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Spin-reorientation phase transition: Magnetic resonance, acoustic and optic properties
Journal of Magnetism and Magnetic Materials 60 (1986) 306-310 North-Holland, Amsterdam
306
SPIN-REORIENTATION PHASE TRANSITION: ACOUSTIC AND OPTIC PROPERTIES *
MAGNETIC
RESONANCE,
E.A. TUROV and A.A. LUGOVOI Institute of Metal Physics, Ural Centre of Ac. SC. USSR, GSP-170 Sverdlovsk
620219, USSR
Received 4 December 1985
The problem dealt within this paper is the study of soft modes near the spin-reorientation phase transition (SRPT) in ferromagnets (and partly in antiferromagnets) when the magnetoelastic interaction is taken into account. A detailed discussion is given for the long wavelength magnetoacoustic modes (MAM) and their influence on various physical properties. The results are: the conditions for linear and nonlinear (parametric) excitations of MAM by an ac magnetic field; MAM contributions to the local magnetic susceptibility, nuclear magnetic and magnetoacoustic resonances; the modulation of the sound velocity by an ac magnetic field; and the magnetic birefrigence of light by sound waves. All these phenomena manifest a sharp increase near the SRPT.
1. Introduction
A spin-reorientation phase transition (SRPT) is the transition induced by temperature, magnetic field or pressure that results in a reorientation of sublattice magnetizations relative to crystal axes without any change of the magnetic structure. The SRPT is accompanied by a mode softening which drives the transition. Near the SRPT point many physical properties have an extraordinary behaviour. (For the review see, e.g., refs. [l-3] and references therein). The present paper gives a systematic consideration of such a phenomenon connected essentially with the magnetoelastic (ME) interaction and coupled ME waves, mainly for the example of a ferromagnet. The results apply to both dielectric and metal magnets if one neglects the conductivity of the latter. Starting with the stability conditions for an ME system, we then relate these conditions to soft modes in such a system. Of particular interest is the case when the softening of a magnon mode is
* Presented at the Intern. Conf. on Magn., 26-30 August 1985, San Francisco, California, USA.
0304-8853/86/$03.50
(North-Holland
transfered the ME interaction to a quasi-phonon (magnetoacoustic) mode (MAM). The linear and nonlinear (parameteric) excitation of these latter modes and their damping are investigated and various phenomena associated with these modes are considered with a single point of view (the soft mode one). In particular, we consider: the contribution of MAMs to the local magnetic susceptibility, the nuclear magnetic and magnetoacoustic resonances, the modulation of the sound velocity by ac magnetic field, and the magnetic birefrigence of light by acoustic waves. All these phenomena manifest a sharp increase just near the SRPT. Some of these phenomena have been discussed previously from different points of view (see, again refs. [l-3] and additionally refs. [4,5]); others are considered here for the first time. Finally, some special features of the phenomena under consideration in antiferromagnets are pointed out.
2. Ferromagneti modal, stability conditions Let us consider a ferromagnet with a ground state magnetization M,(]H]]X in which the energy variation that is bilinear in magnetization and strain perturbations of the plane-wave type has the
0 Elsevier Science Publishers B.V. Physics Publishing Division)
E.A. Turov, A.A. Lugovoi / Spin-reorientation
in (1) only terms where there is some dynamic interaction between the magnetic (m I) and elastic ( Aea8) variables.
form AFZk = fM,( H,,m$
+ 2Bb6m,,Aexy
+ H,,m~)
307
phase transition
+2&mZAeXZ + 2G6(AeXJ2 +2C,,(Ae,,)2+2=Mi(kmL/k)2,
(1)
3. The susceptibility
wherem=M/M,,(m2=1),m,=m-m,={m,,
m, }. Here 3ua/ax,)/2 their ground ment vector).
Let us consider the state near the ET point
Ae,, = ea8 - e$ = (au,/ax, + are the strain variations relative to state values e$ (U is the displaceThen
Hik = H;( ak)2 + HAj + H + HMEi
(i = 2,3)
(2)
are two effective fields which consist of exchange (Hk), anisotropy ( HAi), external (H) and magnetoelastic ( HMEi) contributions; k is the wave vector, a is the interatomic parameter. Cji and Bii are elastic and magnetoelastic (ME) constants. The last term in (1) represents the magnetostatic (volume) interaction. The surface demagnetization fields connected with the shape anisotropy of the sample can be included in HAi. Besides crystallographic and shape anisotropy, the fields HAi include the contribution of ME strains in the amsotropy as well [2]. Then the stability conditions for the state under consideration must be H + HAi > 0, which follows from the positive value of the quadratic form (1) on m I and he,, since HMEZ
=
&i/GM.,
3
HME~
=
&/G,%.
(3)
These fields bring about the spontaneous symmetry breaking effect, or the frozen lattice effect [2]. The equations H,,+H=O
(i=2,3)
HA2 + H = 0 so that H2,, < H30 in (1) (see, (2) at k + 0). The components my is appropriate for an excitation by an ac field k(t) = li(k, w) exp(ikr -
iwt) (L(]Y). Assume firstly that k]]M,-,]]X. The combined solution of the Landau-Lifshitz equation for m and the dynamic equation of the elastic medium for u gives the susceptibility
x,&G)
= ~J44,(ij:,
- w2)/R(kd
(5)
where R(k,w)=(Lj~-W2)(ijtZk--2)-ekijZkW:k +;*-J)(~;pJ’), 0; = 3,,i&) i&k
=
(6)
@ik = Wik -
‘d,k,,i?&
,
wtk
=
irw, s,k,
Wik= YHik9 s:
=
(7)
C,,/P, (8)
Here fi,,,, and ‘iiPk are complex frequencies obtained from solutions of the equation R( k,w) = 0.
(9)
The parameters
r and n describe the energy dissipation in the magnetic and elastic subsystems respectively. From (7) and (2) at the PT point we obtain
(4)
define the spin-reorientation phase transition (SRF’T) points for the phase transitions (ET) or second order, or the lability points for ET’s of first order. The ET may take place as a function of magnetic field, temperature or pressure. The parameters of the SRPT are my or m, correspondingly for i = 2 or 3 in (4). The form (1) is valid both for rhombic crystals and for crystals with higher symmetry (cubic or uniaxial) when it is rhombically broken by external fields or a shape anisotropy. We have included
60 =&,k-0
= l.
4. Quasi-magnon
(10)
mode and ME gap
At k + 0 when the dynamic strains Ae,,ak + 0, it follows from (9) (or from expression (1)) that there is the only one mode with a non-zero frequency: w; = 02&g.
01)
E.A. Turov, A.A. Lugovoi / Spin-reorientation phase transition
308
at the ET point:
At the ET point (& + 1) this reaches to (tin) -
-@ME=Y
*0
%&A3+lHA2~)
*
02)
This is just the ME gap, or frozen lattice effect for the quasi-magnon mode (see, the review paper ref. [2]). For k # 0 in the longwave region, where o,~ s wo, eq. (9) gives for this mode (that is at 6J 3 00) D,,
= omk - )iL\D,, ,
where Q2 Ati:*,
=52 2k03k =
+
tk”rk
7
r( u2k + 6.1~~)= ru3k.
The susceptibility (5) at this frequency equals x,(Q,,)
=
ix;y:y(%k) =i~M~/rfLk.
5. Quasi-phonon (magnetoacoustic)
03)
mode
At frequencies so low that o +Z w. eq. (9) has the single solution fiPk = 52,, - +ihG!,, ,
04)
05)
This means the dynamic elastic modulus C& and the velocity SF of such waves are determined by the expression c,*, = pS,*2 = C&l
-&).
06)
At the FT point C&ak2 and SF&k. From (8) and (2) this special dispersion of the magnetoacoustic waves in ferromagnets manifests itself in extremely narrow regions of wave vector k and magnetic fields H in the vicinity of the ET: H-IH,,(
= (17 + “/YHr&‘%/~~
(18) where w,, = s/a has the value order of the Debye frequency. Because of the small value of HME2 this ratio can become rather large. Nevertheless, the softening of the quasi-acoustic mode has been observed in some easy-plane ferro- and antiferromagnets versus H near the SRPT (see, references in ref. [2]). Note: the ark mode is a magnetoacoustic mode with m I IlulY, and it can be excited by a magnetic field b(]Y. As follows from (5), the susceptibility at the frequency w = 52,, equals:at the frequency o = Qpk equals: x,,&&)
=
ix;#,k)
=
iyMo/(
r +
rl~&,,,/S:
(17)
The reason is that usualy HMEi < l-10 Oe. Besides, the damping of such waves has a maximum
)
$k
9
09)
When comparing (19) with (13) one can see that the absorption peak in the region (17) at the frequency 52,, can be higher than at the frequency 52,, inasmuch as S2pk-=z< a,,,,. 6. The role of magnetostatic
where
AfiPk = ‘& (“I + &/Y&m)-
A$&,&=1
fields
Assume now the vector k forms an angle (Pi with Mo]]x in the XY plane. One can see easily from (1) that for dynamic phenomena all changes compared with the case k]]Mo are reduced to the renormalization of Hzk: Hu --) H2,c
+
4nMosin2q,.
At the same time the, stability conditions (4) remain unchanged. It means the parameter & (8) is now replaced by & =
HM,~/(
H2k
+
4nMosin2e).
(20)
Therefore near the ET point ( HA2 + H = 0) even for the long-wave case, when H2k = HMu, we have & = to = H,,/(H,,, + 4vMosin2g+). to =z 1, with the exception of the region of small angles (PkGJKJ=Z+$ where lo = 1 as before. As a result, for ‘pk > + the
309
E.A. Turov, A.A. Lugovoi / Spin-reorientation phase transition
ME coupling becomes ineffective so that instead of (14) we obtain Gpk = wtk. We should like to draw attention to some interesting effect: if the magnetic field H is modulated (in time or in space) so that the vector M,, is rotated by an angle of the order of +, then the modulus C*66 and the velocity $7 from (16) will show at corresponding modulation. 7. A plane-parallel plate Proceeding to samples of finite size (L) we must note two conditions of the applicability of the above considerations. The first condition ensuring the correctness of the frozen-lattice conception is the inequality S,L_’
< Wh,fE.
(21)
And the second condition is connected with a fact that the modes above were plane waves. In the long-wave limit this will only be true for a planeparallel plate (lamina) with the normal nllk. Otherwise, one must consider magnetostatic (Walker) modes and their coupling with elastic waves. The above formulas are actually applicable with small changes for two cases. For lamina with n]]k]lM,(IWI]X one has simply to replace H by H - 4&f, in all formulas from (1) to (19). The other case is n]lkllY and i&llHllX. Here we can again use the formulas from (2) to (19) by replacing the field HA2 where it appears by HA2 + 4n&,. The essential result is that the point where HA2 + 4+&+H=O becomes a PT point. As k+O .& 2: &, = HME2/( HA2 + 4&f,, + HMEz ) = 1 even though k I M,,. It should be noted that in laminar samples the uniform magnetic field 6(t)]] Y can excite ME oscillations with the frequency s2,, (14) at the discrete values of the wave number k = a(2n + l)L-’
(n = 0,1,2, . . . ),
(22)
where L is the lamina thickness. (The situation here is analogous to the usual spin-wave resonance in films and, as there, the exact calculation of the resonance intensity requites that we take the boundary conditions for m and e,+ into account.)
8. NMR through the magnetoacoustic
mode
The magnetic component of the magnetoacoustic oscillations at the frequency tipk creates a hyperfine field (HF) on the nuclei of the magnetic atoms
where n,, = AX,,,, is a so-called enhancement coefficient. This field can cause NMR if its frequency is equal to the NMR frequency o,, = y&V0 = yH,. For lamina placed in a uniform field hy (t ) there are two forms for the resonance. The first is the forced uniform oscillations of M, which gives q,, = AX,,,, (w = 0) = H,,/Hzo (we assume w, +Z oO). The second just corresponds to excitation of magnetoacoustic waves with the wave number k (22). This last form will be more effective than the former when wave numbers (22)exist for the lamina under consideration for which the condition (17) is fulfilled, provided the corresponding mode numbers do not turn out to be large (since the total bulk oscillating magnetic moment falls with the value of n). Such phenomena have been observed by Petrov et al. [4] for the weak ferromagnet FeBO,.
9. Acoustic NMR One can see from (1) that the elastic acoustic strains Ae,,, cause the rotational oscillations of the magnetization by the angle cp= my = -2&(G6/&)Aexy.
(23)
This creates a HF field on the nuclei h,, = H,,cp which can provide an acoustic NMR. Its intensity increase with & near the PT point.
10. Acoustic modulation of the light Since the acoustic wave turns M through an angle cp (23), then thesis will be a modulation of the magnetic birefrigence of light [5] that will increase near the PT point.
E.A. Turov, A.A. Lugovoi / Spin-reorientation
310
11. Parametric excitation of magnetoacoustic
waves
The magnetization oscillations in such waves are polarized quite close to the axis Y. This means the longitudinal magnetization M, will oscillate with double frequency. Thus, these waves can be excited by a longitudinal field h]]Mo]]x at a frequency w = 2QPK. This again is observable by NMR [4].
phase transition
Dzyaloshinskii’s field. It means the effects under consideration will be fairly large (Ek = 1) under the condition
instead of (17) for ferromagnets. For HME = 1 Oe this condition can be valid up to H = lo3 Oe. Some other new aspects of magnetoacoustic coupling in ferro- and antiferromagnets can be found in ref. [6].
12. Antiferromagnets
References
Since our goal has been to reveal physical features of this phenomena, we have considered the simple case of ferromagnets. However, because of the exchange enhancement of the ME coupling, antiferromagnets are preferable for study. The consideration of antiferromagnets in detail will be elsewhere. We shall only mention the principal feature of those results. We take as an example the easy-plane antiferromagnets (with weak ferromagnetism) cll-Fe,O, and FeBO,. The ME parameter & for them equals
[l] K.P. Belov, A.K. Zvesdin, A.M. Kadomtzeva and R.L. Levitin, Orientation Phase Transitions in Rear-Earth Magnetics (Nat&a, Moscow, 1979) (in Russian). [2] E.A. Turov and V.G. Shavrov, Usp. Fir. Nauk 140 (1983) 429 (in Russian). E.A. Turov, in: The Mechanical Behavior of Electromagnetic Solid Continua, ed. G.A. Maugin. (North-Holland, Amsterdam, 1984) p. 255. [3] R.M. White, R.J. Nemanich and C.P. Herring, Phys. Rev. B25 (1982) 1822. [4] M.P. Petrov, A.V. Ivanov, E.R. Komeev, and G.T. Andreeva, JETF 78 (1980) 1147. [5] N.N. Evtikheev, V.V. Moshkin, V.L. Preobrazhenskii and N.A. Ekonomov, Pis’ma v ZHETF 35 (1982) 31. [6] V.G. Bar’yakhtar and E.A. Turov, in: Magnetic Excitations, ed. AS. Borovic-Romanov and SK. Sinha (North-Holland, Amsterdam, 1986).
5k = G/[
&(ak)2
+ H(H + J&)/2&
+ H,,]
where H, is the uniform exchange field and H, is
306
SPIN-REORIENTATION PHASE TRANSITION: ACOUSTIC AND OPTIC PROPERTIES *
MAGNETIC
RESONANCE,
E.A. TUROV and A.A. LUGOVOI Institute of Metal Physics, Ural Centre of Ac. SC. USSR, GSP-170 Sverdlovsk
620219, USSR
Received 4 December 1985
The problem dealt within this paper is the study of soft modes near the spin-reorientation phase transition (SRPT) in ferromagnets (and partly in antiferromagnets) when the magnetoelastic interaction is taken into account. A detailed discussion is given for the long wavelength magnetoacoustic modes (MAM) and their influence on various physical properties. The results are: the conditions for linear and nonlinear (parametric) excitations of MAM by an ac magnetic field; MAM contributions to the local magnetic susceptibility, nuclear magnetic and magnetoacoustic resonances; the modulation of the sound velocity by an ac magnetic field; and the magnetic birefrigence of light by sound waves. All these phenomena manifest a sharp increase near the SRPT.
1. Introduction
A spin-reorientation phase transition (SRPT) is the transition induced by temperature, magnetic field or pressure that results in a reorientation of sublattice magnetizations relative to crystal axes without any change of the magnetic structure. The SRPT is accompanied by a mode softening which drives the transition. Near the SRPT point many physical properties have an extraordinary behaviour. (For the review see, e.g., refs. [l-3] and references therein). The present paper gives a systematic consideration of such a phenomenon connected essentially with the magnetoelastic (ME) interaction and coupled ME waves, mainly for the example of a ferromagnet. The results apply to both dielectric and metal magnets if one neglects the conductivity of the latter. Starting with the stability conditions for an ME system, we then relate these conditions to soft modes in such a system. Of particular interest is the case when the softening of a magnon mode is
* Presented at the Intern. Conf. on Magn., 26-30 August 1985, San Francisco, California, USA.
0304-8853/86/$03.50
(North-Holland
transfered the ME interaction to a quasi-phonon (magnetoacoustic) mode (MAM). The linear and nonlinear (parameteric) excitation of these latter modes and their damping are investigated and various phenomena associated with these modes are considered with a single point of view (the soft mode one). In particular, we consider: the contribution of MAMs to the local magnetic susceptibility, the nuclear magnetic and magnetoacoustic resonances, the modulation of the sound velocity by ac magnetic field, and the magnetic birefrigence of light by acoustic waves. All these phenomena manifest a sharp increase just near the SRPT. Some of these phenomena have been discussed previously from different points of view (see, again refs. [l-3] and additionally refs. [4,5]); others are considered here for the first time. Finally, some special features of the phenomena under consideration in antiferromagnets are pointed out.
2. Ferromagneti modal, stability conditions Let us consider a ferromagnet with a ground state magnetization M,(]H]]X in which the energy variation that is bilinear in magnetization and strain perturbations of the plane-wave type has the
0 Elsevier Science Publishers B.V. Physics Publishing Division)
E.A. Turov, A.A. Lugovoi / Spin-reorientation
in (1) only terms where there is some dynamic interaction between the magnetic (m I) and elastic ( Aea8) variables.
form AFZk = fM,( H,,m$
+ 2Bb6m,,Aexy
+ H,,m~)
307
phase transition
+2&mZAeXZ + 2G6(AeXJ2 +2C,,(Ae,,)2+2=Mi(kmL/k)2,
(1)
3. The susceptibility
wherem=M/M,,(m2=1),m,=m-m,={m,,
m, }. Here 3ua/ax,)/2 their ground ment vector).
Let us consider the state near the ET point
Ae,, = ea8 - e$ = (au,/ax, + are the strain variations relative to state values e$ (U is the displaceThen
Hik = H;( ak)2 + HAj + H + HMEi
(i = 2,3)
(2)
are two effective fields which consist of exchange (Hk), anisotropy ( HAi), external (H) and magnetoelastic ( HMEi) contributions; k is the wave vector, a is the interatomic parameter. Cji and Bii are elastic and magnetoelastic (ME) constants. The last term in (1) represents the magnetostatic (volume) interaction. The surface demagnetization fields connected with the shape anisotropy of the sample can be included in HAi. Besides crystallographic and shape anisotropy, the fields HAi include the contribution of ME strains in the amsotropy as well [2]. Then the stability conditions for the state under consideration must be H + HAi > 0, which follows from the positive value of the quadratic form (1) on m I and he,, since HMEZ
=
&i/GM.,
3
HME~
=
&/G,%.
(3)
These fields bring about the spontaneous symmetry breaking effect, or the frozen lattice effect [2]. The equations H,,+H=O
(i=2,3)
HA2 + H = 0 so that H2,, < H30 in (1) (see, (2) at k + 0). The components my is appropriate for an excitation by an ac field k(t) = li(k, w) exp(ikr -
iwt) (L(]Y). Assume firstly that k]]M,-,]]X. The combined solution of the Landau-Lifshitz equation for m and the dynamic equation of the elastic medium for u gives the susceptibility
x,&G)
= ~J44,(ij:,
- w2)/R(kd
(5)
where R(k,w)=(Lj~-W2)(ijtZk--2)-ekijZkW:k +;*-J)(~;pJ’), 0; = 3,,i&) i&k
=
(6)
@ik = Wik -
‘d,k,,i?&
,
wtk
=
irw, s,k,
Wik= YHik9 s:
=
(7)
C,,/P, (8)
Here fi,,,, and ‘iiPk are complex frequencies obtained from solutions of the equation R( k,w) = 0.
(9)
The parameters
r and n describe the energy dissipation in the magnetic and elastic subsystems respectively. From (7) and (2) at the PT point we obtain
(4)
define the spin-reorientation phase transition (SRF’T) points for the phase transitions (ET) or second order, or the lability points for ET’s of first order. The ET may take place as a function of magnetic field, temperature or pressure. The parameters of the SRPT are my or m, correspondingly for i = 2 or 3 in (4). The form (1) is valid both for rhombic crystals and for crystals with higher symmetry (cubic or uniaxial) when it is rhombically broken by external fields or a shape anisotropy. We have included
60 =&,k-0
= l.
4. Quasi-magnon
(10)
mode and ME gap
At k + 0 when the dynamic strains Ae,,ak + 0, it follows from (9) (or from expression (1)) that there is the only one mode with a non-zero frequency: w; = 02&g.
01)
E.A. Turov, A.A. Lugovoi / Spin-reorientation phase transition
308
at the ET point:
At the ET point (& + 1) this reaches to (tin) -
-@ME=Y
*0
%&A3+lHA2~)
*
02)
This is just the ME gap, or frozen lattice effect for the quasi-magnon mode (see, the review paper ref. [2]). For k # 0 in the longwave region, where o,~ s wo, eq. (9) gives for this mode (that is at 6J 3 00) D,,
= omk - )iL\D,, ,
where Q2 Ati:*,
=52 2k03k =
+
tk”rk
7
r( u2k + 6.1~~)= ru3k.
The susceptibility (5) at this frequency equals x,(Q,,)
=
ix;y:y(%k) =i~M~/rfLk.
5. Quasi-phonon (magnetoacoustic)
03)
mode
At frequencies so low that o +Z w. eq. (9) has the single solution fiPk = 52,, - +ihG!,, ,
04)
05)
This means the dynamic elastic modulus C& and the velocity SF of such waves are determined by the expression c,*, = pS,*2 = C&l
-&).
06)
At the FT point C&ak2 and SF&k. From (8) and (2) this special dispersion of the magnetoacoustic waves in ferromagnets manifests itself in extremely narrow regions of wave vector k and magnetic fields H in the vicinity of the ET: H-IH,,(
= (17 + “/YHr&‘%/~~
(18) where w,, = s/a has the value order of the Debye frequency. Because of the small value of HME2 this ratio can become rather large. Nevertheless, the softening of the quasi-acoustic mode has been observed in some easy-plane ferro- and antiferromagnets versus H near the SRPT (see, references in ref. [2]). Note: the ark mode is a magnetoacoustic mode with m I IlulY, and it can be excited by a magnetic field b(]Y. As follows from (5), the susceptibility at the frequency w = 52,, equals:at the frequency o = Qpk equals: x,,&&)
=
ix;#,k)
=
iyMo/(
r +
rl~&,,,/S:
(17)
The reason is that usualy HMEi < l-10 Oe. Besides, the damping of such waves has a maximum
)
$k
9
09)
When comparing (19) with (13) one can see that the absorption peak in the region (17) at the frequency 52,, can be higher than at the frequency 52,, inasmuch as S2pk-=z< a,,,,. 6. The role of magnetostatic
where
AfiPk = ‘& (“I + &/Y&m)-
A$&,&=1
fields
Assume now the vector k forms an angle (Pi with Mo]]x in the XY plane. One can see easily from (1) that for dynamic phenomena all changes compared with the case k]]Mo are reduced to the renormalization of Hzk: Hu --) H2,c
+
4nMosin2q,.
At the same time the, stability conditions (4) remain unchanged. It means the parameter & (8) is now replaced by & =
HM,~/(
H2k
+
4nMosin2e).
(20)
Therefore near the ET point ( HA2 + H = 0) even for the long-wave case, when H2k = HMu, we have & = to = H,,/(H,,, + 4vMosin2g+). to =z 1, with the exception of the region of small angles (PkGJKJ=Z+$ where lo = 1 as before. As a result, for ‘pk > + the
309
E.A. Turov, A.A. Lugovoi / Spin-reorientation phase transition
ME coupling becomes ineffective so that instead of (14) we obtain Gpk = wtk. We should like to draw attention to some interesting effect: if the magnetic field H is modulated (in time or in space) so that the vector M,, is rotated by an angle of the order of +, then the modulus C*66 and the velocity $7 from (16) will show at corresponding modulation. 7. A plane-parallel plate Proceeding to samples of finite size (L) we must note two conditions of the applicability of the above considerations. The first condition ensuring the correctness of the frozen-lattice conception is the inequality S,L_’
< Wh,fE.
(21)
And the second condition is connected with a fact that the modes above were plane waves. In the long-wave limit this will only be true for a planeparallel plate (lamina) with the normal nllk. Otherwise, one must consider magnetostatic (Walker) modes and their coupling with elastic waves. The above formulas are actually applicable with small changes for two cases. For lamina with n]]k]lM,(IWI]X one has simply to replace H by H - 4&f, in all formulas from (1) to (19). The other case is n]lkllY and i&llHllX. Here we can again use the formulas from (2) to (19) by replacing the field HA2 where it appears by HA2 + 4n&,. The essential result is that the point where HA2 + 4+&+H=O becomes a PT point. As k+O .& 2: &, = HME2/( HA2 + 4&f,, + HMEz ) = 1 even though k I M,,. It should be noted that in laminar samples the uniform magnetic field 6(t)]] Y can excite ME oscillations with the frequency s2,, (14) at the discrete values of the wave number k = a(2n + l)L-’
(n = 0,1,2, . . . ),
(22)
where L is the lamina thickness. (The situation here is analogous to the usual spin-wave resonance in films and, as there, the exact calculation of the resonance intensity requites that we take the boundary conditions for m and e,+ into account.)
8. NMR through the magnetoacoustic
mode
The magnetic component of the magnetoacoustic oscillations at the frequency tipk creates a hyperfine field (HF) on the nuclei of the magnetic atoms
where n,, = AX,,,, is a so-called enhancement coefficient. This field can cause NMR if its frequency is equal to the NMR frequency o,, = y&V0 = yH,. For lamina placed in a uniform field hy (t ) there are two forms for the resonance. The first is the forced uniform oscillations of M, which gives q,, = AX,,,, (w = 0) = H,,/Hzo (we assume w, +Z oO). The second just corresponds to excitation of magnetoacoustic waves with the wave number k (22). This last form will be more effective than the former when wave numbers (22)exist for the lamina under consideration for which the condition (17) is fulfilled, provided the corresponding mode numbers do not turn out to be large (since the total bulk oscillating magnetic moment falls with the value of n). Such phenomena have been observed by Petrov et al. [4] for the weak ferromagnet FeBO,.
9. Acoustic NMR One can see from (1) that the elastic acoustic strains Ae,,, cause the rotational oscillations of the magnetization by the angle cp= my = -2&(G6/&)Aexy.
(23)
This creates a HF field on the nuclei h,, = H,,cp which can provide an acoustic NMR. Its intensity increase with & near the PT point.
10. Acoustic modulation of the light Since the acoustic wave turns M through an angle cp (23), then thesis will be a modulation of the magnetic birefrigence of light [5] that will increase near the PT point.
E.A. Turov, A.A. Lugovoi / Spin-reorientation
310
11. Parametric excitation of magnetoacoustic
waves
The magnetization oscillations in such waves are polarized quite close to the axis Y. This means the longitudinal magnetization M, will oscillate with double frequency. Thus, these waves can be excited by a longitudinal field h]]Mo]]x at a frequency w = 2QPK. This again is observable by NMR [4].
phase transition
Dzyaloshinskii’s field. It means the effects under consideration will be fairly large (Ek = 1) under the condition
instead of (17) for ferromagnets. For HME = 1 Oe this condition can be valid up to H = lo3 Oe. Some other new aspects of magnetoacoustic coupling in ferro- and antiferromagnets can be found in ref. [6].
12. Antiferromagnets
References
Since our goal has been to reveal physical features of this phenomena, we have considered the simple case of ferromagnets. However, because of the exchange enhancement of the ME coupling, antiferromagnets are preferable for study. The consideration of antiferromagnets in detail will be elsewhere. We shall only mention the principal feature of those results. We take as an example the easy-plane antiferromagnets (with weak ferromagnetism) cll-Fe,O, and FeBO,. The ME parameter & for them equals
[l] K.P. Belov, A.K. Zvesdin, A.M. Kadomtzeva and R.L. Levitin, Orientation Phase Transitions in Rear-Earth Magnetics (Nat&a, Moscow, 1979) (in Russian). [2] E.A. Turov and V.G. Shavrov, Usp. Fir. Nauk 140 (1983) 429 (in Russian). E.A. Turov, in: The Mechanical Behavior of Electromagnetic Solid Continua, ed. G.A. Maugin. (North-Holland, Amsterdam, 1984) p. 255. [3] R.M. White, R.J. Nemanich and C.P. Herring, Phys. Rev. B25 (1982) 1822. [4] M.P. Petrov, A.V. Ivanov, E.R. Komeev, and G.T. Andreeva, JETF 78 (1980) 1147. [5] N.N. Evtikheev, V.V. Moshkin, V.L. Preobrazhenskii and N.A. Ekonomov, Pis’ma v ZHETF 35 (1982) 31. [6] V.G. Bar’yakhtar and E.A. Turov, in: Magnetic Excitations, ed. AS. Borovic-Romanov and SK. Sinha (North-Holland, Amsterdam, 1986).
5k = G/[
&(ak)2
+ H(H + J&)/2&
+ H,,]
where H, is the uniform exchange field and H, is