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Submitted on 1 Jan 1988
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NEW DOUBLE MAGNON MODES IN PLANAR
ANTIFERROMAGNETIC CHAINS IN A FIELD
J. Boucher, M. Remoissenet, R. Pynn, L. Regnault
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Suppl6ment au no 12, Tome 49, dkembre 1988
NEW DOUBLE MAGNON MODES IN PLANAR ANTIFERROMAGNETIC CHAINS
IN A FIELD
J. P. Boucher (I), M. Remoissenet (2), R. Pynn (3) and L. P. Regnault (I)
(I) Centre d9Etudes Nucle'aires de Grenoble, DRF/SPh/DSPE et MDN, 85X, 38041 Grenoble cedez, France
(2) Laboratoire ORC, Universite' de Bougogne, 21100 Dijon, France
(3) MS H 805 LANL, Los Alamos, New Mexico, NM8745 U.S.A.
Abstract. - Planar antiferromagnetic chains in a field provide a good support for the study of non linear excitations (solitons). In recent inelastic neutron scattering measurements performed with TMMC new collective excitations have been found which are explained in terms of non linear magnon processes.
Double magnon (DM) processes in ordered magnetic materials can be observed by neutron inelastic scatter- ing as specific modes, which then can be analysed as a function of both frequency w and wave vector q [I]. In quasi-one-dimensional (ID) magnetic compounds the Van Hove singularities which characterize the magnon density of states enhance the spectral weight of the DM modes relative to the single magnon mode. The DM modes observed to date are a direct consequence of linear magnon theory which describes small oscilla- tions of the spins about their equilibrium position (see inset of Fig. 1). A single magnon with frequency wo re- suits in spin fluctuations in the X or Z directions, per-
pendicular to the spins. The conventional DM modes correspond t o the same fluctuations observed in the Y direction, parallel to the spins. The resulting fre- quency is 2wo. A detailed analysis shows that two kinds of DM'fluctuations are expected in the Y direction, which are associated with the sum (creation-creation process) and the difference (creation-annihilation pro- cess) of the elementary frequencies. In the present work, we report the f i s t observation of DM modes associated with fluctuations in a direction perpendicu-
Fig. 1. - Dispersions of the new double magnons (( ) and
(A)) and of the in-plane ( 0 ) and out-of-plane (*) single
magnons in TMMC.
tar to the spins. We show that these new DM modes result from non-linearities in the spin dynamics.
The results t o be discussed were obtained with (CD3), NMnCl3 (TMMC) an experimental realization of classical quasi 1D planar antiferromagnet. Conven- tional DM modes have been observed previously in zero or low magnetic field [2]. The new experiments were carried out with a strong field (H N 4 - 10 T) applied perpendicular t o the chain axis, on the spec- trometer IN20 at the ILL in Grenoble. The scattering vector Qo was lying in the
Y Z
plane (see Fig. 1). This kind of spectrometer, used in its polarisation analysis configuration, allows the independent measurements of magnetic fluctuations occuring in the field direction X [non-spin-flip (nsf) channel] and in the Y Zlane
[spin-flip (sf) channel]. Here we limit the discussion to the nsf spectra as shown in figure 2a. The intense peak corresponds to the in-plane (IP) single magnon modes associated with oscillations of the angle cp de- fined in figure 1 (the oscillations of the angle B result in out-of-plane (OP) fluctuations: they are observed in the sf channel). Figure 2a also manifests weaker features at higher frequencies. When T is raised one observes a broadening of the single magnon mode and the occurence of a low frequency feature (Fig. 2a). The dispersions of these new high and low frequency fea- tures are shown in figure 1, together with correspond- ing results for the IP and OP single magnon modes. The analysis of these new features is the purpose of the following description.
The magnetic properties of TMMC are well de- scribed by the Hamiltonian:
with J = 6.8 K, D = 0.16 K, g = 2 and S = 512. The spin dynamics can be described by the evolution equa- tions for the two angles 0 and cp defined in figure l . For simplicity the ground state of the spins is assumed to be Bo = 0, cpo 11 0. In the continuum approximation, using equations (3a) and (3b) in reference [3] and after
@ and O are redefined as n/2
+
cp and n/2-
B respec- tively, we obtain in the small angle limit, 0 -+ EB andJOURNAL DE PHYSIQUE
Fig. 2.
-
Single and double mamons observed in the non- - --.--spin-flip (nsf) channel in TM&, at T = 1.4 K (a) and
T = 12 K (b).
with
P
= -2&npw,l( u p
a
-
c$
(+kt)-
n;]
.
The fluctuations observed in the nsf spectra are de- scribed by Sx ( z ,
t )
w (cp ( 2 , t ) cp (0, 0)*) where:accounts for all the possible oscilla,tory modes. The corresponding spectral density S" (q, w ) results in ad- ditive contributions. The first term in ( 1 ) gives rise to the IP single magnon mode (w
>
0) :where n p = l/ [exp ( l i w a / k T )
-
1].
The second term leads to new spectral contributions easily identified as DM modes:s
;
,
( 9 , w ) 4 n ;(s+
+
s-)
/
[w2-
w;(dl2.
with :
p -+ E(P ( E
<<
1) two non-linearly coupled equations:2 2 3
LP = - 2 ~
np9
13+
2~~ (c?p,e,-
~ ~ 0 % )e+
si
= ~ w ~ ( k ) [ n p ( q ~ k + l ) l [ n , ( k ) + i*;]
x+ 2 ~ R p B t p
+
0 ( e 3 ) kxS
{w-
Iwp (4 T k) f we ( k ) l ) ( 2 )Lt8 = - 2 ~ ~ 0 : q ~ ~ / 3
+
E~ ( q-
~c;cp:)
~ ~ 8- AtT
= 1.4 K, the only important contribution is2 2 2
-E npcp 0
-
2eR;cptcp+
0 (c3) expected froms+.
The sum over k i n ( 2 ) is performedwith R p = g p ~ H , Ra = s m ,CO = 455 and where the linear operators L and L' are given by L
+
R: =L'
+
0: = c;6/6z2-
6/6t2. The multiple scale method[4J used to solve these equations consists in expanding the angle variables as p N _ qp1
+
v2cp2+
.
.
(7<<
1)and the differential operators as
and 6 / 6 z + 6/6z
+
v 6 / 6 ~ 1 + v26/6z2+
.
.
where ti, zi (i = 1, 2) define slow variables. Similar expansions apply for 8 and the operators b2/6z2 and
~ ~ / 6 t ~ . TO lowest order in
v
: L q ~ l = 0 and ~ ' 8 1 = 0 yielding for c p l and 01 :91 = B exP (i&)
+
c.c.; 81 = A exp (i$-)+
C.C.with $@ = k z
+
wpt, $, = Ic'z+
w,t and where wp =(0;
+
~ ; k ~ ) ' ' ~ and w, =(02
+
c2
o k ) l 2 ' I 2 a r e t h edispersion relations obtained in the lo~g-wavelength limit. The solutions c p l and 61 are now introduced in the second order equations:
Lp2 = -2 ( ~ 0 2 6 ~ / 6 2 6 2 1
-
~ ~ / 6 t 6 t l ) c p ~ ++2cnp (681ISt) c p l
L'82 = -2 ( ~ ; 6 ~ / 6 z 6 z i
-
b2/6t6t) 81--2~0pcp1(6cpi/6t).
For "off-resonance" solutions, the factors in the right hand side is equal to zero and one obtains for cp2 :
cp2 i [ABF+ e i ( + ~ + + ~ )
-
AB*F- i ( + ~ - - + a ) + ~ .c.1
after substitution of
S
( w ) by the Gaussian instrumen- tal resolution function. For q* = 1-
q = 0.1, the fi- nal result obtained after adding the single magnon and DM contributions is shown by the full line in figure 2a. For the data at T = 12 K one should take into accountthe thermal fluctuations which broaden the elementary modes and a more complicated convolution procedure is used. As seen in figures 2a and 2b a good agreement is obtained between our description ;tnd the data. The dispersion curves for these DM modes can be also de- duced from the theory. The maxima are essentially determined by the singularities of the magnon density of states i.e. for d [w, (q k ) f wp (:Ic)] /dk = 0. One obtains the full lines in figure 1 in reasonable agree- ment with our observation.
[ I ] Cowley, R. A., Buyers, W. L. .J., Martel, P. and
Stevenson, R. W. H., Phys. Rev. Lett. 23 (1969) 86.
[2] Heilmann, I. U., Kjems, J. K., ISndoh, Y., Reiter,
G . F., Shirane, G. and Birgeneau, R. J., Phys.
Rev. B 24 (1981) 3939;
Osano, K., Shiba, H. and Endoh, Y., Prog. Theor.
Phys. 67 (1982) 995.
[3] Wysin, G. M., Bishop, A. and Oitmaa, J., J.
Phys.
