Electromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution.

Transcription

1 arxiv:physis/99536v1 [physis.lass-ph] 15 May 1999 Eletromagneti radiation of the travelling spin wave propagating in an antiferromagneti plate. Exat solution. A.A.Zhmudsky November 19, 16 Abstrat The exat solution of radiation problem of a spin wave travelling in an antiferromagneti (AFM) plate was found. The spin wave with inplane osillations of antiferromagnetism vetor was onsidered. In this ase the magnetization vetor is osillating being perpendiular to the AFM plate and depends on time and plane oordinates as travelling wave does. This model allows to obtain exat analytial expression for Hertzian vetor and, onsequently, the retarded potentials and field strengths as well. It is shown that expressions obtained desribe Cherenkov radiation aused by the travelling wave. The radiated eletromagneti wave is the TEM type if a phase veloity exeeds the speed of light. Otherwise eletri and magneti field values exponentially derease in the diretion normal to the plate. The energy losses were evaluated also. 1 Introdution ItisknownthatnotonlypartilesanbetheCherenkov radiationsouresbut also, so-alled, superlight refletions [1] formed by the motion of partiles number large enough. We want to point out that suh effets an be observed at spin wave propagation on the magneti surfae. In partiular, it is possible to reeive 1

2 the exat solution of radiation problem of a spin wave travelling in an antiferromagneti plate. Spin wave propagation in the antiferromagneti plate Let us onsider planar antiferromagneti (AFM) ontaining two magneti sublatties with magnetizations M 1 and M. Total magnetization of the AFM M = M 1 + M in the ground state is equal zero ( M 1 = M, M 1 = M = M ). We will use the σ model approah based on the equation for the antiferromagnetism vetor l = ( M 1 M )/M [, 3, 4]. The effetive Lagrangian of the σ model for the l vetor reads as [5]: L = αm 1 l ( t l) w( l) d x, (1) where w( l) = 1 β 1l y + 1 β l z ( < β 1 < β ) is anisotropy energy, = γm αδ/ 1. The phenomenologial onstants δ and α desribe the homogeneous and inhomogeneous exhange interations, respetively. The dynami equations for l an be written as Euler-Lagrange equations for the Lagrangian (1). Using l = 1 these equations may be presented in the form [5]: [ ] δl l δ =. () l For the planar AFM it is onvenient to represent the dynamis of unit vetor l by means of the angular variables: l 3 = osθ, l 1 +il = sinθexp(iϕ); (3) wherethepolaraxisisdiretedalongtheeasyaxisoftheafm.theequations of motion for θ and ϕ an be written in the form: ( α θ 1 ) [ θ +αsinθosθ ( ϕ) 1 ( )] ϕ w a t t θ =, 1 In the simplest ase of zero field and zero Dzyaloshinakii interation.

3 α (sin θ ϕ) α ( sin θ ϕ ) w a t t ϕ =. (4) We will searh the solution like travelling wave whih propagate in the XY plane (θ = π/) and ϕ = ϕ(( k r ωt)/k α). In this the ase equations (4) give: ( ) V 1 ϕ +sinϕosϕ =, (5) ξ where V = ω/k, k - wave vetor and r radius-vetor in the magneti plane, ω - frequeny. At small k phase velosity of spin wave inreases infinitely that is why V >, equation (5) has stable harmoni solution and Cherenkov radiation takes plae. Vetors l and l osillate in the AFM plane, thus the magnetization vetor M [ l l] is normal to the AFM plane. Evidently, if ϕ 1 the dependene of magnetization vetor from time and spae variable beomes: M = M exp( iωt+i k r), (6) where M = (,,M ), ω - frequeny and k wave vetor of the travelling wave. In the following (next) setion we will show that dependene (6) allows to reeive the exat solution of the radiation problem. 3 Exat solution of the Cherenkov radiation problem of a spin wave At a given funtional dependene of magnet dipole moment M on the spae and time variable the exat solution of D Alemberian equation for the magneti Hertzian vetor Π m : Π m 1 Πm t = 4π Π m, (7) 3

4 is expressed by the retarded potential: Π m = 1 V M( r,t r r )dv, (8) r r where M isamagnetidipolemomentperareaunit, r isthedistanebetween the origin and the observation point, r is the distane between the origin and the soure point where the magnet dipole moment element is plaed. Carrying out the integration (8) one an find the vetor potential A and salar potential ϕ via: A = rot Π m, ϕ =. (9) And the expression of eletri and magneti fields as usual: E = 1 t rot Π m H = rotrot Π m. (1) It is easy to test that Lorentz gauge is satisfied identially at the made definitions. Taking into aount that a magnet moment in (8) has to ontain a retarded time, we shall obtain: Π m = M V exp( iωt+ iω r r +ik x x )dv r r M = exp( iωt+ik x x) exp( iω (x x ) +(y y ) +z +ik x (x x))dx dy (x x ) +(y y ) +z (11) We assume z axis to be direted along the normal to the plate and x axis oinides with the wave vetor k. Evidently, that k r = kx. 4

5 Let s use the polar oordinates x x = rosϕ and y y = rsinϕ for the further alulation. This gives: Π m = M exp( iωt+ikx) = M exp( iωt+ikx) = π M exp( iωt+ikx) π π exp( iω r +z )exp(ikrosϕ)rdrdϕ r +z exp(ikrosϕ)dϕexp( iω r +z )rdr r +z J (kr)exp( iω r +z )rdr, (1) r +z where J (kr) is the Bessel funtion of the first kind of zero order. In (1) the well-known equality for Bessel funtions was used (e.g in [6]). Let s use the Euler s formula and represent the exponential form (1) in the trigonometri one (by sine and osine): Π m = π M exp( iωt+ikx) + πi M exp( iωt+ikx) J (kr)os( ω r +z )rdr r +z J (kr)sin( ω r +z )rdr (13) r +z Eah integral from (13) an be evaluated exatly (not approximately), by virtue [7, p. 775, (6.737)]: J (kr)os( ω πz N 1( z ω ) r +z, ω )rdr 4 ω > k; = r +z (14) z K1( z k ω ), ω π 4 k ω < k. J (kr)sin( ω πz J r +z )rdr 1( z ω ), ω = 4 ω r +z > k; (15), ω < k. 5

6 Bessel funtion of half-integer order may be expressed through elementary funtions ([6]): N 1(x) = J 1(x) = π πx sin(x), K ± 1 (x) = x exp( x) os(x), (16) πx where N 1(x) and K ± 1(x) are modified Bessel s funtions of half-integer order. It is onvenient to make the following definition: ω, ω q = > k i k ω, ω < k (17) One an return to the exponents and with respet to (16) express (13) in the form: Π m = i qπ M exp( iωt + ikx + iq z ) (18) Expression (18) desribes typial ase of Cherenkov radiation. If wave veloity exeeds the light one then the TEM wave is radiated. In the opposite ase eletromagneti fields exponentially derease in z diretion. Using (1) we readily find the eletri and magneti fields expression: H = i k q π M E = e i q πm ωk exp( iωt+ikx+iq z ) exp( iωt+ikx+iq z ){ q e 1 +k e 3 }, (19) where e 1, e, e 3 - unit Cartesian vetors, signs orrespond to the z > and z < respetively. Also easy the Pointing vetor an be written: S = πωm k q {±q e 3 +k e 1 }, () where signs ± orrespond to the z > and z < respetively. 6

7 4 Conlusion Evidently, that the travelling wave of the eletri dipoles at the plate like (6): P = P exp( iωt+i k r), gives the same solution as (18) with the simple hanges Π m Π e and M P and orresponding expressions for fields: A = 1 Π e t, ϕ = div Π e 5 Aknowledgments I am grateful to Dr. Boris Ivanov for helpful disussion and advie. Referenes [1] B.M.Bolotovsky, V.L.Ginzburg, Usp. Fiz. Nauk 16, 577 (197). [] I.V.Bar yakhtar, B.A.Ivanov, Fiz. Nizk. Temp. 5, 759 (1979) [Sov. J. Low Temp. Phys. 5, 361 (1979)]; Solid State Commun (198). [3] A.F.Andreev, V.I.Marhenko, Usp. Fiz. Nauk (198) [Sov. Phys. Usp. 3, 1 (198)]. [4] H.J.Mikeska. J Phys. C 13, 913 (198). [5] B.A.Ivanov, A.K.Kolezhuk, Fiz. Nizk. Temp. 1, 355 (1995) [Low Temp. Phys. 1, 75 (1995)]. [6] Abramowitz M., Stegun I. Handbook of mathematial funtions with formulas, graphs, and mathematial tables. Washington, D.C.: Government Printing Offie [7] I.S.Gradshtein, I.M.Ruzhik. Tables of integrals, summs, series and produts. Moskow. Phyzmatlit