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On the studies of spin-wave dynamics in ferromagnets by polarized neutron scattering
Physica B 183 (1993) North-Holland
326-330
On the studies of spin-wave dynamics in ferromagnets polarized neutron scattering B.P. Toperverg’, Leningrad Received
V.V. Deriglazov and V.E. Mikhailova’
Nuclear Physics Institute, Leningrad, 5 February
by
Russian Federation
1992
The spin-wave (s-w) contribution to the part of the polarized neutron scattering cross section proportional to the incident polarization is considered theoretically. It is shown that a set of parameters, i.e. s-w stiffness and damping and the constant of dipole-dipole interaction, can be extracted from the analysis of the small-angle scattering pattern.
1. Introduction
As is well known the spin-waves (s-w) well describe the propagation of the transverse fluctuations of the magnetization in ordered Heisenberg ferromagnets at sufficiently low wavevectors q and frequencies w. Strictly speaking, being hydrodynamic type modes, they exist in the limit qr, 4 1 and ot, G 1 where rc and t, are some microscopic scales in space and time respectively. Below the Curie temperature T,, the parameter rc is of the order of-the exchange interaction range, whereas tilt, has the order of the exchange coupling constant .I,,= T,. The s-w spectrum is well defined as E,, = Dq’, where the s-w stiffness D 2: .J,r,‘. The s-w damping r caused by a nonlinear process like magnon-magnon collisions is small and could easily be accounted for under restrictions qrc 4 1 and wt, Q 1 within the frame of perturbation theory. The intermagnon interaction contributes also to the correlations of the magnetization fluctuation components parallel to Correspondence too: B.P. Toperverg, Laboratoire Lton Brillouin, C.E.N., Saclay 91191, Gif-sur-Yvette, France. ’ Guest of the Summer Program at the Brookhaven National Laboratory, Upton, NY 11973, USA. ‘Permanent address: Joint Nuclear Research and Nuclear Energetics Institute, Sofia 1784, Bulgaria.
0921-4526/93/$06.00
0
1993 - Elsevier
Science
Publishers
the total magnetic moment M. This contribution is sufficient to account for in the first nonvanishing order of the perturbation theory expansion either. However, as the temperature increases, i.e. T= IT- T,)T, 1--, 0 the scales r, and t, are renormalizing and one has to consider rc(-r)+ m as a correlation length and t,(T) + 00as a correlation time of the critical fluctuations, respectively. Thus the range of the s-w description is shrunken drastically. On the other hand, at low q or at large distances, the spin dynamics is strongly influenced by the dipole-dipole (d-d) interaction between atomic moments p. Usually the d-d coupling constant w0 = 4~rp*/v~, where V, is the volume per magnetic moment p, is small, i.e. w0 + J,. Nevertheless d-d forces could not be neglected and they become rather essential in the range q s qO, where q,, = (w~/T~)“~v~“~ is defined by the equation E,,(qO> = Dqi = 47rp.M. This definition is quite natural because the s-w spectrum renormalized by the d-d interaction has the form E(4) = {&,(H)(&,(H)
+ 4TPM[l_
(em)‘])>“’?
E,(H)= Dq* + pH. Here H = He - 47~fiM is the internal and He the external field strength, respectively, fi is the demagnetizing factor, e = q/q and m = M/M. Unfortunately, the expression for the spectrum written above is reliably valid with-
B.V. All rights
reserved
B. P. Toperverg et al. I On spin-wave dynamics in ferromagnets
in the range q + q. where the d-d renormalization is small (see ref. [l]). Otherwise one could expect a predominant contribution of the d-d interaction to the s-w damping r which at q G qO may exceed the real part of the spectrum c(q). Thus s-w definitely exists within the frame qO 4 q4r,‘. This apparently brings quite strong demands to the neutron scattering experiment. Here we consider theoretically small-angle polarized neutron scattering by spin-waves in the special ‘inclined’ geometry of the experiment first applied for s-w studies in ref. [2]. As will be shown, the distribution of the scattered polarized neutrons over the plane perpendicular to the beam incidence contains some details that make it possible not only to extract from the data analysis parameters characterizing s-w, but also to control the degree of the validity of the hydrodynamic approach. The results of the theory are applied in ref. [3] for the interpretation of recent experiments.
S(q,w, PO)= KmB{(Sap - e,“e,P) + i(.epByP~+ nae,B - etnP)}
where e, = Q/Q and n = [e, x PO]. Choosing P, = Porn one has S = S, + P,,S, where the first term collects from K”’ the components symmetric with respect to the interchange of indices (Y and p, and the latter one collects the antisymmetric components. The tensor K”‘( q, m, o) contains 9 components which can be expressed via corresponding components of the retarded Greens function G”‘(qo)
=
@(q, w) = (sg,s!,-,)
d24Pd= dL! dw
with the tensor transfer Q: & A’M;
orthogonal
jF(Q>l’s
w, P,,) ,
z
x”‘(q,
w)
where xaP is the tensor of magnetic susceptibility. For Heisenberg ferromagnets, the Green function Gzp consists only of three independent components and can be represented as - rnamP)Gt - ic np”mYG, + mamPG,
(3)
The magnetic scattering cross-section of polarized neutrons is proportional to the Fourier transform in space and time of the spin-correlation tensor
convoluted momentum
, (2)
Gzp = (@
2. The tensor of spin-correlations and the polarized neutron scattering cross-section
327
to the
where G, and G, describe, respectively, symmetric and antisymmetric correlations of the spin projections perpendicular to the direction m. The latter term in eq. (3) corresponds to the longitudinal component of the tensor G‘@. If the vector m is directed along the z-axis, then G, = G,, = Gyy, G, = iG,, = -iG,_. and G, = G,,. In accordance with the general properties of susceptibilities one has G,,,(w) = GF,(-o) and G,(w) = - Gz(-o). Therefore, it is convenient to represent G,,, in the form
(1)
where AM = ye*/m,c*, F(Q) is the magnetic scattering form factor, Q = k’ - k = q + 7, T is a vector of the reciprocal lattice, w = E’ - E is the energy transfer, k and E are the momentum and energy of the incident, whereas k’ and E’ are those of the scattered beam; the function S( q, co, P,) consists of two terms, one independent of the incident polarization P,, and the other proportional to P,:
G,(o) = $G,{2 - w[(w + E, + iT,)-’ + (0 - E, + ic)-‘I}
,
(4)
G,(w) = $ Go{ o[(w + E, + iT,)-’
- (w - E, + ir,))‘]}
,
(5)
where ct,,( q, w) and c,,( q, o) are some real even functions of w. In principle, E, # E, and c f r,. However, as q+- 0, o + 0 they are ap-
B.P. Toperverg et al. I On spin-wave dynamics in ferromagnets
328
proaching the same limit i.e. E, = E, = E,(H) is the s-w spectrum,
gives, at w 6 r, S, = -2(em)’
C = C = &(4r,)*E,(W
z Im G’,( q, 0) ,
ln*(qr,) 6
is the leading term in s-w damping, r, = 1 and G,(q)= -_(S)E;‘(W If the d-d interaction is taken into account, the tensor GaP has all components and in accordance with ref. [l] can be written as
a
2: (S)(l+ ho’?) 1 + w,eiG,
20
(w+iF)2-g2
(8)
’
-2 = &i(H)(l - 211) - (TA)* )
&
F=r(l-A),
A=
%W(l2~,(W(l+
ef) woefG,) ’
G-6 = [G’“P + wO{GfaPGtpY
where A = 1+ W,,epeYGfF” and beyond the ‘dip:!. regi;;’ where ( qorC) < 1, as expected [l], Substitution of the expressions g)-(F)?& eq. (6) with A = 0 immediately gives the s-w spectrum renormaliztition discussed above in the case ,r = 0 and G, = 0. Again, the tensor G@ can be decomposed into two parts, i.e. a symmetric and antisymmetric one. Symmetric components bring a contribution to S,( qw) and antisymmetric to S,( qw), respectively. Substituting eq. (6) and (3) into eq. (2) gives for S,, S, = 2[N(w) + l] Im G,{(eom)* - 2
[(l - ef - e’, 7 &G,
- eLell(e,m)G,,ll (7)
where e,,, e, and e, are the projections of the vector e onto three orthogonal vectors: m, II = [m x e,], j = e, - m(egn) and N(w) is the Planck function. A similar but slightly more complicated expression can easily be derived also for S,. Thus d-d interaction mixes all the components in G”’ : G,, G, and G,. However, their contribution to the scattering pattern can in principle be separated from each other accounting for the fact that at low q the momentum transfer is Q = 7 and using the measurements at different orientations between 7 and m. This allows also to control the influence of d-d interaction for the spin-correlations. In the case of small-angle scattering eq. (7) combined with eqs. (4) and (5)
At w,, = 0, (?a = G, and from eqs. (8) follows the well-known expression for S,. Neglecting longitudinal fluctuations one obtains the d-d renormalization of the s-w spectrum discussed above with some additional contribution from the damping IY Moreover, d-d forces renormalize essentially the effective width of the s-w spectrum changing its dependence on q and making it anisotropic. The contamination of the longitudinal fluctuations brings an additional change in effective spectrum and damping. It should be noted that G,(w) is a complex function of o. However, considering the range where E % r the real part of G, seems to be most important in eqs. (8). On the other hand in the range 9 4 qo, where d-d interaction can not be considered as a perturbation, one has to account d-d forces when calculating both r and G,. At the present time almost no results are known on the topic. Therefore, considering in the next section the small-angle scattering cross-section integrated over energy transfers, we will assume q > qo, ( qrC) G 1 and neglect the contribution of the longitudinal correlator G, to S, at all.
3. The small-angle polarized neutron scattering The most attractive application of polarized neutrons is connected with the possibility to separate the magnetic scattering from other processes contributing to the measured crosssection. However, the term in the cross-section proportional to incident polarization contains two different contributions, i.e. the inelastic
B. P. Toperverg
et al. / On spin-wave dynamics in ferromagnets
magnetic scattering by antisymmetric components of the tensor of magnetic fluctuations and the term corresponding to the interference of the magnetic and nuclear scattering. The latter one is proportional to the factor {(mPO) - (em)(ePO)} and in principle can be subtracted from the data using its angular dependence within the plane perpendicular to the incident-beam direction. The process of our only interest is the inelastic neutron s-w scattering and its contribution to the part of the cross section proportional to P,, is convenient to be represented using eq. (8) as follows:
(em)2 = (z2 - f3’) cos2?P + 0: + 2e, sin 29 [z2+ e"]
x,(x,-e,)
=L 2
R,
(x,
[-4%01 [Z -(2zf?,)']'+ [4y"zeo]2 '
(10)
z=w/2E41,
if2 = E”i(z)(l - 2X) - (pi)")
(i”=
h = pHIDk2,
mo(S)(l- ef) 2E”,(z)Dk2
’
y”(z) = Y(Z)(l
- /i”) 7
y(z) = y0[z2 + 02]E”,(z) ln2Pgo , -y. = ro(kQ2
,
j3 = Dk2q ,
+ R0}1’2,
R,=ei-e2-h,
(11) Y+=Y(z~),
-0, + +(lRol + R0)1’2.
the following dimen-
eo=EIDk24,
zo=z2+ 02+!z,
- e,)‘+ e2 + h ’
R+ = {R; + Y:}~‘~,
z,=
where we have introduced sionless variables:
e, sin 2rY
- eo)’ + e2 (x,
X, = +2-1’2{R, dz (em)”
7
where 0, = 8 cos 4, e,= 8 sin (6, e2 = 0; + 0: and C#Iis the angle between the planes defined by the following couples of vectors: (k, m) and (k, k’). If, for instance, the SANS is measured by a two-coordinate detector and m lies in the horizontal plane, then 0, is the scattering angle in the horizontal and 0, in the vertical direction. As the d-d interaction is neglected, the integral in eq. (10) within the s-w range go(z) < y(z) can be treated analytically and represented as J, = I, - L_, where: I
J, = %
329
r. = 1
and 0 4 1 is the scattering -angle. Within the exchange approxim_ation A = 0, E(Z) = go(z), T(z) = y(z) and Im G, = Im G, is an odd function of o. Thus the integral in eq. (10) could be nonzero only if the factor (em)’ also contains some oddness as a function of z. Actually, this takes place if the angle $ between vectors m and k is not zero or $# in, because at ze1,841 one has
The direct computation of the integral (10) shows that this expression very well approximates the two-dimensional lineshape of J,(e, 4) shown in fig. 1 within the broad range of parameters z, < ye. From fig. 1 it is clearly seen that there exists a cutoff angle 8 = 0, and at 8 > 0, the scattering is dropped to zero. However, it still persists in some range at 8 > $ which results from accounting for the s-w damping. On the contrary, for the ideal spin-waves the scattering at 8 > 0, is forbidden. At 8+ 6Joa square root singularity is developing, i.e. J,(e) = (e, - e)-“‘. This singularity is smearing out by s-w damping. Unfortunately, the integral (10) cannot be evaluated analytically when d-d renormalization is essential. In fig. 2 we have shown the results of the numerical calculations in accordance with eq. (10). The dashed lines in these figures correspond to the calculations at w. = 0, and solid lines show the behavior of J,(e) when d-d forces are switched on. In general, the distributions of J,(e,, e,,) looks similar in both the cases w. = 0 and w. # 0. However, there appears some quali-
B. P. Toperverg et al. I On spin-wave dynamics in ferromagnets
330 G-4
n
of J( q,, q,) upon q, = kB cos 4, q, = being the scattering angle; exchange approximation. Fig. 1. Dependence
kB sin 4, q = /co, 0 = (f3: + 0:}“’
8 Fig. 2. Dependence of J,(q) at (a) C$= 0, (b) C$= 45”; dashed lines correspond to the exchange approximation, solid lines show the influence of the d-d interaction.
tatively new feature in the J,(f3,, 0,,) dependence that makes it possible to check the influence of the d-d forces. Namely, if eY= 0, the weak new singularity is developing at some angle 0 < 0, < 0,. This singularity becomes more pronounced when one scans along the direction in the area detector at the angle 4 # 0. In fig. 2(a) this angle is chosen to be zero, whereas in fig. 2(b) 4 = 45”. Moreover the two-dimensional picture in fig. 1 appears to be distorted by d-d interaction. In particular, as is seen from fig. 2(b) the cutoff angle 0, now is strongly dependent on 4. It should be noted that the degree of distortion is quite sensitive to the angle I,!Ibetween the vectors m and k. All these obstacles are rather promising to study the d-d renormalization of s-w behavior via analysis of the two-dimensional pattern of small-angle polarized neutron scattering. The first attempt of such a kind of analysis is reported in ref. [3].
Acknowledgements
We would like to acknowledge S.V. Maleyev, A.I. Okorokov and R. Kampmann for fruitful discussions. One of us (B.P.T.) deeply appreciates J. Axe, G. Shirane, S. Shapiro and L. Rebelsky for discussions and hospitality at BNL, C. Majkrzak and J. Lynn-at NIST- and M. Grahn for her kind assistance at BNL.
References
111S.V. Maleyev, Sov. Sci. Rev. A 8 (1988) 323. PI A.I. Okorokov, VV. Runov, B.P. Toperverg et al., JETP Lett. 43 (1986) 503. A. Okorokov, V. Runov, B.P. Toperverg, R. Kampmann, H. Eckerlebe, W. Schmidt and W. Lobner, Physica B 180&181 (1992) 262.
[31 V. Deriglazov,
326-330
On the studies of spin-wave dynamics in ferromagnets polarized neutron scattering B.P. Toperverg’, Leningrad Received
V.V. Deriglazov and V.E. Mikhailova’
Nuclear Physics Institute, Leningrad, 5 February
by
Russian Federation
1992
The spin-wave (s-w) contribution to the part of the polarized neutron scattering cross section proportional to the incident polarization is considered theoretically. It is shown that a set of parameters, i.e. s-w stiffness and damping and the constant of dipole-dipole interaction, can be extracted from the analysis of the small-angle scattering pattern.
1. Introduction
As is well known the spin-waves (s-w) well describe the propagation of the transverse fluctuations of the magnetization in ordered Heisenberg ferromagnets at sufficiently low wavevectors q and frequencies w. Strictly speaking, being hydrodynamic type modes, they exist in the limit qr, 4 1 and ot, G 1 where rc and t, are some microscopic scales in space and time respectively. Below the Curie temperature T,, the parameter rc is of the order of-the exchange interaction range, whereas tilt, has the order of the exchange coupling constant .I,,= T,. The s-w spectrum is well defined as E,, = Dq’, where the s-w stiffness D 2: .J,r,‘. The s-w damping r caused by a nonlinear process like magnon-magnon collisions is small and could easily be accounted for under restrictions qrc 4 1 and wt, Q 1 within the frame of perturbation theory. The intermagnon interaction contributes also to the correlations of the magnetization fluctuation components parallel to Correspondence too: B.P. Toperverg, Laboratoire Lton Brillouin, C.E.N., Saclay 91191, Gif-sur-Yvette, France. ’ Guest of the Summer Program at the Brookhaven National Laboratory, Upton, NY 11973, USA. ‘Permanent address: Joint Nuclear Research and Nuclear Energetics Institute, Sofia 1784, Bulgaria.
0921-4526/93/$06.00
0
1993 - Elsevier
Science
Publishers
the total magnetic moment M. This contribution is sufficient to account for in the first nonvanishing order of the perturbation theory expansion either. However, as the temperature increases, i.e. T= IT- T,)T, 1--, 0 the scales r, and t, are renormalizing and one has to consider rc(-r)+ m as a correlation length and t,(T) + 00as a correlation time of the critical fluctuations, respectively. Thus the range of the s-w description is shrunken drastically. On the other hand, at low q or at large distances, the spin dynamics is strongly influenced by the dipole-dipole (d-d) interaction between atomic moments p. Usually the d-d coupling constant w0 = 4~rp*/v~, where V, is the volume per magnetic moment p, is small, i.e. w0 + J,. Nevertheless d-d forces could not be neglected and they become rather essential in the range q s qO, where q,, = (w~/T~)“~v~“~ is defined by the equation E,,(qO> = Dqi = 47rp.M. This definition is quite natural because the s-w spectrum renormalized by the d-d interaction has the form E(4) = {&,(H)(&,(H)
+ 4TPM[l_
(em)‘])>“’?
E,(H)= Dq* + pH. Here H = He - 47~fiM is the internal and He the external field strength, respectively, fi is the demagnetizing factor, e = q/q and m = M/M. Unfortunately, the expression for the spectrum written above is reliably valid with-
B.V. All rights
reserved
B. P. Toperverg et al. I On spin-wave dynamics in ferromagnets
in the range q + q. where the d-d renormalization is small (see ref. [l]). Otherwise one could expect a predominant contribution of the d-d interaction to the s-w damping r which at q G qO may exceed the real part of the spectrum c(q). Thus s-w definitely exists within the frame qO 4 q4r,‘. This apparently brings quite strong demands to the neutron scattering experiment. Here we consider theoretically small-angle polarized neutron scattering by spin-waves in the special ‘inclined’ geometry of the experiment first applied for s-w studies in ref. [2]. As will be shown, the distribution of the scattered polarized neutrons over the plane perpendicular to the beam incidence contains some details that make it possible not only to extract from the data analysis parameters characterizing s-w, but also to control the degree of the validity of the hydrodynamic approach. The results of the theory are applied in ref. [3] for the interpretation of recent experiments.
S(q,w, PO)= KmB{(Sap - e,“e,P) + i(.epByP~+ nae,B - etnP)}
where e, = Q/Q and n = [e, x PO]. Choosing P, = Porn one has S = S, + P,,S, where the first term collects from K”’ the components symmetric with respect to the interchange of indices (Y and p, and the latter one collects the antisymmetric components. The tensor K”‘( q, m, o) contains 9 components which can be expressed via corresponding components of the retarded Greens function G”‘(qo)
=
@(q, w) = (sg,s!,-,)
d24Pd= dL! dw
with the tensor transfer Q: & A’M;
orthogonal
jF(Q>l’s
w, P,,) ,
z
x”‘(q,
w)
where xaP is the tensor of magnetic susceptibility. For Heisenberg ferromagnets, the Green function Gzp consists only of three independent components and can be represented as - rnamP)Gt - ic np”mYG, + mamPG,
(3)
The magnetic scattering cross-section of polarized neutrons is proportional to the Fourier transform in space and time of the spin-correlation tensor
convoluted momentum
, (2)
Gzp = (@
2. The tensor of spin-correlations and the polarized neutron scattering cross-section
327
to the
where G, and G, describe, respectively, symmetric and antisymmetric correlations of the spin projections perpendicular to the direction m. The latter term in eq. (3) corresponds to the longitudinal component of the tensor G‘@. If the vector m is directed along the z-axis, then G, = G,, = Gyy, G, = iG,, = -iG,_. and G, = G,,. In accordance with the general properties of susceptibilities one has G,,,(w) = GF,(-o) and G,(w) = - Gz(-o). Therefore, it is convenient to represent G,,, in the form
(1)
where AM = ye*/m,c*, F(Q) is the magnetic scattering form factor, Q = k’ - k = q + 7, T is a vector of the reciprocal lattice, w = E’ - E is the energy transfer, k and E are the momentum and energy of the incident, whereas k’ and E’ are those of the scattered beam; the function S( q, co, P,) consists of two terms, one independent of the incident polarization P,, and the other proportional to P,:
G,(o) = $G,{2 - w[(w + E, + iT,)-’ + (0 - E, + ic)-‘I}
,
(4)
G,(w) = $ Go{ o[(w + E, + iT,)-’
- (w - E, + ir,))‘]}
,
(5)
where ct,,( q, w) and c,,( q, o) are some real even functions of w. In principle, E, # E, and c f r,. However, as q+- 0, o + 0 they are ap-
B.P. Toperverg et al. I On spin-wave dynamics in ferromagnets
328
proaching the same limit i.e. E, = E, = E,(H) is the s-w spectrum,
gives, at w 6 r, S, = -2(em)’
C = C = &(4r,)*E,(W
z Im G’,( q, 0) ,
ln*(qr,) 6
is the leading term in s-w damping, r, = 1 and G,(q)= -_(S)E;‘(W If the d-d interaction is taken into account, the tensor GaP has all components and in accordance with ref. [l] can be written as
a
2: (S)(l+ ho’?) 1 + w,eiG,
20
(w+iF)2-g2
(8)
’
-2 = &i(H)(l - 211) - (TA)* )
&
F=r(l-A),
A=
%W(l2~,(W(l+
ef) woefG,) ’
G-6 = [G’“P + wO{GfaPGtpY
where A = 1+ W,,epeYGfF” and beyond the ‘dip:!. regi;;’ where ( qorC) < 1, as expected [l], Substitution of the expressions g)-(F)?& eq. (6) with A = 0 immediately gives the s-w spectrum renormaliztition discussed above in the case ,r = 0 and G, = 0. Again, the tensor G@ can be decomposed into two parts, i.e. a symmetric and antisymmetric one. Symmetric components bring a contribution to S,( qw) and antisymmetric to S,( qw), respectively. Substituting eq. (6) and (3) into eq. (2) gives for S,, S, = 2[N(w) + l] Im G,{(eom)* - 2
[(l - ef - e’, 7 &G,
- eLell(e,m)G,,ll (7)
where e,,, e, and e, are the projections of the vector e onto three orthogonal vectors: m, II = [m x e,], j = e, - m(egn) and N(w) is the Planck function. A similar but slightly more complicated expression can easily be derived also for S,. Thus d-d interaction mixes all the components in G”’ : G,, G, and G,. However, their contribution to the scattering pattern can in principle be separated from each other accounting for the fact that at low q the momentum transfer is Q = 7 and using the measurements at different orientations between 7 and m. This allows also to control the influence of d-d interaction for the spin-correlations. In the case of small-angle scattering eq. (7) combined with eqs. (4) and (5)
At w,, = 0, (?a = G, and from eqs. (8) follows the well-known expression for S,. Neglecting longitudinal fluctuations one obtains the d-d renormalization of the s-w spectrum discussed above with some additional contribution from the damping IY Moreover, d-d forces renormalize essentially the effective width of the s-w spectrum changing its dependence on q and making it anisotropic. The contamination of the longitudinal fluctuations brings an additional change in effective spectrum and damping. It should be noted that G,(w) is a complex function of o. However, considering the range where E % r the real part of G, seems to be most important in eqs. (8). On the other hand in the range 9 4 qo, where d-d interaction can not be considered as a perturbation, one has to account d-d forces when calculating both r and G,. At the present time almost no results are known on the topic. Therefore, considering in the next section the small-angle scattering cross-section integrated over energy transfers, we will assume q > qo, ( qrC) G 1 and neglect the contribution of the longitudinal correlator G, to S, at all.
3. The small-angle polarized neutron scattering The most attractive application of polarized neutrons is connected with the possibility to separate the magnetic scattering from other processes contributing to the measured crosssection. However, the term in the cross-section proportional to incident polarization contains two different contributions, i.e. the inelastic
B. P. Toperverg
et al. / On spin-wave dynamics in ferromagnets
magnetic scattering by antisymmetric components of the tensor of magnetic fluctuations and the term corresponding to the interference of the magnetic and nuclear scattering. The latter one is proportional to the factor {(mPO) - (em)(ePO)} and in principle can be subtracted from the data using its angular dependence within the plane perpendicular to the incident-beam direction. The process of our only interest is the inelastic neutron s-w scattering and its contribution to the part of the cross section proportional to P,, is convenient to be represented using eq. (8) as follows:
(em)2 = (z2 - f3’) cos2?P + 0: + 2e, sin 29 [z2+ e"]
x,(x,-e,)
=L 2
R,
(x,
[-4%01 [Z -(2zf?,)']'+ [4y"zeo]2 '
(10)
z=w/2E41,
if2 = E”i(z)(l - 2X) - (pi)")
(i”=
h = pHIDk2,
mo(S)(l- ef) 2E”,(z)Dk2
’
y”(z) = Y(Z)(l
- /i”) 7
y(z) = y0[z2 + 02]E”,(z) ln2Pgo , -y. = ro(kQ2
,
j3 = Dk2q ,
+ R0}1’2,
R,=ei-e2-h,
(11) Y+=Y(z~),
-0, + +(lRol + R0)1’2.
the following dimen-
eo=EIDk24,
zo=z2+ 02+!z,
- e,)‘+ e2 + h ’
R+ = {R; + Y:}~‘~,
z,=
where we have introduced sionless variables:
e, sin 2rY
- eo)’ + e2 (x,
X, = +2-1’2{R, dz (em)”
7
where 0, = 8 cos 4, e,= 8 sin (6, e2 = 0; + 0: and C#Iis the angle between the planes defined by the following couples of vectors: (k, m) and (k, k’). If, for instance, the SANS is measured by a two-coordinate detector and m lies in the horizontal plane, then 0, is the scattering angle in the horizontal and 0, in the vertical direction. As the d-d interaction is neglected, the integral in eq. (10) within the s-w range go(z) < y(z) can be treated analytically and represented as J, = I, - L_, where: I
J, = %
329
r. = 1
and 0 4 1 is the scattering -angle. Within the exchange approxim_ation A = 0, E(Z) = go(z), T(z) = y(z) and Im G, = Im G, is an odd function of o. Thus the integral in eq. (10) could be nonzero only if the factor (em)’ also contains some oddness as a function of z. Actually, this takes place if the angle $ between vectors m and k is not zero or $# in, because at ze1,841 one has
The direct computation of the integral (10) shows that this expression very well approximates the two-dimensional lineshape of J,(e, 4) shown in fig. 1 within the broad range of parameters z, < ye. From fig. 1 it is clearly seen that there exists a cutoff angle 8 = 0, and at 8 > 0, the scattering is dropped to zero. However, it still persists in some range at 8 > $ which results from accounting for the s-w damping. On the contrary, for the ideal spin-waves the scattering at 8 > 0, is forbidden. At 8+ 6Joa square root singularity is developing, i.e. J,(e) = (e, - e)-“‘. This singularity is smearing out by s-w damping. Unfortunately, the integral (10) cannot be evaluated analytically when d-d renormalization is essential. In fig. 2 we have shown the results of the numerical calculations in accordance with eq. (10). The dashed lines in these figures correspond to the calculations at w. = 0, and solid lines show the behavior of J,(e) when d-d forces are switched on. In general, the distributions of J,(e,, e,,) looks similar in both the cases w. = 0 and w. # 0. However, there appears some quali-
B. P. Toperverg et al. I On spin-wave dynamics in ferromagnets
330 G-4
n
of J( q,, q,) upon q, = kB cos 4, q, = being the scattering angle; exchange approximation. Fig. 1. Dependence
kB sin 4, q = /co, 0 = (f3: + 0:}“’
8 Fig. 2. Dependence of J,(q) at (a) C$= 0, (b) C$= 45”; dashed lines correspond to the exchange approximation, solid lines show the influence of the d-d interaction.
tatively new feature in the J,(f3,, 0,,) dependence that makes it possible to check the influence of the d-d forces. Namely, if eY= 0, the weak new singularity is developing at some angle 0 < 0, < 0,. This singularity becomes more pronounced when one scans along the direction in the area detector at the angle 4 # 0. In fig. 2(a) this angle is chosen to be zero, whereas in fig. 2(b) 4 = 45”. Moreover the two-dimensional picture in fig. 1 appears to be distorted by d-d interaction. In particular, as is seen from fig. 2(b) the cutoff angle 0, now is strongly dependent on 4. It should be noted that the degree of distortion is quite sensitive to the angle I,!Ibetween the vectors m and k. All these obstacles are rather promising to study the d-d renormalization of s-w behavior via analysis of the two-dimensional pattern of small-angle polarized neutron scattering. The first attempt of such a kind of analysis is reported in ref. [3].
Acknowledgements
We would like to acknowledge S.V. Maleyev, A.I. Okorokov and R. Kampmann for fruitful discussions. One of us (B.P.T.) deeply appreciates J. Axe, G. Shirane, S. Shapiro and L. Rebelsky for discussions and hospitality at BNL, C. Majkrzak and J. Lynn-at NIST- and M. Grahn for her kind assistance at BNL.
References
111S.V. Maleyev, Sov. Sci. Rev. A 8 (1988) 323. PI A.I. Okorokov, VV. Runov, B.P. Toperverg et al., JETP Lett. 43 (1986) 503. A. Okorokov, V. Runov, B.P. Toperverg, R. Kampmann, H. Eckerlebe, W. Schmidt and W. Lobner, Physica B 180&181 (1992) 262.
[31 V. Deriglazov,