1999lss.fnal.gov/archive/other1/jinr-p17-99-276.pdf2 teophh 3q,q,ekta boma-aapohoba [15] h...
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P17-99-276
BKnAnblfiOn~PH3YEMOCTEH
B tIETblPE EA3HCHbIE nOn5IPH3AUHH
3nEKTPOMArHl1THbIX CPEll
2hv--rlli 1i14middotcJ~$ko~ r$IIlt HanpaBJ1eHO B laquo)fltYPHaJ1 TeXHlPleCKOH qm3HKHraquo
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~ -mail bukinathsunljinrru
1999
HMeIOTCSI BemeCTBa (HanpHMep KPKCTaJIJIHtfeCKHe (1)) 3JIeKTpOMarHHTH1gtIe
CBOHCTBa KOTOPbIX MOgtKHO OnHCbIBaTb npeHMymecTBeHHO paCnpeneJleHHHMI1
IlHnOJIeH MYJIbTHnOJIbHOe pa3JIOgtKeHHe nJIOTHOCTH Takoro pacnpeneJIeHIU
HMeeT pSlll OTJIHtfHH OT pa3JIOgtKeHHSI TOKa 3apSlDOB tfTO npHBOllHT I( HHTeshy
peCHbIM CJIellCTBHSlM OTHOCHTeJIbHO XapaKTepa B3aKMOlleHCTBHH DHIIOJIbHhIX
CHCTeM C BHeIlIHHMH nOJISlMH llOJIHbIH aHaJIH3 3THX pa3JIOgtKeHHH HMeeT ue
TOJIbKO aKalleMHtfeCKHH HHTepeC
H3BecTHo [2 3J tfTO B paMKax 3JIeKTpOllHHaMHKH MaKCBeJIJIa-HopeHIla
MYJIbTHnOJIbHOe pa3JIOgtKeHHe nJIOTIIOCTH 3apSlDa-TOKa COllep)lHT TpH BHlla 60shy
3HCHbIX llHnOJIeH 3JIeKTpHtfeCKHH Q MarHHTHbIH M H (nOJISlpHbIH) TOpOHIlshy
HbIH T 3JIeKTpOMarHHTHaJI cpella B KOTOPOH HeT HI1 CB060DHbIX IIH CBSIshy
3aHHblX 3apSlllOB (p == 0) IIH He3aMKHYTbIX TOKOB (divj = 0) onHCbIBaeTCSI
TOJIbKO MarHHTHbIMH H TOPOHllHbIMH nOJISlpIOaUHSlMH [4J llPH onHCaHltH pacshy
npelleJIeHHSI HaMarHHtfeHHOCTH B mpe BJIaTT H BaHcKonltp (CM IIX KUHry [5] B KOTOPOH UHTHPYIOTCI HX npenIlIecTBeHHHKH) nO-BHllHMOMY BnepBhIe nOll
Ha3BaHHeM HHllYUHpoBaHHbIe 3JIeKTpHtfeCKHe nOnepetfHbIe MOMelITbI BBeJII1
MarHHTHbIe BKJIallbI B TOpOHDHbIe nOJISlpH3allHH llPH KJIaCcHltpHKaIUtH TOllellshy
HbIX rpynn MarHHTHOH CHMMeTpHH KPHCTaJIJIOB Ha BbIDeJIeHHOCTb TIBIa CHMshy
MeTpHH KOTOPbIM 06JIallaeT TOPOHllHbIH IlHnOJIb YKa3bIBaJIOCb B [6]-[9] Marshy
HeTHKH llOnYCKaIOlllI1e cymecTBoBaHHe TOPOHllaJIbHOrO ynopSIllOlleHHJI 6blJIH
Ha3BaHbI TopOHKaMH TIpHMepbI MarHHTHbIX KPHCTaJIJIOB llOnYCKaIOmHX TOshy
pOHllHOe ynopSIllOlleHUe npHBelleHbI B ynoMSlHYTax BbIwe pa60Tax vb 110shy
CJIellHHX pa60T no 3KcnepHMeHTaJIbHOMY H3YlleHHIO TOPOHllHOH nOJISJpH3aUIII1
OTMeTHM ee H3MepeHJU B MarHHTHOM nbe303JIeKTpHKe Ga2-rFcr03 [10] H B
MarHHT03JIeKTpHKe Cr203 [11 J B 1986 r B annapaT 3JIeKTpODIfHaMHKI1 CnJIOllIHbIX cpeJl 6blno BSCJlCnO
ceMeHCTBO aKCHaJlbHbIX TOPOHJlHblX MOMeUTOB [12] 1 nOBOJIbHO CKOPO 3THM
MOMeUTaM 6bIJIO uaHDeno BeCbMa lIpKOe npUMeueHHe B cpH3HKe MOJIeKYJI B 1991 r 6bIJIO YKa3aHO [13] liTO MOJIeKYJIbI pSllIa apOMaTIiTfeCKHX coenHHeHlUl
(aHTpaueu cpeuaHTpeH neUTauell H lIp) npH KPUCTaJIJIlftIeCKOH ynaKOBJ(e
MoryT C03lIaBaTb aKCHaJIbHbIH TOPOHlIHbIH MHKpOlIHnOJIb 3a ClIeT ynoPJlJlOshy
lIeHHSI 3JIeKTpUlIeCKHX llHnOJIeH aTOMOB B COCTaBe BKJIIOlIeHHOrO B CJlOgtKHYIO
MOJIeKyny llliKJIHlIeCKOrO q)parMeHTa 6eH30nbHoro KOJIbua C6 (CM PHc1 )
3TO n03BOJIlmO 06bllCHHTb B3aHMOlIeiicTBHe MHKPOKPHCTaJIJlOB apOMaTHtleshy
CKHX BemeCTB C nepeMeHHbIM MarHHTHbIM nOJIeM [14J TaKOH He06hIlIHbIH BHll
MarHHT03JIeKTpHlIeCKOH lUIlIYKUHH 6bIJI Ha3BaH apoMarHeTH3MoM
3TH ycnexH a TaIOKe HeYlIOBJIeTBOpCuHOCTb cOBpeMeuHbIM COCTOllHHCM
HcropHlecKHM npHIlfHaM f8] 6amcHhIH mmonh tToro ceMeiicTna nllorna o603nalashyeTC~ KaK G
2
2
TeopHH 3qqeKTa BOMa-AapoHoBa [15] H rrOJIBJIeHHe ero aHaJIOrOB (CM Harrp
[16]) 3aCTaBJIJIeT 60JIee CHCTeMaTHqeCKH OTHeCTHCb K BOrrpocaM MYJIbTHrrOJIbshy
HhIX rrpencTaBJIeHHH PalJIHqHhIX KOHqHrypaUHH KOTophIe MoryT B03HHKaTb
HarrpHMep B nHrrOJIbHhIX cpenax H K BhI60py rrapaMeTpOB nJIJI OrrHCaHHJI 3JIeKshy
TpOMarHHTHhIX CBOHCTB KBaHTOBhIX 06beKToB HarrpHMep MOJIeKYJI co CJI02Kshy
HhIM CTpoeHHeM
B HaCTOJImeHCTaTbe HaHneHhI rrOJIHhIe Ha60phI 6a3HCHhIX nHrrOJIeH nJIJI
qH3HqeCKH HaHfuJIee 3HaqHMhIX nHrrOJIbHhIX cpen [9] lIaJIee HCXOnJI H3 MaTeshy
MaTHqeCKoro qaKTa HX cymecTBOBaHHJI YKMaH PJIn B03M02KHhIX He06hIQHhIX
qYHKUHH OTKJIHKa BemecTBa HaxonJImeroCJI B KOHneHCHpOBaHHOM COCTOJIHHH
Ha HeonHoponHhIe HHJIH nHHaMHQeCKHe 3JIeKTpHQeCKHe H MarHHTHhIe rrOJIJI
B 3TOM OTHomeHHH naHHaJI CTaTbJI JIBJIJIeTCJI pMBHTHeM pa60T [12] H [17]
ITocTpoeHHe MynbTHnonbHMx MOMeHTOB
llna cnHHOBMX (llHnOnbHMX) cpell H HX B3ashy
HMOlleHCTBHa C BHelllHHMH nonaMH
IIpeucTaBHM QTO HMeeTCJI HeKaJI cpena (cHcTeMa) HerrpephIBHO pacrrpeneshy
JIeHHhIX 3JIeMeHTapHhIX 3JIeKTpHQeCKHX nHrrOJIeH KOTOPaJI orrHChIBaeTCJI orpashy
HHQeHHOH B rrpOCTpaHCTBe ltpYHKUHeH d rEV B 06meM cJIYQae 3aBHCJIshy
meH OT BpeMeHH lIorrycTHM QTO B TaKoH cpene rro pa3JIHQHhIM rrpHQHHaM -
BCJIenCTBHe BHyTpeHHHx B3aHMoneHcTBHH HJIH rrou BJIHJIHHeM BHemHHX rroJIeH
- MoryT B03HHKaTb H BhIcTpaHBaTbCJI arperaThI pa3JIHQHhIX KOHltpHrypaUHH
COCTOJImHe H3 KOHeQHOrO QHCJIa 3JIeMeHTapHhIx uHrrOJIeH OrrHChIBaJI CBOHshy
CTBa arperaToB C rrOMOmbIO MYJIbTHrrOJIbHhIX rrapaMeTpoB BhIrrHmeM rrepBhIe
QJIeHhI pMJIO)KeHHJI 3HeprHH HX B3aHMoneHCTBHJI C BHemHHMH 110JIJIMH
W = - ( d(r)E(r)d3r = ~r02divE - QE - T(d)rotEshylv 6
-p(d)rotrotE - ~QiAViEj +VjEi) - Tid)Vi(rotE)j + 1 shy+10 (r~2) V)divE + === -QE T(d)rotE + p(d)(Vp - ~E) +
(d) (d)middotmiddotmiddotmiddot+ === -QE + T H + P (J - E) + (1)
rne Q = f d(r )J3r - 06hIQHhIH rrOJIHhIH 3JIeKTpHQeCKHH nHrrOJIbHhIH Moshy
MeHT CHCTeMhI (PHcIa) T(d) = ~ J r x d(r)cPr - aKCHaJIbHhIH TOPOHnHhIH
nHIIOJIbHhIH MOMeHT CHCTeMhI (PHcIb) p(d) = 1~ J[r(rd) - 2r2d]d3r - 110shy
JIJIPHhIH rrOJIOHnaJIbHhIH nHIIOJIbHhIH MOMeHT (PHcIc) BMeCTe OHH COCTashy
BnJIIOT 6MHCHYIO TP0HKY BeKTOPOB nnJI pa3n02KeHHJI rrnOTHOCTH pacrrpeueneshy
HHJI naHHoro BHna KpOMe HHX B (1) HMeIOTCJI r02 = J r EcPr - CKaJIJIPHhIH
3
a)
-
-
c)
f)e)
Puc1
4
MOMeHT KOTOPhIH B pa3JIO)KeHHH TOKa COOTBeTCTByeT BpeMeHHOH npOH3BOlJshy
HOH OT CpelJHeKBalJpaTHtIHoro palJHyca 3apSlJ(a T e lJBOHHOMY CltpepHtIeCKOMY
CJIOIO (PHcld) 2 ri2) = J[2r(rE)+r2Eltf3r - CpelJHeKBaZJpaTHtIHhIH paZJHyC
rtP0lJOJIhHOH tIaCTH lJHnOJIbHoro MOMeHTa Q KOTOPhIH 06pa3yeTcx lJBOHHhIM
UHJIHHlJpHqeCKHM CJIoeM lJHnOJIeH (PHcle) Qij H Ti~d)- 3JIeKTpHtIeCKHH H TOshy
POHlJHhIH KBalJpynOJIbHhIH MOMeHThI onpelJeJIXeMhIe no 06hItIHhIMmiddot npaBHJIaM
Bce B3aHMOlJeHCTBHX MYJIbTHnOJIbHhIX xapaKTepHCTHK C nOJIeM HJIH npOH3BOlJshy
HhIMH OT Hero B BhIpa)KeHHH (1) JIOKaJIbHhI H OTHOCXTCjl K HeKoTOpoH CpelJHeH
TOtlKe ueHTpy CHCTeMhI
ECJIH )Ke B3aHMOlJeHCTBHe dE 3aMeHHTb Ha pH TO nepBhIe tIJIeHhI Pa3JIoshy
)KeHHX 3HeprHH B3aHMOlJeHCTBHx pacnpeneJIeHHX nJIOTHOCTH MarHHTHhIX lJHshy
nOJIeH C BHeIIIHHM HeOlJHOpOlJHhIM MarHHTHhIM nOJIeM npHHHMaIOT BHlJ
w = - f p(r)H(r)d3 r = -MH - T(IJ)rotHshy
P(IJ)rotrotH - -~M(VH + V middotHmiddot) - T(~)V(rotH) + ----gt2 J bull J J J J ---r
==gt -MH - T(IJ)E - p(IJ)(rot - H) ==gt -MH - T(IJ)rotH + p(lJ) ~HCB06 + (2)
BHlJHM tITO COrJIaCHO lJYaJIbHOCTH 3JIeKTpOMarHeT1iI3Ma MOMeHT Q B (1) 3ashy
MeHxeTCX Ha MarHHTHhIH lJHnOJIbHhIH MOMeHT M XapaKTepH3YlOInHH 3KBHshy
BaJIeHTHYIO paMKy C TOKOM T(d) - Ha nOJIXpHhIH TOPOHlJHhIH lJHnOJIb T(IJ) a
BeKTopHhIH nOJIOHlJaJIbHhIH lJHnOJIbHhIH MOMeHT p(d) - Ha aKCHaJIbHhIH p(IJ) KorlJa HOCHTeJIXMH MarHHTHhIX MOMeHTOB XBJIXIOTCX cnHHOBhIe tIaCTHIJhI
HanpHMep HeHTpOHhI TO MYJIbTHnOJIbHoe pa3JIO)KeH~e pacnpelJeJIeHHX HX
MarHHTHhIX MOMeHTOB (HanpHMep B SlJ(pe HJIH B HeHTpOHHOH 3Be3lJe) nOJIyshy
tIaeT nepBHtIHhIH CMhICJI B TeopHH KOHlJeHCHpoBaHHorO COCTOXHHX BemeCTBa
Ha B03MO)KHOCTb TOPOHlJaJIbHOro ynOpSlJ(OtleHHX cnHHOBhIX MarHHTHhIX MOMeHshy
TOB B nepOBCKHTax BnepBhIe YKa3hIBaJIOCb MC )KeJIYlJeBhIM [6 YnopSlJ(Otleshy
HHe )Ke MarHHTHhIX tIaCTHU B CBepXTOHKHX nJIeHKaXB 3aMKHYThIe KOHltpHryshy
paUHH KOTophIe MO)KHO onHChIBaTb TOPOHlJHhIMH MOMeHTaMH no-BHlJHMOMY
Bo06me npelJCTaBJIXeT c060H 06InHH CJIytlaH (CM Hanp [18] H [19]) OTshy
MeTHM tITO cgt6MeHHhIe B3aHMOlJeHCTBH1I Me)KlJy cnHHaMH pacnOJIO)KeHHhIMH
Ha nJIOCKOCTH TaK)Ke npHBOllXT K H3BeCTHOH peIIIeTKe BHxpeH KeCTepJIHuashy
TaYJIeca pacnOJIO)KeHHhIX B aHTHTOp0HuHOM nOpXlJKe
Pa3JIO)KeHHX nOlJ06HhIe (1) H (2) OqeBHlJHO MO)KHO BhInOJIHHTb lJJIX
lJByX OCTaBIIIHXCX ltpYHlJ~eHTaJIbHhIX (TOP0HlJHhIX) lJHnOJIbHhIX pacnpelJeJIeshy
HHH T(r) H g(r) IIOJIytleHHhIe B pe3YJIbTaTe TaKHX pa3~O)KeHHH BHXPH BhIshy
COKOro nopSlJ(Ka BpSlJ( JIH peaJIH3YlOTC1I B KPHCTaJIJIHqeCKHX BemeCTBax llJIX
2TaKYlO KOHltlgtHlypaUHIO BqaCTHOM cnyqae npH paBHOM HynlO BHYTpeHHeM pallHyce HHoshy
llla Ha3hIBaIOT e)KOM
5
3
HX 06pa30BaH~liI HYJKHbI 3K30THtJeCKHe Cpellbl B KOTOPblX 3JIeMeHTapHble Hoshy
CHTeJIH YJKe 06JIauaJOT HanpHMep TOP0HllHblMH llHnOJIliIMH OTMeTHMtJTO
nOCJIellHHe 3KCnepHMeHTbI YKa3b1BaIOT tJTO HeHTpHHO BHllHMO MaHOpaHOBshy3CKHe tJaCTHIUd H 06pa3YIOT cpellbl BblCOKOH nJIOTHOCTH (10HgtcM- ) H3BeCTHO
(CM Hanp [20] H [21]) tJTO TaKHe HeHTpHHo 06JIallaIOT TOJIhKO TOPOHllshy
HblMH llHnOJIhHblMH MOMeHTaMH Ha B03MOJKHOCTh KorepeHTHblX B036YJKlleshy
HHliIX B TaKOH cpelle 06ecnetJHBaIQIllHX B036YJKlleHHe MeJKlly aupOHaMHHa
ypOBHe rpaBHTaUHOHHblX 6b1JIO BnepBble YKa3aHO BM KOPIOKHHblM [22]
BeKTopShIe napaMeTphI
CpenhI
H3JIOJKeHHblH BbllUe MaTepHaJI OTHOCHTCliI K TaK Ha3b1BaeMOB JIHHeHHOH 3JIeKshy
Tp0llHHaMHKe COrJIaCHO ee ypaBHeHHliIM 3JIeMeHTapHble 06beKTbI HanpHMep
llHnOJIH ynpaBJIliIIOTCliI nOJIeM KaKoro-JIH60 THna HJIH TOKOM npH KOHTaKTHOM
B3aHMOlleHCTBHH H He MeHliIIOT npH 3TOM CBOHX napaMeTPOB KpOMe llHnoshy
JIeB MOJKHO BBeCTH If 6oJIee CJIOJKHble 06beKTbI COCTaBJIeHHble H3 llHnOJIeH
KBallpynOJIH oKTynOJIH H Tll IIpocTeBlUHe MaKPOCKonHtJeCKHe KOHCTPYKUHH
TaKoro pOlla MOJKHO npellCTaBHTh HanpHMep B BHlle llByX HJIHTpeX TOKoshy
BblX neTeJIh pacnOJIOJKeHHblX B6JIH3H llpyr llpyra HJIH CBliI3aHHblX B ellHHhIH
KOHTyp PacCMOTPHM HanpHMep npOBOJIOtJHoe KOJIhUO H30rHYTOe B ltpopMe
nponeJIJIepa IIpH npOXOJKlleHHH no TaKOMY KOJIhUY TOKa KaJKllaJI H3 neTeJIh
C03llaeT MarHHTHoe nOJIe H MbI MOJKeM npH6JIHJKeHHO onHCblBaTh HX MarHHTshy
Hble CBOHCTBa C nOMOmhIO MaIHHTHblX MOMeHTOB Torlla 3HepIHliI JIOKaJIhHblX
B3aHMOlleHCTBHH 3THX llByX neTeJIh C BHelUHHM MaIHHTHblM nOJIeM npOH3shy
BOJIhHOH KOHqmrypaUHH 3anHCbIBaeTCSI TaK
(3)
IIoll JIOKaJIhHOCThIO nOllpa3YMeBaeTCliI OTHeceHHOCTh CJIaraeMblX BBelleHHOH
3HeprHH K ueHTpaM neTeJIh BBelleM Teneph KPoMe BeKTopOB Ili H Ilk Koshy
Topble OTHOCliICh K Pa3JIHtJHblM neTJIliIM npHnHCaHbI K Pa3JIHtJHblM TOtJKaM
npOCTpaHCTBa I H 2 KBaupynOJIhHblH MOMeHT CHCTeMbI B BHlle CHMMeshy
TPHtJHOro TeH30pa BTOporo paHra Ilik KaK H B npellblllYIlleM naparpaltPe
napaMeTp Ilik YtJHTblBaeT BKJIall B 3HepIHIO C03llaIOIllHHCliI 3a CtJeT rpauHshy
eHTa HaJIaraeMoro Ha CHCTeMY BHelUHero MarHHTHoro nOJIliI T e 3a CtJeT Heshy
OllHOPOllHOCTH nOJIliI MeJKlly ueHTpaMH neTeJIh Torlla BblpaJKeHHe (3) MOJKHO
npH6JIHJKeHHOnepenHcaTh B BHlle
(4)
6
BHuHM qTO Tenepb BCe qJleHhI pa3JlOJKeHHH OTHeCeHhI K HeKOTOpOH CpeuHeH
TOqKe KOTOpyIO MhI Ha3hIBaeM ueHTpOM CHCTeMhI H B TaKOM npeuCTaBJleHHH
meprHH B3aHMOueHCTBHH H COCTOHT BCH CYTb MYJlbTHnOJlbHoro pa3JlOJKeHHH
npeunOJlOJKHM renepb qTO MaTepHaJI H3 KOTOPOro cueJlaHhI neTJlH rH60K
H MarHHTHoe nOJle opHeHTHpYIOlUee MarHHTHhIe MOMeHThI MOJKeT ueltpopMHshy
pOBaTb KOHTyphI neTeJlb GTO B03MYlUeHHe MhI MOJKeM yqeCTb BBeueHHeM
KBaupynOJlbHOH nOJlHpH3yeMocTH CHCTeMhI a(m) ilonOJlHHTeJlbHaH 3HeprHH
JlOKaJIbHOrO B3aHMoueHCTBHH CHCTeMhI C MarHHTHhIM nOJleM onHChIBaeTCH YJKe
HeJlHHeHHhIM no nOJlIO BhIpalKeHHeM
(5)
TaKHM 06pa30M MbI nOJlyqaeM HOBOe KaqeCTBO B H3yqeHHH CBOHCTB CHCTeMhI
uHnOJleH 0606lUaH CKaJKeM TaK qTO HaJIOJKeHHe nOJlH Hi HHuyUHpyeT B
cpeueuonOJlHHTeJlhHYIO MarHHTHYIO uHnOJlbHYIO nOJlHpH3aUHIO Mle = a~)Hi KOTOPaH B CBOIO oqepeub cnoc06Ha B3aHMoueHcTBOBaTb C nOJleM Hie 3uecb
a) 6yueM Ha3hIBaTb KBaupaTHqHOH MarHHTHOH BOCnpHHMqHBOCTbIO CpeuhI
HTaK MbI BBeJlH HeJlHHeHHOe B3aHMoueHcTBHe yqHThIBaIOlUee HeJlHHeHshy
HhIH OTKJlHK cpeuhI Ha MarHHTHoe nOJle B npHHUHne TeH30p a MOJKeT HMeTb
npOH3BOJlbHOe KOJlHqeCTBO HHueKCOB H B BhIpaJKeHHH uJlH 3HeprHH OH uOJlJKeH
6bITb cBepHyT C COOTBeTCTBYIOlUHM CHMMeTpHqHhIM TeH30pOM COCTaBJleHshy
HhIM H3 BeKTopa HanpHJKeHHOCTH BHeuIHero MarHHTHoro nOJlH ECTecTBeHHo
qTO ueltp0pMaUHH nou06HhIe onHcaHHhIM uJlH CHCTeMhI MarHHTHbix uHnOJleH
MOJKeT HcnbIThIBaTb H CHCTeMa COCTaBJleHHaH H3 3JleKTpHqeCKHX uHnOJleH B
3TOM CJlyqae BhIpaJKeHHH uJlH 3HeprHH HX B3aHMoueHcTBHH C BHelIIHHM 3JleKshy
TpHqeCKHM nOJleM 6yuYT COBeplIIeHHO HueHTHqHhI BBeueHHhIM BhIlIIe a Marshy
HHTHyIO BOCnpHHMqHBOCTb B HHX MhI 3aMeHHM Ha 3JleKTpHqeCKYIO a ebull noshy
CKOJlbKY peaJIbHhIe CHCTeMhI H3 KOTOPhIX COCTOHT JlI06aH cpeua - 3TO aTOMhI
H MOJleKYJlhI HaxOuHlUHeCH B COCTOHHHH HenpephIBHOH BHYTpeHHeH H BHelIIHeH
uHHaMHKH TO BCJleuCTBHe B3aHMOCBH3eH HCTOqHHKOB qepe3 nOJlH B03MOIKHhI
H CMelIIaHHhIe nOJlHpH3yeMocTH HanpHMep MarHHT03JleKTpHqeCKHe C 3Hepshy
rHeH B3aHMOlIeHCTBHa
(6)
HMeH B BHuy OpraHH3aUHIO KOMnbIOTepHOH naMHTH BBeueM nOJlHpH3yeMocTb
TpeTbero paHra OTBeqaIOlUYIO CHMMeTpHH 3ltpltpeKTa XOJlJla H COOTBeT-
CTBeHHO 3HeprHIO B3aHMoueHCTBHH
I (em)E H JWNL -ailel i Ie I (7)C
rue J - BHelIIHHH TOK npoTeKaIOlUHH qepe3 paccMaTpHBaeMYIO CHCTeMY B
qaCTHOCTH 3TOT TOK MOJKeT 6hITb TOKOM CMelUeHHH qTO peaJIH3yeTCH Koshy
rna H3yqaeMaa CHCTeMa pa3MelUaeTca B nOJle KOHueHcaTopa ByueM CqHshy
TaTb qTO TeH30p an) - aHTHcHMMeTpHqeH no HHueKcaM i H k Torua B
7
TeH30pe EiHk MOiKHO BhUleJ1HTh TOJIhKO aHTHCHMMeTpHtJHYIO tJacTh H MhI B
npaBe llml KOHKpeTHOH 3KCnepHMeHTaJIhHOH cHTyaUHH CB1l3aTh C HHM BeKTOp
YMOBa-TIoHHTHHra llJI1l CBeTa npOXOll1lWero tJepe3KpHCTaJIJI C nOl106aIOWHMH
HeJ1HHeHHhIMH CBOHCTBaMH TIoaHaJIOrHH C npellhUlYWHMH pacCY)f(lleHH1lMH
MhI MO)f(eM BBeCTH HHllYUHpOBaHHYIO TOPOHllHYIO nOJI1lpH3aUHIO
1 (em)E H - (em)S11I = -a[ik]1 i k = ajl j (8) c
KOTOPas OTKJIHKaeTC1l Ha npOXOll1lWHH tJepe3 KPHCTaJIJI TOK JI lIeHcTBHshy
TeJIhHO 3neKTpOMarHHTHaH BonHa npOXOll1l HanpHMep tJepe3 qeppoMarHHTshy
HhIH nH3neKTpHK MOiKeT B npHHUHne COBepwaTh B HeM nepeopHeHTaUHIO
IIOn1lpH3aUHH IIpeBpawa1l ero B cna6bIH qeppOMarBeTHK 06JIauaIOWHH TOshy
POHllHOH nOn1lpH3aUHeH KOHKpeTHO TaKHMH CBOHCTBaMH MoryT 06nanaTh
palnHtJHhIe nepOBCKHTbI MaHraHHTbI (CM Hanp [23]) H llPyrHe nOCTaTotJHO
cnOiKBbIe coenHHeHH1l
3KcnepHMeHTaTophI npH H3MepeHHH TOPOHllHOH nOn1lpH3aUHH KPHCTaJIshy
nOB eCTeCTBeHBO nocTynalOT npowe HanpHMep B [10] H3ytJaJIC1l OTKnHK
MOBOKpHCTaJIna F e-Ga-oKcHlla Ha MarHHTHoe none yhBeCTHo tJTO y TaKoro
MaTepHaJIa qeppoMarHHTHhIe CBOHCTBa cotJeTaIOTC1l C nhe303neKTpHtJeCKHMH
(BcnOMHHTe aHaJIOrHIO C yIIpyrHMH neTn1lMH) B 3KcnepHMeHTe H3Mep1lnaCh
senHtJHHa HHnyUHpoBaHHoH 3neKTpHtJecKoH nOn1lpH3auHH npH HanO)f(eHHH Ha
KpHCTann MarBHTBoro IIOn1l H
D _ (em)H LI - ij ) (9)
(em) rne TeH30p aij Ha3bIBaeTC1l TeH30pOM MarHHT03neKTpHtJeCKOH BocnpHHMshy
tJHBOCTH ABTHcHMMeTpHtJHaH tJaCTh TeB30pa a~jm) KaK pal H OIIpenen1leT
TOPOHllHbIH nHnonhHbIH MOMeBT cornacBo COOTBoweBHIO nyanhBOCTH
(em)amiddotmiddot (10)I)
rne eurooijk - aHTHCHMMeTpHtJBbIH ellHBHtJBbIH TeB30p JIeBH-tHBHTa TIon1lpHshy
3aUH1l npHxon1lWaHC1l Ba OllBY 3neMeBTapHYlO 1ltJeHKY KpHCTanna H3ytJaeshy
MOro OKCHlla IIpHBena K 3BatJeBHIO TOPOHllHOro nHIIonhHoro MOMeBTa PaBshy
BOro ITI24JLB A ABanorHtJBbIH 3ltpltpeKT 6bIn IIonytJeB nn1l MarBHT03neKTpHKa
Cr203 IIOMemeBBoro B cHnhBoe MarBHTBoe IIone IIpeBbIwalOwee IIone cnHBshy
qnOII IIepeXOlla
8
4 BKJIanbl IIOJIJIpu3yeMocTeu B BeKTopHble
IIOJIJIpU3auUu cpen U ux MYJIbTUIIOJIbHble
pa3JIOKeHUJI
B HaCTOJlIlleM naparpacpe MhI Pa30BbeM nOJIHhIH CPOPMaJIH3M BKJIanOB BhICWHX
( cMewaHHhIx) nOJIJlpH3yeMocTeH B lIeThIpe 6a3HCHhIe nJIOTHOCTH pacnpeneJIeshy
HHJI nOJIJlpH3aUHH npOH3BOJIbHOH CpenhI P M T(p) H Td) B npHHUHne
npOH3BOnJl naJIee MYJIbTHnOJIbHhIe pa3JIOEeHHJI 3THX nOJIHhIX BhIpaEeHHH nJIJI
3HeprHH HX B3aHMoneHCTBHH C BHeWHHMH nOJIJlpH3aUHJlMH MhI nOJIYlIHM 0606shyIlleHHe npenhInYIllHX CPOPMYJI Ha CJIyqaH HeJIHHeHHhIX cpen B COOTBeTCTBHH
C CPYHnaMeHTaJIbHhIMH B3a~MoneHCTBHJlMH KaEnoH H3 nOJIJlpH3anHH BBeneM
nOJIJlpH3yeMocTH YlIHThIBalOillHe Bce B03MOXltHhIe BapHaHThI HHnYUHpoBaHHJI
naHHOH nOJIJlpH3aUHH BHeWHHMH nOJIJlMH H TOKaMH 0603Hallag KOJIHlIeCTBO
CsepHYThIX HHneKCOB B Ka)KnOM lIJIeHe TOllKaMH BhInHweM CaMhIe 06mHe BhIshy
pa)KeHHJI nJIJI 3JIeKTpHlIeCKoi nOJIJlpH3anHH -C YlIeTOM Bcex BKJIanOB BnJIOTb no
lIeTBepToro nOpJlnKa
P =a~e) E+a~e) EE+a~e) EEE++a~em) EH+a~em) EEH+
+B~m)H+Bm)HH+Bm)HHH+ +Bme)HE+Bme)HHE+ bull (11)
+Y~r) Jm+Yr) JmJm+Yr) JmJmJm++Yer) EJm+Y~mr)HJm++Yemr) EHJm+
+5~r) Je+51r)jeJe+51r) JeJeJe+ +51er) EJe+51mr)HJe++51errmiddotmiddot)EHJe+
rne Jm = ~(i + b) == rotH Je = rotE lIHCJIa npH nOJIJlpH3yeMOCTJlX 060shy3HallaIOT HX TeH30pHhIH paHr H MO)KHO nepeHTH K HenpHBOnHMhIM TeH30PaM
HanpHMep BBOnJl nOJIYCYMMY H nOJIypa3HOCTb a~em) H f3~me) H Tn
nJIJI MarHHTHoH nOJIJlpH3aUHH
M =amp~m)H+ampm)HH+amp~m)HHH+ +ampem)HE+amp~em)HEE+
+p~e) E+pie)EE+pe) EEE++p~me) EH+pme) EEH+ (12)
+i~r) Jm+i~r) JmJm+ir) JmJmJm+bull+i1er)HJm+i~mr) EJm++imer)HEjm+
+S~r) Je+S1r)JeJe+S~r) JeJe Je + +S~er) EJe+S1 mr) HJe+ +S~emr) EHJe+ j
nml TOP0HllHOi nOJIJlpH3ar(HH MarHHTHoro npOHCXO)KneHHJI
T M =atr)Jm+ar) JmJm+~r) Jm JmJm+ +a1re)JmE+~re) JmEE+
+~e) E+i31e) EE+7f4e)EEE+ +per) EJm+~er) EEJm+ (13)
+~m)H+Y1m)HH+~m)HHH+ +Y1rm)JmH+Y1re)JmE+ +~rme)JmHE+bull
+6~r) Je+6r) JeJe+~r) JeJeJe++~e) EJe+~m) HJe+bull+6~me) EHJe+bull j
9
nJISI TOPOHZJHOH nOJISIpH3anHH 3JIeKTpHqeCKoro npOHCXOJKneHHSI
T p =a~) Je+a~)JeJe+a~) JeJeJe++a~e)JeE+a~e) JeEE+
+8~e) E+8e) EE+8e) EEE++8e) EJe+8e) EEJe+ (14)
+i~m) H+i~m)HH+im) HHH+ +i~m) JeH+i~e)JeE+ +ime) Je HE+
+gir)Jm+I~)JmJm+g)JmJmJm+ +g~e)EJm+g~m)HJm+ +Sme)EHJm+
HeTpynHo YleCTb B 3THX BhlpaJKeHHSIX nJISI naHHOro paccMaTpHBaeMoro
KpHCTaJIJIa nYaJIbHhle CHMMeTPHH KOTOphIe npHBenYT K paBeHCTBaM THna (m) f3(e) -r(T) (T) B
f32 = 2 (12 = U2 H T n ce ~KJIaZJhIB nOJISIpH3anHH conepJKalJlHe
BeKTOphI Jm H Je Ha3hIBaJOT HeJIOKaJILHhIMH H B nOCJIenHeeBpeMSI 3KCneshy
pHMeHTaTOphI HayqHJIHCL HX H3MepSITb B HeKOTOphIX cnyqaJIX OTneJIbHO OT
JIOKaJIbHhIX 3laquoplaquopeKTOB [24J EcTeCTBeHHO ITO B npHBene~lHhIX BhIIlle laquoPOPMYshy
JIax qaCTL BeKTOpOB HanpSIJK~HHOCTeH MOJKHO 3aMeHHTL Ha HX npOH3BOnHhIe
PaHr nOJISIpH3yeMocTeH UpH 3TOM B03p~CTeT H 3TH BKJIanhI B COOTBeTCTBYshy
IOlJlHe nOJISIpH3anHH TaKJKe 6yneM Ha3hIBaTb HeJIOKaJILHhIMH B03MOJKHhI H
npyrHe HeJIHHeHHhIe KPOCC-qJIeHhI KpOMe MarHHT03JIeKTpHqeCKHX MexaHQ-
MarHHTHhIe TepM03JIeKTpHqeCKHe H np
Tenepb npOH3BeneM MYJILTHnOJILHhIe pa3JIOJKeHHJI BeKTOPHhlX nOJISIpH3ashy
nHH cpen C yqeTOM CBOHCTB HX nOJISIpH3yeMoCTH llJISI 3TOro BOCnOJIL3yeMCSI
laquopopMYJIaMH nJISI 3HeprHH B3aHMoneHCTBHSI 3THX nOJISIpH3aUHH C BHeIllHHMH
nOJISIMH (1) H (2) HaqHeM C aHaJIH3a 3JIeKTpHqeCKOH nOJISIpH3aUHH TIPHCYTshy
CTBHe B BhIpaJKeHHH nJISI n9JISIpH3anHH BeKTopa 3JIeKTpHIeCKOH HanpSIJKeHshy
HOCTH MonHlaquopHnHpyeT onpeneJIeHHe MYJIbTHnOJILHhIX MOMeHTOB HanpHMep
3JIeKTpHqeCKHH nHnOJILHhIH MOMeHT Q = J P(r )cfr nOJIyqHT n06aBKH 3a CqeT w (e) (e) (emT) TI
nOJISIpH3yeMocTeH 02 03 bull 4 bull bull bull pHBeneM nJISI npHMepa BKJIan OT (eL
O 3
(15)
TaKOH BKJIaZJ MOJKeT 6hITL nOCTaTOqHO BeJIHK eCJIH Cpena 06JIanaeT CHJILshy
HhIMH HeJIHHeHHhIMH CBOHCTBaMH H BeJIHqHHa o~e) cymecTBeHHa KpOMe TOrO
3TOT BKJIaZJ MOJKeT YCHJIHBaTLCSI H 3a CqeT HHTerpaJIa TIepexonSI B ClaquopepHqeshy
CKYlO CHCTeMY KOOpnHHaT MhI MOJKeM nepenHCaTL ero B BHne
2f Ei E kd3r = f E2r dr f ninkdfl (16)
rne n TaKHe HHTerpaJIhI qaCTO BCTpeqalOTCSI B 3JIeKTponHHaMHKe= E lEI H CTaTHCTHqeCKOH MexaHHKe H HX 3HaqeHHSI HaxonSITCSI COrJIaCHO 06meH laquoPopshy
MYJIhI
f linjn nl = ~ aijak 0 1 - aikaj 0 1+ (17) L
10
5
IlPH HaJIHqHH nOnOJIHHTeJIhHhIX YCJIOBHH THna divE = 0 HJIH divH = 0 B3J1THe TaKHX HHTerpaJIOB YCJIOJKHJleTCJI B QgtH3HqeCKOM 3KcnepHMeHTe HaJIashy
raeMhIe Ha KPHCTaJIJI nOJIJI MorYT HMeTh pa3Hoe npOHCXOJKneHHe B qaCTHoshy
CTH MorYT HMeTh palHYIO qaCTOTY TaK 3JIeKTpOOnTHqecKHH 3QgtQgteKT Iloshy
KeJIhCa 06hIqHO Ha3hIBaeMhIH JIHHeHHhIM eCTh 3QgtQgteKT BTOpOro nOpJlnKa TK
JlBJIJleTCJI pe3YJIhTaTOM onHOBpeMeHHoro B03neHCTBHJI 3JIeKTpHqeCKoro nOJIJI
HH3KOH qaCTOThI H CBeTOBOH BOJIHhI B KOTOPOH CYlUecTBeHHOH 06hIqHO JlBJIJIshy
eTCJI 3JIeKTpHqeCKaJI COCTaBJIJllOlUaJI 3HaMeHHThIH 3QgtQgteKT Keppa 06hIqHO
HalhIBalOT KBanpaTHqHhIM nOCKOJIhKY nOKalaTeJIb npeJIOMJIeHHJI KpHCTaJIJIa
npOnOplIHOHaJIeH KBanpaTY CMelUalOlUero HH3KOqaCToTHoro nOJIJI OnHaKO no
qHCJIY nOJIeH CTOJllUHX B BhIpaJKeHHH nJIJI 3HeprHH B3aHMoneHCTBHJI 3TOT
3QgtQgteKT Boo6me roBOPJl TpeThero nOpJlnKa CJIenyeT TaKJKe pa3JIHqaTh CTashy
THqeCKHH H nHHaMHqeCKHH 3QgtctgteKThI IlPH H3yqeHHH MexaHH3MOB nHHaMHshy
QeCKHX 3ltlgtltlgteKToB YQHThIBalOT CYMMOBhIe H PalHOCTHhIeqaCTOThI H OT HX
HanJIeJKalUero BhI60pa MOJKeT CHJIhHO 3aBHceTh BeJIHqHHa 3QgtQgteKTa
HeTPynHo BhInHcaTh Qgt0PMYJIY nJIJI Q yqHThIBalOlUYIO Bce HeJIHHeHHhIe
JIOKaJIhHhIe BKJIa]lhI HCnOJIh3YJI BhIpaJKeHHe (11) B nomIpH3aUHJlX (11) (12) (13) H (14) HeJIOKaJIhHhIe nOJIJlpH3yeMocTH 6yneM 0603HaqaTh TeMH JKe 6yshy
KBaMH qTO U npeJKne HO 3arJIaBHhIMH Torna HanpHMep BKJIan 3a CqeT
HeJIOKaJIhHOH nOJIJlpH3yeMocTH BToporo paHra B nHnOJIhHhIH MOMeHT 6yneT
BhIrJIJlneTh TaK
(18)
TeH30p non HHTerpaJIOM MOJKHO eCTeCTBeHHO Pal6HTh Ha HenpHBOnHMhIe qashy
CTH Torna CJIen ero paBHhIH divE CBeneTCJI K pacnpeneJIeHHIO nJIOTHOCTI1
BHelliHero 3apJlna CHMMeTpHqHaJi qaCTh QgtHKcHPyeT KBa]lpynOJIhHhIe CBOHshy
CTBa neHcTBYlOlUero Ha CHCTeMY 3JIeKTpHqeCKoro nOJIJI a aHTHCHMMeTpHqHaJi
- aKCHaJIhHhIe CBOHCTBa 3TOro nOJIJI
3aKJIIOQeHHe
He npenCTaBJIJleT 60JIhIlIOro Tpyna palBHTh nOJIHhIH QgtOPMaJIH3M non06HhIx
MYJIhTHnOJIhHhIX palJIOJKeHHH IlOJIHaJI napaMeTpH3alIHJI JII060ro QgtYHnaMeHshy
TaJIhHOro B3aHMoneHCTBHJI OKaJKeTCJI Ky6HqeCKOH pellieTKoH 3anOJIHJllOlUeH
npOCTpaHCTBO Boo6lUe roBOPJl HeOrpaHHqeHHO HanpHMep B nOJIOJKHTeJIhshy
HhIX HanpaBJIeHHJlX oceH X Y H Z Ha OCH X KOTOPYIO 6yneM CqHTaTh
rJIaBHOH 6y~eM pacnOJIaraTh aHaJIH3HpYJl KaKYlO-JIH60 CJIOJKHYIO 3JIeKTposhy
MarHHTHYIO CHCTeMY 3HaqeHHJI onHChIBalOlUHX ee MOMeHTOB B nOpJlnKe B03shy
paCTaHHJI l-nOJIhHOCTH B HanpaBJIeHHH OCH Y 6yneM QgtHKCHpOBaTh 3Haqeshy
HHJI BKJIanOB 2n-CTeneHHhIX panHYCOB pacnpeneJIeHHJI KalKnoro MOMeHTa B
nOpJlnKe B03paCTaHHJI CTeneHH n HanOMHHM qTO MYJIhTHnOJIhHhIH aHaJIH3 B
11
ero MaTeMaTHqeCKOH nOllOnJIeKe OCHOBhIBaeTClI Ha TeopHH 0606meHHhIx ltPYHKshy
IlHH H HeCMOTpll Ha 06menpHHlIThIe 3a6JIYJKlleHHlI 0 llOMHHHpOBaHHH MOMeHshy
TOB BeJIHqHHhI pallHYCOB MorYT OKalaTbClI ropa3110 60JIee cymecTBeHHhIMH
npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
Hll8JIbHhIH llBOHHOH CJIOH C He60JIbIllHM 3KcIleHTpHCHTeTOM npeJKlle Bcero 6yshy
lleT OnHChIBaTbClI cpellHeKBallpaTHqHhIM 3apmOBhIM PallHYCOM a nOTOM YJKe
3JIeKTpHqeCKHM KBallPynOJIbHhIM MOMeHTOM C reoMeTpHqeCKoHmiddot JKe TOqKH
3peHHlI pallHYChI onHChIBalOT Mepy OTKJIOHeHHlIKaJKlloro l-noJIbHoro MOMeHTa
OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
eMOCTH Hall KaJKAOH n03HIlHeH MaTpHIlhI ltPHKCHpYlOmeH 3HaqeHHlI MOMeHTOB
HJIH pallHYCOB
OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
KBallpaTHqHO (H Tll COOTBeTCTBeHHO nOpllllKY n-TOpa) ECTecTBeHHo qTO
BeJIHqHHa OTKJIHKa 6YlleT CTpeMHTeJIbHO nallaTb C B03paCTaHHeM nopauKa n H3-3a Hal1HqHlI peJIlITHBHCTCKOro 4gtaKTopa (lc)n B 3HeprHH B3aHMOlleHCTBHlI
n-Topa C BHeIllHHM nOJIeM
IIollqepKHeM TO qTO BhIlleJIeHHe BeKTOPHhIX BKJIallOB HaH60JIee yll06HO llJIj[
3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
HaHO~eXHOJIOrHlIX
ABTOPhI 6 JIarona PHhI MHTII ltIgtYHllaMeHTal1bHaK MeTpOJIOrHlI 3a nOllshy
llepJKKy 3TOH TeMaTHKH no corJIaIIIeHHIO 251
12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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IlHnOJIeH MYJIbTHnOJIbHOe pa3JIOgtKeHHe nJIOTHOCTH Takoro pacnpeneJIeHIU
HMeeT pSlll OTJIHtfHH OT pa3JIOgtKeHHSI TOKa 3apSlDOB tfTO npHBOllHT I( HHTeshy
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3HCHbIX llHnOJIeH 3JIeKTpHtfeCKHH Q MarHHTHbIH M H (nOJISlpHbIH) TOpOHIlshy
HbIH T 3JIeKTpOMarHHTHaJI cpella B KOTOPOH HeT HI1 CB060DHbIX IIH CBSIshy
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qH3HqeCKH HaHfuJIee 3HaqHMhIX nHrrOJIbHhIX cpen [9] lIaJIee HCXOnJI H3 MaTeshy
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B 3TOM OTHomeHHH naHHaJI CTaTbJI JIBJIJIeTCJI pMBHTHeM pa60T [12] H [17]
ITocTpoeHHe MynbTHnonbHMx MOMeHTOB
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QJIeHhI pMJIO)KeHHJI 3HeprHH HX B3aHMoneHCTBHJI C BHemHHMH 110JIJIMH
W = - ( d(r)E(r)d3r = ~r02divE - QE - T(d)rotEshylv 6
-p(d)rotrotE - ~QiAViEj +VjEi) - Tid)Vi(rotE)j + 1 shy+10 (r~2) V)divE + === -QE T(d)rotE + p(d)(Vp - ~E) +
(d) (d)middotmiddotmiddotmiddot+ === -QE + T H + P (J - E) + (1)
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3
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5
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(CM Hanp [20] H [21]) tJTO TaKHe HeHTpHHo 06JIallaIOT TOJIhKO TOPOHllshy
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BeKTopShIe napaMeTphI
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(3)
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(4)
6
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(5)
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cpeueuonOJlHHTeJlhHYIO MarHHTHYIO uHnOJlbHYIO nOJlHpH3aUHIO Mle = a~)Hi KOTOPaH B CBOIO oqepeub cnoc06Ha B3aHMoueHcTBOBaTb C nOJleM Hie 3uecb
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qTO ueltp0pMaUHH nou06HhIe onHcaHHhIM uJlH CHCTeMhI MarHHTHbix uHnOJleH
MOJKeT HcnbIThIBaTb H CHCTeMa COCTaBJleHHaH H3 3JleKTpHqeCKHX uHnOJleH B
3TOM CJlyqae BhIpaJKeHHH uJlH 3HeprHH HX B3aHMoueHcTBHH C BHelIIHHM 3JleKshy
TpHqeCKHM nOJleM 6yuYT COBeplIIeHHO HueHTHqHhI BBeueHHhIM BhIlIIe a Marshy
HHTHyIO BOCnpHHMqHBOCTb B HHX MhI 3aMeHHM Ha 3JleKTpHqeCKYIO a ebull noshy
CKOJlbKY peaJIbHhIe CHCTeMhI H3 KOTOPhIX COCTOHT JlI06aH cpeua - 3TO aTOMhI
H MOJleKYJlhI HaxOuHlUHeCH B COCTOHHHH HenpephIBHOH BHYTpeHHeH H BHelIIHeH
uHHaMHKH TO BCJleuCTBHe B3aHMOCBH3eH HCTOqHHKOB qepe3 nOJlH B03MOIKHhI
H CMelIIaHHhIe nOJlHpH3yeMocTH HanpHMep MarHHT03JleKTpHqeCKHe C 3Hepshy
rHeH B3aHMOlIeHCTBHa
(6)
HMeH B BHuy OpraHH3aUHIO KOMnbIOTepHOH naMHTH BBeueM nOJlHpH3yeMocTb
TpeTbero paHra OTBeqaIOlUYIO CHMMeTpHH 3ltpltpeKTa XOJlJla H COOTBeT-
CTBeHHO 3HeprHIO B3aHMoueHCTBHH
I (em)E H JWNL -ailel i Ie I (7)C
rue J - BHelIIHHH TOK npoTeKaIOlUHH qepe3 paccMaTpHBaeMYIO CHCTeMY B
qaCTHOCTH 3TOT TOK MOJKeT 6hITb TOKOM CMelUeHHH qTO peaJIH3yeTCH Koshy
rna H3yqaeMaa CHCTeMa pa3MelUaeTca B nOJle KOHueHcaTopa ByueM CqHshy
TaTb qTO TeH30p an) - aHTHcHMMeTpHqeH no HHueKcaM i H k Torua B
7
TeH30pe EiHk MOiKHO BhUleJ1HTh TOJIhKO aHTHCHMMeTpHtJHYIO tJacTh H MhI B
npaBe llml KOHKpeTHOH 3KCnepHMeHTaJIhHOH cHTyaUHH CB1l3aTh C HHM BeKTOp
YMOBa-TIoHHTHHra llJI1l CBeTa npOXOll1lWero tJepe3KpHCTaJIJI C nOl106aIOWHMH
HeJ1HHeHHhIMH CBOHCTBaMH TIoaHaJIOrHH C npellhUlYWHMH pacCY)f(lleHH1lMH
MhI MO)f(eM BBeCTH HHllYUHpOBaHHYIO TOPOHllHYIO nOJI1lpH3aUHIO
1 (em)E H - (em)S11I = -a[ik]1 i k = ajl j (8) c
KOTOPas OTKJIHKaeTC1l Ha npOXOll1lWHH tJepe3 KPHCTaJIJI TOK JI lIeHcTBHshy
TeJIhHO 3neKTpOMarHHTHaH BonHa npOXOll1l HanpHMep tJepe3 qeppoMarHHTshy
HhIH nH3neKTpHK MOiKeT B npHHUHne COBepwaTh B HeM nepeopHeHTaUHIO
IIOn1lpH3aUHH IIpeBpawa1l ero B cna6bIH qeppOMarBeTHK 06JIauaIOWHH TOshy
POHllHOH nOn1lpH3aUHeH KOHKpeTHO TaKHMH CBOHCTBaMH MoryT 06nanaTh
palnHtJHhIe nepOBCKHTbI MaHraHHTbI (CM Hanp [23]) H llPyrHe nOCTaTotJHO
cnOiKBbIe coenHHeHH1l
3KcnepHMeHTaTophI npH H3MepeHHH TOPOHllHOH nOn1lpH3aUHH KPHCTaJIshy
nOB eCTeCTBeHBO nocTynalOT npowe HanpHMep B [10] H3ytJaJIC1l OTKnHK
MOBOKpHCTaJIna F e-Ga-oKcHlla Ha MarHHTHoe none yhBeCTHo tJTO y TaKoro
MaTepHaJIa qeppoMarHHTHhIe CBOHCTBa cotJeTaIOTC1l C nhe303neKTpHtJeCKHMH
(BcnOMHHTe aHaJIOrHIO C yIIpyrHMH neTn1lMH) B 3KcnepHMeHTe H3Mep1lnaCh
senHtJHHa HHnyUHpoBaHHoH 3neKTpHtJecKoH nOn1lpH3auHH npH HanO)f(eHHH Ha
KpHCTann MarBHTBoro IIOn1l H
D _ (em)H LI - ij ) (9)
(em) rne TeH30p aij Ha3bIBaeTC1l TeH30pOM MarHHT03neKTpHtJeCKOH BocnpHHMshy
tJHBOCTH ABTHcHMMeTpHtJHaH tJaCTh TeB30pa a~jm) KaK pal H OIIpenen1leT
TOPOHllHbIH nHnonhHbIH MOMeBT cornacBo COOTBoweBHIO nyanhBOCTH
(em)amiddotmiddot (10)I)
rne eurooijk - aHTHCHMMeTpHtJBbIH ellHBHtJBbIH TeB30p JIeBH-tHBHTa TIon1lpHshy
3aUH1l npHxon1lWaHC1l Ba OllBY 3neMeBTapHYlO 1ltJeHKY KpHCTanna H3ytJaeshy
MOro OKCHlla IIpHBena K 3BatJeBHIO TOPOHllHOro nHIIonhHoro MOMeBTa PaBshy
BOro ITI24JLB A ABanorHtJBbIH 3ltpltpeKT 6bIn IIonytJeB nn1l MarBHT03neKTpHKa
Cr203 IIOMemeBBoro B cHnhBoe MarBHTBoe IIone IIpeBbIwalOwee IIone cnHBshy
qnOII IIepeXOlla
8
4 BKJIanbl IIOJIJIpu3yeMocTeu B BeKTopHble
IIOJIJIpU3auUu cpen U ux MYJIbTUIIOJIbHble
pa3JIOKeHUJI
B HaCTOJlIlleM naparpacpe MhI Pa30BbeM nOJIHhIH CPOPMaJIH3M BKJIanOB BhICWHX
( cMewaHHhIx) nOJIJlpH3yeMocTeH B lIeThIpe 6a3HCHhIe nJIOTHOCTH pacnpeneJIeshy
HHJI nOJIJlpH3aUHH npOH3BOJIbHOH CpenhI P M T(p) H Td) B npHHUHne
npOH3BOnJl naJIee MYJIbTHnOJIbHhIe pa3JIOEeHHJI 3THX nOJIHhIX BhIpaEeHHH nJIJI
3HeprHH HX B3aHMoneHCTBHH C BHeWHHMH nOJIJlpH3aUHJlMH MhI nOJIYlIHM 0606shyIlleHHe npenhInYIllHX CPOPMYJI Ha CJIyqaH HeJIHHeHHhIX cpen B COOTBeTCTBHH
C CPYHnaMeHTaJIbHhIMH B3a~MoneHCTBHJlMH KaEnoH H3 nOJIJlpH3anHH BBeneM
nOJIJlpH3yeMocTH YlIHThIBalOillHe Bce B03MOXltHhIe BapHaHThI HHnYUHpoBaHHJI
naHHOH nOJIJlpH3aUHH BHeWHHMH nOJIJlMH H TOKaMH 0603Hallag KOJIHlIeCTBO
CsepHYThIX HHneKCOB B Ka)KnOM lIJIeHe TOllKaMH BhInHweM CaMhIe 06mHe BhIshy
pa)KeHHJI nJIJI 3JIeKTpHlIeCKoi nOJIJlpH3anHH -C YlIeTOM Bcex BKJIanOB BnJIOTb no
lIeTBepToro nOpJlnKa
P =a~e) E+a~e) EE+a~e) EEE++a~em) EH+a~em) EEH+
+B~m)H+Bm)HH+Bm)HHH+ +Bme)HE+Bme)HHE+ bull (11)
+Y~r) Jm+Yr) JmJm+Yr) JmJmJm++Yer) EJm+Y~mr)HJm++Yemr) EHJm+
+5~r) Je+51r)jeJe+51r) JeJeJe+ +51er) EJe+51mr)HJe++51errmiddotmiddot)EHJe+
rne Jm = ~(i + b) == rotH Je = rotE lIHCJIa npH nOJIJlpH3yeMOCTJlX 060shy3HallaIOT HX TeH30pHhIH paHr H MO)KHO nepeHTH K HenpHBOnHMhIM TeH30PaM
HanpHMep BBOnJl nOJIYCYMMY H nOJIypa3HOCTb a~em) H f3~me) H Tn
nJIJI MarHHTHoH nOJIJlpH3aUHH
M =amp~m)H+ampm)HH+amp~m)HHH+ +ampem)HE+amp~em)HEE+
+p~e) E+pie)EE+pe) EEE++p~me) EH+pme) EEH+ (12)
+i~r) Jm+i~r) JmJm+ir) JmJmJm+bull+i1er)HJm+i~mr) EJm++imer)HEjm+
+S~r) Je+S1r)JeJe+S~r) JeJe Je + +S~er) EJe+S1 mr) HJe+ +S~emr) EHJe+ j
nml TOP0HllHOi nOJIJlpH3ar(HH MarHHTHoro npOHCXO)KneHHJI
T M =atr)Jm+ar) JmJm+~r) Jm JmJm+ +a1re)JmE+~re) JmEE+
+~e) E+i31e) EE+7f4e)EEE+ +per) EJm+~er) EEJm+ (13)
+~m)H+Y1m)HH+~m)HHH+ +Y1rm)JmH+Y1re)JmE+ +~rme)JmHE+bull
+6~r) Je+6r) JeJe+~r) JeJeJe++~e) EJe+~m) HJe+bull+6~me) EHJe+bull j
9
nJISI TOPOHZJHOH nOJISIpH3anHH 3JIeKTpHqeCKoro npOHCXOJKneHHSI
T p =a~) Je+a~)JeJe+a~) JeJeJe++a~e)JeE+a~e) JeEE+
+8~e) E+8e) EE+8e) EEE++8e) EJe+8e) EEJe+ (14)
+i~m) H+i~m)HH+im) HHH+ +i~m) JeH+i~e)JeE+ +ime) Je HE+
+gir)Jm+I~)JmJm+g)JmJmJm+ +g~e)EJm+g~m)HJm+ +Sme)EHJm+
HeTpynHo YleCTb B 3THX BhlpaJKeHHSIX nJISI naHHOro paccMaTpHBaeMoro
KpHCTaJIJIa nYaJIbHhle CHMMeTPHH KOTOphIe npHBenYT K paBeHCTBaM THna (m) f3(e) -r(T) (T) B
f32 = 2 (12 = U2 H T n ce ~KJIaZJhIB nOJISIpH3anHH conepJKalJlHe
BeKTOphI Jm H Je Ha3hIBaJOT HeJIOKaJILHhIMH H B nOCJIenHeeBpeMSI 3KCneshy
pHMeHTaTOphI HayqHJIHCL HX H3MepSITb B HeKOTOphIX cnyqaJIX OTneJIbHO OT
JIOKaJIbHhIX 3laquoplaquopeKTOB [24J EcTeCTBeHHO ITO B npHBene~lHhIX BhIIlle laquoPOPMYshy
JIax qaCTL BeKTOpOB HanpSIJK~HHOCTeH MOJKHO 3aMeHHTL Ha HX npOH3BOnHhIe
PaHr nOJISIpH3yeMocTeH UpH 3TOM B03p~CTeT H 3TH BKJIanhI B COOTBeTCTBYshy
IOlJlHe nOJISIpH3anHH TaKJKe 6yneM Ha3hIBaTb HeJIOKaJILHhIMH B03MOJKHhI H
npyrHe HeJIHHeHHhIe KPOCC-qJIeHhI KpOMe MarHHT03JIeKTpHqeCKHX MexaHQ-
MarHHTHhIe TepM03JIeKTpHqeCKHe H np
Tenepb npOH3BeneM MYJILTHnOJILHhIe pa3JIOJKeHHJI BeKTOPHhlX nOJISIpH3ashy
nHH cpen C yqeTOM CBOHCTB HX nOJISIpH3yeMoCTH llJISI 3TOro BOCnOJIL3yeMCSI
laquopopMYJIaMH nJISI 3HeprHH B3aHMoneHCTBHSI 3THX nOJISIpH3aUHH C BHeIllHHMH
nOJISIMH (1) H (2) HaqHeM C aHaJIH3a 3JIeKTpHqeCKOH nOJISIpH3aUHH TIPHCYTshy
CTBHe B BhIpaJKeHHH nJISI n9JISIpH3anHH BeKTopa 3JIeKTpHIeCKOH HanpSIJKeHshy
HOCTH MonHlaquopHnHpyeT onpeneJIeHHe MYJIbTHnOJILHhIX MOMeHTOB HanpHMep
3JIeKTpHqeCKHH nHnOJILHhIH MOMeHT Q = J P(r )cfr nOJIyqHT n06aBKH 3a CqeT w (e) (e) (emT) TI
nOJISIpH3yeMocTeH 02 03 bull 4 bull bull bull pHBeneM nJISI npHMepa BKJIan OT (eL
O 3
(15)
TaKOH BKJIaZJ MOJKeT 6hITL nOCTaTOqHO BeJIHK eCJIH Cpena 06JIanaeT CHJILshy
HhIMH HeJIHHeHHhIMH CBOHCTBaMH H BeJIHqHHa o~e) cymecTBeHHa KpOMe TOrO
3TOT BKJIaZJ MOJKeT YCHJIHBaTLCSI H 3a CqeT HHTerpaJIa TIepexonSI B ClaquopepHqeshy
CKYlO CHCTeMY KOOpnHHaT MhI MOJKeM nepenHCaTL ero B BHne
2f Ei E kd3r = f E2r dr f ninkdfl (16)
rne n TaKHe HHTerpaJIhI qaCTO BCTpeqalOTCSI B 3JIeKTponHHaMHKe= E lEI H CTaTHCTHqeCKOH MexaHHKe H HX 3HaqeHHSI HaxonSITCSI COrJIaCHO 06meH laquoPopshy
MYJIhI
f linjn nl = ~ aijak 0 1 - aikaj 0 1+ (17) L
10
5
IlPH HaJIHqHH nOnOJIHHTeJIhHhIX YCJIOBHH THna divE = 0 HJIH divH = 0 B3J1THe TaKHX HHTerpaJIOB YCJIOJKHJleTCJI B QgtH3HqeCKOM 3KcnepHMeHTe HaJIashy
raeMhIe Ha KPHCTaJIJI nOJIJI MorYT HMeTh pa3Hoe npOHCXOJKneHHe B qaCTHoshy
CTH MorYT HMeTh palHYIO qaCTOTY TaK 3JIeKTpOOnTHqecKHH 3QgtQgteKT Iloshy
KeJIhCa 06hIqHO Ha3hIBaeMhIH JIHHeHHhIM eCTh 3QgtQgteKT BTOpOro nOpJlnKa TK
JlBJIJleTCJI pe3YJIhTaTOM onHOBpeMeHHoro B03neHCTBHJI 3JIeKTpHqeCKoro nOJIJI
HH3KOH qaCTOThI H CBeTOBOH BOJIHhI B KOTOPOH CYlUecTBeHHOH 06hIqHO JlBJIJIshy
eTCJI 3JIeKTpHqeCKaJI COCTaBJIJllOlUaJI 3HaMeHHThIH 3QgtQgteKT Keppa 06hIqHO
HalhIBalOT KBanpaTHqHhIM nOCKOJIhKY nOKalaTeJIb npeJIOMJIeHHJI KpHCTaJIJIa
npOnOplIHOHaJIeH KBanpaTY CMelUalOlUero HH3KOqaCToTHoro nOJIJI OnHaKO no
qHCJIY nOJIeH CTOJllUHX B BhIpaJKeHHH nJIJI 3HeprHH B3aHMoneHCTBHJI 3TOT
3QgtQgteKT Boo6me roBOPJl TpeThero nOpJlnKa CJIenyeT TaKJKe pa3JIHqaTh CTashy
THqeCKHH H nHHaMHqeCKHH 3QgtctgteKThI IlPH H3yqeHHH MexaHH3MOB nHHaMHshy
QeCKHX 3ltlgtltlgteKToB YQHThIBalOT CYMMOBhIe H PalHOCTHhIeqaCTOThI H OT HX
HanJIeJKalUero BhI60pa MOJKeT CHJIhHO 3aBHceTh BeJIHqHHa 3QgtQgteKTa
HeTPynHo BhInHcaTh Qgt0PMYJIY nJIJI Q yqHThIBalOlUYIO Bce HeJIHHeHHhIe
JIOKaJIhHhIe BKJIa]lhI HCnOJIh3YJI BhIpaJKeHHe (11) B nomIpH3aUHJlX (11) (12) (13) H (14) HeJIOKaJIhHhIe nOJIJlpH3yeMocTH 6yneM 0603HaqaTh TeMH JKe 6yshy
KBaMH qTO U npeJKne HO 3arJIaBHhIMH Torna HanpHMep BKJIan 3a CqeT
HeJIOKaJIhHOH nOJIJlpH3yeMocTH BToporo paHra B nHnOJIhHhIH MOMeHT 6yneT
BhIrJIJlneTh TaK
(18)
TeH30p non HHTerpaJIOM MOJKHO eCTeCTBeHHO Pal6HTh Ha HenpHBOnHMhIe qashy
CTH Torna CJIen ero paBHhIH divE CBeneTCJI K pacnpeneJIeHHIO nJIOTHOCTI1
BHelliHero 3apJlna CHMMeTpHqHaJi qaCTh QgtHKcHPyeT KBa]lpynOJIhHhIe CBOHshy
CTBa neHcTBYlOlUero Ha CHCTeMY 3JIeKTpHqeCKoro nOJIJI a aHTHCHMMeTpHqHaJi
- aKCHaJIhHhIe CBOHCTBa 3TOro nOJIJI
3aKJIIOQeHHe
He npenCTaBJIJleT 60JIhIlIOro Tpyna palBHTh nOJIHhIH QgtOPMaJIH3M non06HhIx
MYJIhTHnOJIhHhIX palJIOJKeHHH IlOJIHaJI napaMeTpH3alIHJI JII060ro QgtYHnaMeHshy
TaJIhHOro B3aHMoneHCTBHJI OKaJKeTCJI Ky6HqeCKOH pellieTKoH 3anOJIHJllOlUeH
npOCTpaHCTBO Boo6lUe roBOPJl HeOrpaHHqeHHO HanpHMep B nOJIOJKHTeJIhshy
HhIX HanpaBJIeHHJlX oceH X Y H Z Ha OCH X KOTOPYIO 6yneM CqHTaTh
rJIaBHOH 6y~eM pacnOJIaraTh aHaJIH3HpYJl KaKYlO-JIH60 CJIOJKHYIO 3JIeKTposhy
MarHHTHYIO CHCTeMY 3HaqeHHJI onHChIBalOlUHX ee MOMeHTOB B nOpJlnKe B03shy
paCTaHHJI l-nOJIhHOCTH B HanpaBJIeHHH OCH Y 6yneM QgtHKCHpOBaTh 3Haqeshy
HHJI BKJIanOB 2n-CTeneHHhIX panHYCOB pacnpeneJIeHHJI KalKnoro MOMeHTa B
nOpJlnKe B03paCTaHHJI CTeneHH n HanOMHHM qTO MYJIhTHnOJIhHhIH aHaJIH3 B
11
ero MaTeMaTHqeCKOH nOllOnJIeKe OCHOBhIBaeTClI Ha TeopHH 0606meHHhIx ltPYHKshy
IlHH H HeCMOTpll Ha 06menpHHlIThIe 3a6JIYJKlleHHlI 0 llOMHHHpOBaHHH MOMeHshy
TOB BeJIHqHHhI pallHYCOB MorYT OKalaTbClI ropa3110 60JIee cymecTBeHHhIMH
npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
Hll8JIbHhIH llBOHHOH CJIOH C He60JIbIllHM 3KcIleHTpHCHTeTOM npeJKlle Bcero 6yshy
lleT OnHChIBaTbClI cpellHeKBallpaTHqHhIM 3apmOBhIM PallHYCOM a nOTOM YJKe
3JIeKTpHqeCKHM KBallPynOJIbHhIM MOMeHTOM C reoMeTpHqeCKoHmiddot JKe TOqKH
3peHHlI pallHYChI onHChIBalOT Mepy OTKJIOHeHHlIKaJKlloro l-noJIbHoro MOMeHTa
OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
eMOCTH Hall KaJKAOH n03HIlHeH MaTpHIlhI ltPHKCHpYlOmeH 3HaqeHHlI MOMeHTOB
HJIH pallHYCOB
OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
KBallpaTHqHO (H Tll COOTBeTCTBeHHO nOpllllKY n-TOpa) ECTecTBeHHo qTO
BeJIHqHHa OTKJIHKa 6YlleT CTpeMHTeJIbHO nallaTb C B03paCTaHHeM nopauKa n H3-3a Hal1HqHlI peJIlITHBHCTCKOro 4gtaKTopa (lc)n B 3HeprHH B3aHMOlleHCTBHlI
n-Topa C BHeIllHHM nOJIeM
IIollqepKHeM TO qTO BhIlleJIeHHe BeKTOPHhIX BKJIallOB HaH60JIee yll06HO llJIj[
3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
HaHO~eXHOJIOrHlIX
ABTOPhI 6 JIarona PHhI MHTII ltIgtYHllaMeHTal1bHaK MeTpOJIOrHlI 3a nOllshy
llepJKKy 3TOH TeMaTHKH no corJIaIIIeHHIO 251
12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
2
TeopHH 3qqeKTa BOMa-AapoHoBa [15] H rrOJIBJIeHHe ero aHaJIOrOB (CM Harrp
[16]) 3aCTaBJIJIeT 60JIee CHCTeMaTHqeCKH OTHeCTHCb K BOrrpocaM MYJIbTHrrOJIbshy
HhIX rrpencTaBJIeHHH PalJIHqHhIX KOHqHrypaUHH KOTophIe MoryT B03HHKaTb
HarrpHMep B nHrrOJIbHhIX cpenax H K BhI60py rrapaMeTpOB nJIJI OrrHCaHHJI 3JIeKshy
TpOMarHHTHhIX CBOHCTB KBaHTOBhIX 06beKToB HarrpHMep MOJIeKYJI co CJI02Kshy
HhIM CTpoeHHeM
B HaCTOJImeHCTaTbe HaHneHhI rrOJIHhIe Ha60phI 6a3HCHhIX nHrrOJIeH nJIJI
qH3HqeCKH HaHfuJIee 3HaqHMhIX nHrrOJIbHhIX cpen [9] lIaJIee HCXOnJI H3 MaTeshy
MaTHqeCKoro qaKTa HX cymecTBOBaHHJI YKMaH PJIn B03M02KHhIX He06hIQHhIX
qYHKUHH OTKJIHKa BemecTBa HaxonJImeroCJI B KOHneHCHpOBaHHOM COCTOJIHHH
Ha HeonHoponHhIe HHJIH nHHaMHQeCKHe 3JIeKTpHQeCKHe H MarHHTHhIe rrOJIJI
B 3TOM OTHomeHHH naHHaJI CTaTbJI JIBJIJIeTCJI pMBHTHeM pa60T [12] H [17]
ITocTpoeHHe MynbTHnonbHMx MOMeHTOB
llna cnHHOBMX (llHnOnbHMX) cpell H HX B3ashy
HMOlleHCTBHa C BHelllHHMH nonaMH
IIpeucTaBHM QTO HMeeTCJI HeKaJI cpena (cHcTeMa) HerrpephIBHO pacrrpeneshy
JIeHHhIX 3JIeMeHTapHhIX 3JIeKTpHQeCKHX nHrrOJIeH KOTOPaJI orrHChIBaeTCJI orpashy
HHQeHHOH B rrpOCTpaHCTBe ltpYHKUHeH d rEV B 06meM cJIYQae 3aBHCJIshy
meH OT BpeMeHH lIorrycTHM QTO B TaKoH cpene rro pa3JIHQHhIM rrpHQHHaM -
BCJIenCTBHe BHyTpeHHHx B3aHMoneHcTBHH HJIH rrou BJIHJIHHeM BHemHHX rroJIeH
- MoryT B03HHKaTb H BhIcTpaHBaTbCJI arperaThI pa3JIHQHhIX KOHltpHrypaUHH
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CTBa arperaToB C rrOMOmbIO MYJIbTHrrOJIbHhIX rrapaMeTpoB BhIrrHmeM rrepBhIe
QJIeHhI pMJIO)KeHHJI 3HeprHH HX B3aHMoneHCTBHJI C BHemHHMH 110JIJIMH
W = - ( d(r)E(r)d3r = ~r02divE - QE - T(d)rotEshylv 6
-p(d)rotrotE - ~QiAViEj +VjEi) - Tid)Vi(rotE)j + 1 shy+10 (r~2) V)divE + === -QE T(d)rotE + p(d)(Vp - ~E) +
(d) (d)middotmiddotmiddotmiddot+ === -QE + T H + P (J - E) + (1)
rne Q = f d(r )J3r - 06hIQHhIH rrOJIHhIH 3JIeKTpHQeCKHH nHrrOJIbHhIH Moshy
MeHT CHCTeMhI (PHcIa) T(d) = ~ J r x d(r)cPr - aKCHaJIbHhIH TOPOHnHhIH
nHIIOJIbHhIH MOMeHT CHCTeMhI (PHcIb) p(d) = 1~ J[r(rd) - 2r2d]d3r - 110shy
JIJIPHhIH rrOJIOHnaJIbHhIH nHIIOJIbHhIH MOMeHT (PHcIc) BMeCTe OHH COCTashy
BnJIIOT 6MHCHYIO TP0HKY BeKTOPOB nnJI pa3n02KeHHJI rrnOTHOCTH pacrrpeueneshy
HHJI naHHoro BHna KpOMe HHX B (1) HMeIOTCJI r02 = J r EcPr - CKaJIJIPHhIH
3
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4
MOMeHT KOTOPhIH B pa3JIO)KeHHH TOKa COOTBeTCTByeT BpeMeHHOH npOH3BOlJshy
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rtP0lJOJIhHOH tIaCTH lJHnOJIbHoro MOMeHTa Q KOTOPhIH 06pa3yeTcx lJBOHHhIM
UHJIHHlJpHqeCKHM CJIoeM lJHnOJIeH (PHcle) Qij H Ti~d)- 3JIeKTpHtIeCKHH H TOshy
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Bce B3aHMOlJeHCTBHX MYJIbTHnOJIbHhIX xapaKTepHCTHK C nOJIeM HJIH npOH3BOlJshy
HhIMH OT Hero B BhIpa)KeHHH (1) JIOKaJIbHhI H OTHOCXTCjl K HeKoTOpoH CpelJHeH
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nOJIeH C BHeIIIHHM HeOlJHOpOlJHhIM MarHHTHhIM nOJIeM npHHHMaIOT BHlJ
w = - f p(r)H(r)d3 r = -MH - T(IJ)rotHshy
P(IJ)rotrotH - -~M(VH + V middotHmiddot) - T(~)V(rotH) + ----gt2 J bull J J J J ---r
==gt -MH - T(IJ)E - p(IJ)(rot - H) ==gt -MH - T(IJ)rotH + p(lJ) ~HCB06 + (2)
BHlJHM tITO COrJIaCHO lJYaJIbHOCTH 3JIeKTpOMarHeT1iI3Ma MOMeHT Q B (1) 3ashy
MeHxeTCX Ha MarHHTHhIH lJHnOJIbHhIH MOMeHT M XapaKTepH3YlOInHH 3KBHshy
BaJIeHTHYIO paMKy C TOKOM T(d) - Ha nOJIXpHhIH TOPOHlJHhIH lJHnOJIb T(IJ) a
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Ha nJIOCKOCTH TaK)Ke npHBOllXT K H3BeCTHOH peIIIeTKe BHxpeH KeCTepJIHuashy
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HHH T(r) H g(r) IIOJIytleHHhIe B pe3YJIbTaTe TaKHX pa3~O)KeHHH BHXPH BhIshy
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llla Ha3hIBaIOT e)KOM
5
3
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nOCJIellHHe 3KCnepHMeHTbI YKa3b1BaIOT tJTO HeHTpHHO BHllHMO MaHOpaHOBshy3CKHe tJaCTHIUd H 06pa3YIOT cpellbl BblCOKOH nJIOTHOCTH (10HgtcM- ) H3BeCTHO
(CM Hanp [20] H [21]) tJTO TaKHe HeHTpHHo 06JIallaIOT TOJIhKO TOPOHllshy
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HHliIX B TaKOH cpelle 06ecnetJHBaIQIllHX B036YJKlleHHe MeJKlly aupOHaMHHa
ypOBHe rpaBHTaUHOHHblX 6b1JIO BnepBble YKa3aHO BM KOPIOKHHblM [22]
BeKTopShIe napaMeTphI
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KBallpynOJIH oKTynOJIH H Tll IIpocTeBlUHe MaKPOCKonHtJeCKHe KOHCTPYKUHH
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KOHTyp PacCMOTPHM HanpHMep npOBOJIOtJHoe KOJIhUO H30rHYTOe B ltpopMe
nponeJIJIepa IIpH npOXOJKlleHHH no TaKOMY KOJIhUY TOKa KaJKllaJI H3 neTeJIh
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(3)
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Topble OTHOCliICh K Pa3JIHtJHblM neTJIliIM npHnHCaHbI K Pa3JIHtJHblM TOtJKaM
npOCTpaHCTBa I H 2 KBaupynOJIhHblH MOMeHT CHCTeMbI B BHlle CHMMeshy
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napaMeTp Ilik YtJHTblBaeT BKJIall B 3HepIHIO C03llaIOIllHHCliI 3a CtJeT rpauHshy
eHTa HaJIaraeMoro Ha CHCTeMY BHelUHero MarHHTHoro nOJIliI T e 3a CtJeT Heshy
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npH6JIHJKeHHOnepenHcaTh B BHlle
(4)
6
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meprHH B3aHMOueHCTBHH H COCTOHT BCH CYTb MYJlbTHnOJlbHoro pa3JlOJKeHHH
npeunOJlOJKHM renepb qTO MaTepHaJI H3 KOTOPOro cueJlaHhI neTJlH rH60K
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HeJlHHeHHhIM no nOJlIO BhIpalKeHHeM
(5)
TaKHM 06pa30M MbI nOJlyqaeM HOBOe KaqeCTBO B H3yqeHHH CBOHCTB CHCTeMhI
uHnOJleH 0606lUaH CKaJKeM TaK qTO HaJIOJKeHHe nOJlH Hi HHuyUHpyeT B
cpeueuonOJlHHTeJlhHYIO MarHHTHYIO uHnOJlbHYIO nOJlHpH3aUHIO Mle = a~)Hi KOTOPaH B CBOIO oqepeub cnoc06Ha B3aHMoueHcTBOBaTb C nOJleM Hie 3uecb
a) 6yueM Ha3hIBaTb KBaupaTHqHOH MarHHTHOH BOCnpHHMqHBOCTbIO CpeuhI
HTaK MbI BBeJlH HeJlHHeHHOe B3aHMoueHcTBHe yqHThIBaIOlUee HeJlHHeHshy
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npOH3BOJlbHOe KOJlHqeCTBO HHueKCOB H B BhIpaJKeHHH uJlH 3HeprHH OH uOJlJKeH
6bITb cBepHyT C COOTBeTCTBYIOlUHM CHMMeTpHqHhIM TeH30pOM COCTaBJleHshy
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3TOM CJlyqae BhIpaJKeHHH uJlH 3HeprHH HX B3aHMoueHcTBHH C BHelIIHHM 3JleKshy
TpHqeCKHM nOJleM 6yuYT COBeplIIeHHO HueHTHqHhI BBeueHHhIM BhIlIIe a Marshy
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(6)
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TpeTbero paHra OTBeqaIOlUYIO CHMMeTpHH 3ltpltpeKTa XOJlJla H COOTBeT-
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7
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HeJ1HHeHHhIMH CBOHCTBaMH TIoaHaJIOrHH C npellhUlYWHMH pacCY)f(lleHH1lMH
MhI MO)f(eM BBeCTH HHllYUHpOBaHHYIO TOPOHllHYIO nOJI1lpH3aUHIO
1 (em)E H - (em)S11I = -a[ik]1 i k = ajl j (8) c
KOTOPas OTKJIHKaeTC1l Ha npOXOll1lWHH tJepe3 KPHCTaJIJI TOK JI lIeHcTBHshy
TeJIhHO 3neKTpOMarHHTHaH BonHa npOXOll1l HanpHMep tJepe3 qeppoMarHHTshy
HhIH nH3neKTpHK MOiKeT B npHHUHne COBepwaTh B HeM nepeopHeHTaUHIO
IIOn1lpH3aUHH IIpeBpawa1l ero B cna6bIH qeppOMarBeTHK 06JIauaIOWHH TOshy
POHllHOH nOn1lpH3aUHeH KOHKpeTHO TaKHMH CBOHCTBaMH MoryT 06nanaTh
palnHtJHhIe nepOBCKHTbI MaHraHHTbI (CM Hanp [23]) H llPyrHe nOCTaTotJHO
cnOiKBbIe coenHHeHH1l
3KcnepHMeHTaTophI npH H3MepeHHH TOPOHllHOH nOn1lpH3aUHH KPHCTaJIshy
nOB eCTeCTBeHBO nocTynalOT npowe HanpHMep B [10] H3ytJaJIC1l OTKnHK
MOBOKpHCTaJIna F e-Ga-oKcHlla Ha MarHHTHoe none yhBeCTHo tJTO y TaKoro
MaTepHaJIa qeppoMarHHTHhIe CBOHCTBa cotJeTaIOTC1l C nhe303neKTpHtJeCKHMH
(BcnOMHHTe aHaJIOrHIO C yIIpyrHMH neTn1lMH) B 3KcnepHMeHTe H3Mep1lnaCh
senHtJHHa HHnyUHpoBaHHoH 3neKTpHtJecKoH nOn1lpH3auHH npH HanO)f(eHHH Ha
KpHCTann MarBHTBoro IIOn1l H
D _ (em)H LI - ij ) (9)
(em) rne TeH30p aij Ha3bIBaeTC1l TeH30pOM MarHHT03neKTpHtJeCKOH BocnpHHMshy
tJHBOCTH ABTHcHMMeTpHtJHaH tJaCTh TeB30pa a~jm) KaK pal H OIIpenen1leT
TOPOHllHbIH nHnonhHbIH MOMeBT cornacBo COOTBoweBHIO nyanhBOCTH
(em)amiddotmiddot (10)I)
rne eurooijk - aHTHCHMMeTpHtJBbIH ellHBHtJBbIH TeB30p JIeBH-tHBHTa TIon1lpHshy
3aUH1l npHxon1lWaHC1l Ba OllBY 3neMeBTapHYlO 1ltJeHKY KpHCTanna H3ytJaeshy
MOro OKCHlla IIpHBena K 3BatJeBHIO TOPOHllHOro nHIIonhHoro MOMeBTa PaBshy
BOro ITI24JLB A ABanorHtJBbIH 3ltpltpeKT 6bIn IIonytJeB nn1l MarBHT03neKTpHKa
Cr203 IIOMemeBBoro B cHnhBoe MarBHTBoe IIone IIpeBbIwalOwee IIone cnHBshy
qnOII IIepeXOlla
8
4 BKJIanbl IIOJIJIpu3yeMocTeu B BeKTopHble
IIOJIJIpU3auUu cpen U ux MYJIbTUIIOJIbHble
pa3JIOKeHUJI
B HaCTOJlIlleM naparpacpe MhI Pa30BbeM nOJIHhIH CPOPMaJIH3M BKJIanOB BhICWHX
( cMewaHHhIx) nOJIJlpH3yeMocTeH B lIeThIpe 6a3HCHhIe nJIOTHOCTH pacnpeneJIeshy
HHJI nOJIJlpH3aUHH npOH3BOJIbHOH CpenhI P M T(p) H Td) B npHHUHne
npOH3BOnJl naJIee MYJIbTHnOJIbHhIe pa3JIOEeHHJI 3THX nOJIHhIX BhIpaEeHHH nJIJI
3HeprHH HX B3aHMoneHCTBHH C BHeWHHMH nOJIJlpH3aUHJlMH MhI nOJIYlIHM 0606shyIlleHHe npenhInYIllHX CPOPMYJI Ha CJIyqaH HeJIHHeHHhIX cpen B COOTBeTCTBHH
C CPYHnaMeHTaJIbHhIMH B3a~MoneHCTBHJlMH KaEnoH H3 nOJIJlpH3anHH BBeneM
nOJIJlpH3yeMocTH YlIHThIBalOillHe Bce B03MOXltHhIe BapHaHThI HHnYUHpoBaHHJI
naHHOH nOJIJlpH3aUHH BHeWHHMH nOJIJlMH H TOKaMH 0603Hallag KOJIHlIeCTBO
CsepHYThIX HHneKCOB B Ka)KnOM lIJIeHe TOllKaMH BhInHweM CaMhIe 06mHe BhIshy
pa)KeHHJI nJIJI 3JIeKTpHlIeCKoi nOJIJlpH3anHH -C YlIeTOM Bcex BKJIanOB BnJIOTb no
lIeTBepToro nOpJlnKa
P =a~e) E+a~e) EE+a~e) EEE++a~em) EH+a~em) EEH+
+B~m)H+Bm)HH+Bm)HHH+ +Bme)HE+Bme)HHE+ bull (11)
+Y~r) Jm+Yr) JmJm+Yr) JmJmJm++Yer) EJm+Y~mr)HJm++Yemr) EHJm+
+5~r) Je+51r)jeJe+51r) JeJeJe+ +51er) EJe+51mr)HJe++51errmiddotmiddot)EHJe+
rne Jm = ~(i + b) == rotH Je = rotE lIHCJIa npH nOJIJlpH3yeMOCTJlX 060shy3HallaIOT HX TeH30pHhIH paHr H MO)KHO nepeHTH K HenpHBOnHMhIM TeH30PaM
HanpHMep BBOnJl nOJIYCYMMY H nOJIypa3HOCTb a~em) H f3~me) H Tn
nJIJI MarHHTHoH nOJIJlpH3aUHH
M =amp~m)H+ampm)HH+amp~m)HHH+ +ampem)HE+amp~em)HEE+
+p~e) E+pie)EE+pe) EEE++p~me) EH+pme) EEH+ (12)
+i~r) Jm+i~r) JmJm+ir) JmJmJm+bull+i1er)HJm+i~mr) EJm++imer)HEjm+
+S~r) Je+S1r)JeJe+S~r) JeJe Je + +S~er) EJe+S1 mr) HJe+ +S~emr) EHJe+ j
nml TOP0HllHOi nOJIJlpH3ar(HH MarHHTHoro npOHCXO)KneHHJI
T M =atr)Jm+ar) JmJm+~r) Jm JmJm+ +a1re)JmE+~re) JmEE+
+~e) E+i31e) EE+7f4e)EEE+ +per) EJm+~er) EEJm+ (13)
+~m)H+Y1m)HH+~m)HHH+ +Y1rm)JmH+Y1re)JmE+ +~rme)JmHE+bull
+6~r) Je+6r) JeJe+~r) JeJeJe++~e) EJe+~m) HJe+bull+6~me) EHJe+bull j
9
nJISI TOPOHZJHOH nOJISIpH3anHH 3JIeKTpHqeCKoro npOHCXOJKneHHSI
T p =a~) Je+a~)JeJe+a~) JeJeJe++a~e)JeE+a~e) JeEE+
+8~e) E+8e) EE+8e) EEE++8e) EJe+8e) EEJe+ (14)
+i~m) H+i~m)HH+im) HHH+ +i~m) JeH+i~e)JeE+ +ime) Je HE+
+gir)Jm+I~)JmJm+g)JmJmJm+ +g~e)EJm+g~m)HJm+ +Sme)EHJm+
HeTpynHo YleCTb B 3THX BhlpaJKeHHSIX nJISI naHHOro paccMaTpHBaeMoro
KpHCTaJIJIa nYaJIbHhle CHMMeTPHH KOTOphIe npHBenYT K paBeHCTBaM THna (m) f3(e) -r(T) (T) B
f32 = 2 (12 = U2 H T n ce ~KJIaZJhIB nOJISIpH3anHH conepJKalJlHe
BeKTOphI Jm H Je Ha3hIBaJOT HeJIOKaJILHhIMH H B nOCJIenHeeBpeMSI 3KCneshy
pHMeHTaTOphI HayqHJIHCL HX H3MepSITb B HeKOTOphIX cnyqaJIX OTneJIbHO OT
JIOKaJIbHhIX 3laquoplaquopeKTOB [24J EcTeCTBeHHO ITO B npHBene~lHhIX BhIIlle laquoPOPMYshy
JIax qaCTL BeKTOpOB HanpSIJK~HHOCTeH MOJKHO 3aMeHHTL Ha HX npOH3BOnHhIe
PaHr nOJISIpH3yeMocTeH UpH 3TOM B03p~CTeT H 3TH BKJIanhI B COOTBeTCTBYshy
IOlJlHe nOJISIpH3anHH TaKJKe 6yneM Ha3hIBaTb HeJIOKaJILHhIMH B03MOJKHhI H
npyrHe HeJIHHeHHhIe KPOCC-qJIeHhI KpOMe MarHHT03JIeKTpHqeCKHX MexaHQ-
MarHHTHhIe TepM03JIeKTpHqeCKHe H np
Tenepb npOH3BeneM MYJILTHnOJILHhIe pa3JIOJKeHHJI BeKTOPHhlX nOJISIpH3ashy
nHH cpen C yqeTOM CBOHCTB HX nOJISIpH3yeMoCTH llJISI 3TOro BOCnOJIL3yeMCSI
laquopopMYJIaMH nJISI 3HeprHH B3aHMoneHCTBHSI 3THX nOJISIpH3aUHH C BHeIllHHMH
nOJISIMH (1) H (2) HaqHeM C aHaJIH3a 3JIeKTpHqeCKOH nOJISIpH3aUHH TIPHCYTshy
CTBHe B BhIpaJKeHHH nJISI n9JISIpH3anHH BeKTopa 3JIeKTpHIeCKOH HanpSIJKeHshy
HOCTH MonHlaquopHnHpyeT onpeneJIeHHe MYJIbTHnOJILHhIX MOMeHTOB HanpHMep
3JIeKTpHqeCKHH nHnOJILHhIH MOMeHT Q = J P(r )cfr nOJIyqHT n06aBKH 3a CqeT w (e) (e) (emT) TI
nOJISIpH3yeMocTeH 02 03 bull 4 bull bull bull pHBeneM nJISI npHMepa BKJIan OT (eL
O 3
(15)
TaKOH BKJIaZJ MOJKeT 6hITL nOCTaTOqHO BeJIHK eCJIH Cpena 06JIanaeT CHJILshy
HhIMH HeJIHHeHHhIMH CBOHCTBaMH H BeJIHqHHa o~e) cymecTBeHHa KpOMe TOrO
3TOT BKJIaZJ MOJKeT YCHJIHBaTLCSI H 3a CqeT HHTerpaJIa TIepexonSI B ClaquopepHqeshy
CKYlO CHCTeMY KOOpnHHaT MhI MOJKeM nepenHCaTL ero B BHne
2f Ei E kd3r = f E2r dr f ninkdfl (16)
rne n TaKHe HHTerpaJIhI qaCTO BCTpeqalOTCSI B 3JIeKTponHHaMHKe= E lEI H CTaTHCTHqeCKOH MexaHHKe H HX 3HaqeHHSI HaxonSITCSI COrJIaCHO 06meH laquoPopshy
MYJIhI
f linjn nl = ~ aijak 0 1 - aikaj 0 1+ (17) L
10
5
IlPH HaJIHqHH nOnOJIHHTeJIhHhIX YCJIOBHH THna divE = 0 HJIH divH = 0 B3J1THe TaKHX HHTerpaJIOB YCJIOJKHJleTCJI B QgtH3HqeCKOM 3KcnepHMeHTe HaJIashy
raeMhIe Ha KPHCTaJIJI nOJIJI MorYT HMeTh pa3Hoe npOHCXOJKneHHe B qaCTHoshy
CTH MorYT HMeTh palHYIO qaCTOTY TaK 3JIeKTpOOnTHqecKHH 3QgtQgteKT Iloshy
KeJIhCa 06hIqHO Ha3hIBaeMhIH JIHHeHHhIM eCTh 3QgtQgteKT BTOpOro nOpJlnKa TK
JlBJIJleTCJI pe3YJIhTaTOM onHOBpeMeHHoro B03neHCTBHJI 3JIeKTpHqeCKoro nOJIJI
HH3KOH qaCTOThI H CBeTOBOH BOJIHhI B KOTOPOH CYlUecTBeHHOH 06hIqHO JlBJIJIshy
eTCJI 3JIeKTpHqeCKaJI COCTaBJIJllOlUaJI 3HaMeHHThIH 3QgtQgteKT Keppa 06hIqHO
HalhIBalOT KBanpaTHqHhIM nOCKOJIhKY nOKalaTeJIb npeJIOMJIeHHJI KpHCTaJIJIa
npOnOplIHOHaJIeH KBanpaTY CMelUalOlUero HH3KOqaCToTHoro nOJIJI OnHaKO no
qHCJIY nOJIeH CTOJllUHX B BhIpaJKeHHH nJIJI 3HeprHH B3aHMoneHCTBHJI 3TOT
3QgtQgteKT Boo6me roBOPJl TpeThero nOpJlnKa CJIenyeT TaKJKe pa3JIHqaTh CTashy
THqeCKHH H nHHaMHqeCKHH 3QgtctgteKThI IlPH H3yqeHHH MexaHH3MOB nHHaMHshy
QeCKHX 3ltlgtltlgteKToB YQHThIBalOT CYMMOBhIe H PalHOCTHhIeqaCTOThI H OT HX
HanJIeJKalUero BhI60pa MOJKeT CHJIhHO 3aBHceTh BeJIHqHHa 3QgtQgteKTa
HeTPynHo BhInHcaTh Qgt0PMYJIY nJIJI Q yqHThIBalOlUYIO Bce HeJIHHeHHhIe
JIOKaJIhHhIe BKJIa]lhI HCnOJIh3YJI BhIpaJKeHHe (11) B nomIpH3aUHJlX (11) (12) (13) H (14) HeJIOKaJIhHhIe nOJIJlpH3yeMocTH 6yneM 0603HaqaTh TeMH JKe 6yshy
KBaMH qTO U npeJKne HO 3arJIaBHhIMH Torna HanpHMep BKJIan 3a CqeT
HeJIOKaJIhHOH nOJIJlpH3yeMocTH BToporo paHra B nHnOJIhHhIH MOMeHT 6yneT
BhIrJIJlneTh TaK
(18)
TeH30p non HHTerpaJIOM MOJKHO eCTeCTBeHHO Pal6HTh Ha HenpHBOnHMhIe qashy
CTH Torna CJIen ero paBHhIH divE CBeneTCJI K pacnpeneJIeHHIO nJIOTHOCTI1
BHelliHero 3apJlna CHMMeTpHqHaJi qaCTh QgtHKcHPyeT KBa]lpynOJIhHhIe CBOHshy
CTBa neHcTBYlOlUero Ha CHCTeMY 3JIeKTpHqeCKoro nOJIJI a aHTHCHMMeTpHqHaJi
- aKCHaJIhHhIe CBOHCTBa 3TOro nOJIJI
3aKJIIOQeHHe
He npenCTaBJIJleT 60JIhIlIOro Tpyna palBHTh nOJIHhIH QgtOPMaJIH3M non06HhIx
MYJIhTHnOJIhHhIX palJIOJKeHHH IlOJIHaJI napaMeTpH3alIHJI JII060ro QgtYHnaMeHshy
TaJIhHOro B3aHMoneHCTBHJI OKaJKeTCJI Ky6HqeCKOH pellieTKoH 3anOJIHJllOlUeH
npOCTpaHCTBO Boo6lUe roBOPJl HeOrpaHHqeHHO HanpHMep B nOJIOJKHTeJIhshy
HhIX HanpaBJIeHHJlX oceH X Y H Z Ha OCH X KOTOPYIO 6yneM CqHTaTh
rJIaBHOH 6y~eM pacnOJIaraTh aHaJIH3HpYJl KaKYlO-JIH60 CJIOJKHYIO 3JIeKTposhy
MarHHTHYIO CHCTeMY 3HaqeHHJI onHChIBalOlUHX ee MOMeHTOB B nOpJlnKe B03shy
paCTaHHJI l-nOJIhHOCTH B HanpaBJIeHHH OCH Y 6yneM QgtHKCHpOBaTh 3Haqeshy
HHJI BKJIanOB 2n-CTeneHHhIX panHYCOB pacnpeneJIeHHJI KalKnoro MOMeHTa B
nOpJlnKe B03paCTaHHJI CTeneHH n HanOMHHM qTO MYJIhTHnOJIhHhIH aHaJIH3 B
11
ero MaTeMaTHqeCKOH nOllOnJIeKe OCHOBhIBaeTClI Ha TeopHH 0606meHHhIx ltPYHKshy
IlHH H HeCMOTpll Ha 06menpHHlIThIe 3a6JIYJKlleHHlI 0 llOMHHHpOBaHHH MOMeHshy
TOB BeJIHqHHhI pallHYCOB MorYT OKalaTbClI ropa3110 60JIee cymecTBeHHhIMH
npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
Hll8JIbHhIH llBOHHOH CJIOH C He60JIbIllHM 3KcIleHTpHCHTeTOM npeJKlle Bcero 6yshy
lleT OnHChIBaTbClI cpellHeKBallpaTHqHhIM 3apmOBhIM PallHYCOM a nOTOM YJKe
3JIeKTpHqeCKHM KBallPynOJIbHhIM MOMeHTOM C reoMeTpHqeCKoHmiddot JKe TOqKH
3peHHlI pallHYChI onHChIBalOT Mepy OTKJIOHeHHlIKaJKlloro l-noJIbHoro MOMeHTa
OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
eMOCTH Hall KaJKAOH n03HIlHeH MaTpHIlhI ltPHKCHpYlOmeH 3HaqeHHlI MOMeHTOB
HJIH pallHYCOB
OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
KBallpaTHqHO (H Tll COOTBeTCTBeHHO nOpllllKY n-TOpa) ECTecTBeHHo qTO
BeJIHqHHa OTKJIHKa 6YlleT CTpeMHTeJIbHO nallaTb C B03paCTaHHeM nopauKa n H3-3a Hal1HqHlI peJIlITHBHCTCKOro 4gtaKTopa (lc)n B 3HeprHH B3aHMOlleHCTBHlI
n-Topa C BHeIllHHM nOJIeM
IIollqepKHeM TO qTO BhIlleJIeHHe BeKTOPHhIX BKJIallOB HaH60JIee yll06HO llJIj[
3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
HaHO~eXHOJIOrHlIX
ABTOPhI 6 JIarona PHhI MHTII ltIgtYHllaMeHTal1bHaK MeTpOJIOrHlI 3a nOllshy
llepJKKy 3TOH TeMaTHKH no corJIaIIIeHHIO 251
12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
a)
-
-
c)
f)e)
Puc1
4
MOMeHT KOTOPhIH B pa3JIO)KeHHH TOKa COOTBeTCTByeT BpeMeHHOH npOH3BOlJshy
HOH OT CpelJHeKBalJpaTHtIHoro palJHyca 3apSlJ(a T e lJBOHHOMY CltpepHtIeCKOMY
CJIOIO (PHcld) 2 ri2) = J[2r(rE)+r2Eltf3r - CpelJHeKBaZJpaTHtIHhIH paZJHyC
rtP0lJOJIhHOH tIaCTH lJHnOJIbHoro MOMeHTa Q KOTOPhIH 06pa3yeTcx lJBOHHhIM
UHJIHHlJpHqeCKHM CJIoeM lJHnOJIeH (PHcle) Qij H Ti~d)- 3JIeKTpHtIeCKHH H TOshy
POHlJHhIH KBalJpynOJIbHhIH MOMeHThI onpelJeJIXeMhIe no 06hItIHhIMmiddot npaBHJIaM
Bce B3aHMOlJeHCTBHX MYJIbTHnOJIbHhIX xapaKTepHCTHK C nOJIeM HJIH npOH3BOlJshy
HhIMH OT Hero B BhIpa)KeHHH (1) JIOKaJIbHhI H OTHOCXTCjl K HeKoTOpoH CpelJHeH
TOtlKe ueHTpy CHCTeMhI
ECJIH )Ke B3aHMOlJeHCTBHe dE 3aMeHHTb Ha pH TO nepBhIe tIJIeHhI Pa3JIoshy
)KeHHX 3HeprHH B3aHMOlJeHCTBHx pacnpeneJIeHHX nJIOTHOCTH MarHHTHhIX lJHshy
nOJIeH C BHeIIIHHM HeOlJHOpOlJHhIM MarHHTHhIM nOJIeM npHHHMaIOT BHlJ
w = - f p(r)H(r)d3 r = -MH - T(IJ)rotHshy
P(IJ)rotrotH - -~M(VH + V middotHmiddot) - T(~)V(rotH) + ----gt2 J bull J J J J ---r
==gt -MH - T(IJ)E - p(IJ)(rot - H) ==gt -MH - T(IJ)rotH + p(lJ) ~HCB06 + (2)
BHlJHM tITO COrJIaCHO lJYaJIbHOCTH 3JIeKTpOMarHeT1iI3Ma MOMeHT Q B (1) 3ashy
MeHxeTCX Ha MarHHTHhIH lJHnOJIbHhIH MOMeHT M XapaKTepH3YlOInHH 3KBHshy
BaJIeHTHYIO paMKy C TOKOM T(d) - Ha nOJIXpHhIH TOPOHlJHhIH lJHnOJIb T(IJ) a
BeKTopHhIH nOJIOHlJaJIbHhIH lJHnOJIbHhIH MOMeHT p(d) - Ha aKCHaJIbHhIH p(IJ) KorlJa HOCHTeJIXMH MarHHTHhIX MOMeHTOB XBJIXIOTCX cnHHOBhIe tIaCTHIJhI
HanpHMep HeHTpOHhI TO MYJIbTHnOJIbHoe pa3JIO)KeH~e pacnpelJeJIeHHX HX
MarHHTHhIX MOMeHTOB (HanpHMep B SlJ(pe HJIH B HeHTpOHHOH 3Be3lJe) nOJIyshy
tIaeT nepBHtIHhIH CMhICJI B TeopHH KOHlJeHCHpoBaHHorO COCTOXHHX BemeCTBa
Ha B03MO)KHOCTb TOPOHlJaJIbHOro ynOpSlJ(OtleHHX cnHHOBhIX MarHHTHhIX MOMeHshy
TOB B nepOBCKHTax BnepBhIe YKa3hIBaJIOCb MC )KeJIYlJeBhIM [6 YnopSlJ(Otleshy
HHe )Ke MarHHTHhIX tIaCTHU B CBepXTOHKHX nJIeHKaXB 3aMKHYThIe KOHltpHryshy
paUHH KOTophIe MO)KHO onHChIBaTb TOPOHlJHhIMH MOMeHTaMH no-BHlJHMOMY
Bo06me npelJCTaBJIXeT c060H 06InHH CJIytlaH (CM Hanp [18] H [19]) OTshy
MeTHM tITO cgt6MeHHhIe B3aHMOlJeHCTBH1I Me)KlJy cnHHaMH pacnOJIO)KeHHhIMH
Ha nJIOCKOCTH TaK)Ke npHBOllXT K H3BeCTHOH peIIIeTKe BHxpeH KeCTepJIHuashy
TaYJIeca pacnOJIO)KeHHhIX B aHTHTOp0HuHOM nOpXlJKe
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lJByX OCTaBIIIHXCX ltpYHlJ~eHTaJIbHhIX (TOP0HlJHhIX) lJHnOJIbHhIX pacnpelJeJIeshy
HHH T(r) H g(r) IIOJIytleHHhIe B pe3YJIbTaTe TaKHX pa3~O)KeHHH BHXPH BhIshy
COKOro nopSlJ(Ka BpSlJ( JIH peaJIH3YlOTC1I B KPHCTaJIJIHqeCKHX BemeCTBax llJIX
2TaKYlO KOHltlgtHlypaUHIO BqaCTHOM cnyqae npH paBHOM HynlO BHYTpeHHeM pallHyce HHoshy
llla Ha3hIBaIOT e)KOM
5
3
HX 06pa30BaH~liI HYJKHbI 3K30THtJeCKHe Cpellbl B KOTOPblX 3JIeMeHTapHble Hoshy
CHTeJIH YJKe 06JIauaJOT HanpHMep TOP0HllHblMH llHnOJIliIMH OTMeTHMtJTO
nOCJIellHHe 3KCnepHMeHTbI YKa3b1BaIOT tJTO HeHTpHHO BHllHMO MaHOpaHOBshy3CKHe tJaCTHIUd H 06pa3YIOT cpellbl BblCOKOH nJIOTHOCTH (10HgtcM- ) H3BeCTHO
(CM Hanp [20] H [21]) tJTO TaKHe HeHTpHHo 06JIallaIOT TOJIhKO TOPOHllshy
HblMH llHnOJIhHblMH MOMeHTaMH Ha B03MOJKHOCTh KorepeHTHblX B036YJKlleshy
HHliIX B TaKOH cpelle 06ecnetJHBaIQIllHX B036YJKlleHHe MeJKlly aupOHaMHHa
ypOBHe rpaBHTaUHOHHblX 6b1JIO BnepBble YKa3aHO BM KOPIOKHHblM [22]
BeKTopShIe napaMeTphI
CpenhI
H3JIOJKeHHblH BbllUe MaTepHaJI OTHOCHTCliI K TaK Ha3b1BaeMOB JIHHeHHOH 3JIeKshy
Tp0llHHaMHKe COrJIaCHO ee ypaBHeHHliIM 3JIeMeHTapHble 06beKTbI HanpHMep
llHnOJIH ynpaBJIliIIOTCliI nOJIeM KaKoro-JIH60 THna HJIH TOKOM npH KOHTaKTHOM
B3aHMOlleHCTBHH H He MeHliIIOT npH 3TOM CBOHX napaMeTPOB KpOMe llHnoshy
JIeB MOJKHO BBeCTH If 6oJIee CJIOJKHble 06beKTbI COCTaBJIeHHble H3 llHnOJIeH
KBallpynOJIH oKTynOJIH H Tll IIpocTeBlUHe MaKPOCKonHtJeCKHe KOHCTPYKUHH
TaKoro pOlla MOJKHO npellCTaBHTh HanpHMep B BHlle llByX HJIHTpeX TOKoshy
BblX neTeJIh pacnOJIOJKeHHblX B6JIH3H llpyr llpyra HJIH CBliI3aHHblX B ellHHhIH
KOHTyp PacCMOTPHM HanpHMep npOBOJIOtJHoe KOJIhUO H30rHYTOe B ltpopMe
nponeJIJIepa IIpH npOXOJKlleHHH no TaKOMY KOJIhUY TOKa KaJKllaJI H3 neTeJIh
C03llaeT MarHHTHoe nOJIe H MbI MOJKeM npH6JIHJKeHHO onHCblBaTh HX MarHHTshy
Hble CBOHCTBa C nOMOmhIO MaIHHTHblX MOMeHTOB Torlla 3HepIHliI JIOKaJIhHblX
B3aHMOlleHCTBHH 3THX llByX neTeJIh C BHelUHHM MaIHHTHblM nOJIeM npOH3shy
BOJIhHOH KOHqmrypaUHH 3anHCbIBaeTCSI TaK
(3)
IIoll JIOKaJIhHOCThIO nOllpa3YMeBaeTCliI OTHeceHHOCTh CJIaraeMblX BBelleHHOH
3HeprHH K ueHTpaM neTeJIh BBelleM Teneph KPoMe BeKTopOB Ili H Ilk Koshy
Topble OTHOCliICh K Pa3JIHtJHblM neTJIliIM npHnHCaHbI K Pa3JIHtJHblM TOtJKaM
npOCTpaHCTBa I H 2 KBaupynOJIhHblH MOMeHT CHCTeMbI B BHlle CHMMeshy
TPHtJHOro TeH30pa BTOporo paHra Ilik KaK H B npellblllYIlleM naparpaltPe
napaMeTp Ilik YtJHTblBaeT BKJIall B 3HepIHIO C03llaIOIllHHCliI 3a CtJeT rpauHshy
eHTa HaJIaraeMoro Ha CHCTeMY BHelUHero MarHHTHoro nOJIliI T e 3a CtJeT Heshy
OllHOPOllHOCTH nOJIliI MeJKlly ueHTpaMH neTeJIh Torlla BblpaJKeHHe (3) MOJKHO
npH6JIHJKeHHOnepenHcaTh B BHlle
(4)
6
BHuHM qTO Tenepb BCe qJleHhI pa3JlOJKeHHH OTHeCeHhI K HeKOTOpOH CpeuHeH
TOqKe KOTOpyIO MhI Ha3hIBaeM ueHTpOM CHCTeMhI H B TaKOM npeuCTaBJleHHH
meprHH B3aHMOueHCTBHH H COCTOHT BCH CYTb MYJlbTHnOJlbHoro pa3JlOJKeHHH
npeunOJlOJKHM renepb qTO MaTepHaJI H3 KOTOPOro cueJlaHhI neTJlH rH60K
H MarHHTHoe nOJle opHeHTHpYIOlUee MarHHTHhIe MOMeHThI MOJKeT ueltpopMHshy
pOBaTb KOHTyphI neTeJlb GTO B03MYlUeHHe MhI MOJKeM yqeCTb BBeueHHeM
KBaupynOJlbHOH nOJlHpH3yeMocTH CHCTeMhI a(m) ilonOJlHHTeJlbHaH 3HeprHH
JlOKaJIbHOrO B3aHMoueHCTBHH CHCTeMhI C MarHHTHhIM nOJleM onHChIBaeTCH YJKe
HeJlHHeHHhIM no nOJlIO BhIpalKeHHeM
(5)
TaKHM 06pa30M MbI nOJlyqaeM HOBOe KaqeCTBO B H3yqeHHH CBOHCTB CHCTeMhI
uHnOJleH 0606lUaH CKaJKeM TaK qTO HaJIOJKeHHe nOJlH Hi HHuyUHpyeT B
cpeueuonOJlHHTeJlhHYIO MarHHTHYIO uHnOJlbHYIO nOJlHpH3aUHIO Mle = a~)Hi KOTOPaH B CBOIO oqepeub cnoc06Ha B3aHMoueHcTBOBaTb C nOJleM Hie 3uecb
a) 6yueM Ha3hIBaTb KBaupaTHqHOH MarHHTHOH BOCnpHHMqHBOCTbIO CpeuhI
HTaK MbI BBeJlH HeJlHHeHHOe B3aHMoueHcTBHe yqHThIBaIOlUee HeJlHHeHshy
HhIH OTKJlHK cpeuhI Ha MarHHTHoe nOJle B npHHUHne TeH30p a MOJKeT HMeTb
npOH3BOJlbHOe KOJlHqeCTBO HHueKCOB H B BhIpaJKeHHH uJlH 3HeprHH OH uOJlJKeH
6bITb cBepHyT C COOTBeTCTBYIOlUHM CHMMeTpHqHhIM TeH30pOM COCTaBJleHshy
HhIM H3 BeKTopa HanpHJKeHHOCTH BHeuIHero MarHHTHoro nOJlH ECTecTBeHHo
qTO ueltp0pMaUHH nou06HhIe onHcaHHhIM uJlH CHCTeMhI MarHHTHbix uHnOJleH
MOJKeT HcnbIThIBaTb H CHCTeMa COCTaBJleHHaH H3 3JleKTpHqeCKHX uHnOJleH B
3TOM CJlyqae BhIpaJKeHHH uJlH 3HeprHH HX B3aHMoueHcTBHH C BHelIIHHM 3JleKshy
TpHqeCKHM nOJleM 6yuYT COBeplIIeHHO HueHTHqHhI BBeueHHhIM BhIlIIe a Marshy
HHTHyIO BOCnpHHMqHBOCTb B HHX MhI 3aMeHHM Ha 3JleKTpHqeCKYIO a ebull noshy
CKOJlbKY peaJIbHhIe CHCTeMhI H3 KOTOPhIX COCTOHT JlI06aH cpeua - 3TO aTOMhI
H MOJleKYJlhI HaxOuHlUHeCH B COCTOHHHH HenpephIBHOH BHYTpeHHeH H BHelIIHeH
uHHaMHKH TO BCJleuCTBHe B3aHMOCBH3eH HCTOqHHKOB qepe3 nOJlH B03MOIKHhI
H CMelIIaHHhIe nOJlHpH3yeMocTH HanpHMep MarHHT03JleKTpHqeCKHe C 3Hepshy
rHeH B3aHMOlIeHCTBHa
(6)
HMeH B BHuy OpraHH3aUHIO KOMnbIOTepHOH naMHTH BBeueM nOJlHpH3yeMocTb
TpeTbero paHra OTBeqaIOlUYIO CHMMeTpHH 3ltpltpeKTa XOJlJla H COOTBeT-
CTBeHHO 3HeprHIO B3aHMoueHCTBHH
I (em)E H JWNL -ailel i Ie I (7)C
rue J - BHelIIHHH TOK npoTeKaIOlUHH qepe3 paccMaTpHBaeMYIO CHCTeMY B
qaCTHOCTH 3TOT TOK MOJKeT 6hITb TOKOM CMelUeHHH qTO peaJIH3yeTCH Koshy
rna H3yqaeMaa CHCTeMa pa3MelUaeTca B nOJle KOHueHcaTopa ByueM CqHshy
TaTb qTO TeH30p an) - aHTHcHMMeTpHqeH no HHueKcaM i H k Torua B
7
TeH30pe EiHk MOiKHO BhUleJ1HTh TOJIhKO aHTHCHMMeTpHtJHYIO tJacTh H MhI B
npaBe llml KOHKpeTHOH 3KCnepHMeHTaJIhHOH cHTyaUHH CB1l3aTh C HHM BeKTOp
YMOBa-TIoHHTHHra llJI1l CBeTa npOXOll1lWero tJepe3KpHCTaJIJI C nOl106aIOWHMH
HeJ1HHeHHhIMH CBOHCTBaMH TIoaHaJIOrHH C npellhUlYWHMH pacCY)f(lleHH1lMH
MhI MO)f(eM BBeCTH HHllYUHpOBaHHYIO TOPOHllHYIO nOJI1lpH3aUHIO
1 (em)E H - (em)S11I = -a[ik]1 i k = ajl j (8) c
KOTOPas OTKJIHKaeTC1l Ha npOXOll1lWHH tJepe3 KPHCTaJIJI TOK JI lIeHcTBHshy
TeJIhHO 3neKTpOMarHHTHaH BonHa npOXOll1l HanpHMep tJepe3 qeppoMarHHTshy
HhIH nH3neKTpHK MOiKeT B npHHUHne COBepwaTh B HeM nepeopHeHTaUHIO
IIOn1lpH3aUHH IIpeBpawa1l ero B cna6bIH qeppOMarBeTHK 06JIauaIOWHH TOshy
POHllHOH nOn1lpH3aUHeH KOHKpeTHO TaKHMH CBOHCTBaMH MoryT 06nanaTh
palnHtJHhIe nepOBCKHTbI MaHraHHTbI (CM Hanp [23]) H llPyrHe nOCTaTotJHO
cnOiKBbIe coenHHeHH1l
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nOB eCTeCTBeHBO nocTynalOT npowe HanpHMep B [10] H3ytJaJIC1l OTKnHK
MOBOKpHCTaJIna F e-Ga-oKcHlla Ha MarHHTHoe none yhBeCTHo tJTO y TaKoro
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(BcnOMHHTe aHaJIOrHIO C yIIpyrHMH neTn1lMH) B 3KcnepHMeHTe H3Mep1lnaCh
senHtJHHa HHnyUHpoBaHHoH 3neKTpHtJecKoH nOn1lpH3auHH npH HanO)f(eHHH Ha
KpHCTann MarBHTBoro IIOn1l H
D _ (em)H LI - ij ) (9)
(em) rne TeH30p aij Ha3bIBaeTC1l TeH30pOM MarHHT03neKTpHtJeCKOH BocnpHHMshy
tJHBOCTH ABTHcHMMeTpHtJHaH tJaCTh TeB30pa a~jm) KaK pal H OIIpenen1leT
TOPOHllHbIH nHnonhHbIH MOMeBT cornacBo COOTBoweBHIO nyanhBOCTH
(em)amiddotmiddot (10)I)
rne eurooijk - aHTHCHMMeTpHtJBbIH ellHBHtJBbIH TeB30p JIeBH-tHBHTa TIon1lpHshy
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8
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IIOJIJIpU3auUu cpen U ux MYJIbTUIIOJIbHble
pa3JIOKeHUJI
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( cMewaHHhIx) nOJIJlpH3yeMocTeH B lIeThIpe 6a3HCHhIe nJIOTHOCTH pacnpeneJIeshy
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C CPYHnaMeHTaJIbHhIMH B3a~MoneHCTBHJlMH KaEnoH H3 nOJIJlpH3anHH BBeneM
nOJIJlpH3yeMocTH YlIHThIBalOillHe Bce B03MOXltHhIe BapHaHThI HHnYUHpoBaHHJI
naHHOH nOJIJlpH3aUHH BHeWHHMH nOJIJlMH H TOKaMH 0603Hallag KOJIHlIeCTBO
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lIeTBepToro nOpJlnKa
P =a~e) E+a~e) EE+a~e) EEE++a~em) EH+a~em) EEH+
+B~m)H+Bm)HH+Bm)HHH+ +Bme)HE+Bme)HHE+ bull (11)
+Y~r) Jm+Yr) JmJm+Yr) JmJmJm++Yer) EJm+Y~mr)HJm++Yemr) EHJm+
+5~r) Je+51r)jeJe+51r) JeJeJe+ +51er) EJe+51mr)HJe++51errmiddotmiddot)EHJe+
rne Jm = ~(i + b) == rotH Je = rotE lIHCJIa npH nOJIJlpH3yeMOCTJlX 060shy3HallaIOT HX TeH30pHhIH paHr H MO)KHO nepeHTH K HenpHBOnHMhIM TeH30PaM
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nJIJI MarHHTHoH nOJIJlpH3aUHH
M =amp~m)H+ampm)HH+amp~m)HHH+ +ampem)HE+amp~em)HEE+
+p~e) E+pie)EE+pe) EEE++p~me) EH+pme) EEH+ (12)
+i~r) Jm+i~r) JmJm+ir) JmJmJm+bull+i1er)HJm+i~mr) EJm++imer)HEjm+
+S~r) Je+S1r)JeJe+S~r) JeJe Je + +S~er) EJe+S1 mr) HJe+ +S~emr) EHJe+ j
nml TOP0HllHOi nOJIJlpH3ar(HH MarHHTHoro npOHCXO)KneHHJI
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10
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11
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12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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paUHH KOTophIe MO)KHO onHChIBaTb TOPOHlJHhIMH MOMeHTaMH no-BHlJHMOMY
Bo06me npelJCTaBJIXeT c060H 06InHH CJIytlaH (CM Hanp [18] H [19]) OTshy
MeTHM tITO cgt6MeHHhIe B3aHMOlJeHCTBH1I Me)KlJy cnHHaMH pacnOJIO)KeHHhIMH
Ha nJIOCKOCTH TaK)Ke npHBOllXT K H3BeCTHOH peIIIeTKe BHxpeH KeCTepJIHuashy
TaYJIeca pacnOJIO)KeHHhIX B aHTHTOp0HuHOM nOpXlJKe
Pa3JIO)KeHHX nOlJ06HhIe (1) H (2) OqeBHlJHO MO)KHO BhInOJIHHTb lJJIX
lJByX OCTaBIIIHXCX ltpYHlJ~eHTaJIbHhIX (TOP0HlJHhIX) lJHnOJIbHhIX pacnpelJeJIeshy
HHH T(r) H g(r) IIOJIytleHHhIe B pe3YJIbTaTe TaKHX pa3~O)KeHHH BHXPH BhIshy
COKOro nopSlJ(Ka BpSlJ( JIH peaJIH3YlOTC1I B KPHCTaJIJIHqeCKHX BemeCTBax llJIX
2TaKYlO KOHltlgtHlypaUHIO BqaCTHOM cnyqae npH paBHOM HynlO BHYTpeHHeM pallHyce HHoshy
llla Ha3hIBaIOT e)KOM
5
3
HX 06pa30BaH~liI HYJKHbI 3K30THtJeCKHe Cpellbl B KOTOPblX 3JIeMeHTapHble Hoshy
CHTeJIH YJKe 06JIauaJOT HanpHMep TOP0HllHblMH llHnOJIliIMH OTMeTHMtJTO
nOCJIellHHe 3KCnepHMeHTbI YKa3b1BaIOT tJTO HeHTpHHO BHllHMO MaHOpaHOBshy3CKHe tJaCTHIUd H 06pa3YIOT cpellbl BblCOKOH nJIOTHOCTH (10HgtcM- ) H3BeCTHO
(CM Hanp [20] H [21]) tJTO TaKHe HeHTpHHo 06JIallaIOT TOJIhKO TOPOHllshy
HblMH llHnOJIhHblMH MOMeHTaMH Ha B03MOJKHOCTh KorepeHTHblX B036YJKlleshy
HHliIX B TaKOH cpelle 06ecnetJHBaIQIllHX B036YJKlleHHe MeJKlly aupOHaMHHa
ypOBHe rpaBHTaUHOHHblX 6b1JIO BnepBble YKa3aHO BM KOPIOKHHblM [22]
BeKTopShIe napaMeTphI
CpenhI
H3JIOJKeHHblH BbllUe MaTepHaJI OTHOCHTCliI K TaK Ha3b1BaeMOB JIHHeHHOH 3JIeKshy
Tp0llHHaMHKe COrJIaCHO ee ypaBHeHHliIM 3JIeMeHTapHble 06beKTbI HanpHMep
llHnOJIH ynpaBJIliIIOTCliI nOJIeM KaKoro-JIH60 THna HJIH TOKOM npH KOHTaKTHOM
B3aHMOlleHCTBHH H He MeHliIIOT npH 3TOM CBOHX napaMeTPOB KpOMe llHnoshy
JIeB MOJKHO BBeCTH If 6oJIee CJIOJKHble 06beKTbI COCTaBJIeHHble H3 llHnOJIeH
KBallpynOJIH oKTynOJIH H Tll IIpocTeBlUHe MaKPOCKonHtJeCKHe KOHCTPYKUHH
TaKoro pOlla MOJKHO npellCTaBHTh HanpHMep B BHlle llByX HJIHTpeX TOKoshy
BblX neTeJIh pacnOJIOJKeHHblX B6JIH3H llpyr llpyra HJIH CBliI3aHHblX B ellHHhIH
KOHTyp PacCMOTPHM HanpHMep npOBOJIOtJHoe KOJIhUO H30rHYTOe B ltpopMe
nponeJIJIepa IIpH npOXOJKlleHHH no TaKOMY KOJIhUY TOKa KaJKllaJI H3 neTeJIh
C03llaeT MarHHTHoe nOJIe H MbI MOJKeM npH6JIHJKeHHO onHCblBaTh HX MarHHTshy
Hble CBOHCTBa C nOMOmhIO MaIHHTHblX MOMeHTOB Torlla 3HepIHliI JIOKaJIhHblX
B3aHMOlleHCTBHH 3THX llByX neTeJIh C BHelUHHM MaIHHTHblM nOJIeM npOH3shy
BOJIhHOH KOHqmrypaUHH 3anHCbIBaeTCSI TaK
(3)
IIoll JIOKaJIhHOCThIO nOllpa3YMeBaeTCliI OTHeceHHOCTh CJIaraeMblX BBelleHHOH
3HeprHH K ueHTpaM neTeJIh BBelleM Teneph KPoMe BeKTopOB Ili H Ilk Koshy
Topble OTHOCliICh K Pa3JIHtJHblM neTJIliIM npHnHCaHbI K Pa3JIHtJHblM TOtJKaM
npOCTpaHCTBa I H 2 KBaupynOJIhHblH MOMeHT CHCTeMbI B BHlle CHMMeshy
TPHtJHOro TeH30pa BTOporo paHra Ilik KaK H B npellblllYIlleM naparpaltPe
napaMeTp Ilik YtJHTblBaeT BKJIall B 3HepIHIO C03llaIOIllHHCliI 3a CtJeT rpauHshy
eHTa HaJIaraeMoro Ha CHCTeMY BHelUHero MarHHTHoro nOJIliI T e 3a CtJeT Heshy
OllHOPOllHOCTH nOJIliI MeJKlly ueHTpaMH neTeJIh Torlla BblpaJKeHHe (3) MOJKHO
npH6JIHJKeHHOnepenHcaTh B BHlle
(4)
6
BHuHM qTO Tenepb BCe qJleHhI pa3JlOJKeHHH OTHeCeHhI K HeKOTOpOH CpeuHeH
TOqKe KOTOpyIO MhI Ha3hIBaeM ueHTpOM CHCTeMhI H B TaKOM npeuCTaBJleHHH
meprHH B3aHMOueHCTBHH H COCTOHT BCH CYTb MYJlbTHnOJlbHoro pa3JlOJKeHHH
npeunOJlOJKHM renepb qTO MaTepHaJI H3 KOTOPOro cueJlaHhI neTJlH rH60K
H MarHHTHoe nOJle opHeHTHpYIOlUee MarHHTHhIe MOMeHThI MOJKeT ueltpopMHshy
pOBaTb KOHTyphI neTeJlb GTO B03MYlUeHHe MhI MOJKeM yqeCTb BBeueHHeM
KBaupynOJlbHOH nOJlHpH3yeMocTH CHCTeMhI a(m) ilonOJlHHTeJlbHaH 3HeprHH
JlOKaJIbHOrO B3aHMoueHCTBHH CHCTeMhI C MarHHTHhIM nOJleM onHChIBaeTCH YJKe
HeJlHHeHHhIM no nOJlIO BhIpalKeHHeM
(5)
TaKHM 06pa30M MbI nOJlyqaeM HOBOe KaqeCTBO B H3yqeHHH CBOHCTB CHCTeMhI
uHnOJleH 0606lUaH CKaJKeM TaK qTO HaJIOJKeHHe nOJlH Hi HHuyUHpyeT B
cpeueuonOJlHHTeJlhHYIO MarHHTHYIO uHnOJlbHYIO nOJlHpH3aUHIO Mle = a~)Hi KOTOPaH B CBOIO oqepeub cnoc06Ha B3aHMoueHcTBOBaTb C nOJleM Hie 3uecb
a) 6yueM Ha3hIBaTb KBaupaTHqHOH MarHHTHOH BOCnpHHMqHBOCTbIO CpeuhI
HTaK MbI BBeJlH HeJlHHeHHOe B3aHMoueHcTBHe yqHThIBaIOlUee HeJlHHeHshy
HhIH OTKJlHK cpeuhI Ha MarHHTHoe nOJle B npHHUHne TeH30p a MOJKeT HMeTb
npOH3BOJlbHOe KOJlHqeCTBO HHueKCOB H B BhIpaJKeHHH uJlH 3HeprHH OH uOJlJKeH
6bITb cBepHyT C COOTBeTCTBYIOlUHM CHMMeTpHqHhIM TeH30pOM COCTaBJleHshy
HhIM H3 BeKTopa HanpHJKeHHOCTH BHeuIHero MarHHTHoro nOJlH ECTecTBeHHo
qTO ueltp0pMaUHH nou06HhIe onHcaHHhIM uJlH CHCTeMhI MarHHTHbix uHnOJleH
MOJKeT HcnbIThIBaTb H CHCTeMa COCTaBJleHHaH H3 3JleKTpHqeCKHX uHnOJleH B
3TOM CJlyqae BhIpaJKeHHH uJlH 3HeprHH HX B3aHMoueHcTBHH C BHelIIHHM 3JleKshy
TpHqeCKHM nOJleM 6yuYT COBeplIIeHHO HueHTHqHhI BBeueHHhIM BhIlIIe a Marshy
HHTHyIO BOCnpHHMqHBOCTb B HHX MhI 3aMeHHM Ha 3JleKTpHqeCKYIO a ebull noshy
CKOJlbKY peaJIbHhIe CHCTeMhI H3 KOTOPhIX COCTOHT JlI06aH cpeua - 3TO aTOMhI
H MOJleKYJlhI HaxOuHlUHeCH B COCTOHHHH HenpephIBHOH BHYTpeHHeH H BHelIIHeH
uHHaMHKH TO BCJleuCTBHe B3aHMOCBH3eH HCTOqHHKOB qepe3 nOJlH B03MOIKHhI
H CMelIIaHHhIe nOJlHpH3yeMocTH HanpHMep MarHHT03JleKTpHqeCKHe C 3Hepshy
rHeH B3aHMOlIeHCTBHa
(6)
HMeH B BHuy OpraHH3aUHIO KOMnbIOTepHOH naMHTH BBeueM nOJlHpH3yeMocTb
TpeTbero paHra OTBeqaIOlUYIO CHMMeTpHH 3ltpltpeKTa XOJlJla H COOTBeT-
CTBeHHO 3HeprHIO B3aHMoueHCTBHH
I (em)E H JWNL -ailel i Ie I (7)C
rue J - BHelIIHHH TOK npoTeKaIOlUHH qepe3 paccMaTpHBaeMYIO CHCTeMY B
qaCTHOCTH 3TOT TOK MOJKeT 6hITb TOKOM CMelUeHHH qTO peaJIH3yeTCH Koshy
rna H3yqaeMaa CHCTeMa pa3MelUaeTca B nOJle KOHueHcaTopa ByueM CqHshy
TaTb qTO TeH30p an) - aHTHcHMMeTpHqeH no HHueKcaM i H k Torua B
7
TeH30pe EiHk MOiKHO BhUleJ1HTh TOJIhKO aHTHCHMMeTpHtJHYIO tJacTh H MhI B
npaBe llml KOHKpeTHOH 3KCnepHMeHTaJIhHOH cHTyaUHH CB1l3aTh C HHM BeKTOp
YMOBa-TIoHHTHHra llJI1l CBeTa npOXOll1lWero tJepe3KpHCTaJIJI C nOl106aIOWHMH
HeJ1HHeHHhIMH CBOHCTBaMH TIoaHaJIOrHH C npellhUlYWHMH pacCY)f(lleHH1lMH
MhI MO)f(eM BBeCTH HHllYUHpOBaHHYIO TOPOHllHYIO nOJI1lpH3aUHIO
1 (em)E H - (em)S11I = -a[ik]1 i k = ajl j (8) c
KOTOPas OTKJIHKaeTC1l Ha npOXOll1lWHH tJepe3 KPHCTaJIJI TOK JI lIeHcTBHshy
TeJIhHO 3neKTpOMarHHTHaH BonHa npOXOll1l HanpHMep tJepe3 qeppoMarHHTshy
HhIH nH3neKTpHK MOiKeT B npHHUHne COBepwaTh B HeM nepeopHeHTaUHIO
IIOn1lpH3aUHH IIpeBpawa1l ero B cna6bIH qeppOMarBeTHK 06JIauaIOWHH TOshy
POHllHOH nOn1lpH3aUHeH KOHKpeTHO TaKHMH CBOHCTBaMH MoryT 06nanaTh
palnHtJHhIe nepOBCKHTbI MaHraHHTbI (CM Hanp [23]) H llPyrHe nOCTaTotJHO
cnOiKBbIe coenHHeHH1l
3KcnepHMeHTaTophI npH H3MepeHHH TOPOHllHOH nOn1lpH3aUHH KPHCTaJIshy
nOB eCTeCTBeHBO nocTynalOT npowe HanpHMep B [10] H3ytJaJIC1l OTKnHK
MOBOKpHCTaJIna F e-Ga-oKcHlla Ha MarHHTHoe none yhBeCTHo tJTO y TaKoro
MaTepHaJIa qeppoMarHHTHhIe CBOHCTBa cotJeTaIOTC1l C nhe303neKTpHtJeCKHMH
(BcnOMHHTe aHaJIOrHIO C yIIpyrHMH neTn1lMH) B 3KcnepHMeHTe H3Mep1lnaCh
senHtJHHa HHnyUHpoBaHHoH 3neKTpHtJecKoH nOn1lpH3auHH npH HanO)f(eHHH Ha
KpHCTann MarBHTBoro IIOn1l H
D _ (em)H LI - ij ) (9)
(em) rne TeH30p aij Ha3bIBaeTC1l TeH30pOM MarHHT03neKTpHtJeCKOH BocnpHHMshy
tJHBOCTH ABTHcHMMeTpHtJHaH tJaCTh TeB30pa a~jm) KaK pal H OIIpenen1leT
TOPOHllHbIH nHnonhHbIH MOMeBT cornacBo COOTBoweBHIO nyanhBOCTH
(em)amiddotmiddot (10)I)
rne eurooijk - aHTHCHMMeTpHtJBbIH ellHBHtJBbIH TeB30p JIeBH-tHBHTa TIon1lpHshy
3aUH1l npHxon1lWaHC1l Ba OllBY 3neMeBTapHYlO 1ltJeHKY KpHCTanna H3ytJaeshy
MOro OKCHlla IIpHBena K 3BatJeBHIO TOPOHllHOro nHIIonhHoro MOMeBTa PaBshy
BOro ITI24JLB A ABanorHtJBbIH 3ltpltpeKT 6bIn IIonytJeB nn1l MarBHT03neKTpHKa
Cr203 IIOMemeBBoro B cHnhBoe MarBHTBoe IIone IIpeBbIwalOwee IIone cnHBshy
qnOII IIepeXOlla
8
4 BKJIanbl IIOJIJIpu3yeMocTeu B BeKTopHble
IIOJIJIpU3auUu cpen U ux MYJIbTUIIOJIbHble
pa3JIOKeHUJI
B HaCTOJlIlleM naparpacpe MhI Pa30BbeM nOJIHhIH CPOPMaJIH3M BKJIanOB BhICWHX
( cMewaHHhIx) nOJIJlpH3yeMocTeH B lIeThIpe 6a3HCHhIe nJIOTHOCTH pacnpeneJIeshy
HHJI nOJIJlpH3aUHH npOH3BOJIbHOH CpenhI P M T(p) H Td) B npHHUHne
npOH3BOnJl naJIee MYJIbTHnOJIbHhIe pa3JIOEeHHJI 3THX nOJIHhIX BhIpaEeHHH nJIJI
3HeprHH HX B3aHMoneHCTBHH C BHeWHHMH nOJIJlpH3aUHJlMH MhI nOJIYlIHM 0606shyIlleHHe npenhInYIllHX CPOPMYJI Ha CJIyqaH HeJIHHeHHhIX cpen B COOTBeTCTBHH
C CPYHnaMeHTaJIbHhIMH B3a~MoneHCTBHJlMH KaEnoH H3 nOJIJlpH3anHH BBeneM
nOJIJlpH3yeMocTH YlIHThIBalOillHe Bce B03MOXltHhIe BapHaHThI HHnYUHpoBaHHJI
naHHOH nOJIJlpH3aUHH BHeWHHMH nOJIJlMH H TOKaMH 0603Hallag KOJIHlIeCTBO
CsepHYThIX HHneKCOB B Ka)KnOM lIJIeHe TOllKaMH BhInHweM CaMhIe 06mHe BhIshy
pa)KeHHJI nJIJI 3JIeKTpHlIeCKoi nOJIJlpH3anHH -C YlIeTOM Bcex BKJIanOB BnJIOTb no
lIeTBepToro nOpJlnKa
P =a~e) E+a~e) EE+a~e) EEE++a~em) EH+a~em) EEH+
+B~m)H+Bm)HH+Bm)HHH+ +Bme)HE+Bme)HHE+ bull (11)
+Y~r) Jm+Yr) JmJm+Yr) JmJmJm++Yer) EJm+Y~mr)HJm++Yemr) EHJm+
+5~r) Je+51r)jeJe+51r) JeJeJe+ +51er) EJe+51mr)HJe++51errmiddotmiddot)EHJe+
rne Jm = ~(i + b) == rotH Je = rotE lIHCJIa npH nOJIJlpH3yeMOCTJlX 060shy3HallaIOT HX TeH30pHhIH paHr H MO)KHO nepeHTH K HenpHBOnHMhIM TeH30PaM
HanpHMep BBOnJl nOJIYCYMMY H nOJIypa3HOCTb a~em) H f3~me) H Tn
nJIJI MarHHTHoH nOJIJlpH3aUHH
M =amp~m)H+ampm)HH+amp~m)HHH+ +ampem)HE+amp~em)HEE+
+p~e) E+pie)EE+pe) EEE++p~me) EH+pme) EEH+ (12)
+i~r) Jm+i~r) JmJm+ir) JmJmJm+bull+i1er)HJm+i~mr) EJm++imer)HEjm+
+S~r) Je+S1r)JeJe+S~r) JeJe Je + +S~er) EJe+S1 mr) HJe+ +S~emr) EHJe+ j
nml TOP0HllHOi nOJIJlpH3ar(HH MarHHTHoro npOHCXO)KneHHJI
T M =atr)Jm+ar) JmJm+~r) Jm JmJm+ +a1re)JmE+~re) JmEE+
+~e) E+i31e) EE+7f4e)EEE+ +per) EJm+~er) EEJm+ (13)
+~m)H+Y1m)HH+~m)HHH+ +Y1rm)JmH+Y1re)JmE+ +~rme)JmHE+bull
+6~r) Je+6r) JeJe+~r) JeJeJe++~e) EJe+~m) HJe+bull+6~me) EHJe+bull j
9
nJISI TOPOHZJHOH nOJISIpH3anHH 3JIeKTpHqeCKoro npOHCXOJKneHHSI
T p =a~) Je+a~)JeJe+a~) JeJeJe++a~e)JeE+a~e) JeEE+
+8~e) E+8e) EE+8e) EEE++8e) EJe+8e) EEJe+ (14)
+i~m) H+i~m)HH+im) HHH+ +i~m) JeH+i~e)JeE+ +ime) Je HE+
+gir)Jm+I~)JmJm+g)JmJmJm+ +g~e)EJm+g~m)HJm+ +Sme)EHJm+
HeTpynHo YleCTb B 3THX BhlpaJKeHHSIX nJISI naHHOro paccMaTpHBaeMoro
KpHCTaJIJIa nYaJIbHhle CHMMeTPHH KOTOphIe npHBenYT K paBeHCTBaM THna (m) f3(e) -r(T) (T) B
f32 = 2 (12 = U2 H T n ce ~KJIaZJhIB nOJISIpH3anHH conepJKalJlHe
BeKTOphI Jm H Je Ha3hIBaJOT HeJIOKaJILHhIMH H B nOCJIenHeeBpeMSI 3KCneshy
pHMeHTaTOphI HayqHJIHCL HX H3MepSITb B HeKOTOphIX cnyqaJIX OTneJIbHO OT
JIOKaJIbHhIX 3laquoplaquopeKTOB [24J EcTeCTBeHHO ITO B npHBene~lHhIX BhIIlle laquoPOPMYshy
JIax qaCTL BeKTOpOB HanpSIJK~HHOCTeH MOJKHO 3aMeHHTL Ha HX npOH3BOnHhIe
PaHr nOJISIpH3yeMocTeH UpH 3TOM B03p~CTeT H 3TH BKJIanhI B COOTBeTCTBYshy
IOlJlHe nOJISIpH3anHH TaKJKe 6yneM Ha3hIBaTb HeJIOKaJILHhIMH B03MOJKHhI H
npyrHe HeJIHHeHHhIe KPOCC-qJIeHhI KpOMe MarHHT03JIeKTpHqeCKHX MexaHQ-
MarHHTHhIe TepM03JIeKTpHqeCKHe H np
Tenepb npOH3BeneM MYJILTHnOJILHhIe pa3JIOJKeHHJI BeKTOPHhlX nOJISIpH3ashy
nHH cpen C yqeTOM CBOHCTB HX nOJISIpH3yeMoCTH llJISI 3TOro BOCnOJIL3yeMCSI
laquopopMYJIaMH nJISI 3HeprHH B3aHMoneHCTBHSI 3THX nOJISIpH3aUHH C BHeIllHHMH
nOJISIMH (1) H (2) HaqHeM C aHaJIH3a 3JIeKTpHqeCKOH nOJISIpH3aUHH TIPHCYTshy
CTBHe B BhIpaJKeHHH nJISI n9JISIpH3anHH BeKTopa 3JIeKTpHIeCKOH HanpSIJKeHshy
HOCTH MonHlaquopHnHpyeT onpeneJIeHHe MYJIbTHnOJILHhIX MOMeHTOB HanpHMep
3JIeKTpHqeCKHH nHnOJILHhIH MOMeHT Q = J P(r )cfr nOJIyqHT n06aBKH 3a CqeT w (e) (e) (emT) TI
nOJISIpH3yeMocTeH 02 03 bull 4 bull bull bull pHBeneM nJISI npHMepa BKJIan OT (eL
O 3
(15)
TaKOH BKJIaZJ MOJKeT 6hITL nOCTaTOqHO BeJIHK eCJIH Cpena 06JIanaeT CHJILshy
HhIMH HeJIHHeHHhIMH CBOHCTBaMH H BeJIHqHHa o~e) cymecTBeHHa KpOMe TOrO
3TOT BKJIaZJ MOJKeT YCHJIHBaTLCSI H 3a CqeT HHTerpaJIa TIepexonSI B ClaquopepHqeshy
CKYlO CHCTeMY KOOpnHHaT MhI MOJKeM nepenHCaTL ero B BHne
2f Ei E kd3r = f E2r dr f ninkdfl (16)
rne n TaKHe HHTerpaJIhI qaCTO BCTpeqalOTCSI B 3JIeKTponHHaMHKe= E lEI H CTaTHCTHqeCKOH MexaHHKe H HX 3HaqeHHSI HaxonSITCSI COrJIaCHO 06meH laquoPopshy
MYJIhI
f linjn nl = ~ aijak 0 1 - aikaj 0 1+ (17) L
10
5
IlPH HaJIHqHH nOnOJIHHTeJIhHhIX YCJIOBHH THna divE = 0 HJIH divH = 0 B3J1THe TaKHX HHTerpaJIOB YCJIOJKHJleTCJI B QgtH3HqeCKOM 3KcnepHMeHTe HaJIashy
raeMhIe Ha KPHCTaJIJI nOJIJI MorYT HMeTh pa3Hoe npOHCXOJKneHHe B qaCTHoshy
CTH MorYT HMeTh palHYIO qaCTOTY TaK 3JIeKTpOOnTHqecKHH 3QgtQgteKT Iloshy
KeJIhCa 06hIqHO Ha3hIBaeMhIH JIHHeHHhIM eCTh 3QgtQgteKT BTOpOro nOpJlnKa TK
JlBJIJleTCJI pe3YJIhTaTOM onHOBpeMeHHoro B03neHCTBHJI 3JIeKTpHqeCKoro nOJIJI
HH3KOH qaCTOThI H CBeTOBOH BOJIHhI B KOTOPOH CYlUecTBeHHOH 06hIqHO JlBJIJIshy
eTCJI 3JIeKTpHqeCKaJI COCTaBJIJllOlUaJI 3HaMeHHThIH 3QgtQgteKT Keppa 06hIqHO
HalhIBalOT KBanpaTHqHhIM nOCKOJIhKY nOKalaTeJIb npeJIOMJIeHHJI KpHCTaJIJIa
npOnOplIHOHaJIeH KBanpaTY CMelUalOlUero HH3KOqaCToTHoro nOJIJI OnHaKO no
qHCJIY nOJIeH CTOJllUHX B BhIpaJKeHHH nJIJI 3HeprHH B3aHMoneHCTBHJI 3TOT
3QgtQgteKT Boo6me roBOPJl TpeThero nOpJlnKa CJIenyeT TaKJKe pa3JIHqaTh CTashy
THqeCKHH H nHHaMHqeCKHH 3QgtctgteKThI IlPH H3yqeHHH MexaHH3MOB nHHaMHshy
QeCKHX 3ltlgtltlgteKToB YQHThIBalOT CYMMOBhIe H PalHOCTHhIeqaCTOThI H OT HX
HanJIeJKalUero BhI60pa MOJKeT CHJIhHO 3aBHceTh BeJIHqHHa 3QgtQgteKTa
HeTPynHo BhInHcaTh Qgt0PMYJIY nJIJI Q yqHThIBalOlUYIO Bce HeJIHHeHHhIe
JIOKaJIhHhIe BKJIa]lhI HCnOJIh3YJI BhIpaJKeHHe (11) B nomIpH3aUHJlX (11) (12) (13) H (14) HeJIOKaJIhHhIe nOJIJlpH3yeMocTH 6yneM 0603HaqaTh TeMH JKe 6yshy
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(18)
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CTBa neHcTBYlOlUero Ha CHCTeMY 3JIeKTpHqeCKoro nOJIJI a aHTHCHMMeTpHqHaJi
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MYJIhTHnOJIhHhIX palJIOJKeHHH IlOJIHaJI napaMeTpH3alIHJI JII060ro QgtYHnaMeHshy
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npOCTpaHCTBO Boo6lUe roBOPJl HeOrpaHHqeHHO HanpHMep B nOJIOJKHTeJIhshy
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paCTaHHJI l-nOJIhHOCTH B HanpaBJIeHHH OCH Y 6yneM QgtHKCHpOBaTh 3Haqeshy
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11
ero MaTeMaTHqeCKOH nOllOnJIeKe OCHOBhIBaeTClI Ha TeopHH 0606meHHhIx ltPYHKshy
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npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
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lleT OnHChIBaTbClI cpellHeKBallpaTHqHhIM 3apmOBhIM PallHYCOM a nOTOM YJKe
3JIeKTpHqeCKHM KBallPynOJIbHhIM MOMeHTOM C reoMeTpHqeCKoHmiddot JKe TOqKH
3peHHlI pallHYChI onHChIBalOT Mepy OTKJIOHeHHlIKaJKlloro l-noJIbHoro MOMeHTa
OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
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HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
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OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
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2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
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n-Topa C BHeIllHHM nOJIeM
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3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
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12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
3
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(3)
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(4)
6
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uHnOJleH 0606lUaH CKaJKeM TaK qTO HaJIOJKeHHe nOJlH Hi HHuyUHpyeT B
cpeueuonOJlHHTeJlhHYIO MarHHTHYIO uHnOJlbHYIO nOJlHpH3aUHIO Mle = a~)Hi KOTOPaH B CBOIO oqepeub cnoc06Ha B3aHMoueHcTBOBaTb C nOJleM Hie 3uecb
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HTaK MbI BBeJlH HeJlHHeHHOe B3aHMoueHcTBHe yqHThIBaIOlUee HeJlHHeHshy
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npOH3BOJlbHOe KOJlHqeCTBO HHueKCOB H B BhIpaJKeHHH uJlH 3HeprHH OH uOJlJKeH
6bITb cBepHyT C COOTBeTCTBYIOlUHM CHMMeTpHqHhIM TeH30pOM COCTaBJleHshy
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7
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MOBOKpHCTaJIna F e-Ga-oKcHlla Ha MarHHTHoe none yhBeCTHo tJTO y TaKoro
MaTepHaJIa qeppoMarHHTHhIe CBOHCTBa cotJeTaIOTC1l C nhe303neKTpHtJeCKHMH
(BcnOMHHTe aHaJIOrHIO C yIIpyrHMH neTn1lMH) B 3KcnepHMeHTe H3Mep1lnaCh
senHtJHHa HHnyUHpoBaHHoH 3neKTpHtJecKoH nOn1lpH3auHH npH HanO)f(eHHH Ha
KpHCTann MarBHTBoro IIOn1l H
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(em) rne TeH30p aij Ha3bIBaeTC1l TeH30pOM MarHHT03neKTpHtJeCKOH BocnpHHMshy
tJHBOCTH ABTHcHMMeTpHtJHaH tJaCTh TeB30pa a~jm) KaK pal H OIIpenen1leT
TOPOHllHbIH nHnonhHbIH MOMeBT cornacBo COOTBoweBHIO nyanhBOCTH
(em)amiddotmiddot (10)I)
rne eurooijk - aHTHCHMMeTpHtJBbIH ellHBHtJBbIH TeB30p JIeBH-tHBHTa TIon1lpHshy
3aUH1l npHxon1lWaHC1l Ba OllBY 3neMeBTapHYlO 1ltJeHKY KpHCTanna H3ytJaeshy
MOro OKCHlla IIpHBena K 3BatJeBHIO TOPOHllHOro nHIIonhHoro MOMeBTa PaBshy
BOro ITI24JLB A ABanorHtJBbIH 3ltpltpeKT 6bIn IIonytJeB nn1l MarBHT03neKTpHKa
Cr203 IIOMemeBBoro B cHnhBoe MarBHTBoe IIone IIpeBbIwalOwee IIone cnHBshy
qnOII IIepeXOlla
8
4 BKJIanbl IIOJIJIpu3yeMocTeu B BeKTopHble
IIOJIJIpU3auUu cpen U ux MYJIbTUIIOJIbHble
pa3JIOKeHUJI
B HaCTOJlIlleM naparpacpe MhI Pa30BbeM nOJIHhIH CPOPMaJIH3M BKJIanOB BhICWHX
( cMewaHHhIx) nOJIJlpH3yeMocTeH B lIeThIpe 6a3HCHhIe nJIOTHOCTH pacnpeneJIeshy
HHJI nOJIJlpH3aUHH npOH3BOJIbHOH CpenhI P M T(p) H Td) B npHHUHne
npOH3BOnJl naJIee MYJIbTHnOJIbHhIe pa3JIOEeHHJI 3THX nOJIHhIX BhIpaEeHHH nJIJI
3HeprHH HX B3aHMoneHCTBHH C BHeWHHMH nOJIJlpH3aUHJlMH MhI nOJIYlIHM 0606shyIlleHHe npenhInYIllHX CPOPMYJI Ha CJIyqaH HeJIHHeHHhIX cpen B COOTBeTCTBHH
C CPYHnaMeHTaJIbHhIMH B3a~MoneHCTBHJlMH KaEnoH H3 nOJIJlpH3anHH BBeneM
nOJIJlpH3yeMocTH YlIHThIBalOillHe Bce B03MOXltHhIe BapHaHThI HHnYUHpoBaHHJI
naHHOH nOJIJlpH3aUHH BHeWHHMH nOJIJlMH H TOKaMH 0603Hallag KOJIHlIeCTBO
CsepHYThIX HHneKCOB B Ka)KnOM lIJIeHe TOllKaMH BhInHweM CaMhIe 06mHe BhIshy
pa)KeHHJI nJIJI 3JIeKTpHlIeCKoi nOJIJlpH3anHH -C YlIeTOM Bcex BKJIanOB BnJIOTb no
lIeTBepToro nOpJlnKa
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+Y~r) Jm+Yr) JmJm+Yr) JmJmJm++Yer) EJm+Y~mr)HJm++Yemr) EHJm+
+5~r) Je+51r)jeJe+51r) JeJeJe+ +51er) EJe+51mr)HJe++51errmiddotmiddot)EHJe+
rne Jm = ~(i + b) == rotH Je = rotE lIHCJIa npH nOJIJlpH3yeMOCTJlX 060shy3HallaIOT HX TeH30pHhIH paHr H MO)KHO nepeHTH K HenpHBOnHMhIM TeH30PaM
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nJIJI MarHHTHoH nOJIJlpH3aUHH
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+p~e) E+pie)EE+pe) EEE++p~me) EH+pme) EEH+ (12)
+i~r) Jm+i~r) JmJm+ir) JmJmJm+bull+i1er)HJm+i~mr) EJm++imer)HEjm+
+S~r) Je+S1r)JeJe+S~r) JeJe Je + +S~er) EJe+S1 mr) HJe+ +S~emr) EHJe+ j
nml TOP0HllHOi nOJIJlpH3ar(HH MarHHTHoro npOHCXO)KneHHJI
T M =atr)Jm+ar) JmJm+~r) Jm JmJm+ +a1re)JmE+~re) JmEE+
+~e) E+i31e) EE+7f4e)EEE+ +per) EJm+~er) EEJm+ (13)
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+6~r) Je+6r) JeJe+~r) JeJeJe++~e) EJe+~m) HJe+bull+6~me) EHJe+bull j
9
nJISI TOPOHZJHOH nOJISIpH3anHH 3JIeKTpHqeCKoro npOHCXOJKneHHSI
T p =a~) Je+a~)JeJe+a~) JeJeJe++a~e)JeE+a~e) JeEE+
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HeTpynHo YleCTb B 3THX BhlpaJKeHHSIX nJISI naHHOro paccMaTpHBaeMoro
KpHCTaJIJIa nYaJIbHhle CHMMeTPHH KOTOphIe npHBenYT K paBeHCTBaM THna (m) f3(e) -r(T) (T) B
f32 = 2 (12 = U2 H T n ce ~KJIaZJhIB nOJISIpH3anHH conepJKalJlHe
BeKTOphI Jm H Je Ha3hIBaJOT HeJIOKaJILHhIMH H B nOCJIenHeeBpeMSI 3KCneshy
pHMeHTaTOphI HayqHJIHCL HX H3MepSITb B HeKOTOphIX cnyqaJIX OTneJIbHO OT
JIOKaJIbHhIX 3laquoplaquopeKTOB [24J EcTeCTBeHHO ITO B npHBene~lHhIX BhIIlle laquoPOPMYshy
JIax qaCTL BeKTOpOB HanpSIJK~HHOCTeH MOJKHO 3aMeHHTL Ha HX npOH3BOnHhIe
PaHr nOJISIpH3yeMocTeH UpH 3TOM B03p~CTeT H 3TH BKJIanhI B COOTBeTCTBYshy
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nHH cpen C yqeTOM CBOHCTB HX nOJISIpH3yeMoCTH llJISI 3TOro BOCnOJIL3yeMCSI
laquopopMYJIaMH nJISI 3HeprHH B3aHMoneHCTBHSI 3THX nOJISIpH3aUHH C BHeIllHHMH
nOJISIMH (1) H (2) HaqHeM C aHaJIH3a 3JIeKTpHqeCKOH nOJISIpH3aUHH TIPHCYTshy
CTBHe B BhIpaJKeHHH nJISI n9JISIpH3anHH BeKTopa 3JIeKTpHIeCKOH HanpSIJKeHshy
HOCTH MonHlaquopHnHpyeT onpeneJIeHHe MYJIbTHnOJILHhIX MOMeHTOB HanpHMep
3JIeKTpHqeCKHH nHnOJILHhIH MOMeHT Q = J P(r )cfr nOJIyqHT n06aBKH 3a CqeT w (e) (e) (emT) TI
nOJISIpH3yeMocTeH 02 03 bull 4 bull bull bull pHBeneM nJISI npHMepa BKJIan OT (eL
O 3
(15)
TaKOH BKJIaZJ MOJKeT 6hITL nOCTaTOqHO BeJIHK eCJIH Cpena 06JIanaeT CHJILshy
HhIMH HeJIHHeHHhIMH CBOHCTBaMH H BeJIHqHHa o~e) cymecTBeHHa KpOMe TOrO
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CKYlO CHCTeMY KOOpnHHaT MhI MOJKeM nepenHCaTL ero B BHne
2f Ei E kd3r = f E2r dr f ninkdfl (16)
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MYJIhI
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10
5
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KeJIhCa 06hIqHO Ha3hIBaeMhIH JIHHeHHhIM eCTh 3QgtQgteKT BTOpOro nOpJlnKa TK
JlBJIJleTCJI pe3YJIhTaTOM onHOBpeMeHHoro B03neHCTBHJI 3JIeKTpHqeCKoro nOJIJI
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HeTPynHo BhInHcaTh Qgt0PMYJIY nJIJI Q yqHThIBalOlUYIO Bce HeJIHHeHHhIe
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(18)
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CTBa neHcTBYlOlUero Ha CHCTeMY 3JIeKTpHqeCKoro nOJIJI a aHTHCHMMeTpHqHaJi
- aKCHaJIhHhIe CBOHCTBa 3TOro nOJIJI
3aKJIIOQeHHe
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MYJIhTHnOJIhHhIX palJIOJKeHHH IlOJIHaJI napaMeTpH3alIHJI JII060ro QgtYHnaMeHshy
TaJIhHOro B3aHMoneHCTBHJI OKaJKeTCJI Ky6HqeCKOH pellieTKoH 3anOJIHJllOlUeH
npOCTpaHCTBO Boo6lUe roBOPJl HeOrpaHHqeHHO HanpHMep B nOJIOJKHTeJIhshy
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paCTaHHJI l-nOJIhHOCTH B HanpaBJIeHHH OCH Y 6yneM QgtHKCHpOBaTh 3Haqeshy
HHJI BKJIanOB 2n-CTeneHHhIX panHYCOB pacnpeneJIeHHJI KalKnoro MOMeHTa B
nOpJlnKe B03paCTaHHJI CTeneHH n HanOMHHM qTO MYJIhTHnOJIhHhIH aHaJIH3 B
11
ero MaTeMaTHqeCKOH nOllOnJIeKe OCHOBhIBaeTClI Ha TeopHH 0606meHHhIx ltPYHKshy
IlHH H HeCMOTpll Ha 06menpHHlIThIe 3a6JIYJKlleHHlI 0 llOMHHHpOBaHHH MOMeHshy
TOB BeJIHqHHhI pallHYCOB MorYT OKalaTbClI ropa3110 60JIee cymecTBeHHhIMH
npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
Hll8JIbHhIH llBOHHOH CJIOH C He60JIbIllHM 3KcIleHTpHCHTeTOM npeJKlle Bcero 6yshy
lleT OnHChIBaTbClI cpellHeKBallpaTHqHhIM 3apmOBhIM PallHYCOM a nOTOM YJKe
3JIeKTpHqeCKHM KBallPynOJIbHhIM MOMeHTOM C reoMeTpHqeCKoHmiddot JKe TOqKH
3peHHlI pallHYChI onHChIBalOT Mepy OTKJIOHeHHlIKaJKlloro l-noJIbHoro MOMeHTa
OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
eMOCTH Hall KaJKAOH n03HIlHeH MaTpHIlhI ltPHKCHpYlOmeH 3HaqeHHlI MOMeHTOB
HJIH pallHYCOB
OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
KBallpaTHqHO (H Tll COOTBeTCTBeHHO nOpllllKY n-TOpa) ECTecTBeHHo qTO
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3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
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llepJKKy 3TOH TeMaTHKH no corJIaIIIeHHIO 251
12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
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npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
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60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
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[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
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npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
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[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
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npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
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Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
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npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
eMOCTH Hall KaJKAOH n03HIlHeH MaTpHIlhI ltPHKCHpYlOmeH 3HaqeHHlI MOMeHTOB
HJIH pallHYCOB
OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
KBallpaTHqHO (H Tll COOTBeTCTBeHHO nOpllllKY n-TOpa) ECTecTBeHHo qTO
BeJIHqHHa OTKJIHKa 6YlleT CTpeMHTeJIbHO nallaTb C B03paCTaHHeM nopauKa n H3-3a Hal1HqHlI peJIlITHBHCTCKOro 4gtaKTopa (lc)n B 3HeprHH B3aHMOlleHCTBHlI
n-Topa C BHeIllHHM nOJIeM
IIollqepKHeM TO qTO BhIlleJIeHHe BeKTOPHhIX BKJIallOB HaH60JIee yll06HO llJIj[
3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
HaHO~eXHOJIOrHlIX
ABTOPhI 6 JIarona PHhI MHTII ltIgtYHllaMeHTal1bHaK MeTpOJIOrHlI 3a nOllshy
llepJKKy 3TOH TeMaTHKH no corJIaIIIeHHIO 251
12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
5
IlPH HaJIHqHH nOnOJIHHTeJIhHhIX YCJIOBHH THna divE = 0 HJIH divH = 0 B3J1THe TaKHX HHTerpaJIOB YCJIOJKHJleTCJI B QgtH3HqeCKOM 3KcnepHMeHTe HaJIashy
raeMhIe Ha KPHCTaJIJI nOJIJI MorYT HMeTh pa3Hoe npOHCXOJKneHHe B qaCTHoshy
CTH MorYT HMeTh palHYIO qaCTOTY TaK 3JIeKTpOOnTHqecKHH 3QgtQgteKT Iloshy
KeJIhCa 06hIqHO Ha3hIBaeMhIH JIHHeHHhIM eCTh 3QgtQgteKT BTOpOro nOpJlnKa TK
JlBJIJleTCJI pe3YJIhTaTOM onHOBpeMeHHoro B03neHCTBHJI 3JIeKTpHqeCKoro nOJIJI
HH3KOH qaCTOThI H CBeTOBOH BOJIHhI B KOTOPOH CYlUecTBeHHOH 06hIqHO JlBJIJIshy
eTCJI 3JIeKTpHqeCKaJI COCTaBJIJllOlUaJI 3HaMeHHThIH 3QgtQgteKT Keppa 06hIqHO
HalhIBalOT KBanpaTHqHhIM nOCKOJIhKY nOKalaTeJIb npeJIOMJIeHHJI KpHCTaJIJIa
npOnOplIHOHaJIeH KBanpaTY CMelUalOlUero HH3KOqaCToTHoro nOJIJI OnHaKO no
qHCJIY nOJIeH CTOJllUHX B BhIpaJKeHHH nJIJI 3HeprHH B3aHMoneHCTBHJI 3TOT
3QgtQgteKT Boo6me roBOPJl TpeThero nOpJlnKa CJIenyeT TaKJKe pa3JIHqaTh CTashy
THqeCKHH H nHHaMHqeCKHH 3QgtctgteKThI IlPH H3yqeHHH MexaHH3MOB nHHaMHshy
QeCKHX 3ltlgtltlgteKToB YQHThIBalOT CYMMOBhIe H PalHOCTHhIeqaCTOThI H OT HX
HanJIeJKalUero BhI60pa MOJKeT CHJIhHO 3aBHceTh BeJIHqHHa 3QgtQgteKTa
HeTPynHo BhInHcaTh Qgt0PMYJIY nJIJI Q yqHThIBalOlUYIO Bce HeJIHHeHHhIe
JIOKaJIhHhIe BKJIa]lhI HCnOJIh3YJI BhIpaJKeHHe (11) B nomIpH3aUHJlX (11) (12) (13) H (14) HeJIOKaJIhHhIe nOJIJlpH3yeMocTH 6yneM 0603HaqaTh TeMH JKe 6yshy
KBaMH qTO U npeJKne HO 3arJIaBHhIMH Torna HanpHMep BKJIan 3a CqeT
HeJIOKaJIhHOH nOJIJlpH3yeMocTH BToporo paHra B nHnOJIhHhIH MOMeHT 6yneT
BhIrJIJlneTh TaK
(18)
TeH30p non HHTerpaJIOM MOJKHO eCTeCTBeHHO Pal6HTh Ha HenpHBOnHMhIe qashy
CTH Torna CJIen ero paBHhIH divE CBeneTCJI K pacnpeneJIeHHIO nJIOTHOCTI1
BHelliHero 3apJlna CHMMeTpHqHaJi qaCTh QgtHKcHPyeT KBa]lpynOJIhHhIe CBOHshy
CTBa neHcTBYlOlUero Ha CHCTeMY 3JIeKTpHqeCKoro nOJIJI a aHTHCHMMeTpHqHaJi
- aKCHaJIhHhIe CBOHCTBa 3TOro nOJIJI
3aKJIIOQeHHe
He npenCTaBJIJleT 60JIhIlIOro Tpyna palBHTh nOJIHhIH QgtOPMaJIH3M non06HhIx
MYJIhTHnOJIhHhIX palJIOJKeHHH IlOJIHaJI napaMeTpH3alIHJI JII060ro QgtYHnaMeHshy
TaJIhHOro B3aHMoneHCTBHJI OKaJKeTCJI Ky6HqeCKOH pellieTKoH 3anOJIHJllOlUeH
npOCTpaHCTBO Boo6lUe roBOPJl HeOrpaHHqeHHO HanpHMep B nOJIOJKHTeJIhshy
HhIX HanpaBJIeHHJlX oceH X Y H Z Ha OCH X KOTOPYIO 6yneM CqHTaTh
rJIaBHOH 6y~eM pacnOJIaraTh aHaJIH3HpYJl KaKYlO-JIH60 CJIOJKHYIO 3JIeKTposhy
MarHHTHYIO CHCTeMY 3HaqeHHJI onHChIBalOlUHX ee MOMeHTOB B nOpJlnKe B03shy
paCTaHHJI l-nOJIhHOCTH B HanpaBJIeHHH OCH Y 6yneM QgtHKCHpOBaTh 3Haqeshy
HHJI BKJIanOB 2n-CTeneHHhIX panHYCOB pacnpeneJIeHHJI KalKnoro MOMeHTa B
nOpJlnKe B03paCTaHHJI CTeneHH n HanOMHHM qTO MYJIhTHnOJIhHhIH aHaJIH3 B
11
ero MaTeMaTHqeCKOH nOllOnJIeKe OCHOBhIBaeTClI Ha TeopHH 0606meHHhIx ltPYHKshy
IlHH H HeCMOTpll Ha 06menpHHlIThIe 3a6JIYJKlleHHlI 0 llOMHHHpOBaHHH MOMeHshy
TOB BeJIHqHHhI pallHYCOB MorYT OKalaTbClI ropa3110 60JIee cymecTBeHHhIMH
npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
Hll8JIbHhIH llBOHHOH CJIOH C He60JIbIllHM 3KcIleHTpHCHTeTOM npeJKlle Bcero 6yshy
lleT OnHChIBaTbClI cpellHeKBallpaTHqHhIM 3apmOBhIM PallHYCOM a nOTOM YJKe
3JIeKTpHqeCKHM KBallPynOJIbHhIM MOMeHTOM C reoMeTpHqeCKoHmiddot JKe TOqKH
3peHHlI pallHYChI onHChIBalOT Mepy OTKJIOHeHHlIKaJKlloro l-noJIbHoro MOMeHTa
OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
eMOCTH Hall KaJKAOH n03HIlHeH MaTpHIlhI ltPHKCHpYlOmeH 3HaqeHHlI MOMeHTOB
HJIH pallHYCOB
OTMeTHM qTO B 60JIbIllHHCTBe 3KcnepHMeHTOB npH H3yqeHHH CBOHCTB CHshy
CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
KBallpaTHqHO (H Tll COOTBeTCTBeHHO nOpllllKY n-TOpa) ECTecTBeHHo qTO
BeJIHqHHa OTKJIHKa 6YlleT CTpeMHTeJIbHO nallaTb C B03paCTaHHeM nopauKa n H3-3a Hal1HqHlI peJIlITHBHCTCKOro 4gtaKTopa (lc)n B 3HeprHH B3aHMOlleHCTBHlI
n-Topa C BHeIllHHM nOJIeM
IIollqepKHeM TO qTO BhIlleJIeHHe BeKTOPHhIX BKJIallOB HaH60JIee yll06HO llJIj[
3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
HaHO~eXHOJIOrHlIX
ABTOPhI 6 JIarona PHhI MHTII ltIgtYHllaMeHTal1bHaK MeTpOJIOrHlI 3a nOllshy
llepJKKy 3TOH TeMaTHKH no corJIaIIIeHHIO 251
12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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IlHH H HeCMOTpll Ha 06menpHHlIThIe 3a6JIYJKlleHHlI 0 llOMHHHpOBaHHH MOMeHshy
TOB BeJIHqHHhI pallHYCOB MorYT OKalaTbClI ropa3110 60JIee cymecTBeHHhIMH
npH onHcaHHH KaKoH-JIH60 KOHKpeTHOH CHCTeMhI TaK HanpHMep 3JIl1HnCoshy
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OT 3allaHHSI ero B TOqKe C aKTYal1bHO HYJIeBhIMH palMepaMH [25] lial1ee
H MOMeHThI H pallHYCbI KaK MhI BHllHM H3 MaTepHal1a3TOH pa60ThI MorYT
HMeTb nonpaBKH 3a CqeT HeJIHHeHHOcTH B3aHMOlleHCTBHlI H3yqaeMOH CHCTeMhI
C BHemHHM nOJleM 8TH nonpaBKH MOJKHO paCnOJIOJKHTb BepTHKal1bHO napal1shy
JIeJIbHO OCH Z B nopmKe B03paCTaHHSI qHCJIa HHlleKCOB Y TeH30POB nOJISIpH3Yshy
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CTeMhI 06h1qHO HCnOJIb3YIOTClI JIH60 CTaTHqeCKHe JIH60 H3MeHj[IOmHeClI co
BpeMeHeM rapMOHHqeCKHe nomr MhI06pamaeM BHHMaHHe Ha TO qTO llJIlI
BhIsJBJIeHHj[ BHxpeBhIX CTPYKTYP nepBOro nopmKa - TOP0HllHhIX llHnOJIeH shy
qHCTYIO ltPYHKIlHIO OTKJIHKa (6e3 npHMeCH llPyrHx MYJIbTHnOJIbHhIX B3ashy
HMOlleHCTBHH) MhI 6YlleM nOJIyqaTb HaKJIallhIBaK Ha H3yqaeMYIO CHCTeMY
nOJIlI pacTymHe JIHHeHHO BO BpeMeHH liJIlI BhIlIBJIeHHlI BKJIallOB nOJIOHllal1bshy
HhIX llHnOJIeH HalOBeM HX BHxpeBhIMH CTpyKTypaMH BTOpOro nopauKa HJIH
2-TopaMH [26] Ha110 HaKJIallhIBaTb Ha CHCTeMY nOJIlI pacTymHe BO BpeMeHH
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BeJIHqHHa OTKJIHKa 6YlleT CTpeMHTeJIbHO nallaTb C B03paCTaHHeM nopauKa n H3-3a Hal1HqHlI peJIlITHBHCTCKOro 4gtaKTopa (lc)n B 3HeprHH B3aHMOlleHCTBHlI
n-Topa C BHeIllHHM nOJIeM
IIollqepKHeM TO qTO BhIlleJIeHHe BeKTOPHhIX BKJIallOB HaH60JIee yll06HO llJIj[
3KcnepHMeHTaTopOB npH aHal1H3e KPHCTal1JIHqeCKHX 06beKToB Mal1hIX palMeshy
pOB Eme 60JIee HarJIj[llHO npHMeHeHHe 3TOH MeTOllHKH npH HCCJIellOBaHHH
3JIeKTpOMarHHTHhIX OTKJIHKOB HH3KOpalMepHhlX CTPYKTYP npHMeHlIeMhIX B
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ABTOPhI 6 JIarona PHhI MHTII ltIgtYHllaMeHTal1bHaK MeTpOJIOrHlI 3a nOllshy
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12
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
[1] BeHeBueB IOH rarYJIHH BB nI06HMOB BH CezHemoMazHemunu M HaYKa 1982 224 c
[2] lly60BHK BM lJelliKoB AAMyJlbmUnOJlbHble pa3J10HCeHUJI 6 nJlaCCU14eshy
cnOfl U 6 n6aum060fl meopuu nOJlJI U U34Y14eUue ltlgtH3 3JIeM llaCTHU H aT anpa (3lJASI) 1974 5 B3 c791-837
[3] lly60BHK BM TOCYHJlH nA TOpououble MOMeHmbl 6 ffiu3une 3J1enmpOshy
MazuumHblX U CJla6bX 63auMooeflcm6uiJ ltlgtH3 3JIeM llaCTHU H aT JIltpa (3lJA5I) 1983 14 B5 c1193-1228
[4] lly60BHK BM TOpouoHble MOMeHmbl nan o6UCeffiu3UleCnUe napaMempb
B KH TeopeTHKo-rpYlIlIoBbIe MeTOltbI B qH3HKe TpYltbI TpeTlero ceshyMHHapa IOpMaJIa 22-24 Mag 1985 r T2 M HaYKa 1986 c356-362
[5] BJIaTT llJIlt BaicKolIq B TeopemulecnaJI JIOepHaJI ffiu3una M 1960
[6] )l(eJIYlteB HC 06 OOHOiJ ffiOpMaJlbHoiJ 603MOHCHOCmU c030aHUJI cnOHshy
maHHoiJ nOJlJIpU3aijUU 3a C14em ynopJIOOleHHOZO 06UHCeHUJI 3J1enmpOH06
H3B AH cep ltlgtH3HlleCKU 1969 XXXIII N2 c204-206
[7] Schmid H Int J Magnetism 4 1973 p337
[8] Ascher E in Magnetoelectric interaction phenomena in crystals Eds A Freeman and H Schmid New York 1975 p69
[9J lly60BHK BM KPOTOB CC TyrYllieB BB Manpocnonu14ecnaJI CUMMeshy
mpUJI U MunpOCnonU14ecnaJI npupooa napaMemp06 nopJIona npu monoshy
60M U cnUH-mon060M ynopJIOOleHUU 6 npucmaJlJlax KPHCTaJIJIOrpaqHg 1987 32 N3 c540-549
[10] llOIIOB 10lt1gt 3Be3ltHH AK KanoMueBa AM H ltp MazHumoMenshy
mpu14ecnuiJ 3ffiffienm U mopOUOaJlbHOe ynopJIOO14eHUe 6 npucmaJlJlax
Ga2-xFex03 )l(3TltIgt 1998 114 BbIlI1(7) c263-272
[11] llOIIOB 10lt1gt KaltOMueBa AM 3Be3ltHH AK BenoB 1lB Boposhy6beB rll HHoyijUp06aHue MaZHumHbM nOJleM mopOUOaJlbHOZO MOMeHma
6 MaZHumOMenmpune Cr203 TIHcLMa B )l(3TltIgt 1999 69 BbIlI4 c302shy306
[12] lly60BHK BM TOCYHJlR nA TyrYllieB BB AnCUaJlbHble mopouoHble
MOMeHmbl 6 3J1enmpOOUHaMune U ffiU3une m6epoozo meJla )l(3TltIgt 1986 63 T 90 BbIlI 2 c590-605
13
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
[13] MapueHlOK MA MapueHlOK HM 0 npoucxoJICae1tuu apOMaZ1temU3Ma IIHcbMa B )K3TltIgt 1991 53 B 5 c229-232
[14] ToncToH HA CnapTaKoB AA H06blU 6Ua MaZ1temU3Ma apOMaZ1teshymU3M IIHcbMa B )K3TltIgt 1990 52 B3 c796-798
[15] AqaHacbeB r H Cmapble U 1t06ble npo6AeMbl6 meopuu 3(fi(fie~ma Aap01t06a- EOMa ltlgtH3HKa 3l1A5I 1990 TOM 21 Bhln 1 c172-240
[16] He XG and McKellar BHJ Topological Phase due to Electric Dishypole Moment and Magnetic Monopole Interaction Phys Rev 1993 A47 p3424-3429 AqaHaCbeB rH CPu3UleC~Ue npUAOJICeHUJl COAe1tOshyUa06 ltlgtH3HKa 3l1A5I 1993 24 Bhln2 c512-593
[17] nYOOBHK BM MapueHlOK MA MapueHlOK HM Topoua1taJl nOAJlpU-3aqUJl azpeZUp06aHHbIX MaZ1tUm1tbIX cycneH3UU U ~OMn03um06 ltlgtH3HKa 3neM qacTHU HaT lInpa (3l1A5I) 1993 24 BhlnA cl056-1132
[18] ltlgtHnHnnoB EH KOP3YHHH JIr Hnp HeAU1teUHaJl aUHaMU~a 6uxpe6blX aOMeHHbIX cme1ton 6 MaZHUmHbIX nAe1tnaX 6 tltuponOU o6Aacmu mOAtLtU1t ltlgtH3MeTH MeTaTIJ1oBen 1999 87 No6 c17-23
[19] Usov NA Magnetization curling in soft type ferromagnetic particles with large aspect ratios JMagand MagMat 1999 203 p277-279
[20] Dubovik VM and Kuznetsov VE The Toroid Dipole Moment of the Neutrino IntJModPhysA 1998 13 N30 p5257-5277
[21] Bukina KN Dubovik VM Kuznetsov VE Transition radiation of the neutrino toroid dipole moment PhysLettB 1998 435 N 1-2 p134-138
[22] KOplOKHH BM 3gJgJeICm f(a3uMupa zpa6umaqUJl U peAunm06ble HeUshympU1to JhBecTHlI BY30B ltlgtH3HKa 1996 N 10 c119-120
[23] Sandratskii LM and Kiibler J Role of Orbital Polarization in Weak Ferromagnetism PhysRevLett 1996 76 pA963-4966
[24] npaqeB Bll llepMHHoB CB PaYTHaH cr CaqOHOB Bll IIHcbMa B )K3TltIgt TOM 68 Bhln8 c618-622
[25] Dubovik VM and Tugushev VV Toroid Moments in Electrodynamics and Solid-state Physics Phys Reports 1990 187 NA p145-202
[26] Afanasiev GN Dubovik VM Some Remarkable Charge-Current Conshyfigurations PhysElem Partand AtNucL(Particles amp Nuclei) 1998 29 NA p891-945
PyKonHcb nocTYIIHna B H3laTenbCKHH OTlen bull 18 OKT1I6pll 1999 rola
14
-
P17-99-276EYKI1Ha EH Jly60Bl1K BM BKJIaubI nOnap3yeMocTeH B qeTblpe 6a3I1CHble nOn5Ip3aull 3neKTpOMarHl1THbIX cpen
nepeql1CneHbI Bce BKJIaubI B qeTblpe 6a3l1CHble IUlOTHOCTI1 pacnpeneneHl1a noshynap3aull np0l13BOnbHoH 3neKTpOMafHl1THOH cpenbl 3a CqeT cMernaHHbIX nonapshy3yeMocTeH BnnoTb no TeH30poB qeTBepToro nopanKa PaCCMoTpeHbI KOHKpeTHbIe cp3IqeCKle npHMepbI
Pa60Ta BbmonHeHa B Jla60paTOpl1H TeOpeTI1QeCKOH cpl13l1Kl1 I1MHHEoroshym060Ba Ol15IH
npenpHHT 060enHHeHHoro HHcTHTyra $IllepHblX HccnenOBaHHH lIy6Ha 1999
Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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Bukina EN Dubovik VM P17-99-276 Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media
All contributions to four basis polarizations of an arbitrary electromagnetic medium at the expense of mixed polarizabilities up to fourth rank tensors are preshysented Some concrete examples are considered
The investigation has been performed at the Bogoliubov Laboratory of Theoshyretical Physics lINR
Preprint of the Joint Institute for Nuclear Research Dubna 1999
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