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The temperature dependence of one-magnon light scattering in anisotropic ferromagnets
201 THE TEMPERATURE DEPENDENCE OF ONE-MAGNON LIGHT SCATTERING IN ANISOTROPIC FERROMAGNETS M.G. C O T T A M and A. L A T I F F A W A N G Department of Physics, University of Essex, Colchester, England A Green function formalism is employed to evaluate the temperature dependence of one-magnon light scattering in ferromagnets with single-ion anisotropy. The calculations include both linear and quadratic magneto-optical coupling, and the results are described mainly for the case of spin S = 1. The theory is applied to the ferromagnetic phase of FeCl2 for temperatures up to about 2T~.
1. Introduction A theory is presented for the temperature dependence of one-magnon light scattering in Heisenberg ferromagnets with single-ion anisotropy. The Hamiltonian of the system is assumed to take the general form Ye = -½ E Jo{S~S] + S~Sf + (1 + tr)S~S~} i,i
- g
.nE s: i
- O~
(SD 2 - F ~ i
{($3 2 - (SD2},
(1)
i
where Jij is the exchange interaction between sites i and j, H is an applied magnetic field in the z-direction, and D and F are single-ion anisotropy parameters. The possibility of anisotropic exchange is included by allowing parameter tr to be non-zero. Light scattering from a ferromagnet in the absence of anisotropy effects has been calculated by L o n d o n [1] over a wide range of temperatures. His calculations involved only linear magneto-optical coupling, but more recently Wettling et al. [2] have considered the additional effects of quadratic magneto-optical coupling on light scattering in isotropic ferromagnets at low temperatures T < T¢. The present calculations are concerned with investigating the influence of anisotropy effects on the light scattering spectrum. The results obtained are valid over a wide range of temperatures above and below the critical temperature T¢, and they include linear and quadratic magneto-optical coupling. The results are given here only in outline; detailed calculations will be published elsewhere [3]. 2. General theory Using [2] the differential cross-section for Physica 89B (1977)201-204 © North-HoUand
light scattered into solid angle df~ may be expressed as d2h
2tolto~n2
df~ dto 2
c4Vnl{1 - exp [ - h(toi - to2)/k,T]}
x Im ((A~; A*))F,
(2)
where n~ and n2 are the refractive indices at incident frequency to~ and scattered frequency to2, k is the difference between the wavevectors of the incident and scattered light, and *\ ((Ak; Ak),E is a retarded Green function evaluated at energy E = hto~-hto2+iO +. The onemagnon scattering processes are described by the interaction term Ak, where Ak = ½E ei~"i[K{pAS}- - P*S+~} i
+ C{ps(S;S; + s ; s ; ) + p*(STs; + sTs;)}]. (3) Here K and G are coefficients for the linear and quadratic magneto-optical coupling, respectively. As defined in eq. (3) these coefficients are real for a transparent ferromagnet, whereupon K is proportional to the magnetic circular birefringence (or Faraday rotation) and G is proportional to the magnetic linear birefringence (or C o t t o n - M o u t o n effect), as discussed in [2]. The quantities Ps and PA are symmetric and antisymmetric polarisation factors: z
+
+
z
Ps = e~e2 + e~ e2,
z
+
+
z
PA = e~e2 -- e~ e 2,
(4)
where e~ and e2 are unit vectors in the direction of the electric fields of the incident and scattered light. The one-magnon scattering intensity is deduced by substituting eq. (3) into eq, (2) and evaluating the spin dependent Green functions contained in ((Ak; A*))E. In the special case of F = 0 the only non-vanishing Green functions are of the form ((S+; Sj-)), ((Si +Slz + S~~S~, +" S;))~,
202 ({S/" Sj S~ + S;S; )),~ and ({S/S~ + S~S/; $ 7 S ~ + NfSi })~:, together with the corresponding Green functions with the operators in the reverse ortier. More generally when [ : v ' 0 other spindependent Green functions in ((A~ A~)}~ are also non-zero (e.g. ((S,+:,S', ))~. ((S,' N, .g~Si' " S/))6, etc.), and these must also be evahlated to obtain deh/d~ dw~. The method of calculation inw~lves writing down the equation of motion [4] for each of the required Green functions. The exchange dependent terms may then be simplified by decoupling products of spin operators at different sites using the Random Phase Approximation (RPA):
((S,',,S,:,:
S~ ))~ : (S~)((S,;,: & ))~.
(m ~ n ).
f5~
However, the single-ion anisotropy terms give rise to Green functions in the equations of motion which involve products of operators at the same site and which cannot be decoupled as in eq. (5). This difficulty has been discussed by various authors [5,6], who have suggested modifications of eq. (5) for decoupling operators at the same site, but except in the limit of T tt these approximations are at best only valid for very small D and F. In the present work we avoid any decoupling of the single-ion anisotropy terms. Instead we follow an approach similar to that first employed by Murao and Matsubara [71 to investigate the thermodynamic properties of a S = 1 ferromagnet with uniaxial anisotropy ( F = 0). They found that for this system the Green functions generated by the anisotropy form a closed set of equations and need not be decoupled. In particular using only R P A for the exchange terms it may be shown [7] that for any operator B the Green functions ({St; B))E and ((Si+ Siz_{_ SizSi+ ; B))E are coupled only to each other in their equations of motion. Thus in the present calculations if we take B equal to S i and (Sj S~+ S~S/) in turn we have altogether four coupled equations for S = I, which may then be solved to obtain the four basic Green functions which we have already mentioned as being required to evaluate dZh/daQ dw: in the case of F = 0. The method may be extended to higher spin S and to the case of F # 0. Generally we find that 4S coupled equations are required to solve for d2h/dl'), dw~ when F = 0, and this becomes 8S equations when F # 0. The method
is therefore suitable for relatively low values of the spin S, and we have carried out calculations for S 1 and 5; = I. 3. Discussion of results
For simplicity we coasider first an S-: I ferromagnet with uniaxial anisotropy i F 0). The Green function ((Ak:A*))r describing the oncmagnon light scattering is found to be I,~(E)
4rr( E
E,; )t E
- b.',i )
Lz( E) + ................. , 4rr(E + E,~)~E + E,,)
~6)
where E,I = ~ t t , H + ml(! 4 (r)J(0)-+ E,t.
(7)
E,
(g)
{~m ~J2(0) + [)2 _ m~DJ(O)},~,
I,I(E)-(E x'l~l~H){miK: 2m~KG + m l G z} + D{m2K'- 2 m i K ( ; + m~G:} ,l(O)tnl~(I 4 or)K:
2tnlm2(I + tr)K(;
and L2(E) is simihu- to I<(E) but with E ~ E and G--*- G. Here tnl and m~ denote the the> real averages (S-) and {3{(S~)2) - 2} respectively. which may be ewduated using the Green funclion formalism, and J(O)=Y~iJ O. We have assumed a typical scattering geometry with e~ along the z-axis and e, in the xy-plane, and we have made the usual approximation of taking k--~ 0 since one-magnon light scattering involves only excitations close to the Brillouin zone centre. The excitation energies of the system (for k - 0) are represented by E~ and E,, and they correspond to excitations associated with a change tAm,t= 1 in the magnetic quantum number, as discussed in [7]. In the limit of D ~ 0 these modes have the simple interpretation that E, reduces to the usual spin wave energy (simply gl~BH + o-mlJ(O) for k - 0) whilst E~ is an optical excitation which reduces to the Zeeman energy gl~uH + ( 1 + cr)m J(O). The scattering cross-section is now obtained by taking the imaginary part of (6) at energy E = fioo] - hw2 + iO +, and substituting into eq. (2). In general two Stokes peaks are predicted at frequencies ¢o~ given by oo~-[Eg[/h and w, [E,il/h, and likewise there are two Anti-Stokes peaks at ~o, = w,-~ [E/~[/h and w, + [EO[/h. For example, if E~; and E0 are both positive, the
203 Stokes scattering contribution c o m e s f r o m the first term in eq. (6) and is found to be d2h
tolto~n2
dO dto2 - 4 Vc4n~Eo x a(h~o,-
[L,(Eo)(I + N +) h~o2 -
- LI(Eo)(1 + N
Eo) ) a ( h t o l - hto2 - E o ) ] ,
(10) where N -~ are the thermal occupation numbers {exp (EolkBT)-1} 1 for each excitation. It follows f r o m eq. (10) that at t e m p e r a t u r e T the integrated intensities I * ( T ) and I - ( T ) , associated with the Stokes scattering at frequencies a h - Eo/h and w t - Eo/h respectively, are given by to,to~n2(l + N +-) I~-(T) = [L,(Eo)[. (11)
4hVc4nlEo
A detailed analysis of eqs. (10) and ( l l ) shows that most of the scattered intensity is associated with the usual spin w a v e energy Eo. In particular for low temperatures T , ~ Tc we have m~ ~- m 2 ~- l, w h e r e u p o n it may be verified that the weighting factor Lt(E~) associated with I+(T) is zero, so that in this approximation all the intensity is described by scattering f r o m Eo. The results of eqs. (6)-(11) may be extended to F ~ 0 and to higher values of the spin; details will be given in [3]. Qualitatively we may note that for S = 1 and F ~ 0 the Green function ((Ak; A*))~ m a y still be expressed in the same general f o r m as in eq. (6), except that the expressions for the excitation energies Eft and Eo and for the weighting factors L I ( E ) and L2(E) are more complicated. In the case of S = 3 there are three excitation energies, one of which reduces to the usual spin w a v e energy when D ~ 0 and F ~ 0. It is this excitation which gives rise to the dominant scattering peak, whilst much weaker intensities are associated with scattering f r o m the other two (optical) modes. In the isotropic limit our results simplify to give agreement with previous light scattering calculations [1,2] for the special cases considered by these authors. 4. A p p l i c a t i o n
to FeCI2
The theory has been applied to the romagnetic phase of anhydrous FeCI2 (S Tc = 24 K) in which the anisotropy effects large. This material consists essentially of
fer= 1, are fer-
romagnetically ordered layers, with the intralayer ferromagnetic exchange interactions being much stronger than the weak antiferromagnetic interaction between layers [8]. In moderate applied magnetic fields ( H ~> 11 kOe) the system b e h a v e s as a ferromagnet. The description of FeC12 in terms of an S = I Heisenberg Hamiltonian has been justified by Birgeneau et al. [8], who also deduced values for the exchange and anisotropy p a r a m e t e r s using neutron scattering data. We take J~ = 5.5 cm -~ and ./2 = - 0 . 7 c m J for the nearest and next-nearest exchange interactions within the plane, and J ' = - 0 . 2 cm -~ as the w e a k exchange coupling between layers. The values of the anisotropy parameters are less reliably known than the exchange interactions (see discussion in [8]), and we take here D = 9.4 cm -1, F = 0 and or = 0.3. With the a b o v e choice of p a r a m e t e r s the effective anisotropy field gttBHA at T < Tc is approximately 17cm -j, in agreement with experiment [8]. We also require the ratio G/K of the magneto-optical coupling coefficients in order to evaluate the t e m p e r a t u r e d e p e n d e n c e of the scattered intensity. In principle this ratio (which may be a function of the incident laser f r e q u e n c y too can be deduced from measurements of the linear and quadratic magneto-optical effects, as discussed in [2] for YIG. If it is assumed that only linear magneto-optical coupling is important (G/K = 0) then the calculated t e m p e r a t u r e d e p e n d e n c e of the integrated intensity I-(T) for scattering at to2 = o J ~ - E ~ / h is shown in fig. 1. A qualitatively similar temperature d e p e n d e n c e is predicted if G/K~ O, and this is also illustrated in fig. 1 by the curves for G[K = _0.1. All curves are plotted in terms
1.5
_Q
~
1.0
0.5
z w I.z -- 0 0
G/K= -0.1
01.5
11.01.51 T/T
2.0
c
Fig. 1. Calculated temperature dependence of the intensity ratio 1 (T)/I (0) for FeCI2 in a field H = 14kOe and with GIK taking the values 0.1, 0, -0.1.
2(14 of the r e d u c e d variable I (T)/I (0), where I (0) can be d e d u c e d f r o m eq. (11). An applied field of H = 1 4 k O e has been a s s u m e d . The much w e a k e r intensity I+(T) associated with scattering at a)~- E~/h has also been evaluated, and we find that I+(T)/(I (T) takes values 0, 0.08 and 0.37 for T/T+ = 0, (I.5 and 1.0 respectively (in the case of G/K = 0). At low t e m p e r a t u r e s ( T <~ T D the calculated values of the excitation energies E,~ and E,; c o r r e s p o n d i n g to H 1 4 k O e are 2 0 . 3 c m ~ and 29.1 cm ~, respectively.+ F u r t h e r details of the t h e o r y , particularly for S > I and for F:~ 0 will be given in [3], together with applications to particular materials. W o r k is in progress to extend the t h e o r y to light scattering in anisotropic antiferromagnets.
where there is more scope for c o m p a r i s o n with experiment.
References [I] R. [,oudon, J. Phys. ('3 {It~7il) N72 [21 W. Wettling, M.G. ('~ttam and ,I.R. Sandercock. ,I. Phv', ('8 (1975t 211 [3l M.G. Cottam and A. l,atiff Awang, to, be published. [41 D.N. Zubarev, Soviet P h y s i c s - U s p e k h i 3 f l960) 32(i. [51 F.B. Anderson and H.B. ('allen, Phys. Rev. la6 t1964t 1068. 161 M.E. l+ines. Phys. R e v 156 {1967) 534, [71 T. Murao and T. Malsuhara, ,1. Phys. ,%oc. Japan 2":, (196g) 352. [~1 R.J. Birgeneat,. W.B. Yehm. t+; ('ohetl and L Makov,,k),, Phys. Rev. t'/5 (1¢172~ 2607
1. Introduction A theory is presented for the temperature dependence of one-magnon light scattering in Heisenberg ferromagnets with single-ion anisotropy. The Hamiltonian of the system is assumed to take the general form Ye = -½ E Jo{S~S] + S~Sf + (1 + tr)S~S~} i,i
- g
.nE s: i
- O~
(SD 2 - F ~ i
{($3 2 - (SD2},
(1)
i
where Jij is the exchange interaction between sites i and j, H is an applied magnetic field in the z-direction, and D and F are single-ion anisotropy parameters. The possibility of anisotropic exchange is included by allowing parameter tr to be non-zero. Light scattering from a ferromagnet in the absence of anisotropy effects has been calculated by L o n d o n [1] over a wide range of temperatures. His calculations involved only linear magneto-optical coupling, but more recently Wettling et al. [2] have considered the additional effects of quadratic magneto-optical coupling on light scattering in isotropic ferromagnets at low temperatures T < T¢. The present calculations are concerned with investigating the influence of anisotropy effects on the light scattering spectrum. The results obtained are valid over a wide range of temperatures above and below the critical temperature T¢, and they include linear and quadratic magneto-optical coupling. The results are given here only in outline; detailed calculations will be published elsewhere [3]. 2. General theory Using [2] the differential cross-section for Physica 89B (1977)201-204 © North-HoUand
light scattered into solid angle df~ may be expressed as d2h
2tolto~n2
df~ dto 2
c4Vnl{1 - exp [ - h(toi - to2)/k,T]}
x Im ((A~; A*))F,
(2)
where n~ and n2 are the refractive indices at incident frequency to~ and scattered frequency to2, k is the difference between the wavevectors of the incident and scattered light, and *\ ((Ak; Ak),E is a retarded Green function evaluated at energy E = hto~-hto2+iO +. The onemagnon scattering processes are described by the interaction term Ak, where Ak = ½E ei~"i[K{pAS}- - P*S+~} i
+ C{ps(S;S; + s ; s ; ) + p*(STs; + sTs;)}]. (3) Here K and G are coefficients for the linear and quadratic magneto-optical coupling, respectively. As defined in eq. (3) these coefficients are real for a transparent ferromagnet, whereupon K is proportional to the magnetic circular birefringence (or Faraday rotation) and G is proportional to the magnetic linear birefringence (or C o t t o n - M o u t o n effect), as discussed in [2]. The quantities Ps and PA are symmetric and antisymmetric polarisation factors: z
+
+
z
Ps = e~e2 + e~ e2,
z
+
+
z
PA = e~e2 -- e~ e 2,
(4)
where e~ and e2 are unit vectors in the direction of the electric fields of the incident and scattered light. The one-magnon scattering intensity is deduced by substituting eq. (3) into eq, (2) and evaluating the spin dependent Green functions contained in ((Ak; A*))E. In the special case of F = 0 the only non-vanishing Green functions are of the form ((S+; Sj-)), ((Si +Slz + S~~S~, +" S;))~,
202 ({S/" Sj S~ + S;S; )),~ and ({S/S~ + S~S/; $ 7 S ~ + NfSi })~:, together with the corresponding Green functions with the operators in the reverse ortier. More generally when [ : v ' 0 other spindependent Green functions in ((A~ A~)}~ are also non-zero (e.g. ((S,+:,S', ))~. ((S,' N, .g~Si' " S/))6, etc.), and these must also be evahlated to obtain deh/d~ dw~. The method of calculation inw~lves writing down the equation of motion [4] for each of the required Green functions. The exchange dependent terms may then be simplified by decoupling products of spin operators at different sites using the Random Phase Approximation (RPA):
((S,',,S,:,:
S~ ))~ : (S~)((S,;,: & ))~.
(m ~ n ).
f5~
However, the single-ion anisotropy terms give rise to Green functions in the equations of motion which involve products of operators at the same site and which cannot be decoupled as in eq. (5). This difficulty has been discussed by various authors [5,6], who have suggested modifications of eq. (5) for decoupling operators at the same site, but except in the limit of T tt these approximations are at best only valid for very small D and F. In the present work we avoid any decoupling of the single-ion anisotropy terms. Instead we follow an approach similar to that first employed by Murao and Matsubara [71 to investigate the thermodynamic properties of a S = 1 ferromagnet with uniaxial anisotropy ( F = 0). They found that for this system the Green functions generated by the anisotropy form a closed set of equations and need not be decoupled. In particular using only R P A for the exchange terms it may be shown [7] that for any operator B the Green functions ({St; B))E and ((Si+ Siz_{_ SizSi+ ; B))E are coupled only to each other in their equations of motion. Thus in the present calculations if we take B equal to S i and (Sj S~+ S~S/) in turn we have altogether four coupled equations for S = I, which may then be solved to obtain the four basic Green functions which we have already mentioned as being required to evaluate dZh/daQ dw: in the case of F = 0. The method may be extended to higher spin S and to the case of F # 0. Generally we find that 4S coupled equations are required to solve for d2h/dl'), dw~ when F = 0, and this becomes 8S equations when F # 0. The method
is therefore suitable for relatively low values of the spin S, and we have carried out calculations for S 1 and 5; = I. 3. Discussion of results
For simplicity we coasider first an S-: I ferromagnet with uniaxial anisotropy i F 0). The Green function ((Ak:A*))r describing the oncmagnon light scattering is found to be I,~(E)
4rr( E
E,; )t E
- b.',i )
Lz( E) + ................. , 4rr(E + E,~)~E + E,,)
~6)
where E,I = ~ t t , H + ml(! 4 (r)J(0)-+ E,t.
(7)
E,
(g)
{~m ~J2(0) + [)2 _ m~DJ(O)},~,
I,I(E)-(E x'l~l~H){miK: 2m~KG + m l G z} + D{m2K'- 2 m i K ( ; + m~G:} ,l(O)tnl~(I 4 or)K:
2tnlm2(I + tr)K(;
and L2(E) is simihu- to I<(E) but with E ~ E and G--*- G. Here tnl and m~ denote the the> real averages (S-) and {3{(S~)2) - 2} respectively. which may be ewduated using the Green funclion formalism, and J(O)=Y~iJ O. We have assumed a typical scattering geometry with e~ along the z-axis and e, in the xy-plane, and we have made the usual approximation of taking k--~ 0 since one-magnon light scattering involves only excitations close to the Brillouin zone centre. The excitation energies of the system (for k - 0) are represented by E~ and E,, and they correspond to excitations associated with a change tAm,t= 1 in the magnetic quantum number, as discussed in [7]. In the limit of D ~ 0 these modes have the simple interpretation that E, reduces to the usual spin wave energy (simply gl~BH + o-mlJ(O) for k - 0) whilst E~ is an optical excitation which reduces to the Zeeman energy gl~uH + ( 1 + cr)m J(O). The scattering cross-section is now obtained by taking the imaginary part of (6) at energy E = fioo] - hw2 + iO +, and substituting into eq. (2). In general two Stokes peaks are predicted at frequencies ¢o~ given by oo~-[Eg[/h and w, [E,il/h, and likewise there are two Anti-Stokes peaks at ~o, = w,-~ [E/~[/h and w, + [EO[/h. For example, if E~; and E0 are both positive, the
203 Stokes scattering contribution c o m e s f r o m the first term in eq. (6) and is found to be d2h
tolto~n2
dO dto2 - 4 Vc4n~Eo x a(h~o,-
[L,(Eo)(I + N +) h~o2 -
- LI(Eo)(1 + N
Eo) ) a ( h t o l - hto2 - E o ) ] ,
(10) where N -~ are the thermal occupation numbers {exp (EolkBT)-1} 1 for each excitation. It follows f r o m eq. (10) that at t e m p e r a t u r e T the integrated intensities I * ( T ) and I - ( T ) , associated with the Stokes scattering at frequencies a h - Eo/h and w t - Eo/h respectively, are given by to,to~n2(l + N +-) I~-(T) = [L,(Eo)[. (11)
4hVc4nlEo
A detailed analysis of eqs. (10) and ( l l ) shows that most of the scattered intensity is associated with the usual spin w a v e energy Eo. In particular for low temperatures T , ~ Tc we have m~ ~- m 2 ~- l, w h e r e u p o n it may be verified that the weighting factor Lt(E~) associated with I+(T) is zero, so that in this approximation all the intensity is described by scattering f r o m Eo. The results of eqs. (6)-(11) may be extended to F ~ 0 and to higher values of the spin; details will be given in [3]. Qualitatively we may note that for S = 1 and F ~ 0 the Green function ((Ak; A*))~ m a y still be expressed in the same general f o r m as in eq. (6), except that the expressions for the excitation energies Eft and Eo and for the weighting factors L I ( E ) and L2(E) are more complicated. In the case of S = 3 there are three excitation energies, one of which reduces to the usual spin w a v e energy when D ~ 0 and F ~ 0. It is this excitation which gives rise to the dominant scattering peak, whilst much weaker intensities are associated with scattering f r o m the other two (optical) modes. In the isotropic limit our results simplify to give agreement with previous light scattering calculations [1,2] for the special cases considered by these authors. 4. A p p l i c a t i o n
to FeCI2
The theory has been applied to the romagnetic phase of anhydrous FeCI2 (S Tc = 24 K) in which the anisotropy effects large. This material consists essentially of
fer= 1, are fer-
romagnetically ordered layers, with the intralayer ferromagnetic exchange interactions being much stronger than the weak antiferromagnetic interaction between layers [8]. In moderate applied magnetic fields ( H ~> 11 kOe) the system b e h a v e s as a ferromagnet. The description of FeC12 in terms of an S = I Heisenberg Hamiltonian has been justified by Birgeneau et al. [8], who also deduced values for the exchange and anisotropy p a r a m e t e r s using neutron scattering data. We take J~ = 5.5 cm -~ and ./2 = - 0 . 7 c m J for the nearest and next-nearest exchange interactions within the plane, and J ' = - 0 . 2 cm -~ as the w e a k exchange coupling between layers. The values of the anisotropy parameters are less reliably known than the exchange interactions (see discussion in [8]), and we take here D = 9.4 cm -1, F = 0 and or = 0.3. With the a b o v e choice of p a r a m e t e r s the effective anisotropy field gttBHA at T < Tc is approximately 17cm -j, in agreement with experiment [8]. We also require the ratio G/K of the magneto-optical coupling coefficients in order to evaluate the t e m p e r a t u r e d e p e n d e n c e of the scattered intensity. In principle this ratio (which may be a function of the incident laser f r e q u e n c y too can be deduced from measurements of the linear and quadratic magneto-optical effects, as discussed in [2] for YIG. If it is assumed that only linear magneto-optical coupling is important (G/K = 0) then the calculated t e m p e r a t u r e d e p e n d e n c e of the integrated intensity I-(T) for scattering at to2 = o J ~ - E ~ / h is shown in fig. 1. A qualitatively similar temperature d e p e n d e n c e is predicted if G/K~ O, and this is also illustrated in fig. 1 by the curves for G[K = _0.1. All curves are plotted in terms
1.5
_Q
~
1.0
0.5
z w I.z -- 0 0
G/K= -0.1
01.5
11.01.51 T/T
2.0
c
Fig. 1. Calculated temperature dependence of the intensity ratio 1 (T)/I (0) for FeCI2 in a field H = 14kOe and with GIK taking the values 0.1, 0, -0.1.
2(14 of the r e d u c e d variable I (T)/I (0), where I (0) can be d e d u c e d f r o m eq. (11). An applied field of H = 1 4 k O e has been a s s u m e d . The much w e a k e r intensity I+(T) associated with scattering at a)~- E~/h has also been evaluated, and we find that I+(T)/(I (T) takes values 0, 0.08 and 0.37 for T/T+ = 0, (I.5 and 1.0 respectively (in the case of G/K = 0). At low t e m p e r a t u r e s ( T <~ T D the calculated values of the excitation energies E,~ and E,; c o r r e s p o n d i n g to H 1 4 k O e are 2 0 . 3 c m ~ and 29.1 cm ~, respectively.+ F u r t h e r details of the t h e o r y , particularly for S > I and for F:~ 0 will be given in [3], together with applications to particular materials. W o r k is in progress to extend the t h e o r y to light scattering in anisotropic antiferromagnets.
where there is more scope for c o m p a r i s o n with experiment.
References [I] R. [,oudon, J. Phys. ('3 {It~7il) N72 [21 W. Wettling, M.G. ('~ttam and ,I.R. Sandercock. ,I. Phv', ('8 (1975t 211 [3l M.G. Cottam and A. l,atiff Awang, to, be published. [41 D.N. Zubarev, Soviet P h y s i c s - U s p e k h i 3 f l960) 32(i. [51 F.B. Anderson and H.B. ('allen, Phys. Rev. la6 t1964t 1068. 161 M.E. l+ines. Phys. R e v 156 {1967) 534, [71 T. Murao and T. Malsuhara, ,1. Phys. ,%oc. Japan 2":, (196g) 352. [~1 R.J. Birgeneat,. W.B. Yehm. t+; ('ohetl and L Makov,,k),, Phys. Rev. t'/5 (1¢172~ 2607