Magnetic excitations in insulators with emphasis on nonlinear effects

Physica 137B (1986) 70-82 North-Holland, Amsterdam

MAGNETIC EXCITATIONS IN I N S U L A T O R S WITH E M P H A S I S ON N O N L I N E A R EFFECTS M. S T E I N E R and H. D A C H S tlahn-Meitner-lnstitut. Glientcker.~trasse 100. D-IO00 Berhn, Wannsee. Fed. Rep. (A'rmanv

The aspects of the study of magnetic excitauons in insulators which will be discussed, are: - dispersion of spin waves in the ordered state. - dispersion and damping of spin waves in systems without long range order. . non-linear effects in I-D magnets. The nonlinear effects will be discussed quite extensively in order to show how linear and nonlinear effects in the spin dynamics can be separated. This will give the possibility to discuss some experimental techniques and possibilities which have not been used very widely, like inelastic neutron scattering with full polarization analysis.

I. Introduction In o r d e r to fully a c k n o w l e d g e the i m p o r t a n c e of the neutron scattering studies of excitations in magnetic insulators for the u n d e r s t a n d i n g of the physics of these systems a few aspects are sufficient: A m a g n e t i c system finds its g r o u n d state longe range o r d e r by following the interactions between the magnetic m o m e n t s . The d e t e r m i n a tion of the g r o u n d state does not generally lead to a detailed k n o w l e d g e of the basic interactions, thus not to the H a m i l t o n i a n of the system. If, however, we are able to measure the e l e m e n t a r y excitations and their dispersion in the g r o u n d state, then the p a r a m e t e r s in the H a m i l t o n i a n can be uniquely d e t e r m i n e d . This k n o w l e d g e of the H a m iltonian is of f u n d a m e n t a l i m p o r t a n c e for any d e t a i l e d u n d e r s t a n d i n g of the physics involved. If we u n d e r s t a n d the o r d e r e d state and its excitation we can ask m o r e difficult questions, like: how stable are these excitations against dist u r b a n c e s in the o r d e r e d state? This is of particular interest at the phase transition where the long range o r d e r d i s a p p e a r s . In this situation the meas u r e m e n t of the excitation energy alone is not sufficient, but the line shape has to be studied as well. C o m p a r i s o n with linear theory will give us i n f o r m a t i o n a b o u t the range of validity of the linear theories. This is very i m p o r t a n t because the calculation of t h e r m o d y n a m i c p r o p e r t i e s relies very 0378-4363/86/$03.50 ' Elsevier Science Publishers B.V. (North-Holland Physics Publishing I)ivision)

heavily on the correctness of the linear theory. By choosing the a p p r o p r i a t e systems one can leave the linear regime entering a regime where new n o n l i n e a r excitations become i m p o r t a n t . Here. the basic a n d crucial task is to identify the contrib u t i o n s to the spin d y n a m i c s in o r d e r to get an at least qualitative picture of this physics. This sepa r a t i o n can only be p e r f o r m e d by using the specific p r o p e r t i e s of the magnetic cross-section for unpolarized and p o l a r i z e d neutrons. Needless to say that the a b o v e sketched fields of interest d o not cover the vast a m o u n t of research on magnetic excitations which followed the first e x p e r i m e n t on Fe304 by Brockhouse in 1957 [1]. Needless as well to mention that this work is based on the p i o n e e r i n g work by Shull and coworkers in 1951 [2]. The outline of this p a p e r is as follows: W e will pick up e x a m p l e s which we find interesting in each of the three a b o v e m e n t i o n e d areas and try to describe the present state of research and possible outlines of future directions. We will also m e n t i o n the e x p e r i m e n t a l technique used and if necessary show what kind of i m p r o v e m e n t s are desirable.

2. Excitations in the ordered ground state In o r d e r to c o n s i d e r a system to be in the g r o u n d state it is usually sufficient to fulfill T<<

M. Steiner and H. Dachs / Magnetic excitations in insulators

71

Tc, i.e. the temperature should be much lower than the phase transition temperature Tc. In this case the excitation spectrum can be easily found by linearizing the equation of motion for the ground state given. One then obtains dispersion relations for the excitations which are usually termed magnons. These dispersion relations are given in terms of the microscopic parameters of the Hamiltonian, the interaction energy J and A, and possibly an applied field. Studies of excitations in this linear regime, that is the determination of magnon dispersion relations, has been and still is the classical area of inelastic neutron scattering on magnetic insulators. One can ask why we need to do these difficult experiments to determine all the microscopic parameters in the Hamiltonian, if we need to know the ground state to start with anyhow. The reason is that most ground states are not fully aligned states even at T = 0, and therefore the ground state cannot be calculated from first principles. Therefore comparison between experiment and theory for the excitation is necessary to understand the physics on the basis of a p p r o x i m a t e theories. In addition, the determination of the parameters in the Hamiltonian like

state. Therefore, the knowledge of the spatial anisotropy of the interaction is of fundamental importance for the understanding of magnetic systems. Thus, in conclusion the determination of the magnon dispersion relations in the ordered state, T << T c, allows a classification of the studied system according to spatial dimensionality and to the anisotropy-character or spin dimensionality. This classification is essential for the choice of the appropriate theory and for the study of more details of the spin dynamics as we will discuss in the next chapters. In what we have discussed up to now we have at least implicitely used the classical picture for the magnon as a coherent in phase precession of the spin around a quantization axis. This obviously is correct only for S --, or. It turns out, however, that it works very well even for systems with S = 1 [3]. It cannot be applied to systems with S = 1 / 2 however. Deviations from classical behaviour are most prominent in 1-D systems due to the large fluctuations. It was shown, that there is a quantitative difference in the dispersion relation for the excitations [4] in a 1-D Heisenberg antiferromagnet between the classical and the quantum limits.

H = -2J~_,S,S,+ 1 + AY~.(S,:) 2 - gt~an~s,

classical:

h~q = 4 S I J sin ~rql;

quantum:

h % = ~r I J sin ~rq [.

i

i

i

allows the following classification of the systems: A = 0 Heisenberg system: no preferred direction of the spin; A < 0 lsing system: easy axis for the spin; A > 0 Planar system: easy plane for spins. Since theory can provide most detailed results for the corresponding model Hamiltonians, so classified systems can be compared most effectively to theory. Another important aspect of the determination of the microscopic parameter in the Hamiltonian is the possibility to measure the spatial anisotropy of the interaction energies. This then gives a further classification of the systems: J~ ~ JY ~ J" 3-dimensional J ' --- JY >> J-" 2-dimensional (x, y, z are lattice directions) j x _._j y << j , 1-dimensional. As well known the spatial dimensionality plays a basic role for the phase transition to the ordered

This has been demonstrated experimentally in CuC12 • 2 pyr. [5]. In addition and more spectacular it was shown [6,7] that the excitation is not one single resonance but has higher order states mixed in thus leading to an asymmetric line shape in an inelastic neutron scattering experiment. Again there is experimental evidence for this in CuCI z • 2 pyr. [5]. But due to lack of good realizations and samples for this model no detailed studies exist. However, recently new efforts have been made, both experimentally as well as theoretically; experimentally because new compounds have been synthesized [8,9] and theoretically because of new results on the ground state of quantum spin chains [10] which has consequences on the excitations, e.g. by creating a quantum gap in the spinwave dispersion.

72

M. Sterner and H. Dach.s / Magnetw excitatmns m insulator.~

3. Stability. of magnons Once one understands the ground state and its elementary excitations the gate is open for more difficult problems which occur when the temperature is raised, such that magnons are thermally excited. If the density of magnons is large enough but T < Tc then the simple linear theory for the excitations is insufficient, because the magnons are no m o r e i n d e p e n d e n t , but interact. This m a g n o n - m a g n o n interaction was first studied by Dyson [11] and leads to a renormalization of the magnon energy and a finite lifetime of the magnons, which reflects itself in a broadening of the measured lines. The q-dependence of these effects are particularly interesting. As an example we take the experiments performed on EuO, one of the best realizations of a 3-D Heisenberg ferromagnet to study these effects. Results for the energy renormalization are shown in fig. 1 [12]. The experimental results agree well with the theoretical ones for all temperatures and q's. Fig. 2 shows the q-dependence of the spin wave spectra at two different temperatures close to T~. It is obvious that by approaching Tc the spin wave peaks start to merge at higher and higher q's: while for ( TC.- T ) / T c = 0 . 1 5 the peaks at q = 0.05 are still well resolved, the corresponding q at ( T c - T ) / T c = 0.05 is about 0.15. This strong T-dependence is connected to the strong T-dependence of the order parameter ( M ) and the critical fluctuations in the vicinity of T~. The solid lines are least squares fits of Lorentzian lines convoluted with the experimental resolution to the experimental points. The results shown before have all been obtained at T < Tc, thus in a regime where long range order still persists but with ( M ) << (M)~,, t. The experiments show that linear spin wave theory still works when m a g n o n - m a g n o n interaction is taken into account. One can even decide which interaction process dominates. But what happens when ( M > - - - 0 or T > Tc-? Then the remaining correlations between the spins vary with time and space and the correlation length ~" sets the T-dependent characteristic length scale. One can imagine that some type of excitation can exist within the correlated region, if its wavelength is smaller than ~.

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70

Fig. I. Spin wave energy renormalization in EuO versus 1" for different reduced wavevectors. Solid lines are calculations based upon the D y s o n - M a l e e v theor).. (from [13]).

In three dimensions the theoretical treatment is quite complicated [13]. It predicts that only for large q ' s some types of magnon can persist up to T > Tc. while at small q ' s the excitations become very rapidly overdamped. This has been found in EuO as well experimentally as shown in fig. 3 [14]. That the magnons disappear very rapidly for small q ' s is not surprising, since in 3-D systems

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M. Steiner and H. Dachs / Magnetic excitations in insulators

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direction [15].

are expected to be stable very close to the phase transition only. This makes detailed experimental studies of propagating collective excitations in the critical region T = Tc extremely difficult. Such studies are much easier in low dimensional systems where short range correlations extend over a large temperature range. Another advantage is that such systems can be treated theoretically in more detail than 3-D systems. This is in particular true for 1-D magnets, and we will discuss excitations in these systems in what follows. 1-D magnets have many realisations in nature

73

so that model systems are available for all the theoretical models listed in the introduction. However, only three have been studied in sufficient detail to be discussed here [15]; TMMC, CsNiF 3, CsCoCI 3. All these systems have 3-D ordering at sufficiently low temperatures, but outside that regime they can be considered one-dimensional. In both systems, T M M C and CsNiF 3, the existence of spin wave like excitations has been demonstrated and their dispersion has been measured [16]. From the so determined Hamiltonian one concludes that T M M C is a 1-D Heisenberg antiferromagnet with very weak easy plane anisotropy, while CsNiF 3 is found to be a 1-D easy plane ferromagnet. These results could all be described very well with linear spin wave theories [16]. Detailed studies of the temperature evolution of the spin waves in T M M C and CsNiF 3 yielded very good agreement with the prediction of the linear theory as well: Results for T M M C are shown in fig. 4 [17]. The observed linewidths are found proportional to T in agreement with theory [18]. For CsNiF s the crossover from propagating to overdamped behaviour of the spin waves with q is shown in fig. 5 [19]. Again classical spin wave theory is sufficient to describe the results as quantum calculations do not yield a better description of the results, (broken line in fig. 5). The conclusion from these results for the 1-D Heisenberg antiferromagnet T M M C with very weak easy plane anisotropy and for the 1-D easy plane ferromagnet CsNiF 3 is that even without long range order spin waves in the sense of linear spin wave theory exist for q > qcritical, where qcntical - 1 / ~ ( T ) . This is rather surprising because the number of thermally excited spin waves is very large, so large as to prohibit long range order. No need is found for the introduction of m a g n o n magnon interaction or nonlinear effects. Based on these results one might expect that the application of an external field brings the system even closer to the linear spin wave theory: In T M M C H_L chain introduces a crossover from easy plane to Ising character and thus increases the correlations length (~syplan¢ < ~lsing); in CsNiF 3 H _L chain couples to the order parameter and produces ( M ) 4= 0. Experimental results however do show completely different behaviour.

74

M. Sterner and H. Dachs / Magnetic excitations m insulator.s q

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In TMMC application of the field along the y-axis produces a splitting and mixing of the linear spin waves observed in zero field as can be seen from the field dependence of the zone center modes at 1.8 K shown in fig. 6 [20]. In zero field there are two modes, the IPC-mode (fluctuations in the easy ( x - y ) - p l a n e ) whose energy is zero at the zone center and the OPC-mode (fluctuations along the hard ( z ) axis), which has finite energy at q = 0.

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kO/..b/2g#B. y/

0

>, 1600 to

m

800

...qc 0"0 ~#~

-0.5 0.0 0.5 Energy Transfer (meV)

Fig. 5. The q-dependence of the spin wave-lineshapes in CsNiF~ at 4.2 K. Solid lines are results of the classical theory by Villain.

0

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M. Steiner and H. Dachs / Magnetic excitations in insulators

described by introducing nonlinear m a g n o n - m a g non coupling which introduces a hybridization between one and two magnon states [21]. This is a typical one-dimensional effect due to high density of two spin wave states at low energies. This theory also yields that the field flops the spins such that they are all perpendicular to the field in the easy plane. Thus a crossover to Ising character is produced by the field. In CsNiF 3 the application of the field also led to experimental results which could not all be described by linear spin wave theory. While at low temperatures the field dependence of the spin wave energy and the intensity of the IPC and OPC agreed well with a linear spin wave theory, this theory could not describe the temperature dependence of the spin wave intensity shown in fig. 7 [22]. Further studies at higher energy transfers have shown that two-spin wave processes become important in an applied field as can be seen in fig. 8 [23]. The second peak can be identified as a two-spin wave sum process. This means that two spin waves are created simultaneously, which then evolve independently. This two-spin wave process is only possible because of the easy plane anisotropy; in a Heisenberg system it would be forbidden because of the AS = l restriction for a neutron. From the existence of this sum process one has to conclude, that the difference process,

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Fig. 8. One and two spin wave processes as observed in CsNiF3 at the zone center.

simultaneous creation and annihilation, is possible as well. It will, however, become important at sufficiently high temperature only, where thermal excitation of spin waves becomes significant. From this we get a characteristic temperature for the system: If T is comparable to the gap energy of the spin waves E~ap - ~ then thermal excitation is large. It appears that as long as T < E~p, linear theory with inclusion of higher order noninteracting spin wave processes describes the experimental situation for CsNiF 3 very well, while for T >/E~p other theoretical approaches are needed. The rather surprising conclusion for field induced effects in T M M C and CsNiF 3 is that apparently the magnetic field drives the system out of the regime of linear spin wave theory even when higher order processes are included. This calls for another description of the spin dynamics, which will be discussed in the following paragraph. To end this section on the spin wave stability the following resume can be drawn: In zero field the spin dynamics can be described by linear spin wave theory. Surprisingly in 3-D systems even at T < Tc m a g n o n - m a g n o n interactions are needed to describe the results, while in 1-D systems without long range order this is not necessary. It is found that spin waves are well defined as long as q > qcritical- 1/4. The detailed studies on T M M C and

76

M. Sterner and It. Dachs / Magnetic excitations in insulators

CsNiF 3 showed that quantum corrections appear not important for these S = 5 / 2 and S = 1 systems, respectively. The most important result is that an external field can drive the system in a regime, where the spin dynamics can no more be described by linear spin wave theory. The collective excitations discussed up to now have been spin wave like excitations, but two spin wave processes could be clearly identified as well, but as one leaves the linear regime new collective excitations are expected to appear. We will discuss these new excitations in what follows.

4. Nonlinear spin dynamics If linear spin wave theory becomes insufficient, the best would be to solve the equation of motion of the spins without any restrictions that means allowing spin deviations of any amplitude. The first, who showed how to do this for I - D magnets like CsNiF~ and T M M C was Mikeska [24]. He mapped the equation of motion of these systems on to the sine-Gordon equation. The s-G equation has the great advantage that it can be solved exactly: the solutions are linear excitations, spin waves, and large amplitude solutions. For a ferromagnetic system like CsNiFa the large amplitude solutions are 2~'-solitons as shown in fig. 9a, while for an antiferromagnet like T M M C they are ~rsolitons as shown in fig. 9b. The thermodynamics of such a system is treated in the limit of a very dilute gas of thermally excited noninteracting quasiparticles. Before proceeding we should mention the conditions for the use of the sine-Gordon equation: classical spins strictly confined in the easy plane, continuum limit. Since the s-G equation a2~5

at 2

2 i)2q5

vo ~

m 2

=

sin

- 4', angle between spin and magnetic field direction, t~'0 spin wave velocity, m mass of the soliton is a Lorentz-invariant equation with respect to t, o , one has to stay in a regime where the relativistic effects are negligible. Within this limit it is rather easy to calculate the cross section for neutron scattering.

By

/,,

T~- soh~o r

V

Bx 2~r. 5oh~on

Fig. 9. (a) 2~-soliton in the ferromagnetic chain with eas) plane anisotropy. (b) v-soliton in the corresponding antiferromagnetic chain.

5. I-D easy plane ferromagnet in a symmetry breaking field From fig. 9a it is clear that the neutrons will see the soliton as a moving magnetization cloud. Since the excitation of the soliton by means of neutron scattering is not possible, we expect a quasielastic contribution to the cross section, whose intensity is determined by the density of the thermally excited solitons and whose q-dependence is given by the shape of the soliton, which is analytically known. First experiments [25] have indeed shown a spectrum with a strongly temperature dependent central peak in addition to the spin wave peaks at finite energy transfer as shown in fig. 10. Although the temperature dependence of the width and the intensity did agree rather well with Mikeska's prediction, contributions from the two spin wave

M. Steiner and H. Daehs / Magnetic excitations in insulators

difference process could not be excluded. In order to do the separation one has to consider the cross section quantitatively: since the system has easy plane symmetry and a magnetic field in the easy plane along the y-axis, all spin components in the Hamiltonian

CsNiF3

77

0=(0,0,1.9) H : 5 , 0 ROe

--500 E

~ 300

H= - 2 J E S i S , + , + AE(S,:)2-glzBHYESi .' i

i

•~ 200

i

behave differently and we have to deal with three cross-sections for the solitons. We do not consider the spin waves, because they can always be identified by their energy. Eq. ( l a - c ) are the result obtained for the solitons in the dilute gas and nonrelativistic limit (for the relativistic limit see [26]):

~rqJ2 m }2 Syy(Q, w)oc F(q~, ~o) sinh(~rqJ2m) " { ~rqJ2m } 2, Sx~(Q, ~) oc F(q~, ~o) cosh(~rqd2m) '

cosh(~rqd2m)

/3J e x p l _ 16flmJ

F( q~. ~ ) = voqc

.

) "

(Ib)

Oc)

8/3mJw2

v(~q2 ]

with

m = vgI~BH/2JS ;

vo = 2 ~ S

2•

As we can see the soliton has contributions to all cross sections, although the first two are the dominating ones. They all consist of two parts: F(q~, ~) contains the density of and velocity distribution in the soliton gas, while the second term is the form factor of a single soliton for the corresponding spin component. This form factor is very different for the x- and y-components: the y-component is maximum at qc = 0, while the x-component is zero at q¢ = 0 and maximum at finite q~ (for details see [27]). Since the two-spin wave contribution comes in as a quasielastic contribution too, we need know its cross section: the key point is that it contributes to S yy only. It has a rectangular shape with a cut-off energy E~o = voq~, its integrated intensity in the classical limit is

1 kB T2 12"w = 32~m3J z (1 + ( q J 2 m ) 2 ) "

.10

• 05

0

.05

.I0

Energy (meVl Fig. 10. Temperature dependence of the spectra obtained in C s N i F 3 in an external field of 5 kG: × 6.3 K: © 9.3 K; [] 14 K.

(la)

2

S=(Q, oo)oc F(q c ~0){ Irh~m/qcA

100

(2)

From ( l a ) - ( l c ) and (2) it is clear that the crosssections for solitons and two-spin wave scattering are sufficiently different to separate the two. In addition both, the soliton and the two-sw contributions can be scaled to the Iow-T spin wave intensity and thus the relative strength of the two contributions can be obtained. The way to do the separation is to use the fact that magnetic scattering is suppressed if q I[S. Fortunately in the I - D ferromagnet this can be used very effectively as seen in fig. 11, since the momentum transfer perpendicular to the chain direction does not matter.

"//'/////yy/,"/,////

1=2

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--

H vertical o

6~" ", < S~S~> ~ : + < Sy Sy>

H horizontal

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~,: . 6~: . < S~S~> Fig. 11. Reciprocal space for a I - D ferromagnet. The accessible combinations of correlation functions are indicated•

M. Steiner and H. Dachs / Magnetw excitations in insulator~

78

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(b)

Fig. 12. Background corrected spectra for CsNiF3 for two different field configurations. Cs NiI:~

We are thus able to measure all necessary combinations of the cross-sections to in principle determine each of them. That the cross-sections are very different can be most easily seen in fig. 12a, b where the field has been rotated in the magnet so as to give in a) S::+ S .... and in b) S::+ S'" without changing anything else. From ( l a ) - ( l c ) it is clear that the most sensitive test is the q~.-dependence of the intensity. We therefore show in fig. 13a--c the q~-dependence of the integrated intensity of the central peak at 10 k G and 12 K. In all these the prediction of the soliton and the 2-sw process, if necessary, are shown. S:: has been neglected. In 13a and b the " a g r e e m e n t " between theory and experiment is quite confusing, but 13c, where essentially S ' " is plotted, solves the problem, since here we are in the "sofiton-only" configuration. The scale of the theoretical result has been adjusted at 9 K, the values used for J and A where the ones obtained from the spin wave dispersion ( J / k = 11.8 K; A / k = 4.5 K). The theoretical line at 12 K has no free parameter. The agreement is rather satisfactory and the m a x i m u m in the scattered intensity a r o u n d q~ = 0.05 r.l.u, is clearly seen. The qc-de-

~tt (~, H : 10 k Oe i "6

\

g


\. " \

1

"'~-.

2 SW_ resul !

O

0

i

b

I

.05 10 :15 Reduced wavevector qc [[ Lu. ]

0

CsNiF3 ~H~, H : 10kOe ~=(e.0. l[I T : 12K

c,_

--

Fig. 13. S(q) for CsNiF~ for the three possible configurations: (a) (S'S")+
O =T=12K 1i27,0,2_ qc)

(~T: 9K

I

oN 0

q}

T

T

""-_

' 010

I-

C

O.05 015 Reduced wavevector qc (r.l.u)

M. Steiner and H. Dachs / Magnetic excitations in insulators

pendence of the width agrees very well with the soliton prediction as well. Fig. 13c is thus the form factor of a single soliton which contains about 15 spins. The results in fig. 13 give us a scale for the soliton contribution in the other cross-sections, thus one can determine the 2-SW contribution by subtracting the soliton contribution from the measurements shown in fig. 13a and b. In fig. 14 the scaled results for all configurations have been combined. Because results obtained in different configurations, at different reactors, etc. have been combined there are slight discrepancies at qc = 0 where two of the results should coincide. It is obvious from fig. 14 that the soliton contribution is only 1 / 3 in S yy and therefore /2sw = 21~ot. This is in contrast to the theoretical prediction I~o~> 312s,,. for the experimental conditions. There are several suggestions for improving the simple theory, but it is not possible to select out one condition as the right one on the basis of these data [27]. It should be possible, however, to test the influence of the proposed instability of the solitons [28], because this should increase the S-'~-contribution significantly. In order to obtain higher sensitivity polarized neutron scattering experiments have been performed [29]. In fig. 15 the result of an experiment corresponding to 13c is shown with and without polarization analysis. By comparing 15a and b it is obvious that the measurement of the

Cs Ni F3

M

-~ 13

T:12K

2

H=

10kOe

(3.

{ sxx.sY,

2 E(D L)

c

{ +

I

-..

C

......+

-

U.

.........

E

-

o t._)

200

100 0

- 0.1

0

0.1

"0 Energy (THz)

0.1

(b)

E

300

CD u)

200

(,-)

100 o 0

o "--0.I"

Fig. 15. Spectra obtained from CsNiF 3 obtained at 10 kG, 12 K with (a) unpolarized and (b) polarized neutrons.

spin flip scattering yields a polarization dependence not only of the spin waves but also of the central peak. This means S ~ of the soliton must indeed by nonzero. By changing the configuration such that S Y Y + S :z is measured with polarized neutrons, polarization analysis yields S ~ alone. No indication is found for a higher S ~ contribution than predicted by the simple sine-Gordon theory [30]. Thus we have seen that in CsNiF 3 the soliton contribution could be identified and its intensity was found significantly lower than theoretically expected. No indication for a soliton instability was found. The whole set of data could be described consistently by classical sine-Gordon theory once the intensity scale has been adjusted to the experimental value. This is true too, when the results for the spin waves are included [27].

6. I-D easy plane antiferromagnet in the field

0

o

.c_ 300 ~D

sXX ÷sZZ

r-

--

(ol

SYY .l.S Z z "".......

"tJ

79

o05 0Jo o.15 Reduced wavewdor qr [r.I.u.]

Fig. 14. S(q) for C s N i F 3 in the three configurations all scaled to the Iow-T spin wave intensity.

Here the situation is easier in the sense that the solitons manifest themselves very differently from the ferromagnetic case. As can be seen in fig. 9b the ~r-soliton cuts the chain in two pieces by interchanging the sublattices. This means that the

M. Steiner and H. Dachs / Magnetic excitations in insulator.s

80

density of the solitons determines the correlation length. Thus the soliton induced peak in the 1-D antiferromagnet increases and narrows with decreasing soliton density n~ [31},

S l ( q, oo)= S 2 1~

It/Fq

100

C's

I ~

10

.a

o 10

1

2~, ( r g ( l + q _ /"r 4 )2 + 092 ~j 3 / 2

(3) with

I;¢ = 4n~

_

2all jS2~

5

exp(-aH/T),

(4)

8

a = g~tBS/k.

A typical result of the cross section is shown in fig. 16 indicating the strength of this scattering. It should be noted here, that this is not the scattering from single solitons as in the ferromagnet but from long ordered regions which are terminated by ~rsolitons. The obtained q- and w-widths should collapse on a scaling plot on two lines with the same slope. That this indeed is the case demonstrates fig. 17 very nicely. The slope yields the energy parameter a. The experimental value is 20% below the value calculated in the classical

3000 0o=[0.25,0,I] ~ k, =1.0.4.&1 / ~

TMMC T=2.5K

2000

/ ~ ~ ,

°

.400 .400 ~ o

-0.02

"t4 ?

"'~, FD

0.1

I

~ T

I'~ = -~-~aH e x p ( - a l l ~ T ) ,

m1 0 0 0

:

~_. I L.c

-0.01

0 0.01 Energy (THz)

002

Fig. 16. The quasielastic peak in TMMC at the zone center for different magnetic fields at T = 2.5 K.

10 H/ T (kOe/K)

15

Fig. 17. Scaling plot of the width in energy I"D and in momentum I', for the quasielastic peak in T M M C

sine-Gordon theory. All the classical sine-Gordon calculations assume that the solitons move freely and ballistically. If, however, impurities are put in the chain the soliton motion is modified and becomes diffusive. This has been shown by high resolution spin echo experiments [31 ]. The situation in the antiferromagnetic chain is very favourable for a detailed study of the soliton thermodynamics because one can go from the very dilute limit, large H / T , to higher soliton densities at small H / T . However, the discrepancy between theory and experiment for the energy parameter a has not been explained yet. In concluding the part on the sine-Gordon-like system one can say that it is generally accepted that the spin dynamics consist of spin wave like excitations, including higher order processes, and nonlinear soliton-like excitations. The very good description of the experimental results by means of the sine-Gordon theories is still a surprise and needs further theoretical efforts in view of the very restrictive conditions imposed by the use of the s-G equations. It is worth noticing, that the specific heat measured for CsNiE~ [32] and T M M C [33] do not agree very well with the prediction of s-G theory. Therefore, experimentally all the proposed corrections to the simple classical sine-Grodon theory remain to be studied and clarified. Apparently the use of the polarization analysis is a very effective tool to selectively probe specific parts of the cross section.

81

M. Steiner and H. Dachs / Magnetic excitations in insulators

,i

7. I-D Ising Antiferromagnet

tlt ,

The theoretical basis is more clear for another type of nonlinear excitations and their observation. It was first proposed by Villain [34], that in an 1-D Ising antiferromagnet the movement of domain walls should be observable. These detailed predictions initiated experiments [35] on CsCoC! 3 and CsCoBr 3 both known as very good realisations of I-D Ising antiferromagnets. Villain predicted that scattering from the narrow domain walls should yield a quasi elastic scattering which has a singularity at the q-dependent cutoff energy. The dispersion of this so-called Villain mode is

hC.~Domain

=

4cJ sin ~rq~,

where c is a measure of the transverse coupling strength. In fig. 18 the dispersion observed in CsCoBr 3 by Nagler et al. [36] is shown together with the Villain prediction for the parameters of CsCoBr 3, both in very good agreement. There is a rather simple picture for scattering of the neutron from this type of domain walls, see fig. 19: By flipping one spin up and one down the wall moves by one step. It is exactly this process which is reasonable for the non-spin-flip neutron scattering process of the longitudinal cross section. Therefore neutrons can see the moving walls. It is worth noticing that the spin flip neutron scattering process, the "spin wave" scattering, in this particular case creates two domain walls which then move along the chain because of the transverse coupling. I

'

10

'

'

'

I

I

,

,

,

I

Cs CoBr3 (1.2, 50KO,r/)

-£

/ ~ ~ ' ~

T I--

g

0.5 /

/

0.0 O ~r rt 1.0

~

J : 1.62THz ¢ : 0137 I

~

Wavevector

,

,

it/2 0.5

Fig. 18. Dispersion of the Villain mode in CsCoBr 3. Solid line is calculated using the known values for J and (.

'.t

X i.,.2 I

tH

,

Fig. 19. Movement of a domain wall in the ising antiferromagnet.

This section on nonlinear effects should not be ended without some suggestions for future studies. As mentioned already there is still significant work to be done to understand the sine-Gordon-like systems not only by further studies on the "old" systems TMMC and CsNiF 3 but also and very importantly by searching for new systems wtth different parameters and spin values (in particular S = 1/2, which are far from the classical sineGordon limit). Kosterlitz and Thouless have shown [36] that vortices, very strongly nonlinear excitations, are essential for the phase transition in the 2-D x - y model. There is some experimental evidence that such type of behaviour can indeed be found in real crystals. The observation and identification of vortices would certainly be a very exciting result.

8. Conclusion The study of magnetic excitations in insulators has three major aspects: the determination of the dispersion in order to set up the correct Hamiltonian. the measurement of the line shape of the magnon peaks to study magnon-magnon or, what we did not discuss here, magnon-phonon interactions. A comparison of such results with linear spin wave theory shows how far spin wave theory can be used. It is found that this theory works surprisingly well even in systems without long range order. If, however, discrepancies between linear theory and experiments are found as in 1-D magnets, then careful experiments making use of the characteristics of magnetic scattering for unpolarized and polarized beams can lead to the identification and characterization of new nonline a r excitations. This is a key issue for the further

82

M. Sterner and H. Dachs / M a g n e t t c excitattons tn tnsulatorg

understanding of nonlinear systems, since in such systems the concept of using elementary excitations in conjunction with the superposition principle to calculate the thermodynamics cannot simply be applied as in linear systems. As can be seen from the discrepancies between experiment and theory for the specific heat in nonlinear systems, the calculation of thermodynamic quantities is not yet well advanced. Since it appears difficult to identify the different contributions to quantities like the specific heat and the susceptibility, the only way to obtain a better understanding of nonlinear systems, their spin dynamics and their thermodynamics, seems to be a careful study of the different "elementary" excitations and their behaviour with field, temperature and microscopic parameters. This, however, can only be done by neutron scattering, thus leaving the neutron scatterers with the challenge to make further essential and basic contributions to the field of nonlinear physics, which is not limited to the solid state and the systems discussed here. Since the systems are particularly simple, the chances to solve basic questions here are much better than in other nonlinear systems.

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