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- -

I I n trod uction

In t he oourse of ltudylog the elcctroma~etic gtropertics oj lystcrns wi til distributed eharres and currents there lTiro the lTobshylem of the cho1ce of macroscopic characteristics that prvvidc an adequnte descriptioo for the in t eractio s of thoRO rysterns with

external fields and current s An appropria t e mathematical tool f o) solvinpound such a problem is the formalism of multipol e expansion of

the mIcroscopic charee density FCit) and ourrent density Clt) In the framework of clas1iCal el e ctrodynamics this form3lism i s well eApounded e g in rej 1 bull There it has also been shown that

together with two known families of charged and maenetic multlpole moments the third family that of toroid mul tipol e moments 3lso

appears A formal reason for the latter is the spl ittinr of a vortishycity field (in the given case of a t ransversa oomponent of the

Cllrrent densi t y h(it) i e of the f unction for which oiiA)J ~ O ) in the two compownts

T(~t) wt~yr(~t~ + WUlt[ii(it)] ( 1)

there 1jr Ci t ) and 1 (i t) are respec tively pseudo s cal ar and scalar funct ions We shall cal l thi s sum by t h e Neumann-Debye represhysentation 234 bull The expansion of Ijr(it) ove r radial spherical

harmonics generates a fami l y of ma~eti c multipo~ moment s among

l~ M + r K M eto bull All anal0SOUII expanaion of X generatea a 1alDJ17

wh1ch generat ing is the maeneti c dipole moment M s1ncewith it one may cons t ruc t all hiGher moment s of that family Mshy

~=

of t oroid multipole moment s 11 21 bull A toroid family iG ~enerated by

the toroid dipole moment T BeSides t he expansion of the aharee densi t y y ( i t) generates a fami l y of char~e mul t lpole moments among which a v ect or characteristio is the charge dipo l e moment

~ -P The above v ector quantiti es p M T differ in symmet ry properties ( the behavi our under the operation of coordinate

I and time R invers1on~)from each other and are transshyfo rmed aocording t o table 1

It is seen however that for constructing the complete vector basis of multipole r epresentation of the group of space-t ime inver sions

Table 1 Rreg T the se t of vectors P M ) T is not suffi cient and in principle

p

1 R

+

it is necessary to introduce one more -shy 5axial vector ~ (seee r ) the

tt + symmet ry propert ies of whi ch are preshysen t ed in the lowest l ine of Tablel

r In the classical electrodynamics by ~yell and Lorentz in whichG + +

1(zt)= z ei t SCi - fiCf)) t here i s no place for n vector with such proper ties Neverthele s s as i s shOlm i n this paper there edst s a nwnber of phys ical appli cat ions of the mathema t i cal method of mul tipole expansions where the appearance of the generatinG dipole moment c and the family

of i ts multipoles turn out to be inevitable At a mu~ipole analysi s of a magnetic charee sys tem (se ct2)

the vector C is the toro id moment of t h e marnct ic current It is analoGous ( dual ) to the t oroid moment T of th e el ec tric current However owing to pseu~oscalar pr operties of the de ns1ty J (i t) of magne tic charges G is an axial vector while T i s a polar vecto r

In th e electrodynamics of condensed matt er t he quantity G may be introduced for describine systems with distri but ed charge dipole moments (sect) In this case G i s an anal og of t he i nduced toroid moment lI d in media with distributed marneti o dipole moments 1678 )f) t

Proceeding from the symmetry properties and the analogy with toroid multipoles in the Maxwell-Lorentz el ec tro dynami cs we shall call the vector G the axial t or oid moment and the v ector T the polar toroid mom ent

The introduction of spin necessitates the classif i cation of the toroid moments T and Gf with re spect to the inversion t ransshyformation R(( in the spin space In case it is ne cessary we shall supply the even (sinelet) vectors with index (~) and odd (triplet) ones with index (t) bull

)Note the term induced - i s actual charaoteristic of dipoles arising due to the polarizabil i ty properties (of media) and therefore in the oorre sponding Hami ltonian t h ere appears the addit ional degree of an binducing field But a oircular chain of dipole s is considered in t his context as a primary ob ject and the term middot induced- as appl ied to it eg in the book B is confused

2

In the description of phase transitionR in crystals vectors T

and c (or thei r higher multipoles) can natural ly be t aken to serve as ord er paramete r s t ransforrnine a ccortlil1 to certain i rreduci bshyle representa ti ons o~ the magnetic spac e rroup of a hir~ly symmetric phase Let us di scus s what is the peculiari ty of the t oroid types of orde r ing and for whi ch systems it makes sense to introduce t h ese concepts

From the poi nt of v iew of a fo rmol symme t ry a l ot of the known order parameters are transform ed anal orously to T and G So for ins t ance for descri bine t he spin maenet 3 mul t i pole onlLr parashymeters (spio densities) have been introduced some of whi ch ure annlorous to lit i n transformat ion properties poundhe simpleRt i s the case of 11 two-sublatt i ce antiferromaenet

Analogous to c the ve ctor W transform s whi ch i s dutU to the antisymmetrto ~art of the disto rti on tensor 11 as well as the tlireotor n charncterizing the orientational ordcrine in

12 - Uli quid cry stal 5 bull I n t he theory of s pin nemnt i cs - --GOmL

orcierJaramet ers are i nt r oduced a nalo[ous in symmet ry to G-) t amI T ~ t bull However t ho llescription of these syst ems in terms of to-roid dist r ibu t i ons i s not physically t ransparent thOUGht it may happen to be cons t ructi ve for descrip tion of int er a cti on with an elec t romagn etic fi eld

Separation of t oroid moments in a spe cial gro up of order parashymeters is much more just ifi ed in casu of systems wi th itinerant elecshytrons In particular the polar t oroid moment describes t he orbital antlcrromagnet ic o r~ering rlth fo rmat ion of t he cUrrent-densi t y wave 14 bull The v eclor Tt can be used f or de scribing a number of i t i shynerant antiferro-macne ts wi t h spin-density wave s

The axial toroid moment (j~ dcsc ribes a sp eci f ic char~e orde~l~ like the itinerant antiferroeleetricity ( se e below) Th e vector ~t characterizes the orientat ional ordering i n i tinerant magnets and is connected wi t h the spi n-current density wave (SCDW) 15 bull

The s ch eme of mul t ipo l e expansions all ows a rather cons t ructuve desorlption for the general m~cro scopic properties of a number of systems with intricate distrlbuti ons of charges and current s ( tn particular the response to an el ectromagneti c fie ld) however an i mportant probl em is also t he study of ooncrete quant um-meohanical models real izing different t ypes of multipole str uc t ures An i l lumishynating example is the theory of polar torold orderi ng in cryst als construct ed in r efs 14

In this paper besides a general phenomenological considerashytion of the axial toroid ordering ( sect4) micro sc opic model

of the phase transition with formation of such an ordering is proposhy

oPtl (soc t 5) To be more precise Ie oonsider tYlO band aemimetal s

or s~m1oont1uctors wi th the charge-densi ty wave (CDW) and allowed 1nshyterband trru1sitlon in the orbital moment Earlier 15 another

mJ croscopio model with fortnntion of the (SeDW) W[JS oonsidered thereshy

foreve do not concentratC on its lpeoific feoluTls (a generAl outline tl given in seot4) _

These models clearly illustrate the physical meanine of C (Uld lllow (l bettc1 UDlh-rstnnding of thc tLaturc of formation of G in crystals

b LultipolE_~siOLQL5LsectIstpoundL~~tic~~

In ref 11 it is notiood that in a dual-invariant scheme of elecshy

trodynlUTlic there Ilppears total nyrnmetry between forms of m~tjPOle ~ollrces ald types of fields (speoifically of radiati~on ) 1 bull llote

that n the elec t fOI 11ltnel1 r the ory invalk1lt under R and I inshy

vercions the current of eleotric cbarl1es should be a veotor while lnt of mllncttc charneo a pru~ovector ipound we adopt a conventional

ric ture of space -time properties of fields f and H C ace e ~16 ) Obvious (see ref 17 ) thp tbrmulation of an e1cotroshy

ma~eti c theory with polntlilce magnetic oharges turns our to be difficult (howevlT seo 10 ) 51otmiddot O in particular in file atio

Ut~ H == ~ b(i) oitler the oharGe J t s not 0 number or it is a llIlIlber and parity is nol conseltedln Llle theory In a macroshy

copic formulation of the theory thl~ difficulty does not arise

since in the equation dmiddotd)- H=-Ys(ii) the fun~lon-1 (1 t) may be always considerod t o be odd anO in tOt t =-H - 3j(gt1t) the funotion 1i (ltZ) i ) lo bc an axial vector IIi tll tltls differenshyce of magnetic from electric charres the problem we are interested

in of the mult i pole expansion of denSities9 (it) antl Ic~t)9

is readily solvell One nhould gtg in the oorrespond1ne formulae

of ref 11 to mwe the chanee Je ~y~ and ~ -4gt r~ Let us write the oorresponding formulae denotinc the quanshy

tities having the same form in the eleotric and m~~netic worlds by

t he doubl e symbol ~e In particular tho expansion of J (f)ais wri3ten in the forn Ie 1 Dual -invariant should be not only field equations but al ao

t he equa t i ons of motion of ohareod particles The latter 1s achieved

by int r oducing int o theory of part io1es haVing both electric and magnetic charges 17 bull When e~ = oonst for all types of particles

a scheme l ike that may by a dual transformation be reduoed to a

oonvent i onal one-charge eleotrodynamios

4

2~e(1 t ) = ( 1ft)- 3J--~ (-iK)e yen4W (2ei) F (i) I-yene (k 2t) Id k o T lHHJ ~ r 1( ~eM I )

( 2)

relgtll( 1e (Kl ) Ye (~ ) t -gt

)1

WhBlmiddote chnrge mul ttpole distributions

A

Qe(e)1 J 11 aro Given

(J )

by

QUe (l(2t) = (ltd) 1 bull S0 C~J) r)f (i)d e i C-ik)ljltIT(Jt 1) J~e f k

Hence there follow definitions of tho chnrge multipole rloments

Qtie (0 ) -=-J ~ii I JL e ~)t (V) 0 (i 1) dJl (1)e 1 U-I1 ern j~e J

and their 2n-power radii

J (~ )i 2(gt (t) -= (4ff I r Lt + )l ( ~ JR (i)t) d~

fVo1 Ltd J feW Ve

that oompl ete the mul tipol e p~nmet rizntion of the initial flUlction

YSle(i1) The multipole exp~sion of the current density of marfletic

(electl-1c) cbar~es with a sepnrltted toroid part irl as follows 1

411(2e+1)(ei)p J(T (1t) -= (211( JL (-1 k(1

( Hd)I~e im

~ kTlltJ)(~) H Ve ((2 t) +1 e1( to middot ) (6 )

t ~ L+) (=i) l- Ii ~ Itu t) + kZ T SIe ((2 t)l + ) v VIe e) JjI

1[f-1 T(-) (-+) 1~3e ((2-) 7 (2poundke1 -r li o J

~ I( t

5

I

of the phase transition with formation of such an ordering is proposhy

oPtl (soc t 5) To be more precise Ie oonsider tYlO band aemimetal s

or s~m1oont1uctors wi th the charge-densi ty wave (CDW) and allowed 1nshyterband trru1sitlon in the orbital moment Earlier 15 another

mJ croscopio model with fortnntion of the (SeDW) W[JS oonsidered thereshy

foreve do not concentratC on its lpeoific feoluTls (a generAl outline tl given in seot4) _

These models clearly illustrate the physical meanine of C (Uld lllow (l bettc1 UDlh-rstnnding of thc tLaturc of formation of G in crystals

b LultipolE_~siOLQL5LsectIstpoundL~~tic~~

In ref 11 it is notiood that in a dual-invariant scheme of elecshy

trodynlUTlic there Ilppears total nyrnmetry between forms of m~tjPOle ~ollrces ald types of fields (speoifically of radiati~on ) 1 bull llote

that n the elec t fOI 11ltnel1 r the ory invalk1lt under R and I inshy

vercions the current of eleotric cbarl1es should be a veotor while lnt of mllncttc charneo a pru~ovector ipound we adopt a conventional

ric ture of space -time properties of fields f and H C ace e ~16 ) Obvious (see ref 17 ) thp tbrmulation of an e1cotroshy

ma~eti c theory with polntlilce magnetic oharges turns our to be difficult (howevlT seo 10 ) 51otmiddot O in particular in file atio

Ut~ H == ~ b(i) oitler the oharGe J t s not 0 number or it is a llIlIlber and parity is nol conseltedln Llle theory In a macroshy

copic formulation of the theory thl~ difficulty does not arise

since in the equation dmiddotd)- H=-Ys(ii) the fun~lon-1 (1 t) may be always considerod t o be odd anO in tOt t =-H - 3j(gt1t) the funotion 1i (ltZ) i ) lo bc an axial vector IIi tll tltls differenshyce of magnetic from electric charres the problem we are interested

in of the mult i pole expansion of denSities9 (it) antl Ic~t)9

is readily solvell One nhould gtg in the oorrespond1ne formulae

of ref 11 to mwe the chanee Je ~y~ and ~ -4gt r~ Let us write the oorresponding formulae denotinc the quanshy

tities having the same form in the eleotric and m~~netic worlds by

t he doubl e symbol ~e In particular tho expansion of J (f)ais wri3ten in the forn Ie 1 Dual -invariant should be not only field equations but al ao

t he equa t i ons of motion of ohareod particles The latter 1s achieved

by int r oducing int o theory of part io1es haVing both electric and magnetic charges 17 bull When e~ = oonst for all types of particles

a scheme l ike that may by a dual transformation be reduoed to a

oonvent i onal one-charge eleotrodynamios

4

2~e(1 t ) = ( 1ft)- 3J--~ (-iK)e yen4W (2ei) F (i) I-yene (k 2t) Id k o T lHHJ ~ r 1( ~eM I )

( 2)

relgtll( 1e (Kl ) Ye (~ ) t -gt

)1

WhBlmiddote chnrge mul ttpole distributions

A

Qe(e)1 J 11 aro Given

(J )

by

QUe (l(2t) = (ltd) 1 bull S0 C~J) r)f (i)d e i C-ik)ljltIT(Jt 1) J~e f k

Hence there follow definitions of tho chnrge multipole rloments

Qtie (0 ) -=-J ~ii I JL e ~)t (V) 0 (i 1) dJl (1)e 1 U-I1 ern j~e J

and their 2n-power radii

J (~ )i 2(gt (t) -= (4ff I r Lt + )l ( ~ JR (i)t) d~

fVo1 Ltd J feW Ve

that oompl ete the mul tipol e p~nmet rizntion of the initial flUlction

YSle(i1) The multipole exp~sion of the current density of marfletic

(electl-1c) cbar~es with a sepnrltted toroid part irl as follows 1

411(2e+1)(ei)p J(T (1t) -= (211( JL (-1 k(1

( Hd)I~e im

~ kTlltJ)(~) H Ve ((2 t) +1 e1( to middot ) (6 )

t ~ L+) (=i) l- Ii ~ Itu t) + kZ T SIe ((2 t)l + ) v VIe e) JjI

1[f-1 T(-) (-+) 1~3e ((2-) 7 (2poundke1 -r li o J

~ I( t

5

I

v h erc 2)

f t-) ct=-lgt) ~ (7)T()) ( -1 ) ( - 1 ) 7 l l ) A 0 1

21( el( )

t h ~ IIIni rH~jc lIu l t lpol e cliltrl but lonG

~e () --l ( ie ~ i) 1 ( 8 ) M kl t =- - - -- - - x

e (-iK V4Tt (Ud) fe +1)i

~ r t)l (0)( -i ) ( 7+ ) d3l e 1( 6 e

th e 11(ln r Lir tll i l)l rnoln(lll s ~

41i 1 e y C~) T(-t)d~ (9)Ve (0 +) = )Me ) ~~ + 1 ) (~ +i) 2 U 1e )

the t o ro i a wul t l pole dis tri bu ti ons

(2e-t-1) l~e( L t _ ) 7- (1 0 )

VITe - (-ikl+ 1 ilti1l(2 e~1 ) (e-t 1 l

~ (+) ezl E-l L l( l- 1 k ) t +1CO

Ll1 I Y~1 CZ f~(1)h( Q~ -1)1 Ij ole

~ 1 ~ 1

the to r oid Id u l tipole moment s

) -~ ( ) 2 The bas is of expansion (6) i s intr oduc e d as follows ~ -- =

~~

_ ~ Vr ~O) _ 1- rofif F 7 T-+)= -i whot7 F L --I( e L ) euroIlt - Jere1) L e( 1 ) f1( j( erelV7 e ( ]

and s pher i ca l ve c t o r s a re i ntroduced accmiddotoro i n~ to 1191

~ YHfWl C~ ) 1 ~ ltelm il e ~gt Ye r~ c~raquo

j ~)lf er(II) J3 - b b (1Ti)3 re(- ()-jn -i Qt - H 0 ) X 1lt2 )

( lL ~ k

L l-~)I en] yen L- tLI) (7)1 =8(Ttl ~ ~ (2- f ) e J ml l( ~ e 1( 1 t l( ) j i J

6

~e 0+ _ -JtTi f I f41 -yi + 02 Jf fd I v ~T d~(l )~ ( J)-2 ( 2 e +1l Pf1m ~ R +3 IN dm ) ale

t he lD nGi t ud i nal charpe mul tipole di t ri but 1 011 5 ( I )

~e (1c 2 t) -~le-) f~H~ -jgt (= - ) c1 J(Qem (2 r ~1 )11 1m Ilt Vi ) )I

t he b nrit ud inn l charge rJu ll i]lo l c moments

ell ( 1J ) 31e ~ j f - ( v -gt1 ~ Qe rn (0) ) = J Itli e ee 1 vgtI bull 1(J ) (

() (gt - Ilhc ll the r o ndi U ongt of ( 1)Pc tral expaMlon of JfJle 7t) ~lnd )81 (Z ) ( thc s llilp l e st CV1C i a ha r moni c d epend enoo) v c f u l f 1 l 1ed Lhe l at t e r

d efJ ni tiongt r ed Olce t o fo r hllll a e (J) ( d ) and wy C)lC cxpc Si OllS

( p ) (13) are f unctional -dele nd ent on Oe (k2 t ) CcgtCl ll 1 e of Lh e

~middot- c urr etlt co ngte rvlt i on ~~ e que t Jon of 1011 t h e nlo TlCit uultlol r10 shy

met s QR ( 0 -1- ) - e t in to the cpDJ1 s i nr 01 t he cu rr~n t t gtngt-

ver iDl p l) l ( cli v T=C ) ( e e r Q -~ul 3 (c ) ) 1gt rll lCn 11 0re c oml l t atec1 an d is conid ercrl in r ef III ( and in r efele nce s t he rein) ---gt

fus~d on Ue pro pc t t cs of v e l t o s ph f~r j cal fun c t i on s cre~ 1lt under I - invcndon f it i- n o t lliff l cGll t o e t )bl l h t llt

dis t ri buti ons M~e ( 1l1 - ) n o c reate fie l ds o f th e Ef - l ype

( i n parti cul a r t hey yi el d t e -ra diationeleotrl0 mul t1pole s )

wheroa s char g e (Coul omb) Qt(et) and t oroid T ( 1pound2 f ) d1 s tr1bushy

ti ons p r oduce fiel ds of t h e M e -type (spoc11icall y t9 ~ (o r ) and -r(I(Jt) arc r elpons i ble f or the emission of magnet10 multip oles)

) tu l l if0lpound~cZpoundLliOS of tlipolpound1Q~ a

Sinc e frec m[n tic charres (nn1 l heir current) lre no t I)J

p t lleLrct 1 ( t h e h1to ry of t he prObl em mOlY b e f ound e ~ bullbull ln 120 )

it mirh l e rn tl)t the rt)5u1 1~ of mult i pole expansion i n the dual

el e c tro nynull l u presented in se clion nTC ll s o of 11 pur e schol ast i c

Mtule e st3011 ho ~how thl l t hc fJ ual Rymmetry Mp - QJ ver M t ~ Q~ has J ratheT profo und meanln~ and t h a t the formashy

lism expondrll i n sec t 2 is u seful for des crib l n r some dJpol e

struclud S 1 Lhe e l octrodynamic5 of conden sed mat te r

We shall describe media asl family of elemelltary magne tic

dipoles fL ( [ i J ) N) The magnetio c urrent density

- llgt t) is given by the knoYIll formula 211

7

1~ (i t) = L 1 x Ii cce-zit)) -gt l~T R (i 1) J l 1 f t L I (14)

where the synlbo l -- denotes the t r ansition t o a continuous d1str l butior of thc aagneti c dtpol e mO I11en t d ensity (magnetizati on)

M-1- bull l o~ethcr 11 tIL ( 14 ) i t is conv enient t o introduce n f ormal

f) Uantit y J-Ctt) lt pfieudos colar dist r i but ion density of magnetic charres t- __

J F(i t) - div-M (-r f ) (1 5)

Iote that tJc lIsull llwwel l-lorentz (gtquati olls do no t con t ain t hn qutr1tj ty Mil (=i t ) (the lo~g1 t udinal comp onent of magnetizashy

tlan)

If Yle rep1lcc th c density 2e by9)7 1n fOIlllUla e (J)-( 5) then thel c nri J cs Ilulo s t ltl complete ana1o[y of lDultipola expansions

of t hese Quant LII e s ( th e only dlffe1cl1ce 1s t hat one should se t

i oa == SPF (iZt )d 3r = 0 Clearl y tJle corr esponding for shyl1ul ae will jive oharge mul tll ole moments of the system of llazn~ti c

lipol e(l I e no h that a c omplct e l y ynunotric schelJe of mul t i polc

c)-pan~i (lLl S docs not arise sn opound d ~1gt- 3j (Zt )fJ1(~t)and Moo~ O SO the edly ourren t 3j1 l acs no t contribute to Gem

but it s c ontri but ion t o Tt is nonvminhlne It iR j ust this contribu tion duo t o I7hich t oroidal di stribution s started to be

int rodu c ed into el ectromagneti~Jn ( sec 6 and ref crence~ th ere1n)aUnder ~1t nnro e of lnduoed al e ct)i o fJoments there Was knolVn the

1nduo cive part 0 the to roid moment

(16)T jl (t) = tJ4f1 e S e V MC~ t) dZf (If+ 1) (e-t1) fee m l

It s normali~ation 1s defined as in ref 6 bull It is seen that ( 16)

differs from M~ f or free ourrent s by th=-chane fe -- M So the el ementary dipole by analogy 11th M-L oan be TIlitten

in the fom (1 7)- ~ ~ C-- -- )TF = J 7- -1 X Jltj

A geometrical image in t he case of t he toroi d ( polar) tiipol p is a cl osed ciroular chain of elementary lIU1gne ti c dipoles - F( (Figla)

8

Elect )ic dip ole media wil l be described ~ a $c t of elementarY

Tf dipole s c( 1 In this cl (~l r cilshytion t h e chara c t eri s t i c space scn shy

I e of the lIlul tipole oxpaD[ion is

l arge (LS compured t o th e chrract cshy

ristic scal e oI dipoleR t heiliselve

therefore the l a tter mny lI e consideshy

red pointlike The 01 ect rl n pollri shy~(shy~ation of t h e 31st e p t )

is introduoed i n the Ll SU-ll 1ay

p7(7 t) = ~ J b( - i~ (0) (18)

It 1s clear that the el ectri c

polar12at1on p(~1 ) r1ay clue cO Pig 10 dunl sYHuile try 1111 tat e nul ti pole

moments of rn I ~neti c chargos 7C

i nt r oduce the amiddot101 cUr- ent pO il UO_

vactor wi th rCJpl~ct to t he t ime i nverDior in t he l ~ d ium of dis tri oushy

t ed electri c dipol es Ji-l -gt ---gt--(0)

1cr (Z+ = ~ el f )( vb(i- ~Ct)) - ~t fl (tJ (19)

1

where Fi (-Z i ) 1s the t ransvelse part of t he den s Hy oi electric

djpol ( moments ( polarization) The 10ng1 tUd inal part of polnrization

E (2 t ) ill uP Icribed 1)y tll e scalar denoily of Ute dt Btr ibutioll of elentri c c h-tree s

-raquo ( 20) o C7) t) di V- R (=i)t)

J otgt -0gt (11) ~

lIe emphasize that the axial current 1 cJ (7i) diff er s in U1 t ure from t he polnr CUT lent I-rgt (7t) cnh-ing Into the 1nmell-Lorentz equations

( ~ i) _ Z ct gt(2- 7(o) ~ P(2-t) (21)j ct I 1 t

Unl ike marne U ltation ML Mil both components of polarizationI

( B and -p ) entoT 1nl o the l~~ell-Lorent ~ equations and1 II J-only i n t he static l 1lJlt p eli sa-ppenrB t here

9

GOillS 1KWk to fOl111ula (19) we see that the ~lQ)

effe ctive cur ent 1ct into the definition Oj~ d ciini tion M~ upon int e gration by 11ar ts

l11 --

lcfini tion of thc el e c t ric part in which j ~ lle ru of COU1 8C Eo~ -= 0 Th e case w1th

substitution of the

Ml~ reu uces the

to the conventional ---

is replacetl by PL TJ i s mor e

lntri~l1in [ Substi tlltion 01 (1 9) into the defi nition (11) nnd

t rul Jf er of the d0rivo1t ive V l ead to t he fo rmula

(gt ) 0 _ V ylI e 7 t 7 cl3zlIIJ- ~ TpoundYJ (-t) - t (2e1-1)(e-t1) YU1 PL ( I)

analo rollJ to (16) lIenc e it immediately follows that the elementary

indured ud nl toro Ld dipole moment is equal to

Tcr 1y CZ X cT) (23)

and its reomctric iml~() is a closed chain of electrt dipoles poundz J (FIClb) The last formula denonstrates the simplest possibility of

i~itation in the dipole representashy

tion of symmetry elements absent in ltt -Tet syste~ of pointlike electric charGe s

(In principle th e completeness of

propertIes uncler R anfl I reflections may also be found in ~eshy

d it of el efi1Cntary higher mul tipol es

of the usual tye)

Let us point to onc ~ore possibishy

lity for realizinG symmetry of lr~ in mrtQ1e tic media Recall thtt in the

problem of ~otion of a particle in a central-symrlCtric potential oo rreshy

lati on arisc~ between the vector of

angular omentuO i and moment f This oorrelation is described by the

Runge-Lentz operatorFiglb

A

IT I P (24)

x

This opera tor already appears in consid eing the dynami c symme try

or the nonrelativisti c Kepler problem and it it mor e i mport= t f o

considering the dynamic symmetry of the relat ivis tic Coulomb-pr obshy

lem 22 or motion of a free relativistic particle obeyjn g the

Dirac equation 23 bull

10

-The distribution of a vecto~- like n bein~ present in a rw shy

lt1iuro (of an orbital or spin ori~n) also ne cessitates the introdtu shy

tion of a~ial toroid moments

Consider a medium in which there are distributions 0 th e 0shyment fluxe s desc r i bed in general by a second-rank tensor ( dya d)

nH (2~)ltti reg f5j ) )I-=XJ~2-_

( 1))

We reprc s ent (25) in the forE of symm etri c )1( j and antiYJmiddot lTl e Lr i c

n(~) n C- f)j paTts fhe lat t er is dual to the p ol a r vec t o r Z

=lt1 may be (1 escri0ed ]y =alofY Vii th hOfl it h]s been do ne a bov e 171th

ve ctor p (1t) in electrodipole mcJ i a Let u s int ro (l l1ce the tJ aJ15 shy

verse (axial) current (2amp) r~Q1)-+ J = Lot A~ (t) =[of( L K pgt

n -t

Subs tHut1on of this curren t into f02ll lllal (10) (11) like of jt will produce a mul tipole famil y l n In thi s case the elClltlentashye ry dipole is

~L ~1 )( til (ii t )T- (27)1 1n

An i rlea l [re orlctrl cll iila~C ()f tha t dipole is prCces s ing 10c[11 mOElenLs

on a c ircl e (Fi~l c)

Tn

~~--h~~ Jrtid

I - I I Fl Gmiddot l c

4 Pheilolilelloln~ir11 t he or Cpoundf

~pound~~_1~pound1~_~poundL1g~

pound~tals

C() n ) i u e r a y t cm of it iner ant

e l ed ro lls whi ch eulbi t a s econdshy

-o r der phase transi t ion into a

sta te wIt h t h e a~1 al yen0 c t oT

lruer plrame ter G- even i t h

le e-po et t o t lre i lJV CT ion Pr iori

t o proce ed to pnrtl (ul a r mi croshy

) (oTlie 1110lt1e 15 revelinf lnecbnIi ms

of t ha t onJeJing we dwel l upon

some of it s pllen om en ol ocical conshy

s e qu ence s using only f ormal

~ynm et~y nreum cnt s

II

1~ (i t) = L 1 x Ii cce-zit)) -gt l~T R (i 1) J l 1 f t L I (14)

where the synlbo l -- denotes the t r ansition t o a continuous d1str l butior of thc aagneti c dtpol e mO I11en t d ensity (magnetizati on)

M-1- bull l o~ethcr 11 tIL ( 14 ) i t is conv enient t o introduce n f ormal

f) Uantit y J-Ctt) lt pfieudos colar dist r i but ion density of magnetic charres t- __

J F(i t) - div-M (-r f ) (1 5)

Iote that tJc lIsull llwwel l-lorentz (gtquati olls do no t con t ain t hn qutr1tj ty Mil (=i t ) (the lo~g1 t udinal comp onent of magnetizashy

tlan)

If Yle rep1lcc th c density 2e by9)7 1n fOIlllUla e (J)-( 5) then thel c nri J cs Ilulo s t ltl complete ana1o[y of lDultipola expansions

of t hese Quant LII e s ( th e only dlffe1cl1ce 1s t hat one should se t

i oa == SPF (iZt )d 3r = 0 Clearl y tJle corr esponding for shyl1ul ae will jive oharge mul tll ole moments of the system of llazn~ti c

lipol e(l I e no h that a c omplct e l y ynunotric schelJe of mul t i polc

c)-pan~i (lLl S docs not arise sn opound d ~1gt- 3j (Zt )fJ1(~t)and Moo~ O SO the edly ourren t 3j1 l acs no t contribute to Gem

but it s c ontri but ion t o Tt is nonvminhlne It iR j ust this contribu tion duo t o I7hich t oroidal di stribution s started to be

int rodu c ed into el ectromagneti~Jn ( sec 6 and ref crence~ th ere1n)aUnder ~1t nnro e of lnduoed al e ct)i o fJoments there Was knolVn the

1nduo cive part 0 the to roid moment

(16)T jl (t) = tJ4f1 e S e V MC~ t) dZf (If+ 1) (e-t1) fee m l

It s normali~ation 1s defined as in ref 6 bull It is seen that ( 16)

differs from M~ f or free ourrent s by th=-chane fe -- M So the el ementary dipole by analogy 11th M-L oan be TIlitten

in the fom (1 7)- ~ ~ C-- -- )TF = J 7- -1 X Jltj

A geometrical image in t he case of t he toroi d ( polar) tiipol p is a cl osed ciroular chain of elementary lIU1gne ti c dipoles - F( (Figla)

8

Elect )ic dip ole media wil l be described ~ a $c t of elementarY

Tf dipole s c( 1 In this cl (~l r cilshytion t h e chara c t eri s t i c space scn shy

I e of the lIlul tipole oxpaD[ion is

l arge (LS compured t o th e chrract cshy

ristic scal e oI dipoleR t heiliselve

therefore the l a tter mny lI e consideshy

red pointlike The 01 ect rl n pollri shy~(shy~ation of t h e 31st e p t )

is introduoed i n the Ll SU-ll 1ay

p7(7 t) = ~ J b( - i~ (0) (18)

It 1s clear that the el ectri c

polar12at1on p(~1 ) r1ay clue cO Pig 10 dunl sYHuile try 1111 tat e nul ti pole

moments of rn I ~neti c chargos 7C

i nt r oduce the amiddot101 cUr- ent pO il UO_

vactor wi th rCJpl~ct to t he t ime i nverDior in t he l ~ d ium of dis tri oushy

t ed electri c dipol es Ji-l -gt ---gt--(0)

1cr (Z+ = ~ el f )( vb(i- ~Ct)) - ~t fl (tJ (19)

1

where Fi (-Z i ) 1s the t ransvelse part of t he den s Hy oi electric

djpol ( moments ( polarization) The 10ng1 tUd inal part of polnrization

E (2 t ) ill uP Icribed 1)y tll e scalar denoily of Ute dt Btr ibutioll of elentri c c h-tree s

-raquo ( 20) o C7) t) di V- R (=i)t)

J otgt -0gt (11) ~

lIe emphasize that the axial current 1 cJ (7i) diff er s in U1 t ure from t he polnr CUT lent I-rgt (7t) cnh-ing Into the 1nmell-Lorentz equations

( ~ i) _ Z ct gt(2- 7(o) ~ P(2-t) (21)j ct I 1 t

Unl ike marne U ltation ML Mil both components of polarizationI

( B and -p ) entoT 1nl o the l~~ell-Lorent ~ equations and1 II J-only i n t he static l 1lJlt p eli sa-ppenrB t here

9

GOillS 1KWk to fOl111ula (19) we see that the ~lQ)

effe ctive cur ent 1ct into the definition Oj~ d ciini tion M~ upon int e gration by 11ar ts

l11 --

lcfini tion of thc el e c t ric part in which j ~ lle ru of COU1 8C Eo~ -= 0 Th e case w1th

substitution of the

Ml~ reu uces the

to the conventional ---

is replacetl by PL TJ i s mor e

lntri~l1in [ Substi tlltion 01 (1 9) into the defi nition (11) nnd

t rul Jf er of the d0rivo1t ive V l ead to t he fo rmula

(gt ) 0 _ V ylI e 7 t 7 cl3zlIIJ- ~ TpoundYJ (-t) - t (2e1-1)(e-t1) YU1 PL ( I)

analo rollJ to (16) lIenc e it immediately follows that the elementary

indured ud nl toro Ld dipole moment is equal to

Tcr 1y CZ X cT) (23)

and its reomctric iml~() is a closed chain of electrt dipoles poundz J (FIClb) The last formula denonstrates the simplest possibility of

i~itation in the dipole representashy

tion of symmetry elements absent in ltt -Tet syste~ of pointlike electric charGe s

(In principle th e completeness of

propertIes uncler R anfl I reflections may also be found in ~eshy

d it of el efi1Cntary higher mul tipol es

of the usual tye)

Let us point to onc ~ore possibishy

lity for realizinG symmetry of lr~ in mrtQ1e tic media Recall thtt in the

problem of ~otion of a particle in a central-symrlCtric potential oo rreshy

lati on arisc~ between the vector of

angular omentuO i and moment f This oorrelation is described by the

Runge-Lentz operatorFiglb

A

IT I P (24)

x

This opera tor already appears in consid eing the dynami c symme try

or the nonrelativisti c Kepler problem and it it mor e i mport= t f o

considering the dynamic symmetry of the relat ivis tic Coulomb-pr obshy

lem 22 or motion of a free relativistic particle obeyjn g the

Dirac equation 23 bull

10

-The distribution of a vecto~- like n bein~ present in a rw shy

lt1iuro (of an orbital or spin ori~n) also ne cessitates the introdtu shy

tion of a~ial toroid moments

Consider a medium in which there are distributions 0 th e 0shyment fluxe s desc r i bed in general by a second-rank tensor ( dya d)

nH (2~)ltti reg f5j ) )I-=XJ~2-_

( 1))

We reprc s ent (25) in the forE of symm etri c )1( j and antiYJmiddot lTl e Lr i c

n(~) n C- f)j paTts fhe lat t er is dual to the p ol a r vec t o r Z

=lt1 may be (1 escri0ed ]y =alofY Vii th hOfl it h]s been do ne a bov e 171th

ve ctor p (1t) in electrodipole mcJ i a Let u s int ro (l l1ce the tJ aJ15 shy

verse (axial) current (2amp) r~Q1)-+ J = Lot A~ (t) =[of( L K pgt

n -t

Subs tHut1on of this curren t into f02ll lllal (10) (11) like of jt will produce a mul tipole famil y l n In thi s case the elClltlentashye ry dipole is

~L ~1 )( til (ii t )T- (27)1 1n

An i rlea l [re orlctrl cll iila~C ()f tha t dipole is prCces s ing 10c[11 mOElenLs

on a c ircl e (Fi~l c)

Tn

~~--h~~ Jrtid

I - I I Fl Gmiddot l c

4 Pheilolilelloln~ir11 t he or Cpoundf

~pound~~_1~pound1~_~poundL1g~

pound~tals

C() n ) i u e r a y t cm of it iner ant

e l ed ro lls whi ch eulbi t a s econdshy

-o r der phase transi t ion into a

sta te wIt h t h e a~1 al yen0 c t oT

lruer plrame ter G- even i t h

le e-po et t o t lre i lJV CT ion Pr iori

t o proce ed to pnrtl (ul a r mi croshy

) (oTlie 1110lt1e 15 revelinf lnecbnIi ms

of t ha t onJeJing we dwel l upon

some of it s pllen om en ol ocical conshy

s e qu ence s using only f ormal

~ynm et~y nreum cnt s

II

GOillS 1KWk to fOl111ula (19) we see that the ~lQ)

effe ctive cur ent 1ct into the definition Oj~ d ciini tion M~ upon int e gration by 11ar ts

l11 --

lcfini tion of thc el e c t ric part in which j ~ lle ru of COU1 8C Eo~ -= 0 Th e case w1th

substitution of the

Ml~ reu uces the

to the conventional ---

is replacetl by PL TJ i s mor e

lntri~l1in [ Substi tlltion 01 (1 9) into the defi nition (11) nnd

t rul Jf er of the d0rivo1t ive V l ead to t he fo rmula

(gt ) 0 _ V ylI e 7 t 7 cl3zlIIJ- ~ TpoundYJ (-t) - t (2e1-1)(e-t1) YU1 PL ( I)

analo rollJ to (16) lIenc e it immediately follows that the elementary

indured ud nl toro Ld dipole moment is equal to

Tcr 1y CZ X cT) (23)

and its reomctric iml~() is a closed chain of electrt dipoles poundz J (FIClb) The last formula denonstrates the simplest possibility of

i~itation in the dipole representashy

tion of symmetry elements absent in ltt -Tet syste~ of pointlike electric charGe s

(In principle th e completeness of

propertIes uncler R anfl I reflections may also be found in ~eshy

d it of el efi1Cntary higher mul tipol es

of the usual tye)

Let us point to onc ~ore possibishy

lity for realizinG symmetry of lr~ in mrtQ1e tic media Recall thtt in the

problem of ~otion of a particle in a central-symrlCtric potential oo rreshy

lati on arisc~ between the vector of

angular omentuO i and moment f This oorrelation is described by the

Runge-Lentz operatorFiglb

A

IT I P (24)

x

This opera tor already appears in consid eing the dynami c symme try

or the nonrelativisti c Kepler problem and it it mor e i mport= t f o

considering the dynamic symmetry of the relat ivis tic Coulomb-pr obshy

lem 22 or motion of a free relativistic particle obeyjn g the

Dirac equation 23 bull

10

-The distribution of a vecto~- like n bein~ present in a rw shy

lt1iuro (of an orbital or spin ori~n) also ne cessitates the introdtu shy

tion of a~ial toroid moments

Consider a medium in which there are distributions 0 th e 0shyment fluxe s desc r i bed in general by a second-rank tensor ( dya d)

nH (2~)ltti reg f5j ) )I-=XJ~2-_

( 1))

We reprc s ent (25) in the forE of symm etri c )1( j and antiYJmiddot lTl e Lr i c

n(~) n C- f)j paTts fhe lat t er is dual to the p ol a r vec t o r Z

=lt1 may be (1 escri0ed ]y =alofY Vii th hOfl it h]s been do ne a bov e 171th

ve ctor p (1t) in electrodipole mcJ i a Let u s int ro (l l1ce the tJ aJ15 shy

verse (axial) current (2amp) r~Q1)-+ J = Lot A~ (t) =[of( L K pgt

n -t

Subs tHut1on of this curren t into f02ll lllal (10) (11) like of jt will produce a mul tipole famil y l n In thi s case the elClltlentashye ry dipole is

~L ~1 )( til (ii t )T- (27)1 1n

An i rlea l [re orlctrl cll iila~C ()f tha t dipole is prCces s ing 10c[11 mOElenLs

on a c ircl e (Fi~l c)

Tn

~~--h~~ Jrtid

I - I I Fl Gmiddot l c

4 Pheilolilelloln~ir11 t he or Cpoundf

~pound~~_1~pound1~_~poundL1g~

pound~tals

C() n ) i u e r a y t cm of it iner ant

e l ed ro lls whi ch eulbi t a s econdshy

-o r der phase transi t ion into a

sta te wIt h t h e a~1 al yen0 c t oT

lruer plrame ter G- even i t h

le e-po et t o t lre i lJV CT ion Pr iori

t o proce ed to pnrtl (ul a r mi croshy

) (oTlie 1110lt1e 15 revelinf lnecbnIi ms

of t ha t onJeJing we dwel l upon

some of it s pllen om en ol ocical conshy

s e qu ence s using only f ormal

~ynm et~y nreum cnt s

II

AmonG ongnetic classes VIC mas indicate t~ folloVJine 43 olasses

dmittinr the existence of tto Xinl Vector G that is even nth

respect to time inveuror~lon R 1) 1 3 usul crys8l clas se s D0 t oontailliZ R at lll

c C ~ 1 C$ I C2 C1h C~ )S ) C4 ~ IC6 ) $6 C3h I Cbh ) C3 ( 28)

2) t h e s-le clns(J e s uppl ement gtd by th e o)erat i on R J ) 17 oc t uo~ mazne t~ 0 elo ses

CJc1 ) Cl(C~ ) ) C1iJ(CJ I CH(CJ I CJh (OS t eslC ~) ( 29)

C4 ( ( 2) ~4 (Col) C4IJC~ C~h (Cl)) C~~ (j4)) S6(C3

C lC 3) C((C3) C(C(C~~ G~( Camph(C 3IJ1h

Furt her 0 ~b1l l co n~i d c l olly sylltelLl without nontrivial

t lln3llt i ()I1~ in 1hlch givinc a mIlGn~t i c class is a neoc~snry poundUl~

suffi cien t condition for definine a possibil1 ty ~J the vee tOT G 1 0 ed c t L1Jo il1 the case of t he pol ar vaoto1 T all th e classes

I Jst ecl tbove aTC o t di f f i c1l1 t to 0btd t by usinG the t bl es of

ireducible repre s on t a t l 1I1s of point ~roUs

Etabli shmcnt of t he axinl tOlot d ordering i n cry tnls ma v be

c ~ uncctcd wi t h softeLine of a certain ool l eotive el e ct r on IIlolt1 e ( in

pal ti cular in the model of sect ~ t11 is is a CrM ollcillation lild 1n 1)1 scm) Ie shall call t his mo d e tJe axiru toroid lode oonsitlrgtred earl i er 1241 bull

Let us assume tha t owine to some cirl1uOstnnccs the frelucl1cy

of t he nxial toroid mo do ~ets 9Jlorlalou s l y snall and 0 t endency has

arisen fo r establ1shmg the t Oroid lon-~lT e ard el

It is cOl1ve1l rgtnt to anclyse t he z ene I U properties of such sysshyt ems by the effective-LIlra n c ill ne t holl AS 8 1~rlnc that the symmet ry

group of 1 hiShly synunetllc phas e oongtins a s a subfOU one of the

above-li~ted adl malletic croups we conside) sl1lall lo~ frequenoy

ltJltial t oroid osoi llations above the phil~ e-traJtsition pOint Th

effective kgrongian (wit h no extltrnal fi eld present) has the form

- = k - 1) (JO)

llt -= 1AMG (GY+ )a CG) (Jl)

-4 -- 2U -- ~ G 2 -t G -t- r C~ofG) (32)

12

Ihere M G Da d I 0 1 t gt 0 anll th e RJ1IlLllelrj OL the J~rstem a bove t ile trUlsition poi nt i s cons1cered c ubic I n the lcinctic

ener gy (Jl) we have retaineu the term (G) 2 the ne c ent t y of

whichLo be taken into nacolmt Iril l be eltDl a Lned omewhn t I ntec fhe lnt egtact i(l 11 IIi t11 an externl1 cl ectxOIullYlet io poundL1tl is by oyuneOJ

con3idcrations written in thc fa r m

yen II - ~ ( 3J)f d -gt = shy- G lt-ut- AA t

where If ( Z) t ) is the vector - pot ent i al C is t he light veloci shy

ty I is a co effi cient F orm~a (JJ) may also be 1 itten 05

two equivol ent expressi ons ~ shy

fl pound --gt = - l G B (J4) ~ c

h L E - gtJ~ -wtE ) CJ5 )

= ~ where ~ -0 lt-01 A = - - c A and tho Maxwell equation i s

used for H ie cure of tl c cl etric f i eld E bull 1= ( J6 )uVtE =- C B

From (J3) and (34) it is s e en that the (lynomical m~Gnc t ic sU9cepti bil i t Y b) (w) a t l ow poundr equ enci I)8 W haR t h e fol l owin r

correct ion

(37)J- (w) =- ( ~ y Me W (12 _ ltJLZ

G

where Jl2

G 2MG rL is a n Olilal mode of adpl to ro i d

oscil l atIons VpJlishLl Q(f correspondn t o s ecand-ord er phase

transition Note that t hrouGhou t I e are speaking of the transverse

axial toroid oscillations s ince thc longitudinal oscil lations do

nat inter aat l7i th electr io and mlleneti c fields

Axial toroid mode s could in principle ~eac t to the CUMent of magnetio ohar(es (if thc~ ~ wo uld exist) [(s polar to r oid modes react t o the usu~ electrio ~Uxr(lnt 14

lotelTorthy is a nontrivial frequency dependenc e 1 ((()) w l(the numerator i s proportional to as i n the oase of polar

toroid 0 sc1l1atioJls vhere there occurs analOgOUS t o (37) anomaly in the dynaml u ltliol cotrio susoeptibil i ty 24) Expressi on (37)

i5 valid only in e l ow-frequenoy region whcn the sc cond t erm i n ( 31)

Ji

mixing wi t h lI sllltll rlugnons s J n ce t he efioctive iL1uiltoninn of tl~c

lila be neGlected pound0 find a correct asymptoticgt fOl wraquo 120 i t is necesacy to take thif teJrt into account alld then as it

should be th8 c Oltribulion of 6jCw) va ni s hes at hiGh fccquencies

In the microscopJ c Il oli e l (s ec tiol 5) the second term rets essenLiRl

for J~ j cqucllG ie W - E~where ~ is a chaTaeteristic

o1e-eleGtron eIlCrcy of an order ()f t h e band gap of a semiconductor

Interesti nG eff ects may occur in syst em s in which together ith

1]1 0 ~ial toroid ord 8rinG t~ere is realized nnoth pl typP of the

m1tn e ticlone-ral~c ord e ril~ For illstance in the case of anti shy

fCTr OfJa pnets wI th SDW the a ppearance of axi al tO Teid order cormected

ith SCDW resul ts in a wenk ferromagnetisr1 1 bull In the rase of Ont iferromOrnet contaiilinG in adui t i oL to i ti shy

ncrant electrons loc[tli~ed moment s the axial toroia orueT also

produces in the Larranri~ l nf the system the t erm responsible for

the) weak lerromnVlc till of local moments

f1 pound~ ~ G [I X MJ Os)

wherc L ic the vector of antifelrolilagnctism H is the aver2ge

mn n0Uc noment) ~ 1B a ooefficient Hote that in t he model 1 5 the whole e ffe c t i s of a pll TC - 8Ychange ori Gin and hi1 s no relativ i s tic

smallno gt Ihcr e~_s the comenti onal Dyalo ohin gky-lloriyn mechanism of

weak ferromagnftisn i s COTL1ltc ted Wit ~1 the sl)~_n-01bital o ~middot fl lUGne todishy11 1 pole interactions ~

In the Gose of i ncoliulensurote structure of t he acial tor oi] mo shy

ments G bdow the phls c-transi t ion point t here a r i ses the in-

homogeneous spontaneous polarLI IlU on

(J9) V = - ) UJ-t G middot-hiel (~lre ctl) fo l l ows fro m~I~ rcpre enta t 10n Jf the t ~lrJ of 1 te-rlc~

tion wi th e l ec tri c f ield E by C15) At the ph1 f ( tral s iti)1

tloinl the ttlie llU1svere d ie l e ctri c ~usrc tibility 6 Ej(f) 12 ttl 4ltii Me

( 0)6El (~) C= S2~ ( it )

is (Hv er[cnt On lor1n tvcve ct o r ~ =fo lcfil~rt J~rrm Vl1l l ~G of

t~e t mnsvelse llor JlaJ mOJ~ S2 G( Go)~ OLl t qo)~O()(for qo 1 a ) Axial t oroid excitlltions can intcrl c l wi t h othe r ~ollectiv e

oxcit at iors i n crystals Con s i ler e c a ferromnrnet wi tll l o cal

moment s in tile GTo und rta te of hich t here 1 no xial t ooil

oraeing At t ae name time t~e col l ectivo tor oid oacill a tions are

s ystem conta in t rms 0 the type

Ar CCf) __ MHeH(C) (H) l-gt UeH - )

HQHU) =- (11 G (i)

vrherc 1 is a proportionalit coeificient M is the mailetir moshy

ment Let us Trit e the Bloch eQuation with ncco un t of (41 ) tillC (42)

l or smal l d eviations YYi (~ )of) of tle magne tic Jlomont M from c l lli shy

libriUIJ value Mo to [tfeH) MJ 1m = (4J)

~ -7 (0) -t=M-t-rYl H He-n(G) = HeHM o e~f (44)

--gt(0)~2YYl--(0) +HeH cll ~ ()X~ r) XI( HQ ( 4 )

t - ~ e I (0)Here 0 - TmC ~ is the gyromagw)1ic ratio Ho Y - the

co rtributi on due t o magnetic anisotropy Ihich will not explicitly11 ~I ~ -

bewTittenll ere( see ref ~ ) Settinpoundl cJ=oI nmiddotn ( Y1 i s _1lt 1 K

the unit vector directed alon~ the lIav evector q) m-LMo )~=2) we get for Fourier components YYi w

~ ( r [ ~ Co) ~ ] ~ iw [Ow) ~~t7 (46) 1wmcgt 0 1 HeH)Mo w

--gt and fOT we make use of the eQuation reaul tin from (HyinGGw the c-rective LagranGian (JO) with a c oount of OJ) and (J4)

W 2 ~2 Q-gt -t cu ) _ J - t q) + _ __ I m = 0 (17) ( clMC I W 2 w

The dispersion l aw of maglll1-toroid o scillations is given by the

equJion W( (0) 1s tbe freq u ency of a ferromagnetic resonance)

- 1t2

2 ~ W = (0 LWH (0) t- J ~ t-G JNo ) ( 48)2 1)w

15

H

w2 lt)w ~M- - tl - ~2 ( 49)

G I

Ie shall not write jawn hCTc the general solution to thc c ubic

eluntion (48) lJecllUlC of very cumbersome egtprpssions involved in tho

50lutlon Clclldy the 1l1xlu[ of toroid o3clllatlonll and llklZllocs

1 rnndmcLl at ll1 Rimolnu1La fo eivcn by the oppr07dlL1llte relation

1io M 0 rJ ~() +- (jJ- Co)l ~ S2 - ( C)l H J G 10 ) (50)

~~ ~ (q~) l ~ - t ~~ ) 2 M G

lItc l xlTl( mngllons with xdal toroid o scillat i ons IlaJ Rlso

o~cur 1n ~lIti(rlOJlthOlets COl th (orrespon~inb pnrit j of antiferro_ mlltUetj c s tr l rture

The tiC )middoto ~conimiddot rcalimiddot~tion of thi s intlrtwinning is pOSlJibl

Ln the liiO 1d l~~ IIhere the Udal toroid mode i~ token t o be a SCDW

o cl1 l ntioll ond the frequency J2 G can u e clo se to that of tJle

nnt iferrolnlnutic reSOtlllce 1he bencralizat10ll of formulap (46) (49) to antiferromagne1s i n obvious

Axial t oroid osclllationJ i rl eract wi t h li~h t in a ther

le culinr muuler 0 l resull neVi b~ftnches of polariloD$ =c llrodushy

ced I miood the Llrranlnn of the $ytl)~1 in an clectrom~ctc ~l eld A ( ilt) hn~ the [orl~

i t(f ~ta -i + [~7 (t)- (lofifilcentt0

hryinr C~l) wiLll resptct Lo Efllld A v e alrlve at the systcm

of equatio n Co 111 0 Lltill tOloid rJol~enL and thl ia-wdl cqllations

~ ~ 1M G -- ~IJQ - C -zvt4

G 0 ( 2)

t A- - oW-1 A~ i LiT 1 7A (f -= C- ((2

1ft

For )lovul1 Dodes cqs (51) and (52) yi eld

2t 12 2 w 2 q C tC9Q G ~ [CfCZ ~ poundltkJ J2f-4 fGgt~ ~2cj~

~ pound pound fXgt (~ J)

c = C t 1- 4q( MG V

Ilhen (5 ) ) i s slmpll~led toc~p 1-+ 0

wt -~ 92C~ gto ( 54)

jw~2 ~ J2~ whees fo vz cZ gt~ pound00 J2~ f get the lsynpto tic 1

c ~ (~j )

C ) w --gt rmiddotCl wlmiddot - 1 S2~

Jote t hat IJplicabllity of all the f orIJul te (51)-(~ 5) is llmi shy

led t o the ranee of low frequencies and InJall IJomenla ( in the

mi oro scopic mod 01 sed 5 (v J ( ~ clt t 1 where E i$ n blUld

Ball of a grmiddotlicondictor) I f UJ I C I - E tho1 we ShOUl~ kcep

in LagTangian the t erms with hjGhCr-ordc~ derivative of order aTlshyMe t er ( analoBousl y to the case of polar tOloid oscillations 24 )

Tt() esul t in correct asymptotics for I arca enerries nnd momenta

is 2shy 2

0 ~ J2 a 12CE ( ~6)W 2 ltgt0 bull

1

I n t he system with axinl toroid ordering there may occur intere3shy

tinb nonl i near optical effects In partioular below the transition

pout ther e 00 curs the anomalous contri bution to COIlponcnt$ of tho

ampvratio~ tensor ~ tit that 1 s proflortional to t he elect rlc field E

d1K (57)t1-~en Ge E Yl bull

Note that in the system3 with polar toroid orderug there Iilay al so

occur the anomalouJ contribution to the tenso ~ ~I-lt that is

however propor tional t o the nngnet1c f ield H ( 58)

17

the r e Lore the corresp onding transition Lemle r a t lle ( or ~ hc r l iti cl I

bud Eap E~ n a s omi conduc tor nad el at T = 0 ) i s 10 IJli shyh ll ~ V Ie l))l a lllous b ehavi our of electro-ol1 tical xod IIDlln c t o shy A1nI a rJtl t he s t ate wi t h eD~1 10 ell ergc ticll ly t ll p rlo s t Lwolll n i)l~

- op Ucal c1lJact t r istics o f c-cy stals IJaY help i n i den tif yi n c t o r oid The explj ci t f OlC1 or eficc t Lc intcrac tton cOD Lant pound0 _ tIL no l~i 1l

tTJl1s i t i o s ~ h j struc t ur e s of order l (trallc t e r U~J can bc f ) II](l c La

lhe tensors l i 1r ~he o ae- c l ) ct r on p-u t of ill Iv)U tOIU h5 Axl lL 2 2Q_mo~nt s L crn~~~gLpoundhagLill 2i tLosectVe2 H have thl3 f OJm P

Consi l e-r c LNa- band mod ul of a s emicoruluc Lo l or 8r1ime t 1l vi th 12 ~ 11ri lleGl ~ l rel Il a t tlO igtt Ko o f the Brillouill 7oue Asswlle t h a t t he _~2 I~ -+ _ 1 J pol P~ ( gt1 )

11 1 1- lament gtr Jnt erbad rU p olc trtDs1 tion at poin t flt e qual Jf 1J ltri Sfll LE - E~ =l- EsJ ~ ~ g2 01 ) ( t J Cl rtve fU1Gto n s of band s 1 a nd 2 h a v e the s ame pari ty bu t

U el Oll ~ t tJ ffelmt iJr e d uci bl c rer p s entat~o lls ijl o LXoup of wa ve IIhere E rl i s t h e en e r of - b Cl nd at p oLlt k o PiS ~ ve c tor 1(0 ) lilG the II latru ~l ()IJ 11 0 i n t crb (l nll trw-n si t Lon ovo r t the ma t r i x cl eme n t of t~e mOl lent Uhl b e t ween 01 band t (

t

= 1 2) he o ~i tll mOlgt1eJl t tl l fft r n f rom 7 ero The n oud with t hat GyrlllJet r y and ) hiGhe st band S 1= 12 bull Fu r t h e r t he s it l) 1 t i on l ~ M Ilir1elHl

has be e n con~idero~ 14 To f 6 wh e re i t h a s been shown t hat in t he rlhen the ton~ o r JH 1s [lu r c l y real ( t hi s ho11s e [ lIdt LJ-jgt n -) ~ enc 0 1 t~IC e lt ct-ron- holc pu rin r with all imaGinary SLlpl t 1(lTashy IlIa ch wave fun ctio n s 11 ( 0 C7 It point l(0 C I b0 toJ ll ro~ l)

t 21 LlT t hltr ( -11 ( Lhe orbi t al f errolllrneti~ o f el ectrons or a 111 e t udy of il syt em I1 t h ijfJl1il t onin r ( ~9 ) 1 cn 1ioc ou t by ~ -ji1 ]l llld L~t 110c1 Ihntlt oniln w1l1 be wr itt e n in th e k P oppro shy a u suol Gr e en fun c tioll mct llor1 alld 18 s1al l lIOt 11ICl l upon tIH

7 ll~ i ~f)1I i ~ I t())nal el o ctroVlLlgae t t o fi c l d calcula ti onal t echni qu e ( s e c a detailed eo i-li on of th e 11 8~c1tai1 i r 28 )insulator lOod e l in bull note only pecul i ariti lt 3 tlu~ t o Lle

r eaction to an external elect r omagnetic fi f l rl since ju s t th e e p e cushy- (- - ~ A_)2 + t-~ + e ~ ~l (1 -~ A) (~ - gA) _ liarities allow us to unil er s tand tJI8 t ype of o)( erlnf ~ lmiddoti in[ in

J IYI 1 C f t~ oj ( I -L C JI LlI1 sJster below th lt phase- t r ans ition bull To t hl send 110 Iri t e ~J e

(5))H- cfe ctive LaGr ani an d es c r i binr th e t an s i tion to t le stgtt e witt

eml at T = 0 for t he mOdel of 5ernicorduc tor lith 11 mnll band~l li ( iJ _ ~ A) n-~ fi -~ _1- (Q- ~Ar~J d~ t l -jO~ lt-t c l Imiddotap E~ - E wh ere ~ is of an or d e r of the e citonu ~ lt1- 1 C )1 1 1 bindine enerd Up to the Jt errnsof high et ord e r in the l)aramcte rI ~ (~ 3 ~)6 Re iE lt1 i 6 = - c~ +-Ll in a weak ani slollly changin~

11 ~ gtlte gt 12 21 transverse f i e ld A upon cwnbersome calcul a tion Ie ret

where IYI~ M d 1Yi1 lL e ff c c Livc IlaM of cl lrllll s LJI() h ole ill

z one s 1 nlld E~ _ gt Ll lJalll1 1~ uf l emi c ontluclor ( foy 1111 - leg - K -J mc t lll s f= ~ lt (I ) A(l~) ~nc1 ltft i) ~ e l h( v e dur amI scal ar

(62)

po ten t i ru s of UI~ elcLmiddotorI--1 1 Ur 1(gt111 6 i(ft) 1 the or ce shy -(_ ~ ( ~)Z (~ 2)p yl17I ct er drrcri bin l1l o rarrtd tate be l uw the ~llt11(- i1llH rl t middot f)11 Tn I NVMG 6 h J)cmiddot L1 Re ) ) (63) t he tYtrl -lnnLl c Citonic insula t gtr lYTle mou 6 11 11 enr-~ll

t he tOrSlT tlt1cturc -I f ~ 1 + 1 c (64)

If i t I lf~ J (L (~e y- ~ C~ ~~ VJ44gt)2l(60)

wher e t i Lh e illil ma l i t i1 n vcr-tor roli1noilecl of 1 1111

lJatricer I II Ihtt f ollowo we con1olor the ordcr P-Llt1

der 6

= g E2 ( 65 )N~ t o be Sillrl c t aJ1d rC-Ll 1111-11 co renpollils to the chaT(c lonLt~ o vmve s t ate I-gtc bull It 1 arull1 thlt the f[(cll e Lnte-Dction consshy

t 811 t i s maximal ln the CIlSlt- of llJnsition into the t le IliLh CDI 19

18

)

Dc Z~6 E ( 66)3

c( ~ E~ =-d - (67)

A -= e~ e2 Lj E~ m (68)

-ell c - ~L (_i +_1 )(r

1m S-=12 E~-Es EJ-ES 19 x ~~) (69)

mh EYzN ~ ---- ~ YY

It

VY = rnJil2 1 2 t (70)

where 11 i ~l~ctrol1 IllHSS

-cgt ~

JntrOiUelor tltc notflt I no G -= Yl1Z6 Re wher e nz 1 the unit

Vctor directel DIone tl VIC arrivo nt an eqlrcssion fnalo l ~)ul tf)

the phenomenoloricnl Lnrnniinn (JO) for uniuxial systems

ThUli UI lhc rnil1~0500plc mOdel with Hilrnlltonia (59) beloIt

the errv pIose l rll1Si tion tl ere 111( a Sintl et m1al toroid

1J1uerinf Ttli lll rm us lot ollly to lllustrale the ccncrnl phcnom~_ HoloeJ cll schernp eoo 10crc1 lbovt tut uo -0 consil1s l G~rt-dll tUOtc

pacific p r l1crt i cs or the 5l tf OUI) of the lI1Of t intercstin in

11 ou rpl Llon the proble of illflucnc( of eollnctie excltltlons

111 th e ~yltcn vith tb l~ llton1flll ( ~9 ) on itn Ol tl cal ancl I1la[DCtO shy

opti cal pro wtl i In n IHIcred Jnse bull tll1l1tuu( ccltaLio1- n t Lc

ea$e of Lrn uud stntl 71th DIV --IJ in fact lAidl t oroi1 ocillt shy

1003 wherpHl J1hamiddot l_ l1 1 Imiddot t o ~ I mS~1100$fI ninee nt ltI13l1 lnYlatl shy )MI from fl~111 brl UJO

)~ (+) = ll1 (n I e)lt4gt ( 1 ~(1)) ~ A I (1 + f cent)~~ I ) (71)

6 ~ (-0 = ~ + b ~ (n (7) Re

1lgt1) -t ) = If(t)) (7))

2u

U) 1t C IA (t) ) (71)

~ t-1 ~JgCPU) C7))

Ih~)middot G(1) i n lL Ii ~lt Y oi th ( nJdnl lorn1l J om nl t M(t) t ho snme fo~ the o111l-l nll11nct1 c no ~ Utpl i tll( ( 1 J I J 1

bullLoruia) ofci1lnttonl gt rVe Iltl lmll mOl CA1] 26 11 1 h ~ oscilla ioDs (1 orb1tal r llgJ)O J~ ) u ~ PJ$ IIII _~ r ql)l1tj tr lHOTlorticllll to the lff lfn ~ -tL (In Ll

fo ~ol [l1C 1I ~JlfL~t nrJ er J1~)lll ctc_Jon 1 lgtitn( E1 l t1fJI l

exltouic-iurulotnr l rpc mouel J -~ ~ ) 10 t i Gc ll l L Vm Give J 2OSOUllncc c o 11 l)utjon t l lhlt IU(middotlc cL i UfIilLili111r fIIl

UiellBti c sllsceplib l LJ )1 t h )~ L ja Ju l oll r l 1rO~I1iI t f rnluclIshy

ell S

Au intere5tLn~ ~ituHtio ~l 11-) ( 11) in t]~ r1 1c incJ1JlCIllUol c

aLructill e ut eml (301110n lnttlce) I lCC01-dwlC( dtJ ltonc1ulionll 01

sect t 1clol the LL3osition poinL in U1C Gystcr~ ther ru-iser l _

5J1ontanCI)U5 inhomotcnoll ~ trAll r1~ J101arl ltion PL(~middot ) -- 70t- G(t) In a semjmctali c mn( e1 Id t h lIalJi1 tonicU1 (59) (where c = - E~ 2

tpound ~ j I tIle Fe fl lli- cuoro) 11~ Lh~ 10011 degl nOOffillcnsurnt

strucLure or CIW fJ it T~TC 111131 Tcr i bc t ransltloil

t 01OJ1 (3 1atun Vie have (76)

P-L() = rVlt (~2 Ll~e) gt -= e N~O) N(Q) - mp _ (77)

[II 1 4poundF ~Re

~ 0 _ -~ (70)

~1 ) 6 ~VO Z

allu 10 1 a the Vlave v ectol of 3Up Jstucturc (ULlU th~ Ll~nh1tJ

POL1t av0 - 0 ) IIhlle 1 0Verin~ tepoTnturn therl 1lIlY occur _ onc mol( tranri lion Id til tlt ( ollPCara1CC or orda r PIIrflJUlt er L1 J(Z )

1n addll1ou to 6 ~e (Y ) and Lh middotJ splt1al dlJtribution 01 6~ cn 1l1 compU cd Lo fj~ l~ ) 15 Shi f1 Bu by 112 ( r01 111010 detnil

21) ) I

~

sf r c rt e ~ (- ~

6 [ l ) - 6 r ~~D L (79)

2 1

As has bean ffian tioned in ref 26~tllC appearanc e of IJ (z) ill n sysshyt ell wi t ll iltelUal 1cansi t i on ~lolccl lIitJl rellpect to t he orbi inl

noml t LUll ( f~ l ~ 0 ) signlfie s Lhe appearanc e of orbl till lJJagnc tl uLlrin p The 1 ~IlGJlLtic-uomcnt uennity i 1

M ( 2) - f Li ( t ) (no)I

FrC (19) nnu (aO) it fo lloys t hat in the system wi t h incommcnsushy1 ~ vatc struc t ur e of Pnram(t0r s 0 Ilte and j rIM t here ari ses a p eculi shy

n J~GneLe1 ccL-ric ordcrinC in the ranee of domain wall s CDW and

P1 M ) - F ~ - 1[ 10 -L [~2 I f hOVlev crltf M1 cf(0 ~

~o I t~ 2 thcH P = O bu t M may l)e nonroro and VIC arrive 111 thc case of orhiLal lung- wave f erromagn e t ism

1hus i n the JnIlge of domain walls of i n commensurat e structures

with xd l to roltl n OIlLnb there nay lppltar variolls typcn 01 el ectron

ordering ( ferroelectric ferrol lDle t i c OJ macnetoelec tr1c)

A detail e d nnUrsis of different s LructUl O t nes requires n poOl11

cOllsit] craiion l1Itl JflC ~ u eyollc tho scope of the presen t paper

G Conclusion

Owin~ t o the Ilclwmp- of Jlultipol~ o)rgtansioll lJeing univerlll on e

can in pJinciple lttlro oon3 d e~ mor e compl ex s t ruct urts t hai dic uleu

lbo-no Of (l particular int er e s t ouJd be t he s l udy of (jipol e to r oi d

medl seclf~illl llY It s eL of el cltlen t ary t oroid di p oles 1 f L DC Jlte l einc s e elftlnely exotic ucJ ft nodel may haPlen to b e uSfdul for

study11lf phase t rrmsfOrlH1tions in a number of Iilol ccular crY3tol s

ampcllclnl l y Rpeakine Ising the above schemc one mllY also

construct medi a o[ hlghtr llJul t ipol (tbmiddot1s hLerarchy 0pound Mul l 1polc di s t r i buti ons is consitlered i n Jo )

In particul ru- tlHsC include sy l cl1ls with distributeil fluxcs

of ma gnet io moments ch=acterizin~ by tho G)IlfDelric tensor t~)(~ 1 )n~ i l) r

( see scct J) Howevcr tho ni croscopic description and di s cussion of

propelties of suet mul lipoIc media seem to us to be somewhat earl y

I n thi s paper we have onl y touchod on t he problem of 10nni tudinal -) -) shy

oomponent s of th e quantiti es H p and n Without eoin C into II ) II II

de tail s we only p Oint to a geometrical 1l1ust rati on of el ementary

dist r i butl ons of those quant iti es (Fig2a b)j a pair of magnetic or ~

el e ctr ic clipol e s ( 1-1 1 0r PIt ) directed t owards each o Llier and a

pair of spins precessing around a common axis bu t in opposi te direc_ tion s (n 11 ) bull

22

MOn MJU~) ~) ) )

P I I~

111 e outhors nrc grn~O I ltJ 111 Y1 1 VK IIII Ymiddot Ill ll i I ilh lltltl ~ Ill

he d i scussion of the rccul t o uml u =luJILL t f1IKl Or n t I (Il U )

thanks very mu ch EATolk a chcv and LM 10nlll nltlh fOl vflJ ~1011

i nformation on fundamental s of eloctrodynlll11on wi Lh tilnHntlt 1 nlt r

Re f oren ces

1 Dubovik V M ChesbJov AA FiEEl e lnClulstil LL Y-tllmiddot~ 1 97~ ~

p791 SovJPart Nuol 19 74 5 J1 8

2 l~ors e FM and Feshbach Ilethod of The or Physics tic Grall-U11

NY 195J v II

J Debye p AnnPhys (Leipzig) 09 )0 p 5 7

4 Chandrasekhar S Hydrodynamic s and Ituromaenet1n ~taulllty

Univ Press ()y~ord 19 61 app J 5 Ashe r E I n Magn e t oelectric I nteraction Ph enomena in Cryot a l o

ed by AFreeman HScbmidt NY 1 9 75 pG9

6 Dubovik V U Tosunyan L A Fi s El emChastits At- Yau r a 198) 1 4

p1l9J ( SovJ PartHucl 198J 14 504 ) 7 Bla t t J M Wei skopf VF The oret ical Nuclear Physios Ylloy

N Y 19 5 2

8 J a ck son J D Clas sical Elcct codYnami es Wi ley Il Y 1962

9 Dz ylll o shinsky I E ZhETF 19 64 46 p1 4 20 bull

10 Andreev AF Marchenko V I ZhBTF 1 970 70 1 5 22

ll llue JF Physi cal Properties of Crys t a l s Clarcdon Press (dord

1964

1 2 Landa u LD Lifshi t z BIl Statistical Phys ic s llaulca L 19 78

IJ bull Andreev AF Grishchuk I A ZhETF 19 74 87 p467

1 4 Volkov BA Gorbat sevich Jl KopllYev Yu V Tugushev VV

ZhETF 1981 81 p729 1904 bull

15 Tu[u3hcv VV ZhETF 1984 86 p 22 01

1 6 Zheludev Z S Symmetry and I t s Appl i cations Energoat omiz dat M

198~ (in RussianJ

23

1 7Strruhev 1 1 101111 obi( L L EJ ~c~rOlYilln1CS with tagnetic

Cbarce rhuJra alllt Tgt(lhn1Jra 1Lunk 197~ (in RUGsiroV

JCtllocll~il Yll A loll1chev E A 7 Tomil chil~ LIJ Doclarly AcSo BSSR 1977 XXI 1 988

19AklU p~lll 1-1 J~)~ltcJ~ii VB ~U3Jtw~ Electrodynami c s iViley

rIY 1n gt OCol rJIIIM S U3pi~I auk 19Glt~ 1~ p177 ibid Do180V AD pJa lLJnrl-l1l Ln Llfrltll~ I l ZlectrouynUJics (If Continuous Llcula

N1UkIl I 19[12 1 1 1

2 Foclr V LIvest JI1ld Iauk sssn sor ft 1935 8 plo] ~~ IP1kl1 I i m lo J DynornioOll gtymmetry and Cohercot 3tBtes

of )1HlllwJ ystema ilILa 11 l~7J (in Russim)

middotI Kopaymiddotv YlIV lu~1shcv VV ZhSTF 1985 33 6 p 22~~

5 Itrov BA Phyd(u Properties of JllgIlctordcrinG Cryatalo l irpilra

1 J 1] (J

6 Vollov bA llmHC T r lio~rtYcv YuV ZhETF 7amp l185G

7 l3Iz GL Pllu Egt G E SJl1llDc ry and Deforma Lion Bffects in ernishy

conductors ~ul a l 197gt

23 KO]layv Yu V Hports of FIAN 1975 So pJ

29 Vo1kov iIA lugushcv VV ZhETF 1979 77 p2004

Recei ved by Fubl ishing Department

on August 15 1985

24shy

Jly60B~K BH bull 1TOCYHRH n A bull1Tyrywea B B pound17-85-623 AKcHanbH~e TOpO~AH~e MOMeHT~

B 3n eKTp OAHHaMHKe H tlIH3HKe TBePAor o Ten a

e ic O Yllhll1MO )M t-ITOD I ltlKCHanb HblX TOpOHAH~1t MOMeH-

Tl1U H1 Miol YKil3b1u1eH HiJ eYUlte TBoBa -

T II I~ L uonHoi1 JilPAAIJUOH n n OTHOCTH

Pa6uI lrllitltft 1I(lOIIIl I tI l I thllIlI1l4ltlCKOgtt 4l~3HKH OI1fIH

lJpeITPIt r (j ptllMlI HceneAoaa_A llY(Ra 1985

11JIlIrIt1I VV EI 7-85-623

f i ClIhlclI IIII Hol t t el

Is found wh i ch di f shyf r om t he ones known

i s con s i dered i n i1nd in magne ti c

t he alti a l t oroi d i n teres t i nl] pr ope r t i e s

the ex i s t ence of ha s been

Amiddot fers i nversi ons

rea l izalion 11gtlc tJif1 I ~ mcd i a

mmllli ll J wilve system 11l1I~~ I i tjiHell Some

IM l ieu 1H c hi rge-dcns i ty middotave

III )0111 11 h I 11 I) l IUIOJd 1 Ihl LiJborto r y of Theo r e ti ca l Illys l