dispersion of magnetic excitations in the pseudo one-dimensional induced-moment antiferromagnet...

PHYSICS LETTERS A 1 May 1989 Volume 137, number 1,2

DISPERSION OF MAGNETIC EXCITATIONS IN THE PSEUDO ONE-DIMENSIONAL I N D U C E D - M O M E N T ANTIFERROMAGNET RbFeBr3

A. HARRISON and D. VISSER Oxford University, Inorganic Chemistry Laboratory, South Parks Road, Oxford, OX1 3QR, UK

Received 2 November 1988; revised manuscript received 3 February 1989; accepted for publication 24 February 1989 Communicated by D. Bloch

The dispersion of magnetic excitations in the pseudo 1-D induced-moment antiferromagnet RbFeBr3 has been measured at 4.5 K using inelastic neutron scattering. The dispersion of excitations parallel to the c-axis has half the periodicity usually found in a I-D Heisenberg antiferromagnet, which is a consequence of its induced-moment character.

1. Introduction

There has recently been much theoretical [1-4] and experimental [5-7] interest in the magnetic properties of quantum spin chains described by the generalised Hamiltonian

H= Y.[J,)(S,~Sf +~'~')+~}SfS;)] u

+ ~ D ( S T ) 2 . ( 1 ) i

A variety of combinations of J± , jII and D may be engineered in insulating magnets by changing the magnetic ion, its ligand field environment and the nature of the superexchange pathways [ 8,9 ]. Among materials with small values of D/IJI, the static and dynamic magnetic properties of the one-dimensional ferromagnet CsNiF3 have been studied in great de- tail [10], and the one-dimensional Heisenberg antiferromagnet CsNiCI3 has attracted renewed interest [5,7] after theoretical work [ 11 ] showed that fundamental differences are expected in the dynamic magnetic properties of Heisenberg antiferromagnetic chains with integer or half-integer spin.

When S = 1 and D/IJI is large and positive, the Hamiltonian (1) represents an induced moment system. Examples of such systems are realised in the AFeX3 halides. The rubidium salts, RbFeC13 [ 12-14 ] and RbFeBr3 [ 15,16 ], are quasi one-dimensional ferro- and antiferromagnets respectively, and exhibit

three-dimensional magnetic long-range order below 2.55 and 5.5 K respectively. The corresponding cae- sium salts have smaller exchange constants [ 17-22 ] and show no magnetic long-range order down to 80 mK for CsFeCI3 [23] and 69 mK for CsFeBr3 [24]. Both of the chlorides have been extensively studied, and CsFeBr3 has been the subject of inelastic neutron scattering studies [21,22]. We present here re- suits of an inelastic neutron scattering study of the magnetic excitations in the pseudo one-dimensional induced-moment antiferromagnet RbFeBr3.

RbFeBr3 has a hexagonal perovskite structure and space group P63/mmc at room temperature, with the cell parameters a = 7.422 A and c= 6.304 A. Chains of trigonally-distorted, face-sharing FeB~- octa- hedra lie parallel to the crystal c-axis. At 108 K [25 ] RbFeBr3 undergoes a structural phase transition in which two thirds of the Fe chains (A sites) move by approximately 0.5 A out of the basal plane to form a honeycomb lattice, leaving the remaining third (B sites) in a triangular array. The new space group is P63cm and the unit cell becomes three times larger (a ' = x / 3 a and c' =c) . The distortion does not alter the environment of the iron atoms sufficiently to al- low the A and B sites to be distinguished by M~Sss- bauer spectroscopy [15]. The quasi one-dimensional crystal structure produces anisotropy in the magnetic exchange: the intrachain exchange J~ has been estimated to be an order of magnitude larger than the interchain exchange J2 [ 15 ]. Both J~ and J2

0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

79

Volume 137, number 1,2 PHYSICS LETTERS A 1 May 1989

are antiferromagnetic. The electronic ground state of the Fe 2+ ion is a singlet, I m s = 0 ) , and the first excited state is a low-lying doublet, I ms= + 1 ), which would be at an energy D in the isolated ions. How- ever, interion magnetic exchange or the application of an external magnetic field mixes the excited state into the ground state. For RbFeBr3 the ratio D/JQ (where JQ is the optimum exchange energy, equal in this case to 21Jl I + 3 I J21 ) is sufficiently small for a magnetic moment to be induced in the absence of an applied magnetic field. The inequivalence of the po- sitions of the A and B sites below 108 K leads to an inequality in the interchain magnetic exchange constants, J2(A-A) and J2(A-B). Heat capacity measurements [26] reveal two freezing transitions at low temperatures: a sharp anomaly at 5.16+0.02 K is believed to correspond to a transition from a para- magnetic to a partially disordered (PD) phase, and a broad, weak anomaly at 2.00 + 0.04 K was attrib- uted to a transition from the PD phase to a triangular antiferromagnetic array in the basal plane. In the PD phase, moments at the A sites are frozen in a two-sublattice antiferromagnetic array and moments at the B sites are disordered. The magnetic moment on Fe 2÷ was calculated from powder neutron diffraction data at 1.7 K [15] to be 2.2/zB and 2.7/tB if the phase was assumed to be the PD phase or the 120 ° antiferromagnetic array respectively, but it was not possible to determine which of the two phases was present. The only previous thorough study of the magnetic properties of RbFeBr3 was that of Lines and Eibschutz [16], who used single-crystal magnetic susceptibility data and M6ssbauer spectroscopy and obtained values of -0 .208(33) ,

-0 .008 and 1.04(4) meV for Jl, ,/2 and D respectively: the numbers in brackets describe the un- certainty in the last digit (s).

2. Experimental

The sample of RbFeBr3 used in the experiments was grown by the Bridgman method. It was cylin- drical in shape, measuring 15 mm long by 10 mm in diameter, and was mounted with the (001)N and (110) N reflections in the horizontal scattering plane.

Inelastic neutron scattering experiments were performed on this sample at 4.5 K using the Pluto Tri-

pie-Axis Spectrometer at AERE, Harwell. The dispersion of the magnetic excitations along [0 0 I]N and [1 ] I]N ( l=1-2 ) , and along [h k 1]N and [h k 2]N (h = k = 0-1 ) was measured by the constant-Q method at a wavelength 2=2.351 A ((002) reflection of pyrolytic graphite). Note that the Miller in- dex l is defined relative to the nuclear unit cell, which at low temperatures is equal in size to the magnetic unit cell [ 15 ], and I= 1 corresponds to a reciprocal lattice vector of length n/d A- 1, where d is the sep- aration between neighbouring iron atoms in the chains.

The dispersion of the magnetic excitations along [ h k 1] N (h = k= 0-0.59 ) was also measured to higher resolution using neutrons with 2=2.391 A, ( ( 111 ) reflection of aluminium). The collimation in both cases was 20 ' - 30 ' - 40 ' (monochromator-sample- analyser-detector) and incoherent elastic neutron scattering at Q = ( ~ ] 1 ) N had widths in energy of 0.6 and 0.3 meV with the graphite and aluminium monochromators respectively. A pyrolytic graphite filter was placed in front of the sample to reduce con- tamination of the incident beam by neutrons of wavelength 2/2.

3. Results

The inelastic neutron scattering measurements at 4.5 K showed distinct dispersion of magnetic excitations along and between the magnetic chains (fig. 1 ). The scattering intensity was found to fall off rap- idly as the temperature was raised, but dispersion along the chain axis persisted above TN, showing a small degree of renormalisation. The inelastic neutron scattering peaks were least-squares fitted to Lor- entzian curves convoluted with the Gaussian instrumental resolution function. This had been determined as a function of Q using the program RESCAL at AERE, Harwell. Examples of constant- Q scans and fits to the data are shown in fig. 2.

The most important feature of the results is the periodicity of the excitations along the crystal c-axis which is half that expected for a classical one-dimensional antiferromagnet i.e. the dispersion rises continuously from l - 1 to l= 2 rather than passing through a maximum at l= 1.5 then falling again to some local minimum at •=2.0.

80

Volume 137, number 1,2 PHYSICS LETTERS A l May 1989

3

E (mev)21

a o 1.0 1.5 2.0 l

4 [ _ _ j

3 I ii E (mev)

2

i --k i . -

0 0.0 0-5 1.0

h

Fig. 1. The best fits of the calculated spin-wave and MFA-excitonic dispersion curves to the data. (a) shows the magnon dispersion along (i) [-~ ½ /]N and (ii) [0 0 /]N (1=1--2) and (b) Shows the dispersion within the a-b plane along (i) [h k 1 ]N and (ii) [h h 2IN (h=k=0-1 .0 ) . The errors in the fits to the data are contained within the data points and the experimental resolution width in energy at each value of Q is indicated by the error bars. The curves calculated using the MFA-excitonic model are depicted as heavy solid lines, or dashed lines or dotted lines as the calculated scattering cross-section diminishes from large to small to very small or zero. For the sake of clarity we have only drawn those branches that have significant scattering intensity at some wavevector: all twelve branches are given by Suzuki and Shirai [ 3 ]. The dispersion curve calculated using the spin-wave expression (2) coincideswith thelowest excitonic branch (i) along [3 ~ /]N and (ii) along [0 0 I]N except at l> 1.7, where it is de- noted by a fine solid line. The other fine solid lines (iii) in (a) denote the "mirror" branches discussed in the text.

4. Discussion

M a g n e t i c e x c i t a t i o n s in m a g n e t i c a l l y o r d e r e d sol-

ids a re c o m m o n l y t r e a t e d b y l i n e a r s p i n - w a v e the -

ory. H o w e v e r , t he v a l u e o f D/JQ i n ( 1 ) is n o t s m a l l

for RbFeBr3 , a n d t he h a r m o n i c a p p r o x i m a t i o n u s e d

in s t a n d a r d s p i n - w a v e m o d e l s is e x p e c t e d to b e poor .

N e v e r t h e l e s s , such a m o d e l is usefu l in r e v e a l i n g t he

200

150

0 u 50

0 o 60

Z D 20 0 U

6O ~r~

-£- ~o ~g z ~ r ~ 20 0 U

100

I II

, I

O = (1/31/31.2)N

il tl

Q : ( l t 3 1/3 1.7) N

i I I I

O = ( 1/3 1/3 2.0) N

I I I

2 3 4 5 ENERGY (meV)

8O 0 0 O C ~

~ 60

~ ~0

2O

Q = ( 0 0 1 ) N

I I I

O : ( 0 " ~ 1 0 1 ~ 1 1) N o 150 O ~g

o 5o

0 1 2 I I

3 4

ENERGY (meV)

Fig. 2. Examples of constant-Q scans for RbFeBr3 at 4.5 K. The value of Q is indicated on the figure, as is the instrument's mon- itor count. The solid line is the least-squares fit to the data of a scattering background plus a Lorentzian curve convoluted with the instrumental resolution function.

81


various modes of excitation in RbFeBr3, and in pro- viding a comparison with the dispersion of magnons in a classical Heisenberg magnet with small planar single-ion anisotropy.

Spin-wave theory predicts the following dispersion relation in the extended zone scheme for a magnet with helical ordering vector K [27],

ha~q = 2S{ [.Ix- 1 (J,+x2 + J , - x )1

× (Jx-Jq + D) } '/2 (2)

We expect six branches in the reduced zone scheme for this six sublattice antiferromagnet, some of which have vanishingly small neutron scattering cross sections. Of the branches that are strong, we expect the following form for the dispersion along [0 0 I]N

hogq = 2 S { 4 1 J i 12 s i n 2 [ ~ ( l - l ) ]

9lJ, I I J ~ l { l + c o s [ n ( / - 1 ) ] }

+4DIJ, ] sin2[n(l-1)/2]+9DIj2l} '/2 (3a)

and for dispersion along [~ ~ g]h

ha~q =2S{41Jj 12 sin2[ = ( / - 1 ) ]

+361J, I Ig21 sin2[=(l - 1 ) /2]

+4DIJl [ sin2[=(l-1)/2]} '/2 (3b)

The gap that appears in this model at l= 2 may be observed, for example, in the one-dimensional Hei- senberg antiferromagnet TMMC where D arises from magnetic dipolar interactions [28]. Loveluck and Lovesey considered the effect of raising the ratio D/ 2J, on the magnon dispersion in a one-dimensional easy-plane magnet with Heisenberg exchange [29]: the gap at l= 2.0 for the branch corresponding to in- plane (xy) fluctuations increases and the maximum in the dispersion curve moves from l= 1.5 to I=2.0, remaining at the value for D/2J, > 2.0. We see that expressions (3a) and (3b) reduce to the Loveluck- Lovesey expression for in-plane (xy) fluctuations when J2 is set to zero.

htoq ~J , {sin 2 [ ~ ( l - 1 ) ]

+ (D/I Ji l) sin2[ re( l - 1 ) /2]} ,/2, (4)

where we have rewritten eq. (4.1) of Loveluck and Lovesey's paper in terms of our wavevector representation.

The spin-wave dispersion relation was fitted to the data (fig. la) to give values of -0 .107(32) ,

~.z

0

3

F a 5

I - - - - ~ - - 0 : ( 1/31~31

o:,oo,,.

1.0 12 1.4 16 1.8 20 [

"7 2

Ii

b

t

o o', o'2 o'3 o'~ & 0'6 h

Fig. 3. The dependence on wavevector of the neutron scattering cross-section of magnetic excitations in RbFeBr3. The solid lines are the results of calculations using the MFA-excitonic model, corrected for the magnetic form factor and scattering geometry effects. The crosses or circles are the experimental points. All values are normalised to the scattering intensity at Q= (0 0 l )n.

-0 .024 (7) and 6.59 (2.11 ) meV for J,, J2 and D respectively (Z2= 1.30). The calculated curves appear to describe the dispersion along [ 0 0 l] N and [ ] ] l ] N (1= 1--2) well (fig. la). However, there are several shortcomings in applying this theory to RbFeBr3.

(i) For every branch at [h h I]N the spin wave theory predicts branches of comparable intensity at [h_+ ~ h+ ~ 11~.

(ii) The ratio D/JQ obtained from the fit is ~23(4) . When D/JQ>4 the molecular field approximation (MFA) predicts that a singlet-doublet system has no induced moment. Models that take into account correlations in the orientations of neighbouring moments provide even lower esti- mates for this ratio [ 14 ].

The assumptions on which spin-wave models are based - that the excitations correspond to small de-

82


viations from a known, ordered magnetic ground state - are clearly not valid for low-dimensional induced-moment antiferromagnets such as RbFeBr3: instead we should use models that treat the excitations as excitons (electronic excitations that disperse through relatively weak inter-ion forces) and cal- culate the electronic ground state self-consistently [ 30-33 ]. In order to derive the dispersion relation within such a model it is necessary to decouple prod- ucts of spin operators, converting them into sums of expressions in just one spin operator. This is most easily performed using the MFA, which involves the substitution

s,~sj~, -, s,~ < sj~ ) + sj~ < s,~ >, (5)

where y=x, y, z. Suzuki and Shirai [3] have described the principles of applying the MFA to RbFeBr3: at T= 0 two branches are expected for the excitations in the extended Brillouin zone, corresponding to excitations from the ground state to the levels of the excited doublet. It was assumed that the magnetic exchange in the spin-1 Hamiltonian (1) was Heisenberg in character [ 16 ]. This is not a good assumption because in the fictitious S = 1 Hamilto- nian (1), the magnetic exchange constants incor- porate the orbital contribution to the angular mo- mentum, and the ratios j ( / j i ] and J~/J~ will depend on the relative values of the trigonal com- ponent of the ligand field and the spin-orbit cou- pling parameter. The two ratios were taken to be equal to each other and calculated as 1.3 by Eib- schutz et al. [ 12 ] and derived from inelastic neutron scattering data by Suzuki [ 14] as 0.77 for RbFeC13. For simplicity and comparison with the theoretical work of Suzuki and Shirai [ 33 ] we assume the ratio J;-/Jl j ( i= 1, 2) to be unity. The caseJ/-/J~ # 1 will be considered later [ 35 ]. The neutron scattering intensities of the excitations were calculated in the magnetic zone scheme and showed strong branches corresponding to in-plane (xy) fluctuations at q + K, and very weak branches corresponding to out-of- plane (zz) fluctuations. The appropriate strong branches were least-squares fitted to the data points (fig. 1 ), yielding optimised values of J~, J2 and D of -0 .288 (14 ) , - 0 . 0 3 1 ( 5 ) and 2.55(4) respectively (Z2=1.31). The excitations at [h h 2]N ( h = 0 - 1 ) were ignored in the least-squares-fitting procedure as they had very small scattering cross-sections and their

centres in energy had large uncertainties. In both these figures it is clear that at certain wavevectors - particularly near Q = ( ] 1 1 ) N and Q = ( -~ ] 1 ) ~ -

m o r e than one strongly-scattering excitation should be observed. However, it was impossible to resolve these excitations in the present experiment and it did not prove satisfactory to try to extract two inelastic scattering peaks from one scattering maximum using an estimate of the width in energy of a single excitation. We did not see any of the weaker branches also displayed in figs. la and lb, although the two "mirror" branches, displayed in fig. l a and corresponding to out-of-plane (zz) fluctuations in the Loveluck-Lovesey model, have been observed in the singlet ground state antiferromagnet CsFeBr3 as weak excitations at (1 1 l)N and (] ] I)N ( l = 0 - - 1 ) u s i n g an instrument with a much higher neutron flux [ 21,22 ]. The scattering intensities are compared with the experimental values in figs. 3a and 3b. The ratio Jl/.12 is approximately 11, indicating that RbFeBr3 is not particularly one-dimensional.

For induced-moment, low-dimensional magnets the MFA, which neglects spontaneous fluctuations, is a poor approximation. Correlations between neighbouring moments may cause the local field to deviate substantially from the ensemble average and may be treated to a first approximation using cor- related effective field (CEF) models. This involves the substitution

S,~&~--,s,~[ <s~ ) + a (s ,~- <&~ ) ) ]

+Sj~[ (Siy) +o~(S~ - (Sj~) ) ] . (6)

a is a parameter that describes the nearest-neigh- bour magnetic correlation and must be calculated self-consistently.

The CEF approach has been applied successfully to the static magnetic properties of both RbFeCI3 [ 12 ] and RbFeBr3 [ 16 ] and the dispersion of magnetic excitations in RbFeCI3 [ 14 ]. It would also al- low one to treat the dispersion of magnetic excitations in induced-moment magnets which do not show magnetic long-range order at low temperatures, such as CsFeBr3 [ 21,22 ]: in that particular case we expect the unusual periodicity of the dispersion of the magnetic excitations parallel to the c-axis to have the same physical origin as those in RbFeBr3.

Although the MFA-excitonic model succeeds in

83


describing the form of the magnetic dispersion at a low temperature, the exchange parameters we derive probably differ substantially from the real exchange parameters, and also from those derived using the same model on data taken at higher temperatures using a variety of experimental methods. A reliable set of parameters must describe a variety of magnetic phenomena measured over a wide temperature range. The application of Suzuki's version of the CEF model to inelastic neutron scattering data produced the following values for the exchange parameters and single-ion anisotropy: J j = - 0 . 4 0 ( 1 ) , J ~ = - 0 . 4 3 ( 4 ) , J 2 = - 0 . 0 4 4 ( 3 ) , , / 2= -0 .065(13 ) and D=1 .94(2 ) meV [ 35 ]. The renormalisation of the energy of the most intense inelastic magnetic scattering maximum at (0 0 1 )N was more successfully described using the DCEFA approach, but the predicted value of TN was much worse: 2.1 K compared with 6.7 K for the MFA prediction.

5. Conclusions

The dispersion of the magnetic excitations in the pseudo one-dimensional induced-moment antiferromagnet RbFeBr3 is successfully described by models that treat its induced-moment character ex- plicitly. The unusual periodicity of the dispersion parallel to the crystal c-axis is a natural consequence of such models and is unique among insulating materials with magnetically ordered ground states.

Acknowledgement

The authors wish to express their thanks to Dr. P.J. Walker of the Clarendon Laboratory, Oxford for growing the sample of RbFeBr3, and to Mr. N. Clarke and Mr. P. Bowen at AERE, Harwell for their tech- nical assistance with the experiment. We are most grateful to Dr. N. Suzuki of Osaka University for the loan of his computer programs and for stimulating discussions. We are also grateful to SERC for financial support and to Dr. P. Day of this laboratory for his encouragement and provision of research facili- ties. A.H. would like to thank St. John's College, Ox-

ford for financial support while this work was car- ried out.

References

[ 1 ] J.C. Bonner, J. Appl. Phys. 61 ( 1987 ) 3941, and references therein.

[2] R. Botet, R. Jullien and M. Kolb, Phys. Rev. B 28 (1983) 3914.

[ 3 ] J. Solyom and T.A.L. Ziman, Phys. Rev. B 30 (1984) 3980. [4] P.-A. Lindgard, Phys. Rev. B 30 (1984) 2729. [5l W.J.L. Buyers, R.M. Mora, R.L. Armstrong, M.J. Hogan,

P. Gerlach and K. Hirakawa, Phys. Rev. Lett. 56 (1985) 371.

[6] P. Renard, M. Verdaguer, L.P. Regnault, W.A.C. Erkelens, J. Rossat-Mignod and W.G. Stirling, Europhys. Lett. 3 (1987) 945.

[7] M. Steiner, K. Kakurai, J.K. Kjems, D. Petitgrand and R. Pynn, J. Appl. Phys. 61 (1987) 3953.

[8] W.E. Hatfield, W.E. Estes, W.E. Marsh, M.W. Pickens, L.W. ter Haar and R.R. Weller, in: Extended linear chain compounds, Vol. 3, ed. J.S. Miller (Plenum, New York, 1983) p. 43.

[ 9 ] R.L. Carlin, in: Magneto-structural correlations in exchange coupled systems, eds. R.D. Willett, D. Ganeschi and O. Khan, NATO ASI Series C 140 (Plenum, New York, 1983 ) p. 127.

[10] M. Steiner, K. Kakurai and J.K. Kjems, Z. Phys. B 53 (1983) 117.

[ 11 ] F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [ 12 ] M. Eibschutz, M.E. Lines and R.C. Sherwood, Phys. Rev. B

11 (1975) 4595. [ 13 ] N. Wada, K. Ubukoshi and K. Hirakawa, J. Phys. Soc. Japan

51 (1982) 2833. [ 14] N. Suzuki, J. Phys. Soc. Japan 52 (1983) 3907. [ 15 ] M. Eibschutz, G.R. Davidson and D.E. Cox, AIP Conf. Proc.

18 (1973) 386. [ 16 ] M.E. Lines and M. Eibschutz, Phys. Rev. B 11 ( 1975 ) 4583. [ 17] Y. Takeda, Ann. Phys. Soc. Japan 37 (1974) 276. [ 18 ] H. Yoshizawa, W. Kozukue and K. Hirakawa, J. Phys. Soc.

Japan 49 (1980) 144. [ 19 ] M. Steiner, K. Kakurai, W. Khop, B. Dorner, R. Pynn, U.

Happek, P. Day and G. McLeen, Solid State Commun. 38 (1981) 1179.

[20] P.-A. Lindgard, Physica B 120 (1983) 190. [ 21 ] D. Visser and U. Steigenberger, Annex to the Annual Report

of the Institute Laue-Langevin ( 1984 ). [22 ] B. Dorner, D. Visser, U. Steigenberger, K. Kakurai and M.

Steiner, Z. Phys. 72 (1988) 487. [ 23 ] D.P.E. Dickson, cited in J.A. Baines, C.A. Johnson and M.F.

Thomas, J. Phys. C 16 (1983) 3579. [24] D. Visser, private communication (1987). [25 ] A. Harrison and D. Visser, submitted to J. Phys. C. [26] K. Adachi, K. Takeda, F. Matsubara, M. Mekata and T.

Haseda, J. Phys. Soc. Japan 52 (1983) 2202.

84


[27]A.R. Mackintosh and H. Bjerrum-Moller, in: Magnetic properties of rare earth magnets, ed. R.J. Elliott (Plenum, New York, 1972).

[28] G. Shirane and R.J. Birgeneau, Physica B 86-88 (1977) 639.

[ 29 ] J.M. Loveluck and S.W. Lovesey, J. Phys. C 8 ( 1975 ) 3857. [ 30 ] B. Grover, Phys. Rev. 140 ( 1965 ) A 1944. [31 ] Y.L. Wang and B.R. Cooper, Phys. Rev. 172 (1968) 539.

[32] W.J.L. Buyers, T.M. Holden and A. Perreault, Phys. Rev. B 11 (1975) 266.

[33]P.-A. Lindgard, in: Condensed matter research using neutrons, eds. S.W. Lovesey and R. Scherm, NATO ASI Series B 112 (Plenum, New York, 1984) p. 163.

[34] N. Suzuki and M. Shirai, Physica B 136 (1986) 346. [ 35 ] A. Harrison and D. Visser, submitted to J. Phys. C.

85

dispersion of magnetic excitations in the pseudo one-dimensional induced-moment antiferromagnet...

Documents