NEW NON LINEAR EXCITATIONS IN (CD3)4NMnCl3 (TMMC)

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NEW NON LINEAR EXCITATIONS IN (CD3)4NMnCl3 (TMMC)

J. Boucher, M. Remoissenet, R. Pynn, L. Regnault

To cite this version:

J. Boucher, M. Remoissenet, R. Pynn, L. Regnault. NEW NON LINEAR EXCITATIONS IN (CD3)4NMnCl3 (TMMC). Journal de Physique Colloques, 1989, 50 (C3), pp.C3-29-C3-32.

�10.1051/jphyscol:1989304�. �jpa-00229443�

JOURNAL DE PHYSIQUE

Colloque C3, suppl6ment au n 0 3 , Tome 50, mars 1989

NEW NON LINEAR EXCITATIONS IN (CD,),NMnCl, (TMMC)

J.P. B O U C H E R ( ~ ) , M. REMOISSENET*, R. PYNN** and L.P. REGNAULT

C e n t r e d ' E t u d e s N u c l 6 a i r e s d e G r e n o b l e , D R F / S P h / D S P E

e t

MDN, 8 5 X , F - 3 8 0 4 1 G r e n o b l e C e d e x , F r a n c e

* ~ a b o r a t o i r e O R C , U n i v e r s i t e d e B o u r g o g n e , F - 2 1 1 0 0 D i j o n , F r a n c e ' " M S H 805 L A N L , L o s A l a m o s , N e w M e x i c o , N M 8 7 4 5 , U . S . A .

Resume - Les chaines antiferromagnetiques planaires en presence d'un fort champ magnetique exterieur permettent d'btudier des excitations non lineaires (solitons). De nouvelles excitations collectives observees par diffraction inelastique de neutrons sur le compose TMMC sont expliquees en termes de processus non lineaires de magnons.

Abstract - Planar antiferromagnetic chains in a field provide a good support for the study of non linear excitations (solitons). In recent neutron inelastic scattering measurements performed with TMMC new collective excitations have been found which are explained in terms of non iinear magnon grocesses.

1. INTRODUCTION

Double magnon (DM) processes in ordered magnetic materials can beobserved by neutron inelastic scattering as specific modes, which then can be analyzed as a functlon of both frequency w and wave vector q [ i J . In quasi-one-dimensional (iD) magnetic compounds the Van

Fig. 1

-

Dispersions of the ~ I C W UUUULC ~itagnons ( A and A) and of the in-plane (0) and out-of-plane ( * ) single magnons in TMMC.

("~lso member of Equipe de Recherche CNRS

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989304

C3-30

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Hove slngularities 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 osciliations of the spins about their equilibrium position (see inset of fig.1). A single magnon with frequency wo results in

spin fluctuations in the X or Z directions, perpendiqux to the spins. The conventional DM modes correspond to the same fluctuations observed in the Y direction, parallel to the spins.

The resuiting frequency is 2w0. A detailed analysis shows that two kinds of DM fluctuations are expected in the Y direction, which are associated wlth the sum (creation-creation process) and the difference (creation-annihilation process) of the elementary frequencies. In the present work, we report the first observation of DM modes associated with fluctuations in a direction perpendicular to the spins. A theoretical analysis is also presented, which shows that these new DM modes result from non-linearities in the spin dynamics.

The results to be discussed were obtained with (CD3)4NMnC13(TMMC) an experimental realization of classical quasi 1D planar antiferromagnet. Conventional DM modes have been observed previously in zero or low magnetic field [2]. The new experiments were carried out with a strong field (H E 4 - 10T) applied perpendicular to the chain axis, on the spectrometer IN20 at the ILL in Grenoble. In the present work we limit the discussion to the fluctuation spectra observed in the X direction (see fig. 1). The intense peak of fig. 2a corresponds to the in-plane (IP) single magnon modes associated with oscillations of the angle defined in fig.1 (the oscillations of the angle 0 result in out-of-plane (OP) fluctuations). Fig.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 features are shown in fig.1, together with corresponding results for the IP and OP single magnon modes. The analysis of these new features is the purpose of the following description.

1 3

E (meV1

Fig. 2 - Single and double magnons observed in the X direction in TMMC, at T = 1.4K (a) and T = 12K (b).

2. THEORY

The magnetic properties of TMMC are well described by the Hamiltonian

:

with J

=

6.8K, 4

=

0.16K, g

=

2 and S

=

5/2. The spin dynamics can be described by the evolu- tion equations for the two angles

9

and

cp

defined in fig.1. For simp1 icity the ground state of the spins is assumed to be

60 = 0, q~

= ^0. In the continuum approximation, using Eqs.3a and 3b in Ref. [3] and after

@

and

@I

are redefined as ~ / 2 + (p/2 and m/2 -

⁶

respectively, we obtain in the small angle 1 imit,

6 4 €6

and

cp + ~ c p (E

< 1) two non-1 inearly coupled equations

:

with %

⁼

^{gpBH, Ra}

⁼

^{S m , Co}

⁼

4JS and where the linear operators L and Lh are given by L + @

⁼

^LL' +

⁼

~ $ 6 ~ / 6 ~ ~ - 6 ~ / 6 t ~ . The multiple scale method [4] used to solve these equa- tions consists in expanding the angle variables as

q

-

^qcp +r12cp2+...(-,,

< l) and the differen-

2

1

tial operators as &/St

--t

6/6t +

^rl

^&/Stl + ^q8/6t2 + ... ^and

6/62 +

6/6z + rlb/6zl + v26/6z2 + . . . where ti, zi

(i =

1,2) define slow variables. Similar expansions apply for

6

and the operators

62/6z2

and ti2/6t2. To lowest order in

q :

Lcp,

=

0 and L'O1

=

0 yielding for ql and 6 1

:

with QP

=

kz + ^wpt, a,= k f z + ^{w , t an6 where}

^{up = ( Q i}

+ ^{C;k2)* and}

u, =

(Ri + ^{C;kr2)* are the}

dispersion relations obtained in the long-wavelength limit. The solutions cpl and 61 are now introduced in the second order equations

:

For "off-resonance" solutions, the factors in the right hand side is equal to zero and one obtains for

cp., :

L

i [Qp(k)+~(kf)3 !!*p(Io-@,(kf)l

"2 =

ⁱ

^(ABF'~ - AB*FVe

)

+

^C.C.

with ' F

=

- 2~4w~/([~(k)*~(k~

j 1 2

- ~g(k+k')~ - $1. ^{For a2} one obtains

:

iE~b(k)+*~(k~)l i[*6;k)-~p(k')l

0 2 = ^i.(B2H+e + B ~ H - ~

)

+ ^C.C.

with

:

t h e

exp~essions for

(p2

and 0 2 can be introduced in the third order equations. After removing

the so-called secular terms, two coupled Non linear Schrodinger equations are obtained

:

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where p, p', q, q', r and r ' are f u n c t i o n s o f k, k',

%, %

and Co. As shown i n a d i f f e r e n t context these equations can have s o l i t a r y wave s o l u t i o n s [ 5 ] . A d e t a i l e d d i s c u s s i o n o f t h i s problem w i l l be presented elsewhere. Here we focus on t h e second o r d e r s o l u t i o n s which explains r a t h e r w e l l our r e c e n t experimental r e s u l t s obtained w i t h TMMC.

3. EXPERIMENTS

The f l u c t u a t i o n s observed i n t h e n s f s p e c t r a a r e described by SX(z,t)

-

<cp(z,t)tp(O,O)*>

where :

accounts f o r a l l t h e p o s s i b l e o s c i l l a t o r y modes. The corresponding s p e c t r a l d e n s i t y Sx(q,w) r e s u l t s i n a d d i t i v e c o n t r i b u t i o n s . The f i r s t term i n (1) g i v e s r i s e t o t h e I P s i n g l e magnon mode (w

>

0) *:

STM (q,w)

-

(np + 1) A(w - up)

where np = l/[exp(kop/kT)

-

11. The second term leads t o new s p e c t r a l c o n t r i b u t i o n s e a s i l y i d e n t i f i e d as DM modes : S ~ M ( ~ , W )

-

4 $ ( ~ + + ~ - ) / [ w ~ - u ~ ( ~ ) ] * w i t h :

A t T = 1.4K, t h e o n l y important c o n t r i b u t i o n i s expected from S+. The sum over k i n (2) i s performed a f t e r s u b s i t u t i o n o f S(o) by t h e Gaussian i n s t r u m e n t a l r e s o l u t i o n f u n c t i o n . For q* = I - q = 0.1, t h e f i n a l r e s u l t obtained a f t e r adding t h e s i n g l e magnon and DM c o n t r i b u t i o n s i s shown by t h e f u l l l i n e i n f i g . 2 a . For t h e d a t a a t T = 12 K one should take i n t o account t h e thermal f l u c t u a t i o n s which broaden t h e elementary modes and a more complicated c o n v o l u t i o n procedure i s used. As seen i n f i g s . 2 a and 2b a good agreement i s obtained between our d e s c r i p t i o n and t h e data. The d i s p e r s i o n curves f o r these DM modes can be a l s o deduced from t h e t h e o r y . The maxima a r e e s s e n t i a l l y determined by t h e s i n g u l a r i t i e s o f t h e magnon d e n s i t y o f s t a t e s i . e . f o r d [ s ( q T k ) t % ( k ) ] / d k = 0. One o b t a i n s t h e f u l l l i n e s i n f i g . 1 i n reasonable agreement w i t h our observation.

The discovery o f these DM e x c i t a t i o n s i n TMMC i s an important r e s u l t i n t h e f i e l d o f non l i n e a r e x c i t a t i o n s i n magnetic chains. We b e l i e v e t h a t s i m i l a r e f f e c t s c o u l d be observed i n o t h e r systems, i n p a r t i c u l a r i n two dimensional systems where l o g a r i t h m i c s i n g u l a r i t i e s i n t h e magnon d e n s i t y o f s t a t e s are expected.

REFERENCES

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(1969) 86

2. Heilmann, I.U., Kjems, J.K., Endoh, Y., R e i t e r , G.F., Shirane, G. and Birgeneau, R.J., Phys. Rev. B

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(1981) 3939 ; K. Osano, H. Shiba and Y. Endoh, Prog. Theor. Phys.

67

(1988)

PO6

3. Wysin, G.M., Bishop, A. and Oitmaa, J., J. Phys. C : S o l i d S t a t e Phys. 19 (1986) 221 4 . Kaup,

D.J. and Newell, A.C., Phys. Rev. B (1978)

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5 . Bhakta, J.C., Plasma Phys. and C o n t r o l l e d Fusion

3

(1987) 245