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B ocTanbHbiX cnyqaHx H nbiTancH orpaHH'IHTbCH ccbmKaMH ua KnaccuqecKyiO

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18

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OCTaJibHOro, CB!I3aHHOrO C KOHKpeTHOH cneu;H<t>HKOH cpe,ll;blo 2. 19

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dt ap ' dt aq

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dt (5.2)

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20

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83{ (5.4)

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21

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ox ox (5.7) i-= --

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B006ll.le rOBOp.ll, MO)I(eT J];BHraTbCH. fiOH.IITHO , 'ITO paccMaTpHBaTb Bee 3TH .IIBJie·

HH.II C o6meif TO'IKH 3peHH.II HeKOHCTpyKTHBHO. QJ];HaKO, eCJIH napaMeTp HeJIH·

HeHHOCTH BOJIH ~ MaJI, cneUHij>HKa cpep;bi B 3Ha'IHTeJibHOH CTeneHH OKa3bmaeTC.II

HecymecrBeHHOH, H He3aBHCHMO OT ee CBOHCTB }J;HHaMHKY BOJIH MO)I(HO OllHCbi·

Barb B o6mHx repMHHax: repMHHax JaKOHa J];HcnepcHH w (k), rpynnoBOH H

lj>a30BOH CKOpOCTH, BepO.IITHOCTeH JJieMeHrapHbiX npouecCOB B3aHMO}J;eHCTBH.II

c yqacrHeM rpex, qerbrpex BOJIH H r . .o;. B JTOM nyHKre Mbi npo.o;eMOHcrpHpyeM,

KaK B03HHKalOT 3TH llOH.IITH.II B paMKaX raMHJibTOHOBa lj>OpMaJIH3Ma npH MaJIOM

ypOBHe HeJIHHeHHOCTH ~ .

KaHOHH'IeCKHe nepeMeHHbre ai, at Mbi Bhr6epeM raK, 'ITO OHH 6y.o;yr xapaK­

repH30Barb aMnJIHTYAY BOJIH H 06pamaTbC.II B HYJib B HX OTCYTCTBHe. lfH.o;eKC

j = 1, ... , n xapaKrepH3yer nons:pH3aUHlO BOJIH HJIH HX rHn, Be.o;b, Boo6me roBo­

p.ll, B cpe.o;ax MOryr OJ];HOBpeMeHHO B036y)I(}J;aTbC.II pa3JIH'IHbie BOJIHbi: 3BYK,

CBer H r.n . noilaras: ai, a/MaJibiMH. Pa3JIO)I(HM !J.>YHKUHOHan Jf {a1(r, t) , aj(r, t)} B ps:.o; no creneHHM a1, HyrreBOH 'IJieH J( { 0, 0} Hac He HHrepecyer, raK KaK

OH He BXO}J;HT B ypaBHeHHH }J;BH)I(eHHH : flj( { 0, 0 } joa = 0. lfueHbi nepBO·

ro llOpH}J;Ka Jf1 paBHbi HYJIIO, TaK KaK Mbi C'IHTaeM, 'ITO B OTCYTCTBHe BOJIH cpe­

.o;a HaXO}J;HTC.II B paBHOBeCHH H, CJie}J;OBaTeJibHO, raMHJibTOHHaH MHHHMaJieH

npH a1 = at= 0. TaKHM o6pa30M, pa3JIO)I(eHHe J( Ha'IHHaerc.~~: c 'IJieHOB Broporo

nop.II}J;Ka:

(5 .8)

22

HaH6onee o6man <PopMa JC2 TaKOBa:

n

l: f drdr'{A1J(r, r')a1(r)aj(r') + I, I= 1

+ ~ [B1j(r, r')a1(r)a;(r') + K . c . ]} .

BenH'!HHa JC2 BemecrBeHHa (raMHJibTOHHaH JpMHroB), noJTOMy

A;;(r, r') "=Ajt(r', r), Btf(r, r') = Bft(r', r).

(5 .9)

(5 .10)

KpoMe roro, B npocrpaHCTBeHHO-O,II,HOpo,II,HOH cpep;e A H B JaBHCHT TOJibKO

OT pa3HOCTH KOOpAHHaT H, eCJIH cpep;H 3JleMeHTOB CHMMeTpHH CpeAbl eCTb

HHBepcHH, HBJIHIOTCH '!eTHOH $yHKIJ,HeH 3TOH pa3HOCTH. Torp;a H3 (5 .1 0) cnep;yer, 'ITO

Atf(R) = Aj;(R), B;;(R) = BJt(R). (5.11)

CymecTBeHHoro ynpomeHHH raMHJibTOHHaHa MO:>KHO p;o6HTbCH, coBepwan

npeo6pa3oBeHHe Cl>ypbe

a(k) = ak = ( ~) J dra(r)e-ikr, a(r) = ; akeikr, (5.12)

rp;e V - 06'beM o6paJUa (cpeAbl, B KOTOpoH pacnpocrpaHHIOTCH BOJIHbl).

Mbl C'IHTaeM, 'ITO BOJIHOBOH BeKTOp k HBJIHeTCH ,II,HCKpeTHOH nepeMeHHOH.

ilpH He06XO,II,HMOCTH MO:>KHO 06bi'IHbiM 06pa30M nepeHTH OT CYMMHpoBaHHJI

no k K HHTerpHpoBaHHIO:

(27T)3 l: = v fdk. k

(5.13)

llpeo6pa30BaHHe Cl>ypbe (5 .12) JIBJIHeTCH KaHOHHqecKHM, HO He YHHMOAaJib­

HbiM. 3ro 03Haqaer, qro ypaBHeHHe faMHJibTOHa (5.6) coxpaHner CBOH KaHOHH­

qecKHH BHA, O,IJ,HaKO HOBbiH raMHJibTOHHaH OTJIHqaeTCH OT CTaporo MHO)!(J{Te­

JieM - p;enHTCH Ha o61>eM o6pa3ua:

aak o:JC(ak. aic) ;- = ar oaic

(5 .14)

CymecrBeHHo ro, qro KBap;parHiiHan qacTb raMHJibTOHHaHa oKa3biBaercn

,II,HarOHaJibHOH no BOJIHOBbiM BeKTOpaM. 11MeHHO:

Jf2 = ~ l: [A1;(k)a1(k)aj(k) + ~(B1j(k)a1 (k)a;(-k) + K .c.)], (5.15) i ,J k

Ak = A(k) = f dRA(R) exp ikR, Bk = f dRB(R) exp ikR. (5) 6)

PaJyMeercn, JTO uenHKOM cBH3aHo c npocrpaHCTBeHHOH o,IJ,Hopop;HOCTbiO cpep;bt.

faMHJibTOHHaH (5.15) MO)I(HO ,II,HarOHaJIH30BaTb no "rHnaM BOJIH" i, j C

nOMOIUbiO JIHHeHHOfO npeo6pa30BaHHH

b1(k) = ~ [u1;(k)a;(k) + v1;(k)aj(-k)], (5.17) J

23

" " -MaTpHI.J;hl U H V KOTOpOrO yp,OBJieTBOp.!IIOT YCJIOBH.!IM KaHOHH'IHOCTH (IJ2.8) .

IlpH COOTBeTCTBYIOII.J;eM Bbi6ope 3THX MaTpHU raMHJibTOHHaH Jf2 IIpHo6peTa­

eT BHP,

Jf2 = "L "Lw;(k)b;(k)bj(k). j k

(5 .18)

B IIpOCTeHllleM cnyqae OP,HOH BeTBH KOJie6aHHH i = j = 1 K03<l>qmi.J;HeHTbl rrpe­

o6pa30BaHH.!I (5.17) JiefKO 3aiTHCaTb B .!IBHOM BHP,e

u(k) = [(Ak +wk)/2wk] 112, Wk = w(k) = signAk(Ak -IBk 12 )

112,

v(k) = - (Bkl I Bk I)[(Ak- Wk)f2wkJ11 2, ul- I Vk 12 = 1. (~· 19)

Ilop,'lepKHeM, 'ITO 3HaK Wk coBrrap,aeT co 3HaKOM Ak (B.E. 3axapoB, [106)). OTpHI.J;aTeJibHbie 3Ha'leHH.!I Wk 03Ha'laiOT, 'ITO 3HeprH.!I cpeP,bi yMeHblllaeTC.!I rrpH

B036y:>Kp,eHHH BOJIH. 3TO B03MO:>KHO B OTCYTCTBHe TepMOp,HHaMH'IeCKOfO paBHO­

BeCH.!I. IlpH cymeCTBOBaHHH HeCKOJibKHX BeTBeH JTO CP,eJiaTb He TaK IIpOCTO,

XOT.!I 'laCTOTbl w;(k) MO:>KHO OIIpeP,eJIHTb H He HaXOp,.!I IIpeo6pa30BaHH.!I (5.17).

Jlrr.R: :noro HY:>KHO 3anHcaTb ypaBHeHHH P.BH:>KeHHH (5.14) )J.Ii.R: a;(k):

aa;(k) . • --+ 1 "L [Ali(k)a1(k) + B1;(k)a1 (-k)] = 0, at r

aaj(-k) . • • • -z "L [Ar; (k)a1 ( - k) + B 1;( -k)a1(k)] 0 at r

H, IIOP,CTaBJI.!I.!I a, a* - el iwt, IIOJIY'IHTb anre6paH'IeCKOe "ceKyJIHpHoe" ypaB­

HeHHe P.JI.!I orrpep,eneHH.!I qacToT w:

1

Ar;-wo1; B1; 1--o Bij Aij+wo1; ·

(5.20)

3p,eCb yMeCTHO IIOP,'IepKHYTb, 'ITO ~aHOHH'IeCKHe IIepeMeHHbie b, b *, B OTJIH'IHe

OT KBaHTOBOMeXaHH'IeCKHX 6o3e-orrepaTOpOB, .!IBJI.!IIOTC.!I pa3MepHb!MH BeJIH'IH­

HaMH. Jix pa3MepHOCTb JiefKO OIIpep,eJIHTb H3 (5.18), 3Ha.!I, 'ITO j{ HMeeT pa3-

MepHOCTb ITJIOTHOCTH 3HeprHH, a Wk - 'laCTOTbi:

(Jf) = f"CM-1 · c 2, [w) = C-1 , [bk) = CM-1 ·2 · c-1 .

KaK H cnep,oBano O:>KHP,aTb, pa3MepHOCTb Vb 2 coBrrap,aeT c pa3MepHOCTbiO

IIOCTOJIHHOH IJnaHKa. EcTeCTBeHHO TaK:>Ke , 'ITO Haii.I KJiaCCH'IeCKHH ITOP.XOP,

crrpaBeP.JIHB, Korp,a VI b 12 ~ h, H KBaHTOBOMexaHH'IeCKHe 'IHCJia 3arronHeHH.!I

VI b 12 /h BeJIHKH no cpaBHeHHro c eP.HHHueii.

IIepeMeHHbie b; (k) .!IBJI.!IIOTC.!I HopManbHbiMH rrepeMeHHhiMH JIHHeiiHoii

TeOpHH H ITOJTOMY OC06eHHO yp,o6Hbl p,JI.!I HCCJieP,OBaHH.!I HeJIHHeHHh!X npo6JieM.

CrreUH<PH'IecKHe "nHHeHHhie" TPYP.HOCTH, npHcyii.J;He HCCJiep,yeMOH Mop,enH

cpeP,bl, npeOP,OJieBaiOTC.!I Op,HH pa3: IIpH IIOHCKe nepeMeHHb!X b;. B JTHX nepe­

MeHHbiX JIHHeapH30BaHHhie ypaBHeHH.!I p,BH:>KeHH.!I CTaHOB.!ITC.!I TpHBHaJibHbiMH:

ab;(k) . -a;-+ zw;(k)b;(k) = 0, (5.21)

OHH OIIHChiBaiOT pacrrpOCTpaHeHHe CB060P,HbiX BOJIH C 3aKoHaMH p,HCIIepCHH

w; (k). Bc.R: "nHHeiiHa.R:" HH<l>opMai.J;H.R:, Heo6xcp,HMa.R: P.JI.!I H3y'leHHH HeJIHHeH-

14

HbiX npo6rreM, 3aKJIIO'IeHa B cpyHKUHHX w; (k) . BcH HHcpopMaUHH o B3aHMO­

,ll;eiicTBHH BOJIH CO,ll;ep)I(JITCH B OCTaJibHbiX K03cpcpiill,HeHTaX pa3JIO)I(eHHH j{ B

pH,ll; no CTeneHHM b; :

X= X2 +Xint> Xint = Jf3 +Jf4 + · · · (5.22)

<l>H3H'IeCKHH CMbiCJI Jf3, Jf4 JierKO noHHTb J-13 aHaJIOrHH C KBaHTOBOH MeXaHH­

KOH. faMHJibTOHHaH Jf3 onHCbiBaeT TpeXBOJIHOBbie npoueccbl. B npocToM

CJiyt~ae O,ll;HOrO THna BOJIH

Jf3= !-2 ~ (Vqblb2bj+K .c.)8(1 - 2 - 3)+ 123

+1/ 3 ~ (Uqbibibj+K.c.)8(1+2+3) .

123 (5 .23)

3,ll;eCb H ,ll;aJiee Mbl 6y,ll;eM HCnOJib30BaTb UeJibiH pH,ll; COKpameHHbiX 0603Ha'le­

HHH: bt. b2 , .. . ecTb b (k.) , b (k2) , 8 ( ... ) - CHMBOJI KpoHeKepa, 8(1 + 2 + 3) eCTh 8 (k1 t k2 t k 3 ), 'ITO Bhipa)l(aeT 3aKOH COXpaHeHHH HMnyJihca . .Qanee,

MYJihTHHH,ll;eKc q = (k1 , k2, k 3 ) , TaK 'ITO Vq = V123 = V(k 1 , k2, k 3 ) . HaKoHeU,

~ = ~ ~ - ~ o(k1 +k2 - k3), ~ = ~ . 123 klk 2k3 1 + 2=3 klk2k3 1234 .klk2k3k4

faMHJibTOHJolaH Jf4 OTIHCbiBaeT npoueCChl C y'laCTHeM '!eTbipex BOJIH:

Jf4 = !-2 ~ Wpbibib 3b4 + U! ~ (Gpb 1bib)b.j+K.c.) + 1+2 = 3+4 1=2 + 3+4

+ ~ ~ (Rpb 1b2b3b4 +K.c.) , p=(k1 , k2, k 3 ,k4), 1+2+3+4=0

~ - ~ 8(kt +k2 - k3 - k4)· 1+2=3+4 kl , k, , k3 , k4

(5.24)

Mo)l(eT B03HHKHYTb sonpoc - ,11;0 KaKoro nopH,ll;Ka no b, b• crre,11;yeT pa3naraTb

r aMHJibTOHJolaH Jf? Ha TaKOH 06ll(JoiH BOnpoc MO)I(HO ,ll;aTb CTOJib )l(e 06IUHH

OTBeT : "Jf5 H 'IJieHhi 6orree BhiCOKoro nopH,ll;Ka , soo6me rosopH, y'IHThmaTh

He CJie,ll;yeT". ApryMeHTHpyeTCH OH TaK: nOCKOJibKY paJJIO)I(eHJole B pH,ll; npo­

HCXO,ll;HT no MaJIOMY napaMeTpy, Ka)l(,ll;hiH nOCJie,ll;yiOIUHH 'IJieH MeHbWe npe­

,ll;hl,ll;yll(ero H ,ll;HHaMHKa CHCTeMhl BOJIH 6y,11;eT Onpe,ll;eJIHTbCH caMbiM nepBbiM

'IJieHOM pa3JIO)I(eHJoiH B j{int• T.e. Jf3. 0,ll;HaKO TpeXBOJIHOBbie npoueCChl MOryT

OKa3aTbCH 3anpemeHHh!MH 3aKOHaMH COXpaHeHHH 3HeprHH-HMnyJibca . .QeHCT­

BHTeJibHO, eCJIH 3aKOH ,ll;HCnepCHH BOJIH Wk - KBa,ll;paTH'IHhiH CO ll(eJibiO :

Wk = w 0 + cxk2 , (5 .25)

TO ,ll;JIH MaJihiX k TpeXBOJIHOBhie npouecchi pacna,11;a

w(k) = w(k') + w(k- k') (5 .26)

3anpemeHhi, H6o BOJIHa c MHHHMaJibHOH JHepmeii He MO)I(eT ,11;anee pacna,ll;aTb­

cH. B JTOM cny'lae rosopHT, 'ITO cneKTp .sorrH HBJIHeTcH Hepacna,ll;HhiM. Eoiiee

,ll;eTaJibHhiH aHaJIH3 (5.25) H (5 .26) noKa3hmaeT, 'ITO TpeXBOJIHOBhie npouec­

Cbi (5 .26) 3anpemeHhi npH cxk2 < 2w0 • Mo)I(HO noKa3aTb, 'ITO Hepacna,ll;HhiMH

HBJIHIOTCH TaK)I(e CTeneHHhie 3aKOHbl ,ll;HCnepCHH 6e3 ll(eJIH : Wk "- k!, KOr,ll;a

~ < 1. Bo scex JTHX cny'laHx HY)I(HO yt~eCTb cne,ll;yiOIUHH 'IJieH pa3JIO)I(eHHH

Xint• T.e . Jf4 .

25

Ba:>KHO, 'ITO rrpoueCCbl C ytJaCTHeM 'IeTbipex BOllH, OITHCbiBaeMbie JTHM ra­

MHnbTOHHaHOM, pa3pemeHbl Bcerna. 3TO BH}l;HO H3 3aKOHOB COXpaHeHH.R Jl;n.R

rrpoueccoa pacceHHHH :

w(k) + w(k') = w(k + K) + w(k' + K), (5.27)

KOTOpbie pa3pemeHbi rrpH K -+ 0 rrpH mo6oM JaKoHe Jl;HcrrepcHH. CnenosaTenb­

HO, JC4 H 6yneT orrpenen.RTb J];HHaMHKY CHCTeMbl BOllH C Hepacrra}l;HbiM

crreKTpoM, a cnenywmue tJneHbi paJno:>KeHHH JC5 , JC6 , •.• 6yJl:YT orruchiBaTb

Manhie H, KaK rrpaaHno, HecymecTBeHHbie rrorrpaBKH.

PaJyMeeTcH, peanbHa.R neiicTBHTellbHOCTb H Ha ceii pa3 6oratJe noli cpop­

ManbHOH cxeMbi. HarrpHMep, Jl:n.R CB B <I>M na:>Ke B pacrra}l;HOH tJaCTH crreKTpa

He06XO}l;HMO Y'IHTbiBaTb He TOllbKO JC3, HO H JC4. lleno B TOM, 'ITO raMHnbTO­

HHaH JC3 . B03HHKaeT TOllbKO H3-3a MarHHTHOrO }l;HITOllb-}l;HITOllbHOrO B3aHMO­

}l;eHCTBHJi H HMeeT pell.RTHBHCTCKYIO ManOCTb B cpaBHeHHH C raMHnbTOHHaHOM

JC4 , o6ycnoaneHHbiM o6MeHHbiM aJaHMoneiicTBHeM. B JanatJax o J];HHaMHKe

BOllH Ha ITOBepXHOCTH :>KH}l;KOCTH H3BeCTHbl cnytJaH, KOr}l;a He06XOJ];HMO Y'IHTbi­

BaTb JC5 . TeM He MeHee, ecnH HeT crreuuanbHbiX o6cToHTenbCTB, B pacnanHoH

qacTH crreKTpa nocraTO'IHO ytJHTbiBaTb TOllbKO TpexaonHOBhie rrpoueccbi JC3 ,

a a Hepacrra}l;HOH - tJeTbipexaonHOBbie JC4 .

5.4. ,UHHaMitlleCKlUI TeOpH.R B03MyiUeiiHA - HCKJliO'IeHHe "uepe30HaHCHbiX"

tJneHOB B I'aMHlli>TOHHaHe. 11HTYHTHBHO .RCHO, 'ITO rrpH Hepacrra}l;HOM JaKOHe

Jl;HCrrepcHH, Korna TpexaonHOBbie rrpoueccbi 3arrpemeHbi JaKOHaMH coxpaHe­

HHH, raMHnbTOHHaH B3aHMO}l;eHCTBHJi JC3 , OITHCb!BlliOJUHH J:rH rrpoueCCbl, B

KaKoM-TO CMbicne oKaJ~;>maeTcH HecymecTBeHHbiM. CeiitJac Mbi rroKa:>KeM,

'ITO B JTOM cnyqae MO:>KHO rrepeHTH K HOBbiM KaHOHH'IeCKHM rrepeMeHHbiM

Ck, Ck TaKHM, 'ITO JC3{Ck, Ck} = 0. flpH 3TOM KBa}l;paTHtJHa.R tJaCTb raMHnbTO­

HHaHa coxpaHHeT rrpe:>KHHH BHil:

(5.28).

a B 'IeTbipeXBOllHOBOM raMHnbTOHHaHe B3aHMO}l;eHCTBH.R B03HHKHYT }l;OITOJlHH­

TellbHbie cnaraeMbie, KBa}l;paTH'IHbie ITO MaTpH'IHb!M JJleMeHTaM JC3 { bk, bic} B CTapbiX rrepeMeHHb!X. 3Ta Uellb J];OCTHraeTC.R KBa3HnHHeHHbiM rrpeo6pa30-

aaHHeM

bk = Ck- I:{Uk12cici8(k+ 1 +2)(wk +w1 +w2)-1 + 12

+ CVk, 12CiC2 - 2 v2,Hc[c2)8 (k- 1 - 2)(wk - WJ - w2)-1}. (5.29)

3Jl;eCb k =: - k. flpeo6pa30BaHHe (5 .29) JiBll.ReTCJi KaHOHH'IeCKHM rrpH6lliDKeH­

HO, C He06XO}l;HMOH TO'IHOCTbiO. B 3TOM MO:>KHO y6e}l;HTbCJi C ITOMOJUbiO COOT­

HOilleHHH (fl2.6) H (fl2 .7).

flO}l;CTaBJl.RJi (5.29) B raMHllbTOHHaH (5.22), nerKO y6e}l;HTbC.R, 'ITO }l;eHCT­

BHTellbHO JC2 { Ck, Ck} COXpaHHeT BH}l; (5 .28) , JC3 = 0, a MaTpH'IHbie JlleMeHTbl

JC4 rrpuo6peTaiOT norronHHTenbHbie cnaraeMhie . Mbi BbiiTHilleM HX TOnbKO nnH HaH6onee cymecTBeHHbiX Jl:n.R Hac B nanbHeiimeM rrpoueccoa pacceHHHH 2 -+ 2:

JC4= ~ 2,; T1234cicicJc4, 1+2=3+4 '

26

u <1 + 2)12ut3 +4)34 T =W -2--------2

P P w(3+4) + w3 + w4 w(1+2)- wl- w2

v1 ,3<1- 3) v4,2<4- 2) v ;,4<2-4) v3,1 <3-I) -2---------2--------

w(4-2) + w2 - w4 w(3-l)' + wl- w3

v2·,3<3 -2) v4,1 <4-1) Vt,4<1-4) v3,2<3- 2) -2---------2--------

w(3 - 2) + w2- w3 (5.30)

3,D;eCb (i ± i) = kl + kt . nonoJIHHTeJlbHbie cnaraeMbie B Tp OllHCbiBaiOT npouecCbi

pacceHHIDI, B03HHKaiOUIHe BO BTOpOM nOpH,D;Ke TeOpHH B03Myi.lleHHH llO Tpex­

BOJIHOBbiM npOUeCCaM. IlpH 3TOM Ha npOMe)!(yrO'IHOM nane B03HHKaeT "BHp­

TyanbHaH" Bbi~eHHaH BOJIHa, ,D;nH KOTOpOH He BbinOnHHeTCH ycnOBHe pe30-

HaHCa. B noli HHTepnpeTaUHH Ka)!(,ll;OMY cnaraeMoMy B (5.30)' MO)!(HO conocTa­

BHTh KapTHHKy:

1 3 1 J 1 J 1 3

X=~ +2>----< 2 4 2 4 2 *

+2 )--{+ 2 4-

+:X +Z X +z 'X +z X' 2 4 2 4- 2 3 2 J

(5.31)

B 3aKJIIO"'eHHe He06XO,ll;HMO HaUOMHHTb·, "'TO BCH OllHCaHHaH npoue,D;ypa npHH·

UHTIHanbHO OCHOBaHa Ha TOM, tiTO npeo6pa30BaHHe (5.29) 6nH3KO K JIHHeHHO·

MY. "'TO B03MO:li<HO, ecnH "'a"TOTHbie 3HaMeHaTenH He o6pamaroTCH B Hynb. A ,D;nH

JTOrO He06XO,ll;HMO, "'T06bi ClleKrp BOJIH 6bm HepaCUa,D;HbiM. llpyrHMH CJIOBaMH,

KaHOHH"'eCKHMH npeo6pa30BaHIDIMH MO)!(HO HCKJIIO"'HTb TOJlbKO Hepe30HaHCHbie

lJJieHbi raMHnbTOHHaHa B3aHMO,D;eHCTBIDI, ,D;nH KOTOpbiX 3aKOHbi COXpaHeHIDI

3HeprHH·HMnyJlbca He BbinOJIHHIOTCH. MO)!(HO TaK)!(e CKaJaTh, "'TO HaHn}"llllHMH

B KBa3HJIHHeHHOM CMbiCJie KaHOHH"'eCKHMH nepeMeHHbiMH HBnHIOTCH TaKHe,

,[),.JIH KOTOpbiX raMHnbTOHHaH He CO,D;ep)!(HT cnaraeMbiX, COOTBeTCTByrollUIX 3anpe-

meHHbiM npoueccaM.

§ 6. faMHnbTOIDI8H CDHHOBbiX BOJIH B clleppOMarHe111K8X

Mbi ~e OTMe"'anH B nepBOH rnaBe, "'TO KnaCCH"'eCKHM 061>eKTOM ,D;nH H3Y"'e­

t1HJI HenHHeiiHoii AHHaMHKH CB HBnHeTcH Ky6H"'eCKHH clleppHMarHeTHK - )!(eJie-

30-HTTpHeBbiH rpaHaT - )I<J-ff. IlpH HH3KHX TeMnepaTypax OTHOCHTeJibHbie ,ll;BH­

)f(eHIDI MarHHTHbiX IJO,D;pemeTOK ( OUTH"'eCKHe MarHOHbi) He B036~,D;eHbi, fi03TOMY MO)!(HO C"'HTaTb, "'TO )I<J-ff HBJIHeTCH <J>eppoMarHeTHKOM. J1MeHHO

no:noMy Mbi Ha"'HeM c raMHJlbTOHHaHa CB B <I>M c Ky6H"'eCKOH CHMMeTpHeii.

pHeHTHpyHCb Ha 3KCnepHMeHT, 6y,D;eM C'IHTaTb, BO-nepBbiX, "'TO BHeiliHee Mar-

1111TIIOe none H HBnHeTcH npocrpaHCTBeHHO-OAHOPOAHbiM. Bo-BTOpbiX, "'TO

27

vopa3eu HMeeT <t>opMy 3JIITHrrcoH,IJ;a. Torna MarHHTHOe rrorre BHYJPH Hero TaK)I(e

6y,ll;eT rrpOCipaHCTBeHHO-O,ll;HOpO,ll;HbiM [3) :

(6.1)

r,ll;e M - HaMarHHtleHHOCTh, a N a{J - TeH30p pa3MarHH'IHBaiOll.IHX K03<l><l>HUHeH­

TOB, CYMMa ,ll;HaroHaiThHhiX 3ITeMeHTOB KOIOporo paBHa e,ll;HHHUe. HaH6orree

TeXHOITOrJ.ftlHbl B H3rOTOBITeHHH o6pa3Ubl B <t>opMe c<t>epbl, ,ll;ITHHHOrO UHITHH,ll;pa

H IIJIOCKOrO ,ll;HCKa. ).lTUI c<t>epbl, ,ll;JIHHHOrO (B,!J;OJlb OCH Z) UHJIHH,ll;pa H ,ll;JI.R

ll1IOCKOrO (rrorrepeK OCH Z) ,ll;HCKa HMeeM COOTBeTCTBeHHO

Na{J = 1/3 Oa(J, (6.2)

Nz =Nzz = 0, Na{J = 72oaf.l• a, (3 = x, y,

Nz =Nzz = 1, Na{J = 0, a, (3 =X, y.

(6.3)

(6.4)

J1 HaKOHeU, orpaHJ.ftlHMCH CIT}"'aeM, KOr,ll;a IIOJie H OpHeHIHpOBaHO IIO O,ll;HOH H3

rrraBHbiX OCeH 3111IHIICOH,IJ;a, KOIOpaH B CBOIO O'lepe,ll;b COBJla,l];aeT C O,ll;HOH H3 OCeR

CHMMeipHH KpHCTarrrra: OCbiO 'leTBepTOrO (100), IpeThero (111) HJIH BIOporo

rropH,ll;Ka ( 11 0). 3Toro ,ll;OCTaTO'IHO ,ll;JIH Toro, 'IT06bi HaMarHJ.ftleHHOCTb M B He­

B036y)l(,ll;eHHOM <I>M 6hma rraparrrrerrbHa H. 6.1. CneKTp CDHHOBLIX BOTIH. IJpHBe,ll;eM 3,ll;eCb pe3yiThTaT Bbi'IHCITeHHH 'laCTO­

Tbl CB, rrpo,ll;erraHHoro B rrocrre,ll;yroll.IHX rraparpa<t>ax:

wi = Ai -I Bk 12, Bk = YlwM sin2 8e 2

i'P + f3wa, (6.5)

Ak = WH- wMNz + awa + Wex(ak) 2 + 72wM sin2 8. (6.6)

3,ll;eCb 8 H '{J - IIOITHpHbii.f' H a3HMYTaiTbHbiH yrrrbl BeKIOpa k B c<t>epHtieCKOH CH­

CTeMe KOOp,ll;HHaT, OCb KOIOpOH OpHeHTHpOBaHa B,ll;OITb M, yrorr '{J OTC'IHTbiBaeT­

CH OT HarrpaBJieHHH (100) B IIJIOCKOCTH, rreprreH,ll;HKYITHpHOR M, WH - Kpyro­

BaH 'laCTOTa rrpeueCCHH MarHHTHOrO MOMeHTa BO BHeiiiHeM IIOJie:

WH = gH, (6.7)

r,ll;e g = J.l.o/h - MarHeTOMeXaHJ.ftleCKOe OTHOUieHHe ,ll;JIH 3JieKipOHa, J.l.o - MarHe­

TOH Eopa, g = 27T · 2,8 Mru/3 - BeiTHtiHHa pa3MepHaH, ee He crre,ll;yeT rryTaTh c

6e3pa3MepHbiM g-<t>aKIOpOM, 1IpHMepHO paBHbiM ,ll;BOHKe y 31IeKipOHa.

qaCTOTbi WM H Wa XapaKiepH3YIQT MarHHTHOe ,ll;HIIOJib-,ll;HIIOJibHOe B3aHMO­

,ll;eHCTBHe H KpHCTaiTITOrpa<t>HtieCKYIO aHH30Ip01IHIO:

WM = 47TgM,

Wa =gHa.

(6.7)

(6.8)

CBH3h rrorrH aHH30ipOIIHH Ha c o6merrpHHHTOH KOHCTaHTOH Ky6HtieCKOR aHH30-

IpOIIHH 'leTBepioro rropH,ll;Ka K orrpe,ll;erreHa B § 10 <t>opMyrroii (10.3). Ko3¢­

<l>HUHeHThi a H (3 3aBHCHT OT OpHeHTaUHH MarHHTHOrO IIOJIH H (H COBila,l];aiOll.lero

C HeH HarrpaBITeHHH HaMarHJ.ftleHHOCTH) OTHOCHTeJibHO KpHCTaiTJIOrpa<t>HtieCKHX

oceif ( Ta6rr. 6.1). qaCTOTa Wex xapaKiepH3yeT 06MeHHOe B3aHMO,ll;eHCTBHe, IIO IIOpH,ll;Ky BeJIH­

'IHHbl hwex - Tc, r,ll;e Tc - TeMrrepaiypa MarHHTHOro ynopH,ll;O'IeHHH, o6br<rnO

Ha3hiBaeMaH TeMnepaiypoif KIDpH, a - nocTOHHHaH peUieTKH. BerrH'IHHy

Wexa2 /2gM HHOr,ll;a Ha3hiBaiOT KOHCTaHTOH HeO,ll;HOpO,ll;HOrO 06MeHa H 0603Ha­

'laiOT 6yKBOH a.

28

T a 6 n H u a 6.1

Ko3$11JuUHeHnl, xapaxTepH3YIO~e BKJJ8,D; 3HeprHH aHH30TpOIIHH a .cneKTpCB (6.5) H (6.25)

OpHeHTa­UHll Ha­MarHH'IeH­HOCTH

[100] [111]

[110]

1 - 2/3

- 1/2

0 0

- 3/2

-9 6

9/2

0603Ha'leHIDI B <j>opMyrre (6.5) CIIel(HaiibHO BblfipaHhl TaK, 'ITOfibl 6bl1IO

yp;o6Ho cpaBHHBaTb BKIIaAbi pa3IIH'IHbiX BJaHMop;eiicTBHH B 'laCTOTY CB wk (a BIIOC1Ie/];CTBHH H B KOJ<j><j>Hl(HeHTbl Jfint). B Ka'leCTBe XapaKTepHOrO.MaCWTa-

6a yp;o6HO Bbi6paTb 'laCTOTY WM H COOTBeTCTByrol.llee eH MarHHTHOe IIOIIe 41TM. ,llJul )l(}lf rrpH KOMHaTHOH TeMrrepaType 47TM= 1,75 K3, Hex = Wexfg = 390 K3,

Ha = 0,084 K3. IlpH .aHarrH3e JKcrrepHMeHTaiibHbiX p;aHHbiX r:rorreJHO IIOMHHTb,

'ITO nomo H = 1 K3 coorneTCTByeT IIHHeiiHa.ll qacToTa v = w/21T, paBHa.ll

2,8. 109 ru, KpyroBa.ll qacTOTa 1,76. 10 10 c-1 H TeMIIeparypa 0,13 K. BH/l;HO,

'ITO B Bbi6paHHOM MacWTa6e Ky6H'leCKaH aHH30TpDIIH.II ·:>IUif .IIBII.IIeTC.II MaiiOH :

wafwM = 0,05. Marra oHa TaK)I(e H B pap;e APY!'HX Ky6H'leCKHX <DM.

B rrepBoM rrpHfirrH)I(eHHH TaKHe KpHCTa111Ibr MO)I(liiO c'IHTaTb H30TpOIIHbiMH.

3TO, 0/];HaKO, He 03Ha'laeT, 'ITO KpHCTaiiiiOrpa<j>H'leCKOH aHH30rpOIIHeH MO)I(HO

IIOIIHOCTbiO rrpeHefipe'lb. Brrocne/];CTBHH Mbl yBH/l;HM, 'ITO HeKOTOpble .llpKHe rrpo­

.IIBIIeHIDI • HenHHeiiHoii p;HHaMHKH CB rrpHHUHIIHarr&Ho CB.II3aHbr c HaiiH'IHeM

aHH30rpOIIHH.

06MeHHaH 'laCTOTa Wex cymecTBeHHO rrpeB&rwaeT WM. TaK AII.II >Kl1f

Wex/WM :::::> 220. 113 <j>OpMy1Ibl (6.5) BH/];HO, 'ITO MO:lKHO OIIpep;e1IHTb XapaKTep­

HOe 3Ha'leHHe BOIIHOBOrO BeKTOpa kM, IIpH KOTOpOM cpaBHHBaiOTC.II BK1Ia/];bl

/];HI101Ib-/];HI101IbHOro H 06MeHHOro B3aHMO/l;eHCTBHH

(6 ,9)

.Ilrr.11 )I{Yif akM = 0,07 H COOTBeTCTBeHHO kM = 7 ' 105 CM-

1• B xapaKTepHbiX

JKcrrepHMeHrax no rrapaMerpH'leCKOMy . B036~p;eHHIO CB qacTora HaKa'IKH

1Ie)I(HT B /l;Haiia30He 10 HIIH 36 ffu ( 1 ffu = 109 fu), rrpH 3TOM B 3aBHCHMOCTH

OT aenH'IHHbi H BOIIHOBOH BeKTOp CB MO)I(eT H3MeH.IITbC.II OT 5 · 103 p;o

5 ' 106CM-

1, C1Ie/];OBaTe1IbHO, A1J.II. HaC OKaJbiBaiOTC.II cymeCTBeHHbiMH KaK /];H­

II01Ib-/];HI101IbHOe, TaK H 06MeHHOe B3aHMO/l;eHCTBIDI. 3aBHCHMOCTb (6.5) 'laCTO­

Tbl CB OT k .IIBII.IIeTC.II AOBOIIbHO cnO)I(HOH. Ee MO:lKHO H3o6pa3HTb (pHc. 6.1) B BH/];e CeMeHCTBa IIOBepXHOCTeH IIOCTO.IIHHOH 'laCTOTbl. IlpH Wk ~ WM HX <j>op­

Ma 61IH3Ka K c<j>epH'leCKOH:

Wk = WH - WMNz + Wex (ak)2 + ~WM sin2 8. (6.10)

IlpH Wk - WM B03HHKaeT rrepeT.II)I(Ka Ha 3KBaTOpe IIOBepXHOCTH Wk = const, H

OHa HaiiOMHHaeT raHTe1Ib. IlpH eme MeHbWHX k 3Ta IIOBepXHOCTb HaiiOMHHaeT

ABa CKpyrrreHHbiX KOHyca, corrpHKacaiOI.liHXC.II ocrpH.IIM~. KaK BHAHO, rrpH

k ~ kM Ba)f(HYIO pOJib HrpaeT 3aBRCRMOCTb Wk OT HarrpaBITeHIDI k IIpR k ~ 0. 3Ta HeaHaiTRTR'lHOCTb o6yciTOBITeHa ,IJ;aiTbHO,JJ;eHCTBYIOll{HM xapaKTepOM ,li;RllOJib·

,JJ;RnOITbHOro B3aRMO,IJ;eHcTBIDI. 3aKawJRBa.H aHaiTR3 JaKOHa ,JJ;RcnepcRR CB

(6.5), Mbi XOTRM c,JJ;erraTb eme ,JJ;Ba 3aMeqaHRH. Bo-nepBbiX, cneKTp CB RMeeT

merrb. MRHRMaiThHyro qacToTy RMeroT CB c k ~ 0, 8 = 0. B npeHe6pe)l(eHRR aHR-

30rponaeH:

Wmin = WH- wMNz = g(H- 47TN2 M). (6.11)

KaK BR,IJ;HO R3 (6.1), 3Ta qaCTOTa COOTBeTCTByeT npeu;eCCRR MarHRTHOrO MOMeH­

Ta BO BHyTpeHHeM none H;: W M = gH;. IlpR H < 41TN zM <PopMyrra ( 6.11) y)l(e He cnpaBe,IJ;JIRBa. 113 ( 6.5) npR 3TOM

CITe,JJ;yeT, qTO wi CTaHOBRTC.!l OTpRI.J;aTeiTbHOH IIpR k ~0, 8 = 0 R, crre,JJ;OBaTeJlbliO,

a

PH c. 6.1. CeMe~CTBo noBepxHocTe~ nocToliHHo~ tJaCTOThi AJlll H3oTponHoro <fleppoMame­THKa: BHYTpeHHee none Ht = H ·-4rrNzM Bbi6paHo paBHbiM 0,1 oT HM= 4rrM (a); H; = HM (6). flo BepTHKanH OTflOJKeHa BeflH'IHHa (ak11) 2, no ropH30HTanH - (ak1) 2. llH<flpbl Y

KpHBbiX - OTHOWeHHe Wk(W M

qacTOTa CB CTaHOBRTCH qacTo MHRMOH. B CBOIO oqepe)J.b :no 03HaqaeT JKcno­

HeHI.J;RaiTbHbiH poeT aMllJIRTY,JJ; CB, a6o ak(t) <n epx (- iwk t). TaKRM o6pa30M,

IIpR H < 41TN 2 M OAJIOpOAJIO HaMarHaqeHHOe COCTOHHRe <DM OKa3biBaeTCH He­

ycToHqRBbiM, R o6pa3eU pa36RBaeTCH Ha ,JJ;OMeHbi. BTopoe 3aMeqaHRe KacaeTCH

o6rraCTR npRMeHRMOCTR <PopMyiTbi (6.5). 0Ha cnpaBe,IJ;JIRBa TOJibKO npR kL ~ 1, r,JJ;e L - xapaKTepHbiH pa3Mep o6pa3u;a. Ilpa MeHbiiiRX k caMo noHHTRe BOIT­

HOBoro BeKTOpa Tep.HeT CMbiCIT, TaK KaK C06CTBeHHbie KOJle6aTeJlbHbie MO)J.bl

o6pa3I.J;a y)l(e He HBITHIOTC.!l llJlOCKRMR 6eryll{RMR BOITHaMR. B <DM TaKRe MO)J.bl

npRHHTO Ha3biBaTb yoKep08CKUMU .MOOa.MU (KOITe6aHRH HaMarHaqeHHOCTR) •

0HR HBITHIOTC.!l peiiieHReM MarHRTOCTaTaqeCKOfO ypaBHeHIDI

div(H + 4mW) = 0 (6.12) C COOTBeTCTBYIOll{RMR rpaHR'lHbiMR yciTOBIDiMR Ha llOBepXHOCTR o6pa3Ua [3, 7, 34). IlpOCTeHIIIeH yoKepOBCKOH MO,li;OH HBIT.HeTC.!l OAJIOPOAH!UI npeu;eCCIDI

HaMarHaqeHHOCTR - Oil. MO)I(HO cKa3aTh, qTo Oil - :no CB c k = 0. Ee co6-

CTBeHHaH qacTOTa Wo B npeHe6pe)l(eHRR KpRCTaiTITOrpa<PaqecKOH aHR30Tp0llReH

paBHa

w~ = [wH- wMNz + lh.wM(Nx + Ny)]2- [lh.wM(Nx - Ny)]2

• (6.13)

Oco6eHHO npocToe Bbipa)f(eHRe norryqaeTCH ,IJ;JIH c<Pepbr: w 0 = gH. Bbipa)f(eHIDI

,IJ;ITH qacTOT ,ll;pyrRX yoKepOBCKRX MO,IJ; MO)I(HO HaRTH B [3, 7) . 30

6.2. OueHKa seJIH'IHHLI Ma'fPH'IHLIX 311eMeHTOB r&MHJILTOHHBHB B38HMo­

a;eACTBHJI. Tipe)l()];e BCero paCCMOTpHM BKnap; MarHHTHOrO )J;HnOnb·)J;HnOnbHOrO

B3aHMO)J;eHCTBIDI. KaK XOpOillO H3BeCTHO [3), MOTHOCTh ero JHeprHH W M eCTb

81rWM =Hit, (6.14)

rp;e HM . - :no MarHHTHOe none, coJp;aaaeMoe HaMarHH'IeHHOCTbro M. 113 ypaa­

HeHHH MarHHTOCTaTHKH (6.12) cnep;yeT, 'ITO

HM =- 47TM. . (6.15)

3Ta Ol.(eHKa OT11HtlaeTCH OT TOIJHOrO pe3y11bTaTa (6.1), npHBep;eHHOrO 6e3 Bbl·

aop;a, TOmKo <flaKTopoM N 01tJ :::::: 1, YIJHT&raaromHM <flopMy o6paJua. 113 ( 6.14), (6.15) cnep;yeT

(6.16)

Tipep;nonmKHM Tenep&, 'ITO B <l>M pacnpoCTPaHHeTcH CB, aMnnHTyp;a KOTopoif

xapaKTepH3yeTCH yrnoM npeueccHH MarHHTHoro MOMeHTa ~ k . PaJnaraH W M B

pHp; no h, nonyqHM

WM = Wo + W2 + W3 + ... "'=' Wo + 27TM 2 (A~k + B~~ + qz + .. . ). (6.17)

3p;ec& 6e3pa3MepHbie IJHcnoabre KQJQJQJHUHeHThi A, B, C nopHp;Ka ep;HHHUbi,

H6o npH ~ - 1 BCe '111eHbi pHp;a p;OmKHbl 6biTb Op;HOrO nOpHp;Ka BenHIJHHbl.

C p;pyroif CTOpOHbl, KBaJJ;paTHIJHOe no h cnaraeMOe )J;OmKHO HMeTb BH)J;

JC2 = w I ak 12

• 3Aecb xapaKTepHaH 'laCTOTa w He MO)I(eT 6&ITb HH 11eM HHbiM,

KpOMe KaK 'laCTOTOH npeueCCHH MarHHTHOrO MOMeHTa B none HM, T.e.

w = WM. CpaBHHBaH w2 H J(2' HMeeM

~k "'='ak(2g/M)112• (6.18)

ilO)J;CTaBnHH h B pa3nO)I(eHHe ( 6 .17) )J;JIH W M H onycKaH HecymeCTBeHHble

'IHcnoabte KOJQJQJHUHeHThi B H C, nonyqaeM oueHKH )J;JIH p;Hnon&-p;Hnon&Horo

BKnap;a B MaTPH'IHbte :meMeHTbi TPeX· H 'leTbtpexaonHoaoro raMHnbTOHHaHa

B3aHMO)J;eHCTBHH:

VM "'='wM(2gjM) 112 ~wM(g/2M)1 f2, (6.19)

(6.20)

Cnpasa OT CTPenKH npHsep;eHbi TO'IHbie Bbtpa)l(eHHH )J;JIH MaTpH'IHbiX JneMeHTOB.

0TMeTHM, 'ITO OUeHKH (6.19), (6.20) (c TOIJHOCTbiO )J;O IJHCen) MO)I(£10 nOnYIJHTb

npOCTO H3 C006pa)l(eHHH pa3MepHOCTH . .IlrrH 3TOrO H3 BenH'IHH, XapaKTepH3YIO·

mHX Jap;aqy, - HaMarHH'IeHHOCTH M H MarHHTOMexaHHIJecKoro OTHorneHHH g -

He06XO)J;HMO nOCTPOHTb KOM6HHaUHH M 01 g (J, HMeiOli.(He pa3MepHOCTH MaTPH'I·

HblX JneMeHTOB V H W. MO)I(HO nOKaJaTh, 'ITO OCTaJibHbie B3aHMO)J;eHCTBHH - 06MeHHO(! H KpHCTa11·

norpa<flH'IecKoif aHH30TPOnHH - He npHBO)J;HT K TPeXBonHOBbiM npoueccaM.

OueHKY )J;JIH HX BKnap;a B JC4 MO)I(HO nOnYIJHTb H3 (6.20), 3aMeHHH W M

Ha COOTBeTCTByroiUHe xapaKTepHble IJaCTOTbl

Wex "'=' Wex(ak)2 g/2M, W0 "'=' gw0 /2M. (6.21)

Tiony'leHHblX 3/];eCb Ol.(eHOK MaTpH'IHbiX :meMeHTOB raMHJibTOHHaHa B3aHMU·

p;eiicTBHH 6btBaeT p;ocTaTO'IHO )J;JIH Bbi'IHcneHHH rpy6Lrx xapaKTepHCTHK CB,

HHTerpanbHO 3aBHCHIUHX OT MaTPHIJHbiX :meMeHTOB JCint, HanpHMep, TeMnepa-

31

TypHoii JaBHCHMOCTH HaMarHaqeHHOCTH, qacTOTbi H JaTyxaHHH CB. 0AHaKo

npyrHe HeJIHHeHHbie xapaKTepHCTHKH CIIHHOBOH AHHaMHKH, 0 KOTOpbiX 6yAeT HATH peqb B IIOCJieAyiOJUHX rraparpa<t>ax, MOryT KapAHHaJibHO JaBHCeTb OT TOH· KHX CBOHCTB MaTP~biX JJieMeHTOB JCint - AeTaJieH HX yrJIOBbiX JaBHCH· MOCTeii H T.II. IIo:noMy ocTaeTcH eAHHCTBeHHhiH, XOTH H He oqeHb rrpHHTHbiH rryTb - aKKypaTHO Bbi~CJIHTb JCint, yqHTbiBaH BCe CKOJibKO·HH6yAh JaMeTHbie B3aHMOAeHCTBHH: 06MeHHOe, AHIIOJibHOe, aHH30TPOIIHIO H T.A. Bee JTO 6yAeT IIpOAeJiaHO B § § 8, 9. 3AeCb Mbl IIpHBeAeM TOJibKO pe3yJibTaTbl Bbi~CJieHHH .

6.3. TpeXMaruomu.dt raMHJibTOIDiaH. BbiqacrreHHH rroKa3biBaiOT, qTo KOJ<l>· <l>HJ.J.HeHTbi raMHJibTOHJ:taHa JC 3, JaAaBaeMoro <t>opMyrroii (5 .23), HMeiOT crre­

AYIOJUHH BRA:

V1 23 = *[(V3 + V2)u1u2u3 + (v; + v;)u; u2u3 + (V1ui + v;u 1) * ' * (u2u3 + u2u 3) + ui(V2u 2u3 + V3u2u 3) + u1(V;u2u3 + Vju2u3)],

U123 =*[Vtul(u2u3 +u2u3)+ V2u2(ulu3 +ulu3)+

+ V3u 3(u1u 2 +u1u2)+ v;u1(u2u3 +u2u3)+

+ v;uiutu3 +ulu3)+ v3·u3(ulu2 +ulu2)],

V1 = V(kt) , .. . , V(k)=- Vkz(kx + iky)/k 2 = = - V sin (J k cos (J k exp(i~Pk), V = wM (g/LM) 112

,

(6.22)

(6 .23)

Uk H Uk - KOJ<P.<t>HJ.J.HeHTbl JIHHeifHoro KaHOH~eCKOrO AHarOHaJIH3YIOJUero rrpeo6pa30BaHHH .(5.19), B KOTopoM qacTOTa CB wk H KoJ<t><t>aJ.J.HeHTbi Ak H

Bk 3aAaiOTCH <t>opMyrraMH (6.5). BHAHO, qTO no IIOPHAKY BeniMiHbl vl,23

orrpeAerrHeTCH rrpocTOH ou;eHKOH ( 6.19) . .llnH AaJibHeiiwero, OAHaKo, 6yAyT cymecTBeHHbi yrrroBbie 3aBHCHMOCTH Vt, 23. a qacTHOCTH TO, qTo Vt, 23 o6-

pamaeTcH B HYJib, ecrra ace norrHOBbie aeKTOpbi k 1 , k 2 , k3 opaeHTHpOBaHhi BAOJib HJIH rrorrepeK M, T.e. (Jk = 0 HJIH (Jk = 1T/2.

6.4. qeTblpeXM8I'HOIDibdt 1"8MHJibTOIDIIlH B38HMOAeACTBHJI. 06mee Bbipa)l(e­HHe AJ1H MaTp~biX JJieMeHTOB JC4 (5.24) CJIHiliKOM CJIO)I(HO, qT06bi ero 3AeCb rrpHBOAHTb. IIo:noMy Mbi paccMOTpHM HecKOJibKO rrpeAeJibHbiX crryqaea. l1MeH­HO TeX, KOTOpbie IIOHaA06HTCH B AaJibHeHilieM . HaqHeM C CHTyaJ.J.HH, KOrAa WM

Marro B cpaBHeHHH c wH HJIH Wex (ak) 2• IlpH JTOM Ak ~ I Bk I, H B JC4 MO)I(HO

He AeJiaTb u- u-rrpeo6pa30BaHHH, cqHTaH, qTo ak ~ bk. TorAa

w12. 34 = El2 + E34 + ~(c13 + cl4 + c23 + c24) ­

- ~(D 1 + D2 + D 3 + D4) + Wa,

Ekk' =- Wex a2 k. k'gfLM, ckk' = C(k- k'),

C(k) = (gjM)wM(kzfk)2, C(O) = wMNz gjM,

Dk =I Bk ig/M= (g/M)wM(k+/kl , k+ = kx + iky ,

Wa = (jwagfLM.

3HaqeHHH {j rrpHBeAeHbi B Ta6JI. 6.1,

Gl 234 = _1/3. (gfM)(B2 + B3 + B4). '

32

(6.24)

(6.25)

0rKa3aBUIHCh or npe)J,nono)l(eHIDI Ak ~ I Bk I, Mhi JanHUieM Bhlpa)l(eHIDI ,IJ,n.l!

W 12, 34 B cnyqae, KOf)J,a BOliHOBbiC BCKTOpbi k I , k2 , k3 H k4 y)J,OBliCTBOpH!OT

CllCl(HanhHhiM COOTHOUICHIDIM

Sw(k, k') = W(k, -k; k', -k') = W1(uiui· + VfVf•) + 4W2 ukuk'vkvk' ­

- ~(g/M){vkuk ' [ukuk ·(Bk + 2Bt•) + vkvk ·(Bk' + 2Bt)l +

+ Vt •Uk [ukuk·(2Bk + Bk ·) + VkVk ·(2Bk' + Bk))} ,

Tw(k, k') = ~ [W(k, k' ;k, k') + W(k, -k'; k, -k')) = =.2 W1 Reukuk'vkvk' + W2 (uf +I vk l2)(uf• + lvk•l 2) -

- ~(g/M) [(uf + I vk 12) Re(Bk + 2Bk' )v;. uk +

+ (uf • +I Vk•I 2 )Re(2Bk + Bk·)vkuk·],

W1 = [wex(ak)2- wM)g/2M+ Wa,

W2 = wM(Nz- l)g/2M + Wa·

(6.26)

(6.27)

(6.28)

KpoMe roro, B 'ICThipeXBonHosoe B3aHMO)J,eiicTBHe cyUieCTBCHHbiH BKna)J,

BHOCHT 'lnCHbi TpeXBOliHOBOfO raMH1IhTOHHaHa BO BTOpoM nopH)J,KC TCOpHH 803-

MYIUCHHH. 06max <t>opMyna ,IJ,nH :noro BKna)J,a npHBe)J,eHa s § 5. 3)J,ecb · Hac

HHTepecyer ee qacTHbiH cnyqaii npH k2 = k 3 , k 1 = k4 ; k 2 = -k1 , k4 = -k3 :

S12=Sw(k1 ,k2 )-2U• - U~22 (w0 +2wkr 1-

011

- 2Re v~ll Voi2 (wo - 2wkrl - 4 V\ 25 Vi 15/w5- 4 vl 26 Vi 15fw6, ' ' ' ' (6.29)

T12 = Tw(k1 , kz) - 1il 21 61 2(w6 + 2w1r 1 -I U125l 2(w5 + 2wkr 1 -

-I v6,t212(w6- 2wkrl -I v5,t21z(w5- 2wkrl-

- - 2 2 - 4Re(V1,10 V2, 20)/w0 - 2(1 V1,25 1 +I V2,1sl )/w5 -

- 2(1 vt2612 +I v2t612)/w6 . ' '

(6.30)

3,1J,ech k 5 = k 1 - k2 , k 6 = k 1 + k 2 , w0 - qactoTa Oil, H npe)J,llonaraeTCH, 'ITO

w (kJ) = w (k2 ) = w (k). BTopoe H TpeThe cnaraeMbre B (6.30) o6ycnosneHbi

BJaHMO)J,eiicTBHeM nap CB c ±k1 H ±k2 qepe3 "sHpTyanhHYJO" Oil, nocne)J,HHe

)J,Ba cnaraeMbiX - sJaHMO)J,CHCTBHeM qepe3 "BHpTyanhHbre" CB c k = k 1 ± k 2 •

Ilo nopHAKY senH'IHHbi cnaraeMbre (6.29), (6.30) TaKHe )l(e, KaK H Henocpe)J,- ·

CTBCHHbiH BK1Ia)J, )J,Hll01Ib-)J,Hll01IbHOfO B3aHMO)J,CHCTBHH B W, T .e. llOpHro<a 1T g2

B ll01IHOM COOTBCTCTBHH C OUCHKOH (6.20). 0)J,HaKO B Ba)I(HOM 'laCTHOM C1IY·

qae, KOr)J,a k 1 1M H k 2 1M (T.e. 8 (k 1 ) = 8 (k2 ) = rr/2), 'lnCHbi TpeThero no­

pli)J,Ka B006IUe He BHOCHT BK1Ia)J, B S (k, k') H T (k, k') . 8a)I(HO OTMCTHTh, 'ITO B <l>opMynax (6.24)- (6.28) 'laCTOTa aHH30TpOllHH W0

YMHO)I(aCTCH Ha 'IHC1IOBOH KOJ<l><l>HUHCHT 9 + 11, B TO B peMH KaK W M )J,C1IHTCH

OOTIOJiaM. TaKHM 06pa30M, OTHOCHTC1IhHbrH BK1Ia)J, Wa B cpaBHCHHH C WM

YBC1I H'IHB3CTCH B 20 pa3, 'ITO npHBO,IJ,HT K CH1IhHOH KpHCTannorpa<l>H'IeCKOH

8HH30 TpOllHH 'ICTbipeXB01IHOBOfO B3aHMO)J,CHCTBHH B "nO'ITH H30TpOllHbiX"

$M C ManbiM OTHOUICHHCM Wa/WM· HanOMHHM, 'ITO B KnaCCH'ICCKOM $M -)I{Jilf Wa/WM ""< 1/20.

. B.c. J I bO OB 33

B Bbtpa:>KeHHH )J.JIH W P BXOAHT pa3MarHH'IHB3,IOIUHH <t>aKrop Nz (z - HanpaB­neHHe HaMarHH'IeHHOCTH) o 3ro npHBO,tU{T K 3aBHCHMOCTn W OT cl>opMbl o6-pa3IIa H HanpaBneHHH HaMarHH'IeHHOCTHo TaK, )J.JIH H:opManbHO HaMarHH'IeHHoro ,u;HcKa Nz = 1, )J.JIH KacarenbHO HaMarHH'IeHHoro AHCKaNz = 00 Bee nepe'IHcnes­Hoe ,u;aer 3KCnepHMeHraropy pe.u;KyiO B03MO:>KHOCTb H3MeHHTb no CBOeMy :>Ke­naHHIO KOJ<l><l>HIIHeHTbi raMHnbTOHHaHa B3aHMo,u;eflcrBHH CBO .llrrn y.u;o6crsa pa6oraiOIUHX c 3KcnepHMeHraropaMH reopeTHKOB MO:>KHO ,u;a:>Ke HanHcaTb sa pYlJKaX ycTaHOBKH, KaKHe HMeHHO K:>Jcl><l>HIIHeHTbl raMHnbTOHHaHa OHH H3· MeHHIOTo

§ 7 o faMWU.TOHH8B CIIHHOBhlX BOJUI B aHTH<l>eppoM8I'HeTHK8X

7 .lo CneKTp CB B ACI»Mo Ilpocref!UIHe A<l>M HMeiOT .u;Be MarHHTHbte no.u;pe­rneTKH H coorBeTCTBeHHo ,u;Be BeTBH CBo KBa.u;paTH'IHan liaCTb raMHnbTOHHaHa HMeer CTaH,u;apTHblH BH,!J; (5o18):

Jfz = l:: (wkakak + nk bicbk) 0 (7 01) k

Mbt npHBenH ee )J.JIH Toro, IJT06bt .u;oroBopHTbCH o6 o6o3Ha'leHHHX liacroT CB B .u;Byx BeTBHX Wk H nk H HOpManbHblX KaHOHH'IeCKHX nepeMeHHblX ak> bk, B KOTOpblX KBa,u;paTH'IHaH liaCTb raMHnbTOHHaHa ,tU{aroHanbHao B O.u;HOOCHblX A<l>M c aHH30TponHeH rHna "nerKaH ocb" (A<l>M JIO) none KpHcTannorpa<l>H· lieCKOH aHH3orponHH Ha cTpeMHTCH y,tiep:>Karb HaMarHHlieHHOCTH I]apannenbHO 3TOH OCH ( ee 06bl'IHO Ha3bmaiOT OCbiO Z) 0

flo aHanorHH c. <DM 1Mo:>KHo 6hmo 6bt npe.u;nono:>KHTb, liTO 'laCTOTbi CB c k -+ 0 ,u;on:>KHbl COOTBeTCTBOBaTb npeiieCCHH HaMarHHtleHHOCTH no,u;perneTOK B none Ha, Toeo Wo =no= Wa, r.u;e

Wa=gHao (7o2)

3TO, O,u;HaKO, HeBepHOo lleno B TOM, liTO Ha HaMarHHtleHHOCTb O.u;HOH no,u;pe­rneTKH M 1 , opHeHTHpOBaHHOH BBepx, .u;eflcTByeT none aHH30TponHH H a 1 , opH­eHTHpoBaHHoe raK:>Ke BBepx: Hat = Ha . Ha BTopyro no.u;perneTKY (M2 = -Md .u;eflcTByeT .u;pyroe none Hal = -H a 1 0 B pe3ynbTaTe no.u;perneTKH "xouT" npeueccHpOBaTb B pa3Hble CTOpOHbl. flpH 3TOM HeH36e:>KHO ,[J;On:>KHa HapyrnaTbCH aHTHnapannen&HOCTb M1 H M2, 'leMy npenHTCTByeT CHnhHoe o6MeHHoe B3aHMO· .u;eflcTBHe Me)I(,[J;y no,u;perneTKaMHo B pe3ynbraTe oKa3bmaeTcH, 'ITO [3]

(703)

3.u;ecb Wex = gH ex xapaKTepH3Y~T BenHtiHHy aHTH<l>eppoMarHHTHoro o6MeHa Me:>K,u;y no,u;perneTKaMHo lfHor,u;a )J,JIH TaKOH xapaKTepHCTHKH HCnon&3yeTCH 6e3pa3Mepsan "o6MeHHaH KOHCTaHTa" B: Hex = BMO

EcnH K AM<l> JIO npHnO:>KHTb BHeUIHee MarHHTHoe none B.u;onb ocH c, TO 3K· BHBaneHTHOCTb no,u;perneTOK, npHBO,u;HmaH K COBna,u;eHHIO 'laCTOT Wo H n0 ,

HapyrnaeTCHo IlpH noM

n~ = 2WaWex + W~, WH =gH.

(7.4)

B KpHTHtleCKOM none H = (2HaH:x) 112 'lacToTa w 0 o6pamaeTCH B sym H pa3BHBaercH HeyCTOHlJHBOCTb CB, npHBO,u;HmaH K CMeHe OCHOBHOrO COCTOH· HHH - onpoKH.u;btBaiDliO no.u;perneroK [3] 0

34

B OAHOOCHbiX A<I>M c aHHJ.orponHeif THDa "nerKaH nnocKOCTb" (A<I>M JIII) none aHH3orponHH pa3Bopa'IHBaer MOMeHTbiM1, M2 B nnocKOCTh, nepneHAH­Kynxpuyro OCH C_. Bo3MO)I(HOCTb nO'ITH 6ecnpeDXTCTBeHHbiX KOnefiaHHH MO· MenroB B 3TOH nnocKocTH npHBOAHT K TOMY, 'ITO OAJia H3 qacror CB npH k -+ 0 oKa3bmaercH Manoif - nopx,ll;Ka gH. Bbi'IHcneHHH, KOTOpbie Mbi npo~enaeM B q~eeyromeM naparpa<Pe, noKa3bmaror, 'ITO

wi = wH(wH + wD) + 2wex(Wn + wph) + vfk~ + v~kf, Ui = 2WaWex +Wk.

(7.5)

3~ecb H 1 c, v1 H v2 - npo~onbHaJI H nonepequax cKopocrH CB, no nopH,ll;KY senH'IHHbi v1,2 :=::: Vexa, r~e a - nocroXHHax peweTKH, wD = gHn, r~e Hn­none ll3xnoiiiHHCKoro, B03HHKaiOIUee H3-3a cneUH<P~ecKoro penXTHBHCTCKoro B3aHMO~eifcTBHJI Me)I()Jy no~pewerKaMH. BenHIJHHa H D o6bNHO nopH,ll;Ka ue­CKOnbKHX KHn03pcre~. B pH~e AM <I> H3-3a CHMMeTpHH H D = 0. OcraBIIIHecH ~a cnaraeMbiX B (7 .5) Wn H wph o6ycnosneHbi oqenh ManbiM B3aHMo~eif­CTBHeM :meKTpOHOB C MepHbiMH COHHaMH H C KpHCTann~eCKOH peiiieTKOH coorsercrseHHo. OAJiaKo B <PopMYne (7 .5) 3TH qacroTbi yMHO:>KlliOTCJI ua caMyiO 6onhiiiYIO qacrory Wex, TaK 'ITO BKn~ 3THX ManbiX B3aHMO~eHCTBHH B qacrory wk MO)!(eT oKa3aThCJI snonue 3aMeTHbiM. B A<I>M 3TOT <PenoMeH npH­HJITO Ha3bffiaTb o6MeHHblM ycu/leHUeM MilllblX 83QUMOOeUCT8UU. qaCTOTa Wn COOTBeTCTByeT npeu;eCCHH MarHHTHOrO MOMeHTa JneKTpOHa B J<l><PeKTHBHOM MarHHTHOM none Hn, o6ycnosneHHOM csepxroHKHM B3aHMo~eHCTBHeM JneK­rponoB c H~poM: .

Wn =gHn =A([z)/h. (7.6)

3~eCb A - KOHCTaHTa CBepXTOHKOrO B3aHMO~eHCTBM, (/z) - CpeAHee 3Ha'le­HHe Mepuoro cnHHa, o6paruo nponopUHOHanhHoe reMOeparype. HanpHMep, ~JI Mep Maprauu;aHn::::: (10/T) 3 H "Mepuax menb" Hi= 2Hexiln OKa3bi­Baercx nopH,ll;Ka uecKonbKHX KHno3pcre~ npH T = 4 K (ra6n. 7.1).

Mo)!(eT B03HHKH}'Tb sonpoc - orKy~a npH H = 0 6epercH .IIIenh B cneKrpe

CB, paBHaJI gH 11 = (2wexwn) 112 , ecnH B 3TOH CHryau;HH 3a~aqa o6na~aer aKCH­anbHOH CHMMerpHeH ·oTHOCHTenhHO nosopora MoMeHTOB no~peweroK B ner­KOH nnoCKOCTH? Orser B TOM, 'ITO qacrora npeu;eCCHH Jl~epHbiX cnHHOB nn 3Ha'IHTenbHO MeHhiiie, '!eM 'laCTOTbl npeu;eCCHH 3neKTpOHHOH no~CHCTeMbi gH t:. , gH D, ... TI03TOMY Jl~epHbie CnHHbl H3·3H CHnhHOH HHepUHOHHOCTH no'ITH He ~pHHHMaiOT yqaCTM B 3neKTpoHHbiX ~H)!(eHHJIX, BbiCTpaHBaJICb B~Onh Cpe~­Hero 3Ha'leHHX naMarH~eHHOCTH no~peweroK. TaKHM o6pa30M, B03HHKaer HapyweHHe aKCHanbHOH CHMMeTplUI KpHCTanna, npHBO~IUee K noHBneHHIO

Ta6nHua 7.1

4hHOMeHonorH'IecKHe KOHCTaHThl, onpe~eJlliJO~e cneKrp CB B AGIM nn

KpHCTIIJUI a, A H~T, K3·K

MnC03 32,4 0,39 320 3,07 4,4 2,5 5,8 CsMnF3 53 ,6 0,28 350 2,4 0 6,4

3* 35

3<J><J>eKTHBHhiX MarHHTHhiX noneii Hn 1 = Hn, Hn2 = - Hn, .o;eHCTBYIOJ.UHX Ha

no,o;peweTKH M1 , M2 cooTBeTCTBeHHo. A 3TH nonH B cooTBeTCTBHH c <J>opMY­

noii (7 .3) ,!J;JIJI A<I>M JIQ npHBO):OIT K npeu.eccHH JneKTpOHHhiX CIIHHOB C tJaCTO-

TOH (2wexwn) 112. To )l(e caMoe npoHCXO.o;HT H c MarHHTOCTpHKUHeii, npHBO­

nHlll.eH K O,!l;HOOCHOH ,o;e<J>opMali.HH KpHCTanna, KOTOpaH He ycneBaeT cne,o;HTh

3a npeu.eccHeii MOMeHTOB no,o;peweTOK . 3Ta ,o;e<J>opMali.HJI B CBOIO oqepe,!l;h

npHBO.o;HT K noHBneHHIO 3<J><J>eKTHBHhiX noneii H ph 1 = -H ph 2 = H ph B nerKoH:

nJIOCKOCTH. llone H ph MaJIO - OHO nponOpli.HOHaJibHO KBa,o;paTy KOHCTaHTbi

MarHHTOCTpHKUHH. O.o;HaKo B pe3yn&Tare o6MeHHoro ycHJieHHJI MarHHTOCTpHK­

li.HJI MO)I(eT npHBeCTH K 3aMeTHOMY BKna)zy B qacToTy CB. B Ky6HtJecKHX A<I>M (HanpHMep, B RbMnF3 ) Bhrpa)l(eHHH ,!J;JIH qacTOT CB HMelOT BH,o;

2-3 2 2k2 wk - WaWex + w A + u '

ilk= Wk- %waWex + w1, + u2 k 2•

(7 .7)

3,o;ec& npe,o;nonaraeTCJI, 'ITO Hll [100] Hero 3HatJeHHe 6on&we , tJeM none onpo­

KH,!l;hiBaHHH (%HaHex) 112.

Bo Bcex Bhrpa)l(eHHHX ,o;nH .qacTOThi CB B A<I>M Mhi He YtJHThiBaJIH ,o;HIIonh­

,o;HIIon&Hoe B3aHMo.o;eiicTBHe . .Ueno B TOM, 'ITO MarHHTHhiH MOMeHT M = M 1 -M2

B A<I>M, B03HHKaiOIUHH no,o; ,o;eHCTBHeM BHeWHero MarHHTHOro nonJI H penHTH­

BHCTCKHX B3aHMO,!J;eHCTBHH, JIBJIHeTCH MaJihiM. M/M0 ~ H0 /Hex· B cooTBeT­

CTBHH C 3THM OTHOCHTeJibHhiH BKna,o; ,o;HnOJih-,o;HIIOnhHOfO B3aHMO,!l;eHCTBHJI B

qacTory CB oKa3&IBaeTC.II, KaK npaBHJio, MaJihiM no napaMeTpy H 0 /Hex, 'ITO

,!J;JIJI 6onbWHHCTBa A<I>M COCTaBn.lleT 10-3 7 10-4• TeM He MeHee, 3TO B3aHMO­

,o;eHCTBHe MO)I(eT OKa3aThC.II cymecTBeHHbiM npH H3y'leHHH TOHKHX HenHHeH­

HhiX CBOHCTB CB B A<I>M. 7.2. fBMmn.ToRHBH B33HMo~J;eAemHJI CB B ACI>M Jill. HaH60JlhWHH RHTepec

,!J;JIJI JKcnepHMeHTOB no HenHHeHHbiM CBOHCTBaM CB npe,o;cTaBnHIOT A<I>M Jill

H Ky6HtJeCKHe · A<I>M, HMeiOIUHe HH3Kone)l(aWHe BeTBH CB , qacroTa KorophiX

nona,o;aeT B y,o;o6Hhm ,o;Hana30H HH)I(e 50 rru.. HaH6onee npocraH CHTyaUHH

B03HHKaeT B A<I>M Jill, y KOTOphiX BropaH BeTBh CB ne)I(HT 3HatJHTen&Ho BhiWe

nepBOH : nk ~ Wk . 11MeHHO ,!l;JIH A<I>M Jill Mhl H BhiDHWeM pe3yJihTaTbl Bhi­

'IHCJieHHH ,!l;JIH HHTepecyiOmeii Hac qacTH JCint:

JC3 = :E [~ VJ 1)b1aiaj .+ VJ2>a1biaj + K .C.) + 1=2+3 .

JC4 =~ :E Wpaiaia3 a4 , p=(k1 ,k2,k3 ,k4) , 1 +2=3+4

VJl) = -l/.i[(2gWex)/(Mw2w3.Q1)) lf2wH(.Q1 + w2 + w3),

VJ2 ) = - l/.i[(2gWex)/(Mw1 W3il2)] lf 2wH(wl - il2 + W3) ,

Uq = -14[(2gwex)/(Mw2w 3n 1)] lf 2wH(.Q 1 - w 2 - w3 ),

(7.8)

(7 .9)

(7 .10)

Wp = (9wex/8M) (wi + w~ + wj + wl- w(1_ 3) - w(2_3) - w(1+2)) X

X (w1w 2w3w4)- 112 • (7.11)

TaK )l(e, KaK H paHbWe, Mbl HCIIOJib30BaJlH COKpameHHYIO 3aiiHCb

n 1 = D(kd, w1 = w(k;), w0 _3) = w(k 1 - k 3 ).

HarroMHHM, 'ITO rrpH orrHcaHHH rrpoueccoB pacceHmur rHrra 2 -+ 2 HY)I(HO }"'H­

TbiBaTb He TOJibKO raMHJibTOHHaH Jf4 , HO H raMHJibTOHHaH Jf3 BO BTOpOM IIO­

pHnr<:e reopHH B03MYIUeHHH. Mo)I(HO rroKa3aTb, 'ITO rro rrapaMerpy wkfwex HaH6onee cymeCTBeHHbiM 6yner BKnan npoueccoB c }"'acrHeM BHpryanbHOH

BOJIHbl H3 BepxHeH BeTBH, raMHJibTOHHaH KOTOpbiX HBHO BbillHCaH B (7.8). !1CIIOJib3YH <t>opMYJIY (5 .30), liOJI}"'aeM Bb!pa)l(eHHe ,!l;JIH 34>4>eKTHBHOfO MaT-

pwmoro JJieMeHra qerbrpeXBOJIHOBbiX rrpoueccoB :

Tp = Wp- ~{V~!l2 V~!j;[w (k5 ) - 2w(kW 1 +

+ Us2tuh4[w(ks) + 2w(kW 1 + [V~2J3 V~2J; + ' '

+ v< 22• v< 2 > ]w- 1 (k6 ) + [v< 2 > v<2 P + v< 2 >_: v< 22 ]}w- 1 (2k) . (7 .12) 3,61 2,64 2,73 4,71 3,72 1,74

3necb I coorBercrByer - k 1 , k 5 = k 1 + k 2 , k 6 = k 1 - k 3 , k 7 = k 2 - k3 . 113

<PopMyn (7.10)- (7 .12) ·MO)I(HO yBH,!l;eTb , 'ITO ,!l;HaroHanbHbrH BKnan B Tp , B03HHKalOIUHH H3-3a Jf3 , roro )l(e IIOpHnr<:a BeJIH'IHHbl, 'ITO H Wp . 06mee Bbl­

pa)l(eHHe ,!l;JIH Tp noBo,1bHO rpoM03nKO. Mbr BbillHWeM ero B HaH6onee HHTe­

pecHoM ,!l;JIH Hac cn}"'ae, Korna w 1 = w 2 = wk H k 1 = k3 , k 2 = k4 HJIH

ki +k2 ·=0:

S(kk') .= T(kk, k' k') , T(kk') = T(kk' , kk') ,

S(kk') = T(kk') = - 2 ex w 2 + w 2 0 .

w [ 3.Q2

- 4w2(k) ]

g 8w 2 (k) 0 H .Q~ - 4w2 (k)

(7 .13)

B 3aKJIIO'IeHHe JToro rraparpa4>a yKa)l(eM Ha O,!l;HO o6croHrenhCTBo , Koropoe

BHHMareJibHbiH 'IHTaTeJlb ,!l;OJI)I(eH 6bm ,!l;aBHO 3aMeTHTb : CB B Acl>M nrr ropa3,!1;0

6onee HemmeHHbi, 'leM B cl>M. neHCTBHTeJibHO, qacrOTbl CB B A<I>M nrr H <I>M

O,!l;HOfO IIOpHnr<:a BeJIH'IHHbl Wk ~ 21T • ( 10 7 30) J.:Tu IIpH IIOJIHX H IIOpH,!l;Ka

HecKOJibKHX KHJI03pcren. B ro )l(e BpeMH B <I>M Tp rropHnr<a gwk/2M ~ 1rg2,

a B Acl>M nrr Tp llOpHnr<agwe xf2M ~g 2 B/2. B MnC03 KOHCTaHra B = Wex/M = = 800 (eM. ra6n . 7.1) , B CsMnF3 B ~ 103 . CnenoBarenbHO , qerbrpexMarHoH­

Hoe B3aHMO,!J;eHCTBHe CB B Acl>M nrr Ha ,!l;Ba rropHnr<a CHJibHee , 'leM B <I>M.

11raK, B § § 5-7 JTOH rnaBbi Mbl 3arrHcanH: raMHJibTOHOBbi ypaBHeHHH ,!l;BH­

)I(eHHH )1.1IH cna6o HeJIHHeHHbiX BOJIH H rrpe,!l;'bHBHJIH B fOTOBOM BH,!l;e raMHJibTO­

HHaH CB B 4>eppo- H aHrH<t>eppoMarHerHKax . qHrarenH, He HHrepecyiDmHecH

peuerrraMH MarHHTHOH KYXHH, Moryr rrporrycrHTb BTOpyro IIOJIOBHHY 3TOH

maBbl ( § § 8, 9) H rrepeHTH K rperbeH. C OCTaJibHbiMH Mbi paCCMOTpHM B CJie­

,!J;YJOIUeM rraparpa4>e ypaBHeHHH ,!l;BH)I(eHHH ,!l;JIH CB H H3Y'IHM rrpocreHWHH crro­

co6 Bb!'IHCJieHHH K034>4>HUHeHTOB CIIHH-BOJIHOBOfO raMHJibTOHHaHa.

§ 8. YpaBHeHHJI ,!l;BH)I(eHHJI H KaHOJU111ecKHe nepeMeHHbie

8.1. YpaBHeHHJI .ln}H)I(eHHJI MarHH1Horo MOMeHTa. IIocnenoBarenbHbiH Bbi­

Bon ypaBHeHHH ,!l;BH)I(eHHH MarHHTHOro MOMeHra M ( r) II pH KOHe'IHbiX reMIIe­

parypax - 3TO '1pe3Bb!'laHHO CJIO)I(HaH 3a,!l;aqa ,!l;a)l(e ,!l;JIH ,!l;JIHHHOBOJIHOBbiX

Kone6aHHH. PeweHHe JTOH rrpo6neMbi 03Hatrano 6bi , B qacrHocrH, qro Mbi orrpe­

nenHJIH TeMIIeparypHbie 3aBHCHMOCTH MaTpH'IHblX 3JieMeHTOB B3aHMO,!l;eH-

37

CTBHH, qacroTbi CB, HaiiDIH HX JaryxaHHe H eme Macey none3HbiX Bemeii: . IlpH

nonbrTKe nonyqHTb Bee , :no crporo H nocne)J;oBarenbHO Mbi 6br JacrpHJIH B

caMOM Haqa.rre nyTH H 3a6billH 6bi 0 HaiiieH KOHeiiHOH UeJIH - nOJiyqHTb npOCTOH

H noHHTHbiH MeTOJJ: onHcaHHH HeJIHHeii:Hoii: .u;HHaMHKH CB .npH HH3KHX H npoMe­

)KYTO'IHbiX TeMUeparypax. IlOJTOMY Mbl C)J;eJiaeM HeCKOJibKO UJIOXO KOHTpOJIH·

pyeMbiX, HO JaTO npoCTbiX, B BbiCIIIeH CTeneHH "ecTeCTBeHHbiX" H CTOJlb )Ke

rpa)J;HUHOHHbiX )J;onymeHHH. IlpOUHTHpyeM )J;JIH 3TOrO MOHOrpacpHlO (3), C. 44: "onaro)J;apH CHJibHOMY 06MeHHOMY B3aHMO)J;eHCTBHlO Me)K)J;y CnHHaMH OT)J;eJib·

HbiX aroMOB cpeppoMarHeTHKa ero MarHHTHbrH MOMeHT c 6onbiiiOH creneHbro

TO'IHOCTH HBJIHeTCH ")KeCTKHM", eCJIH TOJlbKO TeMUeparypa cpeppoMarHeTHKa

)J;OCTaTO'IHO HH3Ka". I1HbiMH CJIOBaMH, MO)J;yJlb BeKTOpa UJIOTHOCTH MarHHTHOrO

MOMeHTa JlHUib B OlleHb cna6oii: CTeneHH MO)KeT 3aBHCe'Tb OT BpeMeHH.

Il03TOMY H3MeHeHHe CO BpeMeHeM UJIOTHOCTH MarHHTHOrO MOMeHTa )J;OJI)K·

HO B nepBOM npH6JIH)KeHHH HOCHTb xapaKrep npeueccHH, T.e. npoHCXO.u;HTb

no JaKOHY

aM - = g[Hef(r, t), M(r, t)), at . (8.1)

r)J;e Hef(T, t) - HeKOTOpbiH BeKTOp, KOTOpbrH Mbl 6y)J;eM Ha3bffiaTb 3¢¢e·K· TU6HblM MflZHUTHblM nolleM. Tolfflo raK )Ke, paccy)K)J;aH Ha npHHHTOM HaMH

ypoBHe crporocrH, MO)KHO yrBep)K)J;arb, liTO ypaBHeHHH JJ:BH)KeHHH (8.1) )J;OJI)I(Hbl 6biTb KOHCepBaTHBHbiMH, T.e. coxpaHHTb nonHyro JHeprHlO cl>M W. lleii:crBHTeJI&Ho, ecnH HH'Iero, KpoMe CB, oUHcbmaeMbiX ypaBHeHHeM (8.1), B HaiiieH MO)J;eJIH HeT, TO HX nOJIHaH 3HeprHH )J;OJI)I(Ha COXpaHHTbCH. IJOCKOJlbKY

W HBJIHeTCH cpYHKUHOHa.JIOM OT HaMarHH'IeHHOCTH,

. dW bW oM(r, t) W=- = f dr.

dt bM(r, t) ot

IJO)J;CTaBJIHH ClO)J;a oMjot H3 ypaBHeHHH )J;BH)KeHHH (8.1), HMeeM

dW = g f ( b W [ M H f] \,~r. dt bM ' e f

llJIH TOrO IIT06bi 3TOT HHTerpa.rr paBHHJICH HYJilO npH Jll060M BH)J;e JaBHCHMOCTH

M(r), Heo6xo)J;HMO, IIT06bi BeKrop Hef 6bm napa.rrneneH BeKrop y 8WfbM. Ko3cpcpHUHeHT nponopUHOHa.JibHOCTH MO)KHO ycraHOBHTb, paccMa1 pHBaH Ka­

KYIO-HH6YJJ:b npocryro CHTyaUHlO C 3apaHee H3BeCTHbiM ypaBHeHHt:M )J;BH)Ke­

HHH, HanpHMep OJJ:HOpOJJ:HYIO npeuecCHlO MarHHTHOrO MOMeHTa B O)J;HOpOJ];HOM

none H . Tor)J;a W - 3To JHeprHH B3aHMOJJ:eii:cTBHH c BHeiiiHHM noneM :

W= -HfM(r)dr.

BbiiiHCJIHH <l>YHKUHOHa.JibHYID npoH3BOJJ:HYID no npaBHJIY (I11.2), nonyqHM

8W/8Ma = - Ha .

(8.2)

(8.3)

3TO 03Hattaer, ttTO HCKOMbiH K03cpcpHUHeHT nponopUHOHa.JibHOCTH eCTb MHHYC

eUHHHUa:

Hef = - b W/bM. (8.4)

38

TaKHM o6pa30M, Mbi npHXOAHM K ypaBHeHHHM )J.BH)I(eHHJI

oM(r, t) = g [ 8 W • M]. at l>M

(8.5)

Pa3yMeeTCJI, ra HenpHey)l()l.eHHaJI rrerKOCTh, C KOTOpOH Jl COBMeCTHO C aBTO·

paMH MOHOrpacpHH (3) npOH3BOAHJI MaHHnyJIHUHH, npHBemr.rne K ypaBHeHHIQ

(8.5), OTHIQ,IJ;b He rapaHTHpyer HX npaBHJihHOCTH. COBMeCTHbiMH YCHJIHJIMH

Mbi ynpHTaJIH llO,IJ; KOBep MHOrO TOHKHX H CJIO)I(HbiX BOllpOCOB. HeKOTOpbie

H3 HHX 6y,o;yr 06Cy)l()l.aTbCJI 3Ha'IHTeJibHO ll03)1(e. CeiitJaC )l(e Mbi 6y,o;eM HCXO·

AHTh H3 ypaBHeHHH (8.5), 6y,o;y'IH TBep,o;o YBepeHHbiMH B TOM, 'ITO OHH ,o;aror

xopornee nepBoe npH6JIH)I(eHHe ,o;rrH onHcaHHJI CB.

8,2, KaHoJU111ecKHe nepeMelllllde ,l1;1IJl CIDUIOBhiX BOJDI B cpeppoMarHeTHKax.

B ypaBHeHHH (8.5) nepeii,o;eM K KpyroBbiM nepeMeHHbiM

aM+ M±=Mx±iMy, --- = ±2igMz, M;=M5-M~- · (8.6) ot

HanpaBJieHHe OCH Z pa3yMHO Bbi6paTb B,IJ;OJib paBHOBeCHOrO 3HatJeHHJI HaMar­

HHtfeHHOCTH. Tor,o;a npH MaJibiX aMnJIHTy,o;ax KOJie6aHHH MarHHTHOrO MOMeHTa

BeJIH'IHHbi M± 6y,o;yr MaJibi, a Mz 6y,o;er 6JIH3KO K ,o;rrHHe BeKropa M, r.e. M0 .

CpaBHHBaJI (5.7) H (8.6), BHAHM, 'ITO 3TH ypaBHeHHH B JIHHeHHOM noM± npH-

6JIH)I(eHHH HMeiOT raMHJibTOHOB BH.o;, eCJIH B KatJeCTBe KaHOHH'IeCKHX nepeMeH·

HbiX a H a* B3J!Th CJie,o;yiOllUie BeJIHtJHHbi:

M_(2gM0 )- 112, M..,(2gM0 )- 112.

IJOJTOMY KaHOHH'IeCKHe nepeMeHHbie pa3yMHO HCKaTb B BH,IJ;e

M+=af(a*a)(2gM0 ) 112, M_=a*f(a*a)(2gM0 )112, f(O)=l. (8 .7)

IJo,o;CTaBJIJIJI (8.7) B (8.6), ll01IY'IHM ypaBHeHHe ,11;1IJI a, KOTOpoe ,IJ;OJI)I(HO COBna­

,o;aTb c KaHOHHtJeCKHM (5 .7). B pe3ymrare norryqaercn AHcpcpepeHUHaJibHoe

ypaBHeHHe ,11;1IH /{I a 12), e,IJ;HHCTBeHHbiM peiiieHHeM KOTOpOrO, y,o;OBJieTBO·

pHIOW:HM YCJIO.BHIQ f (0) = 1, HBJIJleTCJI

f(x) = [1 - gx/2M0 ] 112 .

TaKHM o6pa30M, Mbi Bbipa3HJIH "ecrecrBeHHbie nepeMeHHbre " <DM Mz. Mm tiepe3 KaHOHHtfeCKHe: ( ca c:t a \ "' '2 ~J..I..:,

M+=a[:gMo(l-ga*a/2Mo)]1f2, "'o._~ f ~ ~ ).: Vl~ - a ~8~) tb"'}·

M _ =M+, Mz =M0 -ga•a.

3HeprHH <DM W, Bbipa)l(eHHaJI qepe3 KaHOHH'IeCKHe nepeMeHHbie, CTaHOBHTCJI

cl>YHKUHeii faMHJibTOHa (raMHJibTOHHaHoM) :X:(a*, a) . Herpy.o;Ho BH,o;eTh, 'ITO

npeo6pa30BaHHe (8.8) HBJIHercn KrraccHtJeCKHM aHarroroM ripeo6pa3oBaHHJI

rorr&IIITeHHa L - IlpHMaKOBa [2,7]. BnepBble ,11;1IJI aHaJIH3a HeJIHHeHHbiX npouec­

COB B <DM OHH HCll01Ib30BaJIHCh E. UlrreMaHHOM H ,o;p. (1960 (234)). Pa3yMeercn, :nor Bbl6op KaHOHH'IeCKHX nepeMeHHbiX He HBJIJiercn e.o;HHCT·

BeHHbiM._ TaK MO)I(HO noKa3arb (eM. 3a,o;aqy 2.1), tiro ,o;rrH ypaBHeHHH (8.5) cyw:ecrByror ,o;pyrHe KaHoHH'IeCKHe nepeMeHHbie, tiepe3 Koropbie BeKrop M Bbipa)l(aercH crre,o;yiOllUIM o6pa30M: '

Mz + iMx = Mo [1 + glb* - b 12 /2Mo) 112exp {i(b* + b)(g/2M0 ) 112 },

My= i('hgMo)1f2(b*- b). (8.9)

39

3TH q,opMYJibl HBJIRIOTCH KJiaCCH'!eCKHM aHaJIOfOM IIpell;CTaBJieHHH CIIHHOBblX

orrepaTopo B qepeJ 6oJe-orrepaTopbi, rrpeJJ.JIO)I(eHHoro B pa6oTe · B .r. EapbHXTa­

pa, ,Il.A. 5!6JIOHCKOfO (1975 (55]) . CpaBHHBan <t>opMYJibi (8.8) H (8.9) , MO)I(HO ysHp;eTb, 'ITO rrepeMeHHbie

foJibiiiTeH:Ha - flpHMaKosa a•, a H EapbRXTapa - .H6rroHcKoro b, b• cosrra­

p;aJOT B JIHHeH:HoM rrpH6JIH)I(eHHH. Borrpoc o rrpeHM}'mecTBe Toro HJIH HHoro

rrpep;cTaBJieHHH HaMarHHlJeHHOCTH '1epe3 K aHOHHlJeCKHe rrepeMeHHbie IIpH pe­

rueHHH KOHKpeTHbiX 3ap;aq Mbi o6cy~J;HM HH)I(e . 3p;ecb )l(e Mbi eme pa3 rrop;­

qepKHeM caMy B03MO)I(HOCTb IilllpOKOfO 1IpOH3BOJia IIpH Bbi6ope rrepeMeHHbiX,

B KOTOpbiX ypaBHeHHH IJ;BH)I(eHHH HMeiOT KaHOHH'IeCKHH BH/]; .

§ 9. CIIHH·BOJDIOBOA raMHJibTOHHaH B cpeppoMarHentKax

u ero auepnu1 B npu6niDKeHHH cnnoiiiHoA cpe,II,LI

9.1. OnepaTop faMHJibTOHa. KaK xoporno H3SecTHo, B CIIHHOBOH CHCTeMe

MarHHTO/];HJJieKTpHKOB HaH6oJiee CHJibHbiM HBJIHeTCH 06MeHHOe B3aHMO/];eH­

CTBHe, o6ecnelJHBaJOmee caMo cymecTBOBaHHe MarHHTHoro nopHAKa. OcHoB­

HaH lJaCTb 3TOfO B3aHMOIJ;eHCTBHH OIIHCbmaeTCH reibeH6eprOBCKHM raMHJibTO­

HHaHOM Xex (2.13). B AHHaMHKe JJ.JIHHHbiX CB Ba)I(HyJO pom HrpaeT TaK)I(e

AHfiOJib-AHIIOJibHOe B3aHMop;eH:cTBHe Jfdd (3.1). Heo6xo~J;HMO YlJHTbiBaTb

TaK)I(e B3aHMO/];eHCTBHe MarHHTHbiX MOMeHTOB 3JieKTpOHOB C 0/l;HOpO/l;HbiM

BHeiiiHHM MarHHTHbiM rrorreM H:

Xm = -2p.o'E-(HSa)· (9.1) a

B § 3 Mbl IIOKa3aJIH, 'ITO CIIHH-op6HTaJibHOe B3aHMOIJ;eHCTBHe rrpHBO/];HT K

IIOHBJieHHIO 34>4>eKTHBHOH 3HeprHH MarHHTHOH adH30Tp01IHH. B OAJ{OOCHblX

KpHCTaJIJiaX ( CM. (3.3))

XJ2> = A< 2 >'E, (Sax);. (9.2) a

B <l>M C Ky6HlJeCKOH CHMMeTpHeH ( CM. (3 .4))

XJ4 > = A(4 )'E ((Saxt + (Say)4 + (Saz)4

). a

(9.3)

B 3TH Bbipa)l(eHHH BKJIIO'IeHbl TOJibKO lJJieHbl BTOporo H lJeTBepTOfO IIOp HIJ;KOB

IIO CIIHH·Op6HTaJibHOMy B3aHMO/];eHCTBHIO. J1Horp;a YlJHTbiBaiOT eme 6oJiee CJia-

6bie lJJieHbi rnecToro rropn~J;Ka . B OAJiOOCHbiX reKcaroHaJibHbiX K pHCTaJIJiax

OHH 1IpHBOIJ;HT K aHH30Tp01IHH B "6a3HCHOH IIJIOCKOCTH", rreprreH~J;HKYJIHpHOH maBHOH ocH CHMMeTpHH rnecToro nopn):U(a .

fiOJIHbiH raMHJibTOHHaH CIIHHOBOH CHCTeMbl <l>M BKJIJOqaeT B ce6H, BOo6me

rosop.11, see nepe'IHCJieHHbie crraraeMbie :

J(=J(ex +Jfdd +J(m +J(a· (9.4)

B 3TOM Bbipa)l(eHHH He YlJHTbiBaeTCH B3aHMO/];eHCTBHe CIIHHOBOH IIO,I:i;CHCTe­

Mbi KpHCTaJIJia C ero p;pyrHMH IIO/];CHCTeMaMH - <t>oHOHaMH, 3KCHTOHaMH H T./];.

9 .2. 3aepllUI cpeppoMarHentKa B npu6JIIDKeHHH cnnonmoA cpe,rn.I. B JTOM

IIYHKTe Mbi o6cyAHM KrraccuqecKoe rrpH6JIH)I(eHHe AJI.II 3Hepruu <I>M B rrpe)J.Jio­

JIO)I(eHHH, 'ITO MafHHTHbie MOMeHTbl 3JieKTpOHOB Ha COce/];HHX Y3JiaX IIO'ITH

naparrrreJibHbi. <l>opMaJibHO Mbl /];OJI)I(Hbl IIOCTYIIHTb CJiep;yJOmHM o6pa30M:

40

BO-IIepBbl.X, JaiiHcaTb KJiaCCH'IeCKOe Bbipa)!(eHHe AJUI 3HeprHH <l>M , COOTBeT­

CTBYlOIUee raMHJibTOHHaHy (9.4). 80-BTOpb!X , BHeCTH IIOHHTHe ycpe]J,HeHHOH

IInOTHOCTH MarHHTHOfO MOMeHra, "pa3Ma3biBaH " MarHHTHbiH MOMeHT :meKTpO­

Ha no :meMeHrapHoii Hl!eiiKe. 8-rperbHX, rrepeiim or cyMMHpOBaHHH no yJnaM

K HHrerpHpoBaHHIO no rrpocrpaHcrBy . llpome Bcero 3TO rrpo}J.enaTb MH roii

qacrH raMHnbTOHHaHa, KOTOpaH CO]J,ep)!(HT O}J,HOKparHOe CYMMHpOBaHHe ITO

YJJiaM. J1MeHHO H3 (9 .1)- (9 .3) nerKO IIOJiyqaeM H3BeCTHbie Bbipa)!(eHHH }J.nH

WmH Wa :

Wm = - fHM(r)dr, WJ 2 > = '!4(A.< 2 >v0 /Jl.~)JM;(r) dr,

WJ 4 > = p., < 4 >v~/16Jl.:)f[Mi(r) + M: (r) + Mi(r)] dr. (9.5)

3}J.eCb M(r) - rrnorHocrb MarHHTHoro MOMeHra, v0 - ofueM 3JieMeHrapHOH

HtieHKH. HecKonbKO 6onee cno)!(Ho BbiTJIH}J.HT rrpeo6pa3oBaHHe Bbipa)!(eHHH

(5.1) }J.nH 06MeHHOH 3HeprHH (eM. 3ii}J,aqy 2.2) : 3M; 3M1 Wex = 1/J.o:ikf-- -- dr, (9.6) ax; axk

O:;k = elsJl.~) ~ J(rr,)r;(n)rk(n). (9 .7) n

3 }J.eCb r;(n) - i-H IIpOeKIJ,HH (i =X, y, z) KOOp}J.HHaTbi Tn MarHHTHbl.X arOMOB

B pewerKe. llpH BbiBO}J,e Bbipa)!(eHHH (9 .6) , (9 .7) Ppe}J.Ilonaranocb, qro pac­

croHHHe [, Ha KOTOpOM cymeCTBeHl!O H3MeHHeTCH 06MeHHbiH HHTerpan, Man6 B cpaBHeHHH C paCCTOHHHeM, Ha KOTOpoM cymecrBeHHO MeHHeTCH !lf(r) .

OrMeTHM, qro Bbipa)!(eHHe (9.6) }J.nH o6MeHHoii JHeprHH o6na}J,aer 6omwoii

0 6Il.IHOCTbiO. 8 3HatiHTeJibHOH Mepe OHO He JaBHCHT OT MO}J.eJibHbl.X rrpe]J,IIOn<))!(e­

HHH : BeJIH'IHHbi CIIHHa , JaBHCHMOCTH 06MeHHoro HHTerpana OT paCCTOHHHH Me)!(­

iJ.Y aroMaMH H r .rr. Ero MO)!(HO paccMarpHBaTb KaK <PeHOMeHonorHtiecKoe Bbipa­

)!(eHHe }J.nH 3HeprHH MarHHTHOH Heo}J.Hopo}J.HocrH. UeiicrBHrenbHO , ecnH rrpe}J.­

rrono)!(Hrb , 'ITO, BO-IIepBbiX , 3Ta 3HeprHH He JaBHCHT OT OpHeHTaiJ,HH HaMarHH­

qeHHOCTH OTHOCHTeJibHO KpHCTannorpa<lJHtie CKHX OceH , BO-BTOpbiX, Cpe}J,H :me­

MeHTOB CHMMeTpHH KpHCTanna eCTb HHBepCHH , B-TpeTbHX, JaBHCHMOCTb 3Hep­

rHH OT HaMarHH'IeHHOCTH KBa]J,paTH'IHaH, TO, IIOJib3YHCb C006pa)!(eHHHMH CHM­

MeTpHH, MO)!(HO rronyqHTb Bbipa)!(eHHe (9 .6) . UoKaJarenbcrBo noro rrpocroro

yrBep)!(}J.eHHH CO}J.ep)!(HTCH , HaiipHMep, B MOHOrpa<lJHHX (3,39), H Mbl ero

IIp'iBO}J.HTb He 6y}J.eM. OrMeTHM ronhKO , 'ITO rrepBoe H3 C}J.enaHHbiX rrpe}J.Ilono­

)!(eHHH }J,OCTaTO'IHO o6mee H CJie}J,yer H3 IIpHpO}J.bi 06MeHHOrO B3aHMO}J.eHCTBHH,

HHBapHaHTHOrO OTHOCHTeJibHO BpameHHH BceX CIIHHOB . Bropoe rrpe}J,IIOJIO)!(eHHe

M bi C}J,eJianH IIOTOMy, 'ITO 601IbWHHCTBO MarHHTOyrropH}J.O'IeHHbl.X }J,H3JieKTpH­

KOB HMeHHO raKoBo . qro Kacaercn rrocrre}J.Hero rrpe}J.IIOJIO)!(eHHH , ro, crporo

roBopn , oHo crrpaBe}J,JlHBO ronhKO MH cnyqan S = 1/2, Kor}J.a orreparop o6-

MeHHoro B3aHMO}J,eHCTBHH HMeer reii3eH6eprOBCKHH BH}J, (2.13) . llpH S > 1/2 IIpH6JIH)!(eHHe (2.13) HHOr}J,a OKa3biBaeTCH He}J,OCTaTO'IHbiM , H He06XO}J,HMO

yqHTbiBaTb }J,OIIOJIHHTeJibHOe cnaraeMOe , raK Ha3biBaeMbiH " 6HKBa]J,paTHbiH

o6MeH", OIIeparop KOTOporo BbillHCaH B II. 2.2. EMy COOTBeTCTByer }J,OIIOJI­

HI-fTeJibHOe cnaraeMOe B 3HeprHH MarHHTHOH HeO]J,HOpO}J.HOCTH :

4 _

1 _ 3Ma 3M13

Wex - Yz o:;kfMa -- Mp -- dr , ax; axk

(9 .8)

Zi;k = 1/(8116)~ J(a) a;ak . (9.9) a

41

Bbipa)!(eHHe (9 .9) rronyqeHo B ripe)J,IloJio)!(eHHH, 'ITO cymecTBeHHbi B3aHMO·

neiicTBH.II TOJibKO 6JIH)!(aHlllHX coceneii . B o6meM cnyqae Bbipa)!(eHHe (9.8) MO)!(HO IlOJiyqHTb H3 coo6pa)!(eHHH CHMMeTpHH : HHBapHaHTHOCTH OTHOCHTeJib·

HO BpameHHH M H HaJIH'IH.II HHBepCHH.

9.3. 3nep111Jl ;z:umoJIJr,[UIIIO.JIIdiOrO B38HMOAeACTBIUI. faMHJibTOHHaey JCdd (3 .1) COOTBeTCTByeT MaKpOCKOilH'IeCKa.ll 3HeprH.II

Wdd = ~v~ ~ R;~ [(MaMb)R~b - 3(MaRab) (MbRab)] . (9.10) a*b

fiepexon K HHTerpHpOBaiDUO B (9.10) He CTOJib TpHBHaJieH , XOT.II OKOH'IaTeJib·

HOe Bbipa)!(eHHe AJI.II W dd .IIBJI.IIeTC.II npOCTblM :

Wdd = -~fdr(MHm + 41TM 2 /3 + f3ucMtMk], (9 .11)

a2

( 1 ) 13 =vo~ - - · ik . k b 3R~b3Rab Rab

(9.12)

3necb H m - cTaTH<IecKoe MarHHTHoe none, C03AaBaeMoe HenpepbiBHbiM pac­

rrpeneneHHeM MarHHTHOfO MOMeHTa M(r) . .HCHbiH H llOApo6HblH BbiBOA 3THX

Bbipa)!(eHHH conep)!(HTC.II B MOHOrpa<l>HH (3] . fio3TOMY Mbl ero npHBO.D;HTb

He 6yAeM , orpaHH'IHBlliHCb TOJibKO o6cy)!(AeHHeM <l>H3H'!eCKOf0 CMbiCJia 3THX

Bbipa)!(eHHii. llepBoe cnaraeMoe B (9.11) - no xopowo H3BeCTHoe Bbipa)!(eHHe

AJI.II MarHHTHOH 3HeprHH IlpH HerrpepbiBHOM pacnpeAeJieHHH MarHHTH~IX AH·

IlOJieH B npocTpaHCTBe , OCTaJibHbie YlJHTbiBaiOT .D;HCKpeTeyiO CTpYJ<TYPY Be·

meCTBa. 0HH B03HHKaiOT . H3·3a pa3HH11bl Me)!(Ay HCTHHHblM 3Ha'leHHeM Mar­

HHTHOfO IlOJl.ll B MeCTax pacnoJIO)!(eHH.II AHllOJieH H ycpe.D;HeHHOH BeJIH'IHHOH H m.

BTOpoe cnaraeMoe B (9. 1) YlJHTbiBaeT no o6cTO.IITenbCTBO npH6nH)!(eHHo,

wanorH'IHo TOMY, KaK no nenaeTC.II npH Bbi'IHcneHHH nonpaBKH nopeHII -

ilopeHIIa K .D;H3JieKTpH'!eCKOH npoHHIIaeMOCTH KpHCTaJIJia . 11MeHHO, IlpH Bbl·

'IHcneHHH MarHHTHoro rro11.11 H, neiicTByromero Ha ,z:umonb, C'IHTaeTc.H, 'ITO on

HaXO.D;HTC.II BHYTPH c<PepH'!eCKOH llOJIOCTH (c<l>epbl nopeHua) B CnJIOWHOH cpe­

ne c HerrpepbiBHbiM pacrrpeneneHHeM HaMarHH'!eHHOCTH M. Mo)!(Ho noKa3aTb

(eM. HH)!(e pa3nen o pa3MarHH'IHBaromHX <PaKTopax), 'ITO

41T H=Hm +- M. (9 .13)

3 C yqeToM noro noH.IITHO, 'ITO ADa rrepBbiX cnaraeMbiX B (9 .11) ecTb npocTo

3HeprH.II B3aHMOAeHCTBH.II .D;HllOJieii -~ (MH) . HaKoHeii, nocne.D;Hee cnaraeMoe

B (9 .11) YlJHTbiBaeT OTJIH'IHe peanbHOH .D;HCKpeTHOCTH KpHCTaJIJia OT npH61IH·

)!(eHHOH MOAeJIH co c<Pepoii nopeHua. B Ky6H'IeCKHX KpHCTannax H3 coo6pa­

)!(eHHH CHMMeTpHH f3tk = (30tk· C npyroif CTOpOHbl, H3 (9.12) CJieAyeT, 'ITO

Sp(31k = 0 H , cneAOBaTenbHo, (3 = 0. 3To 03HaqaeT, 'ITO B Ky6H'IeCKHX KpHCTan­

n ax rrpHfinH)!(eHHa.ll MOAem co c<Pepoii nopeHua naeT TO'IHbiH pe3yJihTaT . B on­

HoocHbiX )!(e KpHcTannax

f3;k = /3;0 ik • f3z i= f3x = f3y' (9 .14)

H rrocne.D;Hee cnaraeMoe B (9.11) He o6pamaeTC.II B Hym. Ero MO)!(HO rrpeACTa·

BHTb B BHAe

(3z - f3x J 2 f3x J 2 - Mz (r)dr - - M (r) dr. 2 2

(9.15)

42

Ilepsoe cnaraeMoe 3~ec& cosna,o;aeT no cpopMe c B&Ip3)KeHHeM (9.5) AlUI Wa(l) H npe~CTaBnJieT, TaKHM o6pa30M, ,[UIIIOnb-,[UIIIOnbHbiH BKna,IJ; B :meprHIO KpHC­TannorpacpH'JeCKOH aHH30TpOIUJH. BnpeA& Mbl 6y~eM C'IHTaTb, 'ITO OHO y:>Ke

yqTeHo B (9.5} cooTBeTCTBYIOIUeH nepeHopMHpOBKOH KoJ4>4>HUHeHTa A( 2).

OcTaaum:ecJI cnaraeM&Ie B (9 .11) H (9 .15) ' nponopUHOHan&Hbie M 2 (r) ' npe~­CTaBnJIIOT COOOH He HHTepecyiOIUHH HaC nOCTOJIHHbiH BKna,o; B 3HeprHIO W, He 3aBHCHIUHH OT opHeHTaiJ;HH M. TaKHM o6pa3oM, B (9.11) MO:>KHO oCTaBHT& TOn&Ko nepsoe cnaraeMoe. )lnH Toro 'IT06bi Bbipa3HT& ero Ton&Ko qepe3 Ha­MarHH'IeHHoCT&, Heo6XO~MO pewaTb ypaBHeHHJI MarHHTOCTaTHKH

rotH=O, divB=O (9.16)

c cooTBeTCTBYIOIUHMH rpaHH'IHbiMH ycnoBHHMH. B ~&HeiiweM Hac 6y~T HHTepecoaaT& ~a cnyqaH:

Jl!IOCKaJI 60/IHQ:

M(r) = (M) + [mkexp(ikr) + K.c.],

npH'IeM ~HHa CB 2rr/k MHoro MeHbWe, 'leM pa3Mep o6pa3u,a. 06pa:Je14 w.teer rftop.My 311/luncouoa , a M(r) = M = const . KaK xopowo H3-

secTHO, ypaBHeHHH (9.16) pewaiOTCH TO'IHO B o6oHX cnyqaHx. B nepaoM cnyqae

Hm(r) = [Hm(k)exp(ikr) + K .c.] , Hm(k) = -4rrk(kmk)/k2• (9.17)

Bo BTOpoM cnyqae

Hm ,t = -4rrN1kMk, SpN1k = 1, (9.18)

r~e N,k - TeH30p "pa3MaiHH'IHBaiOIUHX cpaKTOpoB", 3aBHCJIIUHH OT cpopMbl JnnHnCOH,!¥\. Tenep& MO:>KHO Bbipa3HTb JHepnuo ~on&-~Hnon&Horo B3aHM:o­~eHCTBM · (9.11) qepe3 M. IIpe,n;cTaBHM: ~H :noro M(r) B B~e pa3no:>KeHwi no DnOCKHM: BOnHaM:

M(r)=<M>+ 1: mk exp(ikr). k

(9.19)

3~ec& Mbi C'IHTaeM, 'ITO mk = 0 npH k = 0, a cpe~ee 3Ha'leHHe HaMarHH'IeHHoCTH no o6pa3u,y onpe~eneHo q,opMYnoii

V<M>= f M(r)dr,

a KOTopoii V - o5&eM o6pa3u,a. IIo~cTasnu pa3no:>KeHHe n&I (9.17), (9.18) H3aTeMB (9.11),nonyqHM:

Wdd = 2rr VNtk <Mt ><Mk) + 2rr V 1: (kmk) (km_k)/k 2•

k

(9 .. 20)

(9 .19) B <l>OPMY-

(9.21)

HanoMHHM:, 'ITO npH BbiBo~e :noro B&Ip3)KeHHH npe~onaranoc&, 'ITO MHHHM:ani.­HbiH mu~eHHbiH pa3Mep o6pa3u,a MHoro 6on&we, 'leM 2rr /k. 3To 03Ha'laeT, 'ITO Mbl He Y'IHTbmaeM yoKepoBCKHe MO~bl - COfiCTBeHH&Ie MarHHTOCTaTH'JeCKHe THnbl Kone6aHHH HaMarHH'IeHHocni c xapaKTepHbiM pa3MepoM nop11~a pa3Mepa o6-pa3u,a (eM., HanpHMep, § 11 B (3]), E~CTBeHHa.JI yoKepoBCKa.JI MO~a, KO.TO­paH yqHTbmaeTcx nepBbiM cnaraeMbiM B (9.21), - :no o~opo~aH npeu,eccM HaMarHH'JeHHOCTH - orr. Bnpo'leM, 3TOT ~e<l>eKT Bblp3)KeHHJI (9 .21) He CnHW­KOM cymeCTBeH, HOO B OOnbWHHCTBe 3KCnepHMeHTOB, KOTOpbie HaC 6y~T HH­TepecOBaTh B ~anbHeHWeM, yoKepOBCKHe MO~bl (3a HCKniO'IeHHeM OIJ) He BO:l· 6y~aiOTCJI .

43

Pe3IOMHpyn pe3yn&rar&I noro rraparpacpa, MO)I(HO CKa3aTb, trro 3HeprHn cpep­poMarHeTHKa W- no cyMMa

(9.22)

B KOIOpOH 3HeprHn B3aHMO)l;eHCTBHfl: C BHelliHHM IlOJieM W m, 3HeprHn KpHCTaJI­JIOrpacpH'IeCKOH aHHJorpomm: W a , o6MeHHafl: 3HeprHn W ex R 3HeprHn )l;HilOJib­.n;urron&Horo B3aHMo.n;eiicrBHn W dd orrpe.n;enniOrcn B&Ipa:>KeHMMH (9.5), (9.6) H (9 .21) COOTBeTCTBeHHO. Terrepb y HaC y:>Ke BCC rOTOBO )l;Jlfl: TOro, trr06bi Oll­pe.n;eJIHTb 3aKOH .n;ucrrepcHH CB H MarpH'IH&Ie 3JieMeHTbi raMHJibTOHHaHa B3aHMo­.n;eiicrBHn. 3TH Bbi'IHCJieHHfl: He Co.n;ep:>KaT HHKaKHX IlpHHLUffiHaJibHbiX 3arpy.n;He­HHH, H HX rrpe.n;naraercn c.n;enar& caMocronreJibHO, pell.IHB 3a.n;aqu 2.3, 2.4.

3a.n;aliH

3aAB'Ia 2.1. IToKa:JaTb, 'ITO WUl ypaBHeHHH Bnoxa (8.5) KaHOHH'IeCKHMH nepeMeHHbiMH KOOp/J;HHaTa - HMnyllbc HB.11HIOTCH

q = Mx/g, p = op = arctg Wx/Mz) . (9.23)

YKa3anue. ITpHMLIM ,D;Hcp<l>epeH~poaaHHeM c yqeToM cBH3H M' (p, q) = M 02 yoe,D;HThcH,

'ITO (8.5) B nepeMeHHLIX (9.23) npHBO,II,liTCH K BH,lzy (5.3). Jlanee, no OIIHCaHHOII: B § 5 CXeMe MOlKHO nepeATH OT p, q K KOMMeKCHLIM nepeMeHHbiM b, b*, 'lepe3 KOTOpble BeK­TOp M 6yAeT BLipalKaThcH no cpopMynaM (8.9) .

3aAB'Ia 2.2. J.icnollb3YJI reA3eH6eproacKHA raMHllbTOHHaH Xex (2.13), nOJIY'IHTh Bblpa­lKeHHH (9.6), (9. 7) WUl 06MeHHoA 3HeprHH B KOHTHHyallbHOM npH6nHlKeHHH.

YKa3anue. lleACTBoBaTh no cxeMe, H3nolKeHHoA B § 9.2, H 3aTeM, pa3nOlKHB M B PHA no (r - T'), npOHHTerpHpOBaTb no 'laCTHM, npeAOOnaraH, 'ITO KpHCTann HMeeT !leHTp HHBepCHH.

3aAa'la 2.3. BLI'IHcnHTb Ko3$cp~eHThl cnHH-BonHoaoro raMHllbTOHHaHa a cl>eppoMai'He-rnKu. •

YKa31lnue. YAo6Ho AeACTBoBaTh B cneAYIOIIIeA nocneAoBaTellbHocrn : npeACTaBHTh Ha­Mai'HH'IeHHOCTh B BHAe pa3nOlKeHHJI no nnocKHM BonHaM, AaJiee Bblpa3HTb 3HeprniO cJ>M (9.22) B BHAe PHAa no CTeneHHM Mk (t), 3aTeM B nonyqeHHOM BblpalKeHHH nepeATH K ne­

peMeHHLIM rollblliTeAHa - ITpHMaKOBa ak, ai (8.11). HaKOHe!l, COBepliDITb u - v-npeo6pa-

30BaHHe (5.17) K KaHOHH'IeCKHM nepeMeHHhlM bk, bk• ,D;HarOHanH3YI0lllHM ero KBaAPaTH'I­HyiO 'laCTh.

3aAB'Ia 2.4, Bbi'IHCnHTb 3aKOH ,D;HcnepcHH H KoJ<l>cpH~eHThl raMHllbTOHHaHa B3aHMOAeA­CTBHJI CB B Aci>M c aHH30TpOIIHeA THna "nerKaH nnocKoCTh". J.icxo,D;HTh H3 <l>eHoMeHonorn­

'lecKoro BblpalKeHHH AJUi3HeprHH AcJ>M niT B ABYXllOAPeWeTO'IHOA MOAenH :

W = 'h f dr { [BM' +a, (oL/or) • +a, (oM/or) 'I + + bMz' + aLj - 2(3 (n [ML)) - 2MH}.

n = r/r. (9.24)

3Aec& M,, M, - HaMarHH'IeHHocrn noAPeWeTOK, M 1 =M, =M0 , M =M, +M,, L =M, - M 1 ,

(J > 0, a l, 2 > 0 - 06MeHHbie KOHCTaHTbl, (J - KOHCTaHTa ll3HnOWHHCKOrO ( 7 8), a > 0, b -KOHCTaHTbl aHH30TpOnHH.

YKa31lnue. YAo6Ho AeACTBoBaTh B cneAYIOIIIeA nocneAOBaren&Hocrn : cHa'lana H3 ycno­BHH MHHHMyMa 3HeprHH (9.23) HaATH KOHcpHrypa!lHlQ MOMeHTOB M~, M~ B OCHOBHOM COC­TOJIHHH, 3aTeM B AByx BCllOMOraTellbHbiX CHCTeMU KOOpiJ;HHaT, Ka)I(AaH H3 KOTOpbiX OpHeH­THpOBaHa no COOTBeTCTBYIOIIIeA HaMai'HH'IeHHOCTH Mt (i = 1, 2), nepeATH K KaHOHH'Iec­

KHM nepeMeHHblM (8.8) Cj (r) , Ct(r) (i = 1, 2) B KalKAOA OOApeweTKe TaK lKe, KaK H

B cJ>M, AaJiee~peATH K npeACTaBneHHIO Cl>yp&e C(r) -+ Ck, H nepeATH K CHMMeTpH30BaH­

HLIM nepeMeHHhlM .j2 d± = C1 ± C1 , npH 3TOM KBaApaTH'IHaH 'laCTb raMHllbTOHHaHa OKalKeT­

CH KBa3H,D;HarOHallbHOA, nepeMeHHble d+, d.t 6yAyT OIIHCbffiaTh OAHY BeTBb CB, ad_ , d~ ­APYrYIO. noroM c noMOIIIb!O npeo6pa3oBaHHH (5.17) B KalKAOA BeTBH ,D;HaroHanH30BaTh KBaApaTH'IHble 'laCTH raMHllbTOHHaHa, HaKoHe!l, nonyqHTb BblpalKeHHH WUl Wk, [/.k H X jnt B

3THX nepeMeHH&lX. • 31l.Me'lllHUe. CyweCTBeHllbiM M OMeHTOM npH U - v-npeo6pa30BaHHH 6yAeT TO, 'ITO

u > 1, v > 1, a To apeMll KaK I u 11 ~ I v 12 =· 1. 3To npHBOAHT K coKpameHHIO B Xint ca-

Mb!X 6om.lliHX cnaraeMblli, nponop~OHllJibHbiX Bu 4' Bu 1 v% H 3aCTaBJVIeT npOBOAHTJ> rpo­

M03)J;I<He Bbi'IHCJieHHll O'leHb aKKyparno, yp;epliCHBaH B HCXO)J;HOM BblplllKeHHH A1iJI :;mepi'HH

MaJible 'lJieHi>I, BKJ!a)J; KOTOpbiX llOCJie YMHOlKeHHJI Ha CTelleHH U H V MOlKeT CTaTb C}'meCT­

BeHHbiM. BbiCKa3b!BaJIOCb MHeHHe, 'ITO ManiHTIIOe )J;HnOIILHoe B3aHMO)J;eitCTBHe MOlKeT Ta­

I<HM o6pa3oM npHBecTH K aHnMaJILHO 6om.woMy BKJ!ap;y B Xmt . Op;uaKo Bbi'IHcneHHJI

B .C. JlyTOBHHOBa, B.JI. Ca4J?HoBa [ 146] noKa3aJIH, 'ITO 3Toro He npoHcxop;HT.

II p H n o )I( e H H e 1. BapHilWIOIDILJe npoH3BO)l,HLie .

Mbl He XOTHM 31J;eCb npHBOp;HTb CKOJibKO-HH6yp;L crporoe onpep;eJieHHe 3Toro llOHJITHJI,

KOTOpOe YAOBJieTBOpHJIO 6LI MaTeMaTHKOB HJI!i XOTJI 6bi ¢lH3HKOB-TeOpeTHKOB. )lrol npoc­

Tb!X neWeXOIJ;OB Mbl C06HpaeMCJI llOJICHHTb HeCKOJILKO llpaBHJI Bbl'lHCJieHHJI BapHa~OHHbiX npoH3BOAHbiX o/o f (f) . OHH cneiJYIOT H3 Toro, 'ITO o (of (r) o6o6maeT noHHTHe qacrnoA

npOH3BO)J;HOA o /of (f n) (r n - )J;HCKpeTIIblA Ha6op TO'leK) Ha CJIY'laA KOHTHHYaJibHOrO '!Hc­

Jia nepeMeHHblx.

l. llyCTb I - cnep;yiOnuDt ¢lyHK11,HoHan o T ¢lyuK~ f (r) :

I= ff(f) 'P (f)dr, (lll.l)

rp;e '{)(f) - BeCOBaH ¢JyH~. )lrol Bbi'IHCJieHHJI o//of (r) MOlKHO MbiCJieHHO 3aMeHHTb ee

Ha 'laCTIIYIO npOH3BOp;HyiO, Op;HOBpeMeHHO 3aMeHHB IniTerpHpOBaHHe CYMMHpOBaHHeM H

llOJIY'lHB OKOH'laTeiiLHblA pe3yJILTaT, nepeATH o6paTHO OT p;HCKpeTIIOA K HenpepbmHoit ¢lop­

Me 3anHCH. HanpHMep,

o a -- f '{) (f')f(r ' )dr' =.-

1( -) I: '{) (rm)f(rm) =

of (f) a 'n m - of(rm) =I:'P(rm)-f() =I:'{J(rm)o(rm,rn)='{J(rn)='{J(r).

m a rn m 3TOT pe3ym.TaT MOlKHO npeiJ;CTaBHTb CHMBOJIH'leCKOA ¢JopMyJIOA

of(r')/of(r)=o (r-r'). (ll1.2)

2. EcJIH B ¢JyHK11,HOHaJie Ha ¢lyHKLUOO f p;eACTBYIOT AH¢J!JjepeH~aJibHble onepaTOpbi,

TO MJI TOro, 'lT06bl BOCllOJib30BaTLCJI ilpaBHJIOM (lll.2) , Heo6XOAHMO npep;sapHTeiiLHO

" nepe6poCHTb HX HaneBo", BbmOJIHJIJI HHTerpHpOBaHHe no 'laCTJIM. HanpHMep,

-0

-- f'PVfdr'= -0-f[V'{Jdr'=-V'{J(r) .

of(r) of(r) (lll.3)

3, 0CTaJibHble npaBHJia Bbi'IHCJieHHJI ¢JyHK11,HOHaJibHOA npOH3BO)J;HOA, KOTOpble HaM

MOryT ll0ija)J;06HTLCJI, COBlla,!l,lllOT C ilpaBHJiaMH A1iJI 'lacTHbiX ilpOH3BOp;HbiX. HanpHMep,

___J_ !' (r ") = 2[ (r) oftr) . of (r) of (r')

(lll.4)

II p H n o )I( e H H e 2. Y C!IOBHJI KaHOHif1DIOCTH npeo6pa30BaHHR

1. llyCTb ai (r, t), al (r , t) - KaHoHH'leCKHe nepeMeHHbie, TaK 'ITO HX ypasHeHHJI p;sHJKe­

HHJI HMeiOT KaHOHH'leCKHA BHA (5.6). BBeiJ;eM HOBble nepeMeHHble bt (r, t), bt(r, t) C no­

MOILU>!O npeo6pa30BaHHA, He cop;eplKa!UHX JIBHO BpeMJI :

b; = f; { aj. al}' bj = rt { aj. aj"}' (02.1)

3p;ecb f; - HeKoTopbiA ¢lYHKUHOHan. BbNHCJIHM

ab; ~ f , ( ob; ox ob; ox ] -=~ dr -i--- +i , at j oaj (r') oal (r') oal (r') oaj (r')

~ = I: dr" [ _Ei_ obz (f") + ____Ei_ obi (f") J oai (f') 1 f obz (r") oai (f') obi (f") oai (r') '

i ab; = I:fdr ' [____Ei_{b· (r)b · (f ' )} +___i!i_ {b · (r)b · (f ' )} ] · at i obj (f ' ) 1 I aa• obj (f) 1 I aa• (02.2)

45

MLI BBeJIH 3AeCh o6o3Ha'leHHJI AIDI cKo6oK 11yaccoHa:

{ r r ' } = I; f dr" [ of (r) og (r') - of (r) og (r') ) f G ) g ( ) aa • 1· oa~ (r" ) o .a (r") oa · (r") oa1~ (r" ) ·

I I I

(02.3)

11oTpe6oBaB, '!To6h! ypaBHeHHJI ABIDKeHHJI nepeMeHHhiX b, b • (112.2) coBnap;aJIH c Ka­HoHH'IecKHMH ypaBHeHHHMH (5 .6), nOJIY'IaeM ycnOBHe KaHOHH'IHOCTH npeo6pa30BaHHA (02.1) :

{b;bj} = o, {b; (r)bj (r ' )} =-oqo (r - r'). (02.4)

3nf YC1IOBWI JlBIDIIOTCJi KrraCCH'IeCKHM aHarroroM mipeCTaHOBO'IHhiX COOTIIOWeHHlt )l;1IJI

6o3e-onepaTopoB. 2. 11yCTh a/ (k) , al(k) - KaHoHH'IeCKHe nepeMeHHhle, 3ap;aHHhie B k-npep;CTaBneHHH.

HeTpYAHO y6e.o;HThCll. 'ITO ycrroBHe KaHOHH'IHOCTH npeo6pa30BaHHH

bj (k) = bjk {a/ (k) , aj (k)} - -- - - (ll2.5)

HMOOTBHA

{aj (k) , a; (k ' )} bb • = oii 6(k - k'),

{aj (k), a; (k')} bb 0 = 0.

3p;eCh { .•. } - CK06KH llyaccoHa

• [ of; (k) ogj (k' ) {!; (k) , Kj (k )} bb. = r P· obJ (k" ) obi (k" )

(02.6)

of; (k) ogi (k ') ].

obj (k" ) obj (k" ) · (112.7)

HeKOTOpoe OTJIH'IHe cpopMyn (112.6) H (112.7) OT (112.3) H (112.4) B03HHK110 H3-3a TOro, 'ITO BOJIHOBOlt BeKTOp k JlBIDieTCH y Hac p;HCKpeTHolt BeJIH'IHHOlt.

3. l'aCCMOTpHM rrHHeltHYIO JaMeHy nepeMeHHhiX BH)l;a (5.17). 8h!'IHCIDIJI CK06KH 11yacco-Ha (112.6), (112. 7), nonf'IHM ycrroBHe KaHOHH'IHOCTH 3Toro npeo6paJOBaHHH :

~ [Utj (k)VIj (k) - Vtj (k)vq .(k)) = o il• I :

~ [uq (k) vq ( - k) - Vfj (k)uq (- k)] = 0. I

(02.8)