpH He6onbWOM npeBbiWeHHH nopora cneKTp AK COCTOHT H3 OllHOH nHHHH, C poCTOM ypoBHll MOWHOCTH npoHCXO.IlHT yBenH'IeHHe 'IHCJla nHHHH H CMeWeHHe HX B CTOpoHy 6onee BbiCOKHX 'laCTOT. IlpH 6onbWHX npeBbiWeHHl'IX

Han noporoM cneKTp HMeeT wyMoBoif xapaKTep. 2. Ilopor AK o6bi'IHO npeBbiWaeT nopor napaMeTpH'IecKoro B036y)f(neHHH

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npH BBeneHHH B KpHcTann HeonHoponHocTeif. 3. Ha6nronaeTcH rnraHTCKaH KpHcTannorpa<l>H'IecKaH aHH30TponHH CBOHCTB

AK, 3Ha'IHTenbHo npeBbiWarowaH aHH30TponHro cneKTpa CB. TaK, HHTeHCHB­

HOCTb AK B )J{l1f npH OpHeHTaUHH HaMarHH'IeHHOCTH Bnon& oCH [I II] npeBbi­waeT HHTeHCHBHOCTb 3THX KOne6aHHif BnOn& OCH [I00) llpH6nH3WienbHO B

IOO pa3. Bonpoc o <l>H3H'IecKo~ npHpone AK 6bm onaoif H3 ocHOBHbiX npo6neM B 3a­

naqe o napaMeTpH'IeCKOM Bo36y)f(!leHHH CB . . )J;u1 o6nHCHeHHH npHpollbi AK npHBneKanHc& pa3nH'IHbie rHUoTe3bi [I7]. Ceifqac, O.llHaKO, Bee OHH npencTaB­nHIOT Ton&Ko HCTOpH'IeCKHH HHTepec. B paMKax S -TeopHH AK HaxonHT ecTeCT­

BeHHoe 06'bliCHeHHe KaK pe3ynbTaT pa3BHTH.Il OllHC3HHOH B rn. 5 HeyCTOH'IHBOC· TH CTaUHoHapHpro COCTO.IIHH.Il OTHOCHTen&HO B036y)f(!leHHll KK. EcrrH XOT.Il 6bi WI.Il onHOH MOnbi (c HOMepoM m) BbillOnHeHbi ycnOBH.Il HeyCTOH'IHBOCTH Sm (2Tm + Sm) < 0, TO B paMKax S-TeopHH CHCTeMa IICB He HMeeT ycTOH'IH­BbiX CTaUHOHapHbiX COCTO.IIHHH. IlpH 3TOM MO)f(HO npenCTaBHTb UBe B03M0)1(­HOCTH - "Bb16poc" CHCTeMbl 3a npenenbl llpHMeHHMOCTH S -TeOpHH (npH 3TOM non)f(Ha CHn&HO B03paCTH aMnnHTYAA.P<;>36y)f(!leHHbiX BOnH) HnH ycraHOBneHHe Kone6aHHH BOKpyr CTaUHOHapHoro t0CTO.IIHH.Il. 3TH Kone6aHH.Il, ecrrH OHH pa3-BHBaiOTCll, H Ha6nronaroTcH Ha on&rre KaK aBTOKone6aHH.Il HaMarHH'IeHHoCTH. 113 o6WHX coo6pa)f(eHHif cnenyeT O)!(HnaT&, 'ITO AK MoryT HOCHTb KaK pery­

n.llpHbrn, KaK H xaoTH'IeCKHH xapaKTep, B nocnenaeM crryqae HX MO)f(HO TpaKTO­BaT& KaK "BTOpH'IHYfO" Typ6yneHTHOCTb C BpeMeHHbiM MaCWTaOOM, MHOrO

6on&WHM MacWTa6a "ocHOBHoif" Typ6yneHTHOCTH- nepHona CB.

KaK BH,llHO H3 <l>opMynbi (23.6), HeyCTOH'IHBOCTb CTaUHOHapHoro COCTO.IIHH.Il .IIBnlleTCll 'IHCTO anepHO.IlH'IeCKOH (Re Qm = 0) , ll03TOMY BTOpH'IHa.ll Typ6yneHT· HOCT& .IIBn.lleTc.ll CHn&HOH . 3To nenaeT KpaifHe 3aTpYnHHTen&HbiM aHanHTH'IecKoe

peweHHe Bonpoca o HenHHeifHoif cra.llHH pa3BHTHH KK H o xapaKTepe BropH'Ilioif

ryp6yneHTHOCTH, ecnH TaKOBall B03HHKaer. Il03TOMY npencraBmleTC.Il uene­coo6pa3HbiM 'IHCneHHOe MonenHpoBaHHe BTOpH'IHOH ryp6yneHTHOI:TH Hll 3BM .

IlpH noM pacqer pean&HOH CHTY3UHH, HanpHMep nn.11 )J(J1f, norpe6oBan 6bi or­poMHbiX 3arpar MaWHHHoro BpeMeHH H BpHn nH uenecoo6pa3eH. 3To 3acraan.11er

146

o6pamaTbC.JI K lffiCJieHHOMY 3KCilepHMeHry Ha ynpoweHHbiX MOIJ:eJI.IIX OCHOB·

Horo cocTo.IIHH.II. B pa6orax B.C.lliBoBa,CJI. Mywepa, C.C. CTapo6HH~ [152}, BJI. fpaHKHHa H np. [70} onHcaH lffiCJieHH&rn 3KcnepHMeHT no B036Y>KJleHHIO

AK Ha MonenH "nayx n}"'eii", B KoTopoii Clffi:Tanoc&, 'ITO CB cocpenoTo~eH&r npH ll:BYX cl>HKCHpDB3HHbiX yrrrax fJ 1 ,;, rr/2 fJ.2 = rr/4. 3Ha'lJeHH.II K03cl>cl>HUHeH­

TOB S0 H T0 Bbi6HpanHcb 6JIH3KHMH K Bbi'ffiCJieHH&IM ll:JI.II >KI1f npH opHeHTaUHH

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'ITO B TaKOH MonerrH ycraHaBJIHBaiOTC.II AK aMnnHTYIJ: H 4>a3 BOJIH Ha rryqax

P H c . 31.1. 3aBHCHMOCTL aMnnHTYJJ.bi nap

Ha nyqax OT BpeMeHH npH HeyCTOINHBOCTH

Hynesol! MOJJ.bl (N1 H N2 OTBe'laiOT fl}"'aM No c 8 = rr/2 H 8 = rr/4)

0

(pHc. 31.1). 3aBHCHMOCTb 'lJaCTOTbi 3THX AK OT ypoBH.II HaKa'lJKH"i<a'lJeCTBeHHo

corrracyeTC.II C aHaJIOrH'IHOH 3aBHCHMOCTbiO, 06bl'IHO Ha6JIIOII.lliOWeHC.JI B rra6o­

paTOpHOM 3KcnepHMeHTe. Ha MonenH nayx rry'leii 6bm, KpoMe Toro, nponerraH

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npH m #: 0. IlpH 3TOM H3}"'aJIC.JI BOllpoC 0 llOBeneHHH CYMMapHOH aMnnHTYIJ:bi

BOJIH Ha HeJIHHeHHOH CTaiJ:}m pa3BHTH.JI 3TOH HeyCTOH'ffiBOCTH, HHTepeCHbrH llO·

TOMy, 'ITO Ha JIHHeHHOH CTaJ],HH H3MeHeHH.JI cyMMapHOH aMIDIHTYil:bi He npOHC·

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(p- Pc) B6JIH3H npenemHoro UHKrra o6pa3yerc.11 y3KHH CJIOH, 3anonHeHHbiH

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nr,. (70] . .IJ.n.11 3TOro Ha OIJ:HH H TOT :>Ke 3KpaH 3anHCbffiaJIHCb 3aBHCHMOCTH

x '(t) ll:JI.II pa3Hbrx HMilYJibCOB HaKa'IKH Oll:HOH cepHH c cl>HKCHpoBaHHOH Han­

KpHTH'IHOCTbiO H npyrnMH yCJIOBH.IIMH 3KcnepHMeHTa. IlpH noM ll:JI.II pa3Hbrx

HMllYJibCOB H3·3a cl>rrYKTYaUHH CHCTeMa BOJIH Ka:>Kil:brH pa3 CTapTyeT (npH t = 0) c pa3JIH'IHbiMH Ha'lanbHbiMH yCJioB.IIMH. IlpH HanKpHTH'IHocT.IIX p = 2,5 nB

llOCJieiJ:OBaTeJibHbie KpHBbie X11(t) COBllaji,lliOT npyr C npyrOM. IJpH llpeBbiUie ­

HHH p > 2,5 nB ' Korna llO.JIBJI.JieTCJI pe3KOe yii.DipeHHe cl>Ypbe-cneKTpa AK '

KpHBbre x "(t) ll:JI.II pa3HbiX HMnyJibCOB Halffi:HaiOT pacxoll:HTbC.II. BHa'lane pacxo-

ll:HMOCTb 6rrH3KHX TpaeKTOpHH cKa3bmaeTc.ll rrHUib npH 6orrbii.DIX BpeMeHax.

10* 147

----' C_ v - hV=5y

10 20 Zyt p H C. 31. 2. 3aBHCHMOCTb CYMMapHolt aMnnHTYAbl nap Ha n}"'ax OT BpeMeHH npH HeyCTolt­

'IHBOCTH MOAb! m = 2

tJ 3aTeM riO Mepe ysenH'IeHHH Hall,KpHTH'IHOCTH "BpeMH pa36eraHH.!I" CTaHOBHTC.!I

cpasHHMhiM c nepHOUOM AK r R: (3 + 5) Ao H nocnenosaTenhHhie KpHBhie

x" (t) o6pa3yiOT lliHpoKyro nonocy.

Pe3IOMHpyH, MO)I(HO cKa3aTh, 'ITO '!HcneHHhrn 3KcnepHMeHT Ha MonenHx

noKa3hmaeT, 'ITO B paMKax ypasHeHHH S-TeopHH pa3BHTHe BHyTpeHHeii Heyc­

TOH'IHBOCTH ocHOBHoro cocroHHHH npHBOUHT K aBToKone6aHHHM. Csoiicrna

3THX AK - 3aBHCHMOCTh 'laCTOThl H cneKTpanbHOro COCTaBa OT MOWHOCTH Ha­

Ka'IKH - conocTaBHMhi co csoiicrsaMH peanbHhiX AK, Ha6nronaeMhiX B na6o­

paTopHOM 3KcnepHMeHTe. B 'laCTHOCTH, npH ManhiX Hall,KpHTH'IHOCTHX B na6o­

paTopHoM H 'IHcneHHOM 3KcnepHMeHTax AK HBnHIOTCH nepHO.llH'IeCKHMH .

06pa3UOM TaKoro UBH)I(eHHH B cpa30BOM npocTpaHCTBe HBnHeTCH ycTOH'IHBhiH

npenenhHhrn UHKn. B o6oHx THnax 3KcnepHMeHTOB nepexon K cTOxacTH'IeCKHM

AK c pocToM Ha.o;KpHTH'IHOCTH npoHCXO.llHT He c noMOWbiO no6asneHHH HOBhiX

THfiOB )J;BH)I(eHHH Ha HeCOH3MepHMhiX l!aCTOTaX, a COllpOBO)I()J;aeTC.!I ylliHpeHHeM

y)l(e cyweCTBYIOWHX CfieKTpanbHhiX nHHHH C KpaTHhiMH 'laCTOTaMH H CBH3aH

c noTepeii ycToH'IHBOCTH cpa30BhiX TpaeKTopHii , npHBoUHweii K HX pa36eraHHIO.

nanee, KaK B '!HcneHHOM, TaK H B na6opaTOpHOM 3KCnepHMeHTax pa3BHTHe AK He BnHHeT cywecTBeHHO Ha cpe.o;HHH ypoBeHb JlCB. liHcneHHhiH 3KCnepHMeHT

fiOKa3bmaeT, KpOMe TOrO, 'ITO ripH pa3BHTHH HeyCTOH'IHBOCTH Ha Hynesoji MOUe

npOHCXOU.!IT rny60KHe, a npH pa3BHTHH HeyCTOH'IHBOCTH Ha BhiCIIIHX MO)J;aX -

MenKHe KOne6aHHH CYMMapHOH aMfinHTYUbl BOTIH H BMeCTe C HeH H Ha6nro.o;ae­

MOH BenH'IHHhl x" 0 J1ro6orihiTHO, 'ITO B03HHKHOBeHHe AK B 'IHCneHHOM 3Kcrre­

pHMeHTe, KaK rrpaBHTIO, COrip0BO)I()J;an0Cb yMeHbiiieHHeM cpe.o;Hero 3Hal!eHHH

X' . 3Ta 3aKOHOMepHOCTb Ha6nro.o;aeTC.!I H B na6opaTOpHOM 3KCriepHMeHTe .

148

IlonY'IeHHbie Bhlrne pe3YJihtaTbi no3BomuoT cnenaTb npencKa3aHHe OTHOCH­

TeJihHO Toro, B KaKHX cnytta.11x AK llOJI)I(Hhi Ha6monaThC.II B peanhHhiX 3KcnepH­

MeHTax. ITpHBC)lCM 3HatteHH.II KoJ<P<l>HUHCHTOB s m H T m Jlfl.ll )KJ1T' Bbi'IHCJieH­

HbiC CJl. MyrnepoM, C.C. Crapo6HHUeM H B.C . JlbBOBhiM [152] Jlfl.ll rnrrntt­

HOH 3KcnepHMCHTaJibHOH CHTyaUHH: Nz = 1/3 (c<l>epa)' Wp = 9,4 rru,

k = 0 (H = H c)' WM = 49 rru, Wa = 0,23 rru (KOMHaTHa.ll TeMnepaTypa)'

ra6n. 31.1.

BH,!J,HO, 'ITO He TOJibKO 3Ha'ICHH.II, HO H 3HaKH K03<l><l>HUHCHTOB 3aBHC.IIT OT

opHeHTaUHH HaMarHH'ICHHOCTH. IIoncTaBJI.II.II Ta6JIH'IHbie naHHhie B KpHTepHH:

HeycTOH'IHBOCTH, MO)I(HO y6eUHThC.II, 'ITO B ''nerKoM" HanpaBneHHH [ 111]

HMCCTC.II HCYCTOH'IHBOCTb OTHOCHTCJibHO HYJICBOH MO)lbi, B TO BpeM.II KaK B

"TpyllHOM" HanpaBJICHHH [100] Bee MOllbi B paCCMOTpeHHOH CHTyauHH yc­

TOH'IHBbi. Ha onbiTe npH H = H c B "Tpy,!I,HoM" HanpaBJICHHH AK neH:CTBHTCJih­

HO OTCYTCTBYIOT BllJIOTb llO npeBbiiiiCHHH p2 ~ o + 7 llD, COOTBeTCTBYIOIIIHX

BTOpoMY nopory, B "nerKOM" )I(C HanpaBJICHHH HaMIO)laiOTC.II HHTCHCHBHbie AK npaKTH'IeCKH cpa3y Ja noporoM p1 {17] .

YcnoBH.II B03HHKHOBeHH.II aBTOKone6aHHH B pa3JIH'IHhiX 3KcnepHMeHTaJih­

HhiX cHzyaUH.IIX nonpo6Ho npoaHaJIH3HpoBa~bi B.B. 3ayTKHHhiM H C.C. CTapo-

6HHUCM [87]. BapbHpy.ll .B IIIHpoKHX npenenax K03<l><l>HUHCHThi raMHJihTOHHaHa

S m, T m (H3MCH.II.II HaMarHH'ICHHOCTb C llOMOWbiO TCMllepaTypbi H KOHUCHTpaUHH

npHMCceH, H3MCH.II.II <PopMy o6pa3ua - c<Pepa, UHJIHHllp, llHCK - H HanpaBJieHHe

HaMarHH'ICHHOCTH), OHH, B qaCTHOCTH, llOKa3aJIH, 'ITO HHTCHCHBHbiC AK B03HH­

KaiOT TOJibKO npH HCYCTOH'IHBOCTH HYJICBOH MO)lbl ITO KpHTCpHIO (23 .7), B TO

B peM.II KaK HeycTOH'IHBOCTb MOll C m -=F 0 Bcer)la COllpoBO)I()laCTC.II B03HHKHOBC­

HHCM AK MaJIOH aMnJIHTYllbi. llpyrne Ha6JIIO)laCMbiC oeo6eHHOCTH AK TaK)I(C

HaXO)l.IIT CCTCCTBCHHOC 06'b.IICHCHHC B paMKaX S -TeopHH.

B llOCJIC)lHee BpeM.II B CB.II3H C ycnexaMH TCOpHH )lHHaMH'ICCKHX CHCTCM B

H3Y'ICHHH nepeXO)la K Typ6yJICHTHOCTH B paMKaX llpellCTaBJICHHH 0 CTOXQCTU­

'ICCKOM (HJIH CTpGHHOM) QTTpaKTOpe B0306HOBHJIC.II HHTepec K H3Y.'ICHHIO

CBOHCTB AK B CHCTeMe DCB. B qaCTHOCTH, B pa6oTe r . fH6COHa H !J . ,ll)l(a<l>pHca

[68] neTa]IbHO HJyqeH nepexon oT nepHOllH'ICCKHX AK K xaocy B )1{11f (c npH­

MeCbiO raJIJIH.II) npH napaMeTpH'ICCKOM B036y)I(JleHHH CB MCTOllOM nonepeq­

HOH HaKa'IKH Ha qacTOTe Wp/2rr = 1 ,3 rru. 11ccnenoBaHbi <Pa30BbiH nopTpeT AK H rjjyHK~UJl noCileOOBGHUJl JlyaHKape B 3aBHCHMOCTH OT HallKpHTH'IHOCTH .

06Hapy)l(eHbi 6H<l>ypKaUHH YllBOeHH.II nepHona AK. DoKa3aHo, 'ITO B <Pa3oBOM

npocrpaHCTBC 3TOH CHCTCMbi CCTb ·HeCKOJibKO CJIO)I(HbiX aTTpaKTOpOB, nepeXOll

Me)I(Jly KOTOpbiMH conpoBO)I(JlaCTC.II rncTepe3HCOM . EH<l>ypKaUHH YllBOCHH.II

UHKJia AK 6hmH o6Hapy)l(eHbi TaK)I(e HMa3aKH (1984 r., a) npH rrapanneJihHOH

HaKattKe CB B A<I>M CuCl2 2H2 6 Ha qacroTe Wp/2rr = 8,9 rru npH TeMnepaType

1 ,4 K H HallKpHTH'IHOCTH p > 102

. BnocnencTBHH aHanorn'IHhie pe3yJihTaTbi

T a 6 n H 1..1 a 31.1

Ko:J<flcl>Hl\HeHTbt S m, T m B eAHHHI.\BX 2rrg 2 AJIH }KHr

0pHeHT8UHil To So T2

= r _2 s2 s_2

[100) 0,28 0,52 0,11 O,Dl -0,36 (111] -0,75 0,30 0,05 O,Dl -1,27

- -· -------·--· -- -------149

6bmH norryqeHbi X. 51Ma3aKH H np. [239, 240] H npH napaMeTpH'IeCKOM B03-

6y)I(Uemm CB MeTonoM naparrrrerrbHOH HaKa'IKH B KBa3HnBYMepHoM <I>M

(CH3 NH3 ) 2 CuC14 . Qqem HHTepeCHbie pe3yrrbTaTbi no HccrrenoBaHHIO nepexona

K xaocy AK B A<I>M JID norry'leHbi A.M. CMHpHOBbiM [208]. K co)l(aJJeHHIO,

ceifqac eme paHO rOBOpHTb 0 TOM, KaKOB "cyXOH OCTaTOK" OT 3THX pa6oT ," BO­

nepBbiX, UJUI TIOHHMaHHJI npo6rreMbl CTOX3CTH3aUHH B paMKaX o6meif TeOpHH

):(HHaMH'IeCKHX CHCTeM (KpoMe O'leBHUHOro YTBep)I(UeHHJI, 'ITO HeJJHHeHHble

CllHHOBbie BOJJHbl- O):(Ha H3 CHCTeM, K KOTOpGH OHa npHMeHHMa) H, BO-BTOpbiX,

nJJJI <i>H3HKH HeJJHHeHHbiX CTIHHOBbiX BOJJH C06CTBeHHO . Il03TOMY B 3TOH KHHre

Mbi He 6yneM o6cy)I(UaTb HHTepecHyro npo6rreMy nepexona K xaocy npH AK

CTIHHOBbiX BOJJH C TO'IKH 3peHHH o6meif TeopHH UHHaMH'IeCKHX· ~CTeM. 0TMe­

THM TOrrbKO, 'ITO 'IHCrreHHoe MonenHpoBaHHe nepexona K xaocy B AK Ha Mo­

nennx c He6orrblllHM 'IHCJJOM ypaBHeHHH: npoBOUHJJOCb E.B . ACTalllKHHoH:,

A.C. MHxaH:noBbiM [51]. OnHaKo Bonpoc o coorneTCTBHH Jrnx MoneneifJKC­

nepHMeHTY, B KOTOpoM B036y)l(naiOTCJI CB B lllHpoKoH: o6rracrn k-npoc'I­

paHCTBa, OCTaeTCJI OTKpbiTbiM.

B 3aKnro'leHHe Hano cKa3aTb, 'ITO onHcaHHaH Jnecb npocTan TeopHH AK

npencKa3bmaeT, 'ITO nopor HX B03HHKHOBeHHn ha coBnanaeT c noporoM na­

paMeTpH'IecKoro B036y)I(UeHHJI hth• a qacTOTa ·aBTOKOJJe6aHHH llpH h = h0 paBHa

Hynro. B JKcnepHMeHTax )l(e Ha6rrronaeTcH ha > hth H HeHyneBaH Ha'larrbHan

'laCTOTa.

B MOHOKpHcTarrrrax )1{.11f nopor AK ha o6bi'IHO_Bbillle, '!eM hth. Ha 0,1 + l.nB,

a Ha'laJJbHaH 'laCTOTa He KoppenHpyeT C BeJJHtlHHOH nopora H B 3aBHCHMOCTH OT

TIOCTOHHHOrO MarHHTHOrO llOJJH H3MeHHeTCH, B npenerrax 104 + 105 fu [ 17) . 3TH <PaKTbl MO)I(HO o6,ljJICHJITb BJJHHHHeM cna6oro HeJJHHeHHOrO 3aTyxa"HHH,

He H3MeHJIIOUiero cymecrneHHO BerrH'IHHbl X1

H X11

, UpHCYTCTBHeM B 'KpHC­

TaJJJJaX cnyqaHHbiX HeOUHOponHOCTeH, OTCYTCTBHeM TO'IHOH aKCHaJJbHOH CHM­

MeTpHH, o6paTHbiM BJJHHHHeM o6pa3ua Ha pe30HaTOp H T.TI. ,UrrJI BbiJICHeHHJI OT­

HOCHTeJJbHOrO BKnana 3THX MeXaHH3MOB Hed6XOUHMO UaJJbHeifwee Teopern­

'leCKOe H JKcnepHMeHTaJJbHoe H3}"1eHHe AK.

§ 32. ABTOKOIIe6aHHH H K0111l3llC

npH napaMeTpH'IeCKOM B036y)I(UeHHH Y3Koro naKeTa BOIIH

PaHee Mbl C'IHTaiiH, 'ITO nopor napaMeTpH'IeCKOH HeycTOH'IHBOCTH MHHHMa­

neH nJJH nap BOJJH C BeKTOp3MH ±k, 3aTIOJJHJIIOUlHX JJHHHIO HJJH DOBepXHOCTb

B k-DpoCTpaHCTBe. IlpH 3TOM CYMMa 4>a3 B nape HBJJJieTCJI ):(HHaMH'IeCKOH ne­

peMeHHOH, a HHUHBHnyaiibHbie <Pa3bl BOIIH xaoTH3HpOBaHbl. 3necb Mbl OUHllleM

CHTyaUHIO, Korna nopor B036y)l(neHHJI MHHHMaiieH niiJI enHHCTBeHHOH napbl

±k0 , HanpHMep, UpH naparrrreJJbHOH HaKa'IKe B OnHOOCHbiX <l>M C aHH30TpOTIHeH

THlla "nerKaH llJJOCKOCTb " . IlpHHUHDHaiTbHOH OC06eHHOCTbiO 3ana'IH HBJIJieTCH

Y30CTb B036y)I(UaeMbiX B k-npoCTpaHCTBe DaKeTOB BOJJH. 3TO He TI03BOJJHeT

DOJib30BaTbCH CTaTHCTH'IeCKHM OTIHcaHHeM, KaK B S -TeOpHH, OUHaKO nenaeT

B03MO)I(HbiM ynpOCTHTb TO'IHbiH raMHJJbTOHHaH B3aHMOUeHCTBHH DO npyroMy

napaMeTpy - yJocrn naKeta • .llrrn JToro Janaqy Heo6xon~~Mo c<PopMyriHpoBaTb

Ha H3biKe BOJIH orn6aromHx.

32.1. YpaoHeHHSI UJlSI MeUlleHHo MeHSIIOmHXCSI aMD1Imyn H ''yHHoepcanbH3SI"

HeycroA'IHBOCTb OUROA napbi. 11CTI01lb3YH K3.HOHH'IeCKHe ypaBHeHHH nBH)I(eHHH

C TO'IHbiM raMHJJbTOHHaHOM H npennonaraH, .'ITO B k-npOCTpaHCTBe B036y)I(UeHbl

ISO

yJKHe naKeTbi BonH

ak =A(k- ko) + B(k - ko),

nony<rnM TaK)I(e, KaK H B rn. 3, ypaBHeHH.II nn.~~: orn6aroWHx A (r) , B (r) <i>Ypbe·KoMnoHeHT A (K) H B (it) :

[t(:t +vV)+ ~L]A= -i-yA+hVB"+ +(wk. - ~ Wp + T lA 1

2 + 2S I B I2 )A,

[; ( : r - v v) + ~ £] B = -ir B + h v A· +

+(wk. - ~wp+TIBI2 +2SIA I2)B,

S = T(ko ,ko, ko , ko), T= T(ko ,ko , ko, ko),

(32.1)

YpaaHeHH.II (32.1) HMeroT TpHBHamHoe peweHHe I A (r) I = I B (r) I =A 0 , rAe

A (r) =A 0 exp( - i'l' t) , B(r) = Aoexp( -i 'IT 2 ),

2IA 0 12 =N, SN=((hV)2- -y2

)112 , sin{'l' 1 +'1'2 )=-y/hV, (32.2)

COOTBeTCTBYIOWee B036y)l()];eHHIO CTO.IIllleH BOnHbi ±k0 . TioKa)l(eM, liTO pewe­HHe (32 .2) npaKTHI!eCKH BcerT{.a HeyCTOH'nfBO OTHOCHTenbHO HapacTaHH.II MOllY· n.IIUHH aMnnHTYA H 4>a3 BOnH orn6a!OUIHX:

oA(r, t), oB(r. t) cr. exp [iitr + v(it)t] .

Ha .113biKe HCXO)J.Horo raMHnbTOHHaHa 3TO 03Hal!aeT HeyCTOH'nfBOCTb no OTHOllle· HHIO K l!eTbipeXBOnHOBbiM npoueccaM

2w(±k0 ) = w(±k0 + K) + w( ±k0 -K),

w(ko) + w(- k0 ) = w(ko +It)+ w(- k 0 -K).

(32.3)

{32.4)

TipH 6omll1Hx K BenHtrn:Hbi k 0 + K H k0 - K, KaK npaBHnO, He MOryT OAHO· BpeMeHHO nonaCTb B o6rraCTb k-npoCTpaHCTBa, r)J.e 33TYXaHHe CKOMneHCHpoBaHO

. HaKai!KOH H nopor MO.uyn.IIUHOHHOH HeycTOH'nfBOCTH (32.3) OKa3bmaeTC.II Bbl· COKHM. B 3TOM cny11ae cyrueCTBeHHbl TOnhKO npoueCCbl (32.4), B KOTOpbiX BOnHbl B3aHMO)J.eHCTBYIOT napaMH ±k, liTO COOTBeTCTByeT npH6JIH)I(eHHIO S ·TeO· pHH. AHanH3 Bblpa)l(eHH.II nn.~~: HHKpeMeHTa (32.9) noKa3bmaeT, liTO BKnaA npo­ueccoB (32.3) BO B3aHMO)J.eHCTBHe BOnH Man, ecnH

K.!ko ~ [(h v- r)/wJ 1'

3• (32.5)

3TO HepaBeHCTBO OUeHHBaeT MHHHM3nbHOe paCCTO.IIHHe Me)l()];y napaMH B AHHa· MHIIeCKOM pe)I(HMe, llpH KOTOpoM HX ewe MO)I(HO OnHCbffiaTb B paMKaX S-TeO­pHH. TipH K, MeHblllHX, l!eM (325), cyrueCTBeHHbl o6a npouecca (32.3) H (32.4), H cne.uyeT paCCMaTpHBaTb napbl ±k0 no OTHOllleHHIO K po)I()J.eHHIO l!eTbl· pex CB.II3aHHbiX CB ± (k0 + It) H ± (k0 -It) ; cooTBeTCTByrowee AHcnepCHoHHoe

151

COOTHOIUeHHe HMeeT BH)J.

(r+v)2 -r2 =-(Kvf-r2 WK 2- l/4({-K 2

)2

-

-*LK2(2S + T)N- N 2S(2S + T) ± {(Kv)2 ((LK 2)

2 +

+ 2LK2(2S + T)N + 4N2 S(S + T)] +

+N2(SLK 2 +NS(2S+T)] 2 } 112 .

3necb Mbl yi{]IJ{ 3aBHCHMOCTb V k OT k:

V(k0 + K) = V(1 + * WK 2),

(32.6)

CnenyeT Q)IG{.[J,aTb, 'ITO HeyCTOH'lliBOCTb 6yner pa3BHBaTbC.II B OCHOBHOM IJ.n.ll

BonHOBbiX BeKropoB It B6nH3H noBepXHocTH:

±Kv+*LK2 =0. (32.7)

3ro 03Haqaer, qro It noqrH nepneHnHKYn.llpeH v H HaH6onee qyBCTBHTenbHOH

no orHorneHHIO K H3MeHeHHIO HanpaBneHH.II "K BenH~HoH: . B (32.6) 6yner

CKan.llpHoe npoH3BeneHHe K v. llo:noMy • 6yneM HCKaTb MaKCHManbHbiH HHKpe·

MeHT, BapbHpy.ll (32 .6) romKo no It v. llony~M B pe;1ynhTare, qTO MaKCHManb­

HbiH HHKpeMeHT Vm )J.OCTHraerC.II Ha noBepXHOCTH

{2'Kv)2 = [L K2 + (2S + T)N] 2 - T2N 2 -

4N2 (SLK 2 +S(2S1tT)N] 2

(LK 2 + {2S + T)N] 2 - T 2N 2

KOTOpa.ll npH 60TibiUHX L K 2 Mano OTnHqaeTC.II OT (32 .7) , H paBeH

T2S2N4 (-v+v )2 ,2 _ ,2WK2 +-----------

1 m - 1 --, (LK 2 +{2S+T)N] 2 - T 2N 2

(32.8)

{32.9)

Bropoe cnaraeMoe (32 .9) B S -reopHH orcyrcrByer. CpaBHHBa.ll ero npH

6onhiUHX 3HaqeHH.IIX K c nepBbiM cnaraeMbiM, nonyqaeM oueHKY (32.5) npH­

MeHHMOCTH S ·TeOpHH. lloaeneHHe HHKpeMeHTa npH L K 2 ~ 'Y 3aBHCHT OT COOT·

HOIUeHH.II 3HaKOB S, T H (2 S + T ) . PaccMOTpHM BHaqane cnyqaH: T > 0, S > 0. )J;J.11 Hero Bbtpa:>KeHHe {32.9)

cnpaBe)J.nHBO npH K > K o, K o /k0 :::::.SN/w. llpH K :::::. K o

(r+vm)2 -r2 =T2N 2S/(T+S). (32.10)

llpH K < K o HHKpeMeHT Ha~HaeT pe3KO yMeHblllaTbC.II, TipH K ~ Ko

(r+vm)2 -r2 =(Kv)2TSN2

npH It II v (pHc. 32.1). Ecnu 3TH ycnoBH.II He BhmonH.IIIOTC.II, ro Bbtpa:>KeHH.II

(32.8), (32.9) cnpaBe)J.nHBbl TOnbKO npH K > K 1 , rne

LK~ = [(T2 +S 2)

1 12 +IS I-(2S+T)] N.

llpH MeHbiUHX K HHKpeMeHT MaKCHManeH Ha nnocKocrH "K 1 v H paBeH

(r + vm? - r 2 =- 1/4 rLK 2 + (2S + T)N F +

+(l/4T 2 -S 2 )N2 + ISN(LK 2 +(2S+T)N] i-r2 WK 2 . (32.11)

152

flpH K = 0 OTCJOAa CIIeAyeT

(r + Vm)2

- r 2 = - 2S(2S + T)N 2 ' Vm = 0. (32.12)

flepBaH BeTBb COOTBeTCTByeT "c06CTBeHHOH" HeycTOH'IHBOCTH B S ·TeOpHH,

BTOpaH - COCTOHHHIO 6e3pa31IJil!HOf0 paBHOBeCHH no OTHOWeHHIO K H3MeHeHHIO

pa3HOCTH <j>a3 '/)o - '/)o napbi. flpoHCXO)K,lleHHe BTOpoH MOIJ.bl CBH3aHO C AHHaMH·

'leCKHM xapaKTepOM CTaUHOHapa - OTCYTCTByeT XaOTH3aU:HH pa3HOCTeH <j>a3,

6e3pa31IH'IHOe paBHOBeCHe B03HHKaeT H3·3a npOCTpaHCTBeHHOH 0/l.HOpOIJ.HOCTH,

f f

a

p H C. 3 2. 1. 3aBHCHMOCTb HHKpeMeHTa HeyCTOAtiHBOCTH Vmax, f = v :nax + 2-y Vmax OT K .

S < 0, 2S + T > 0 (a) ; S < 0, T > 0, 2S + T > 0 (6)

TaK KaK pa3HOCTb <Po - <Po onpe11.e11HeT ll01IO)KeHHe B npOCTpaHCTBe Y31IOB CTOH·

'leH B01IHbl. flpH Ma1lb1X K JTa MOAa OCTaHeTCH YCTOH'IHBOH, eCIIH

T> 0, (2S + T) > 0. (32.13)

11HTepeCHO, 'ITO T> 0 - yciiOBHe OTCYTCTBHH CaMo<j>oKyCHpOBKH MOHOXpO­

MaTJilleCKOH BOIIHbi B KOHcepBaTHBHOH cpe11.e. YcrroBHH (3 2.f3) AOCTaTO'IHO

H 11.1IH "co6CTBeHHOH" ycTOHtiHBOCTH. flpH MaiibiX K llOBell.eHHe (32.11) 3aBHCHT

OT 3HaKa S(2S + T) . IlpH S(2S + T) > 0

(-y+vm)2 -r2 = -\4 (LK 2)

2 -l/J.LK 2 TN. (32.14)

EcrrH S(2S + T) < 0, TO

(-y+vmi - r 2 = - 14 (LK 2 i -l/J.LK 2 (4S+T)N - 2S(2S+T)N 2• (32.15)

f1011.p06Hee paCCMOTpHM CIIytiaH "co6CTBeHHOH" yCTOHtiHBOCTH napbl

S(2S + T) > 0. Tor11.a 11.1IH ycTOH'IHBOCTH npH MaiibiX K Heo6xOAHMO, 'IT06br

T> 0, KaK JTO BHAHO H3 (32.14). 0TCJOAa C He06XOAHMOCTbiO CIIeAyeT S < 0 (HHa'le pearrH3yeTcH HeycTOH'IHBbiH crrytiaii T > 0, S > 0), nonoMy 2S + T > 0. flpH JTOM HHKpeMeHT OllHCbiBaeTCH <j>OpMyiiOH (32.14) npH K2 > K > 0, LK~ = = I 2S + TIN. EcrrH K 1 > K > K 2 , TO HHKpeMeHT npH T > 0, S < 0, (2S + 7) < < 0 onHcbiBaeTcH <j>opMyrroii (32.15). 113 Hee crre11.yeT, 'ITO npH I SNI ~r HHKpeMeHT cTaHOBHTCH no1IO)KHTe1IbHbiM y)Ke npH LK 2 = 2LK ~ (pHc. 32.2). OH

IJ.OCTHraeT MaKCHMa1IbHOfO 3Ha'leHHH

(r + vmi - 'Y2 = - 'Y2 WK i + 1/J. Is I ((T 2 + S 2)

1'

2 - Is I )N 2 (32. 16)

npH p = K i . CtiHTaH W ~ ko 2, L ~ wk02

, norryqaeM ou:eHKY 11.1IH nopora He­

ycTOH'IHBOCTH :

(32.17)

Ou:eHHBaH (r/w) ~ 10- 2 + 10-3, norr}"iHM 11.1IH nopora HeycTOHtiHBOCTH napbi

153

3Ha'leHHH (hth v- -y)/'y ~ 10-4 7 w-s . .llnH TOro 'IT06bi napa OCTaJiaCb YCTOH­

'IHBOH npH npeBbiiiieHHJIX B HeCKOJlbKO ,D;eii,H6eJIJI (h V - 'Y ~ 'Y), He06XO,D;HMO,

'IT06bi IT I~ (0,1 7 0,01)S. TaKHM o6pa30M, MOHoxpoMaTH'IeCKaH CTOH'IaH

CB npH napanneJibHOH HaKa'IKe npaKTH'IeCKH Bcen:~;a HeycTOH'IHBa.

XapaKTep pa3BHTHH HeycTOH'IHBOCTH cylUeCTBeHHO JaBHCHT OT napaMeTPOB

raMHJibTOHHaHa T H S. · 3a HCKJIIO'IeHHeM cnyqaH T> 0, S > 0, KOTOpbiH Mbi

B ,D;aJibHeHIIIeM o6cy)I(,D;aTb He 6y,o;eM, HHKpeMeHT HeycTOH'IHBOCTH, KaK y)l(e

OTMe<ianocb, MaKCHMaJieH Ha noBepXHOCTH Ko 1 k 0 , a npH y,o;aneHHH OT no­

BepxHOCTH 6biCTp0 y6biBaeT. 3TO npHBO,D;HT K TOMy, 'ITO OCHOBHbie CBOHCTBa

t/J

rr/2

0

-rr/2 1

0,5 yt 2,0

PH c. 32.2. 3aaHCHMOCTb <t>aJLI <II (0, t) a ueHTpe naKeTa AnH p = 4, T = - S . KpHBLie J,2, JCOOTBeTCTByiOT k=2 ; 3; 3,5

HeJIHHeHHOH CTa,D;HH pa3BHTHH HeycTOHtiHBOCTH MO)I(HO npocJie,D;HTb, IJpJib3'yHCb

.o;ByMepHbiMH ypaBHeHHHMH, B KOTOpblX A H B JaBHCHT TOJlbKO OT KOOp,D;HHaT

X Hy, OpTOfOHaJibHbiX V.41Jb!p3)1(eHHH (32.14) H (32.15) ,D;JIH HHKpeMeHTa COOT·

BeTCTBYIOT B03MYIIIemiHM BH,D;a oA (r , t) ± oB(r, t). B cnyqae T> 0, s < 0 BH,D;­

HO, 'ITO B03MYIIIeHHe oA = - oB HMeeT Janac ycTOHtiHBOCTH H, KaK llOKa3aHO

B.C. JlbBOBbiM, A.M. Py6eH'IHKOM [ 151], npH :noM A (r, t) = B(r, t) H Ha HeJIH­

HeHHOH cTa,D;HH ,o;BH)I(eHHH. OrpaHH'IHBaHCb ,D;JIH npocTOTbi cnyqaeM w" > 0, MO)I(HO, H3MeHHH MaCIIITa6, npeo6pa30BaTb ypaBHeHHe (3 2.1) K BH,D;Y

i ( :t + 'Y )A - hVA • + ~M = [(2S + T) I A 12

- TA~]A. (32.18)

3,o;eCb Mbl onpe,o;eJIHJIH 3Ha'leHHe Wk H3 ycJIOBHH BHeiiiHeH YCTOHtiHBOCTH (2.12). 0

3THM Bbi6HpaeTCH HaH60Jiee yCTOHtiHBaH CTOH'IaH BOJIHa, ,D;JIH KOTOpOH, KaK

MO)I(HO llOKa3aTb, o6JiaCTb llOJIO)I(HTeJibHOfO HHKpeMeHTa B k-npOCTpaHCTBe

opraHH'IeHa , K ~ k 0 • 11JyqeHHe HeJIHHeHHOH CTa,D;HH pa3BHTHH HeyCTOHtiHBOCTH

nJIOCKOH CTOH'IeH BOJIHbl B ,D;aJibHeHIIIeM 6y,o;eT npOBO,D;HTbCH B paMKax BbiBe,D;eH­

HbiX 3,D;eCb ypaBHeHHH (32.18). ITpoBe.o;eHHbiH B.C. JlbBOBbiM, A.M. Py6eH'IHKOM [ 151] aHaJIH3 ,o;aeT ocHo­

BaHHH C'IHTaTb, 'ITO BCe CTaii,HOHapHbie COJIHTOHbl B ypaBHeHHHX (32.18) He­

yCTOH'IHBbl. TaM )l(e paccMoTpeHa TaK)I(e Ha'laJibHaH HeJIHHeHHaH CTa,D;HH :noif

HeycTOHtiHBOCTH, npH'IeM OC060e BHHMaHHe y,o;eJIHeTCH CJiy'laiO, KOr,o;a HHKpe­

MeHT aHOMaJibHO MaJI. 0H HHTepeceH TeM, 'ITO HeJIHHeifHoe B3aHMO,D;eHCTBHe

Ha'IHHaeT HrpaTb CylUeCTBeHHYJO pOJib npH BeCbMa MaJibiX aMUJIHTy,o;ax. 0Ka3a­

JIOCb, 'ITO TeM He MeHee OHO He OrpaHH'IHBaeT pOCTa Ha'laJibHOfO B03MylUeHHH,

a JIHIIIb CHJibHO JaMe,D;JIHeT ero. AMnJIHTy,o;a ero npH :noM pacTeT KaK t 1 /2

.

B pe3yJibTaTe o6paJyeTcH yJKHH B k-npocTPaHCTBe naKeT BOJIH c K ~Ko ~ ko:

15-4

~K - Ko . TaKOe COCTO.IIHHe .IIBJI.IIeTC.II CHJibHO Typ6yneHTHbiM, K ero H3}"1eHHIO

Mbi npHcTynHM B cnep;ywmeM llYHKTe.

32.3. Cpe.~n~He xapaKTepHcmKH aBTOKone6amdt napbl. BHal!ane oueHHM lliH·

pHHY B036yx<p;eHHOH B k·npoctpaHCTBe o6naCTH npH npOH3BOJibHOH Ha)J;KpHTH'I·

HOCTH. 113 S -TeOpHH cnep;yeT, 'ITO B o6cy)I(AaeMOM H3MH CJI}"'ae, KOr,ria Vk MaKCHMaJibHa B TO'!Ke k0 , naKeT napaMeTpH'!eCKH B036y)I(AeHHbiX BOJIH penaK­

CHpyeT K CTO.II'IeH MOHOXpOM3TH'!eCKOH BOJIHe C k = k 0 , eCJIH HH)J;HBH)J;yaJibHbie

Q>a3bl BOJIH MO)I(HO C'IHTaTb CJI}"'3HHbiMH·. ,Un.11 CJI}"'aHHOCTH Q>aJ )J;OCT3TO'IHO,

'IT06bl Q>a3bl p;syx BOJIH B naKeTe paCXO)J;HJIHCb Ha BeJIH'IHHY nOp.!Ip;Ka e)J;HHHUbi

33 BpeM.II, MeHblllee xapaKrepHOrO BpeMeHH HeJIHHeHHOrO B33HMO)J;eHCTBH.II.

3To npoHcxop;HT B naKere c (~k) 2 ~ "~ = SA~/w" . IlonoMy naKer c M ~ Ko

cyx<aerc.11 .no pa3MepoB " - Ko , a ero cpe,nHll.II aMnnHryp;a penaKcHpyer K senH­

l!HHe A0 . EcnH )l(e M ~ Ko, ro TaKOH naKer HeycTOH'IHB c HHKpeMeHTOM

(32.14) OTHOCHTeJibHO B03MYIUeHHH C K -K0 H, cnep;OBaTeJibHO, 6yp;er paClliH·

p.!ITbC.II )J;O ~k- Ko - (SA~/w") 1 12 . Ilop;qepKHeM, 'ITO BO BCeH o6naCTH ryp6y­

neHTHOrO ABH)I(eHH.II HHKpeMeHT HeycTOH'iHBOCTH (32.14) noJIO)I(HreneH B y3-

KOM cnoe it11 <SAUV s6JIH3H nnocKocrH It 1 v, r.e. paccMarpHBaeMaH HaMH

ryp6yneHTHOCTb .IIBJI.IIeTC.II llO'ITH AByMepHOH:

(K 11 /K.L)2 ~ SA-~/wk ~ 1. (32.19)

CpeAHHH ypoBeHb ryp6yneHrHOCTP. A 2 He MO)I(er CHJibHO ornH'!aT&C.II or se­

JIH'IHHbl A~, onpep;en.IIeMOH Q>opMynoii (32.2) . .UeiicTBHTeJI&HO, KaK OTMe'la­

nocb Bbillle, MOHOXpOMaTH'!eCKa.II llJIOCKa.II BOJIHa YCTOH'iHBa OTHOCHTeJibHO KO·

porKOBOJIHOBbiX _B03MYIUeHHH c "~ k0 , ron&KO ecnH ee aMnnHryp;a pasHa A 0 •

Ol!eBH,nHO, 'ITO raKa.II HeycrOH'IHBOCTb coxpaH.IIeTc.II H AJI.II napbi, npoMop;ynH­

poBaHHOH c "~k0 , nonoMy, ecnH A2 3aMerno OTJIH'Iaerc.II or At TO B036y:>KAe­

Hbi KOpOTKOBOJIHOBbie MO)J;yJI.IIUHH, 'ITO npOTHBOpel!HT llOJI}"'eHHOMy Bbillle

yTsep:>KAeHHJO o6 Y30CTH naKera. TaKHM o6pa30M, pa3BHTHe Heycroii'IHBOCTH

nnocKOH BOJIHbi npHBOAHT K cHJibHOH KBa3H)J;ByMepHOH ryp6yneHTHOCTH BOJIH

Mop;ynHUHH A (r, t) co cpeAHHM ypoBHeM

(32.20)

rny6HHoii Mop;yn.IIUHH nop.11,nKa ep;HHHUbi, xapaKrepHOH qacroroii JJ:BH)I(eHH.II

h V - 'Y H XapaKTepHbiM M8Cll1Ta60M B K00p)J;HH8THOM npOCTpaHCTBe r .L ~

~ K(i 1 ~ k01 ( w/SA~) 112

• Mo)I(HO cKa3ar&, 'ITO B03HHKaeT AHHaMH'IecKa.~~ cmiH­

IOHHa.II CTpyKrypa C AJIHHOH KOrepeHTHOCTH nOp.!Ip;Ka pa3MepOB COJIHTOHOB,

't.e. A0 1 ( w" /S) 112

, Koropa.~~ cymecrseHHO MeH.IIerc.II B npocTpaHCTBe 3a

speMH (hV-r)- 1 .

32.4. Pa3pyweiDie COJIHTOHOB H MexaiDI3M orpaHHqeHJUI llMlUIH'fYALI Kon­

nancoM. B rex o6nacr.11x, rp;e aMnnHryp;a conHTOHOB B npouecce ryp6yneHrHoro

)J;BH)I(eHH.II OKa3anac& aHOMaJibHO 6omrnoii, A ~ A 0 , B ypaBHeHH.IIX )J;BH)I(eHH.II

MO)I(HO npeHe6pel!b 3aTyxaHHeM H H3K8'1KOH, T3K KllK 38 xapaKrepHOe BpeMH

3a)J;al!H (SA 2 ) -1

CHCTeMa ITB He ycneer CYIUeCTBeHHo o6MeH.IITbCH :meprHeH

c repMocraroM H HaKal!KOH. B JTOM npH6JIH)I(eHHH ypaBHeHH.II (32.1) onHCbi·

Bll!OT HeCT3UHOHapHOe llOBep;eHHe napbi BOJIH B KOHCepBaTHBHOH cpep;e. Ilpe)l(p;e

Bcero noKa)l(eM, 'ITO .IIBJieHHe Konnanca xapaKrepHo He ron&KO AJI.II orH6awmeii

MOHOXpOM8TH'IeCKOH BOJIHbl, HO H )J;JI.II napbl BOJIH . ,Un.11 JTOrO OTMeTHM, 'ITO

Henocpep;CTBeHHbiM Bbi'IHCJieHHeM MO)I(HO y6ep;HTbCH B TOM, 'ITO ypaBHeHH.II

155

(32.1) (npH 'Y = h = 0) HMeror cnep;yromHH HHrerpan ABIDKeHJrn:

I= ~w"f(IVA 12 +I VBI 2 )dr+ ~T f(IAI 4 +I Bl 4 )dr+ 2S fiA I2 IBI 2 dr. (32.21)

,Uanee Mbi noKIDKeM, qro ero JHaK so MHOrOM onpep;enner JB01IJOUHJO cHcre­Mbi. PaCCMOTpHM Anll JTOfO BTOpyro np01i3BOAHYJO OT cymecrBeHHO nona>KH·

re1lbHOH aen~HHbi

(32.22)

Henocpep;crBeHHbiM Bbi~cneHHeM c noMOIUbJO (32.1), a KoropoM no napa­

Merpy (SA~/w) ~ 1 MO)J(HO npeHe6pe~ BTOpbiMH npOH3BOp;HbiMH no Z, TaK )J(e, KaK B 11. 12.4 rn. 3, MO)J(HO nOKaJaTb, qro 32R/3t2 = 2J, OTKyp;a cnep;yer

R (t) = It 2 + 2o:t + ~. (32.23)

rp;e o: H ~ - nocrO.fiHHbie HHrerpHpOBaHHJI. SHABo, qro npH I < 0 Ja KOHe~oe BpeMJI R (t) craHOBHTCJI orpHuarenbHbiM, qro nporHBOpeqHr (32.22). 3ro OJHa­qaer, qro peweHHe ypaBHeHHH (32.1) "pa3pywaercn" Ja KOHe~oe BpeMn,

T.e. nOJIB11JieTCJI OC06eHHOCTb npH T = 0. PaccMorpHM JIB1IeHHJI, npoHcxoA.~ImHe npH Konnance conHTOHOB. QqeBHAHO,

qro npH HJyqeHHH 3B01IIOUHH K0111IancHpyromero C01IHTOHa MO)J(HO He yqHTbiBaTb anHJIHHe HaKaqKH H JaryxaHHJI. YpaaHeHHe (32.4) nepeXOAHT npH JTOM B He-1IHHeHHOe napa6onHqeCKOe ypaBHeHHe, CBOHCTBa peweHHH KOTOporo nop;po6HO HJyqanHCb B CBJIJH c Jap;aqeii o caMo<t>oKycHpOBKe caera. KaK noKaJaHo B.E. 3axapoBbiM H B.C. CbmaxoM [108], aMnnHryp;a B ueHrpe conHrOHa 6bicrpo

pacreT CO BpeMeHeM: ~~0, t) "' (r' - t 0 )-2f3 . IlpH 3TOM pap;Hyc C01IHTOHa

yMeHbwaercn raK, qro B npouecc cxnonbmaHHJI Jaxaarbmaercn crporo onpe­p;eneHHoe KOnHqecrao JHeprHH

Ie = w f I A l 2 dr = 1,86w"w/12S + Tl. (32.24)

IlpH yaen~eHHH aMll1IHTYAbi B01IH B K0111IanCHpyromeM C01IHTOHe CTaHO· BHTCJI cymecTBeHHbiM HenHHeHHOe JaTyXaHHe, KOTOpOe npHBOAHT K 6biCipOMY BbiropaHHro ·JHeprHH conHTOHa . OueHHM J<t><t>eKTHBHOe HenHHeiiHoe JaryxaHHe

'Ynz ~ IeK~/rwA~ ~ 1/r, (32.25)

rp;e r - BpeMJI Me)J(Ay ABYM.fl nocnep;oBarenbHbiMH KOMancaMH B o6nacrH

C pa3MepoM K(j2 . J.1J C006piDKeHHH pa3MepHOCTH MO)J(HO nonyqHTb, qro T ~ ~ (h vr I llpH h >he.

Yqer 3T0f0 MeXaHHJMa AHCCHnaUHH JHeprHH npHBOAHT K TOMy, qro cpep;H.fiJI aMnnHryp;a ryp6yneHTHbiX nynhcaUHH craHOBHTCJI Memwe A 0 , a BocnpHH~H­BOCTb X 11

C pOCTOM aMn1IHTYAbi h He y6biBaer, a BbiXOAHT Ha nnaTO, COBnap;aJO· II

mee no nopHAKY aen~HHbi c x B MaKCHMyMe.

IIepcneKTHBHbiM Merop;oM JKcnepHMeHTanbHoro HJyqeHHll CH1IbHOH conHTOH· HOH ryp6yneHTHOCTH napaMerp~eCKH BOJ6~p;eHHbiX CB HBnnercn HJMepeHHe cneKrpanbHOH ll1IOTHOCTH 31IeKTpOMarHHTHOfO H31IyqeHHJI <l>M Ha qacTOTax, 61JH3KHX K qaCTOTe HaKaqKH Wp .

IlpH h < he, KOrp;a K0111IanCbi pep;KH, cneKrpanbHall ll1IOTHOCTb UiyMa 6nH3Ka

K rayccoBOH c IIIHpHHOH I w- Wp I - -y(p- 1) 112. IlpH h >he JaMeTHbiH BK1Iap;

B H31IyqeHHe BHOCJIT K0111IancHpYJOIUHe C01IHTOHbi, y KOTOpbiX <t>aJa napbl 'lr(r, t.) "orpbiBaercn" or <l:>aJbi HaKaqKH H HaqHHaer 6blcrpo BpamaTbCll, np11

156

JTOM: >¥- (t - to)- 113 • nblcrpoe BpameHHe <PaJbi npHBO)l;HT K yumpeHHIO

CfleKrpa H3n)"'eHIUI (h2 )w, no KOTOpOM)' MO)I(}{O perHCTpHpOBaTb KOnnanCbl.

MO)!(HO nokaJaTb, liTO (h 2 )w -I w - Wp 1- 7/ 4 .

HenHHeifHoe JaryxaHHe, orpaHHI.IHBaromee aMnnHrygy B Konnance A <A max ,

gomKHO o6pe3aTb Ill)'MO~oe HJn)"'eHHe H<!- 1.1acrore

SAmu ~I Wmax- Wp 1.

3aga'IH

33.A3'13 6.1. BLNHCJIHTh 33.BHCHMOCTh OT BpeMeHH norJioiUeHHOit MOIUHOCTH Ha JIHHeitHoit

CTa.u;HH pa3BHTHJI napaMerpH'IeCKOit HeyCToit'IHBoCTH npH MrHOBeHHOM BKJIIO'IeHHH

H3K3'1KH.

YKa:umue. McnonL3oBan. yp3.BHeHHll (29.2) B JIHHeitHoM npu6nulKeHHH.

0T8eT:

T

F n,(T) = f 11 (T' ) exp{- Pn. T, ) dT', 0

r ne l.(T) - ¢lyHKUHJ[ oeCOeM OT MHHMoro apryMeHTa Pn, 'Yn,fhVn,, Tn, = 2hVn,t .