one- and two-magnon excitations in a one-dimensional antiferromagnet in a magnetic field

PHYSICAL REVIE% B VOLUME 24, NUMBER 7 1 OCTOBER 1981

One- and two-magnon excitations in a one-dimensionalantiferromagnet in a magnetic field

I. U. Heilmann and J. K. KjemsRisyf National Laboratory, Roskilde, Denmark

Y. Endoh,' G. F. Reiter, and G. Shirane

Brookhaven National Laboratory, Upton, New York 11973

R. J. BirgeneauDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 21 January 1981)

%'e have carried out a comprehensive experimental and theoretical study of the inelastic

scattering in the one-dimensional near-Heisenberg antiferromagnet (CD3)4NMnC13 (TMMC)at low temperatures, 0.3 «T «2.5 K, in magnetic fields varying between 0 and 70 kOe; thefield is applied perpendicular to the chain axis. In zero field at long wavelengths we observe two

sets of excitations, a low-energy acoustic branch corresponding to spin motion within thedipolar-determined easy plane and a high-energy optical branch corresponding to oscillations outof the plane. For magnetic fields greater than 30 kOe and T = 2 K we observe as many as fourdistinct excitations —the two one-magnon modes plus two sharp excitations at higher energies.Our theroretical analysis suggests that the two higher-energy modes correspond to two-magnon

processes in the longitudinal response function. The theory, which is done within the harmonic

approximation expanding out to fourth order in the magnon operators, gives a good qualitative

description of the data but underestimates the two-magnon intensities by a factor of 2 or 3. %ealso observe a marked anticrossing of the one- and two-magnon branches; this latter result

shows that anharmonic effects are quite important in the spin dynamics. Finally at T =0.3 Kand zero field we observe a gap of 0.1 meV in the acoustic spin-wave dispersion relation due toa very small in-plane anisotropy field of 71 +30 Oe.

I. INTRODUCTION

The spin-dynamical properties of one-dimensional(1D) magnetic systems continue to be the focus ofmuch research activity. ' Further, a considerable partof the experimental information is being provided bythe neutron scattering technique. Recently, specialinterest has been directed towards the properties ofmagnetic chains in the presence of an applied mag-netic field. ' For example, as shown by Mikeska, '

the classical 1D easy-plane ferromagnet in a magneticfield is equivalent to a sine-Gordon system, and con-sequently possesses exactly solvable nonlinear breath-er and soliton solutions, in addition to the low-lyinglinear spin-wave states. Much effort has been devot-ed to the possible experimental observation of suchnonlinear excitations in certain quasi-1D magneticmaterials. ' ' " However, before such nonperturba-tive effects can be properly characterized, one mustfirst understand completely the behavior in thequasiharmonic regime including, for example, multi-ple magnon excitations and interactions. '8 As weshall see, very rich behavior is indeed observed forthe spin dynamics of a 1D Heisenberg antiferromag-

net in a magnetic field. %e find, however, that theprincipal features of the finite energy excitations maybe understood within the spin-wave approximationprovided that one includes two-spin-wave processes.

In the present paper we present a detailed neutronscattering study of the finite energy spin dynamics oftetramethylammonium magnanese chloride,(CD3)4NMnC13, (TMMC) in magnetic fields between0 and 70 kOe applied perpendicular to the directionof the chains. A similar but less extensive study was

previously carried out on the S = —1D magnet, di-

chlorobispyridine copper, CuC12 ~ 2N(CSD5) (CPC) "It was shown there that the field dependence of thespin-wave excitation energies in CPC was in agree-ment with exact quantum-mechanical calcula-tions, ""rather than with exact classical spin-wavetheory. Since the spin value of the Mn2+ ion is large,S = —,we originally anticipated that TMMC would

provide a test of the theory based on classical spinmodels, thus providing a counterpart to the quantumsystem CPC. Based on previous neutron studies and

. other measurements, ' TMMC has generallybeen considered the best realization of the 1D anti-ferromagnetic classical Heisenberg XY model.

3939 O1981 The American Physical Society

3940 I. U. HEILMANN et al. 24

From the outset of the original studies of TMMC itwas realized, however, that a slight deviation fromisotropic Heisenberg symmetry should be expecteddue to the magnetic dipole-dipole coupling. Thisanisotropic interaction favors spin directions in the xyplane perpendicular to the chain axes, and thus givesrise to a crossover from Heisenberg to XY behaviorwhen the temperature is lowered. Effects originatingfrom this anisotropy have been observed in the low-

temperature susceptibility, specific heat, NMRlinewidths, and electron paramagnetic resonancespectra. " In a two-sublattice antiferromagnet with aneasy-plane anisotropy there are typically two branchesof spin-wave-like excitations": one associated within-plane spin-fluctuation correlations (IPC) andanother associated with out-of-plane (OPC) correla-tions. The IPC component is expected to showbehavior similar to that of the isotropic Heisenbergchain': a well-defined resonance is foun'd forwave vector q much larger than the inverse correla-tion length ~, ~hereas the excitation broadens andbecomes overdamped in the small-q limit. ' " Thisis a consequence of the lack of long-range order inthe in-plane spin components and has been demon-strated experimentally in previous inelastic neutronscattering studies on TMMC' and CsNiF3. " In con-trast to this, the OPC component should be sharpeven in the q =0 limit, and concomitantly for the an-tiferromagnetic case, a q =0 gap energy is predict-ed. The existence of both IPC and OPC com-ponents in the ferromagnet CsNiF3 has been verifiedexperimentally. " We shall, of course, be interestedin similar effects in TMMC.

An unusual feature which must be taken into ac-count in the present study is the strong field depen-dence of the 3D ordering temperature T~ ofTMMC. " The paramagnetic and the antiferromag-netic (AF) phases of TMMC are divided by twosecond-order transition lines in the HT plane, "onefor the field along the in-plane easy axis and the oth-er for the field in the plane perpendicular to the easyaxis; the former shows spin-flop bicritical behaviorwhile the latter, which is shown in Fig. 1, has T~ in-

creasing uniformly with increasing H.In a recent publication we reported preliminary

results of inelastic neutron scattering studies ofTMMC at both zero and finite magnetic field appliedperpendicular to the chain axis. ' It was demonstrat-ed that TMMC indeed displays a sharp OPC com-ponent with a q =0 gap, as expected, thus givingdirect evidence for the importance of the dipolar an-isotropy. The observations at higher fields turnedout to display complex features and it was earlier sug-gested by us that these effects may be caused by thefield-induced 3D ordering. However, as will bedemonstrated in this paper, the 3D ordering has nomeasurable effect on the observed spin-dynamicalproperties, apart from permitting a small (0.1 meV)

TMMC PHASE DIAGRAMl I I I I

80-

60—

I

0.84

20—

I -I.o 2.0

T(K)

FIG. 1. Upper critical boundary for three-dimensional or-dering for TMMC with a field perpendicular to the chain

. axis. The open squares and triangles are from Ref. 3. Thefilled circles represents our own results. The inset shows thestrength of the Bragg scattering in zero field in the immedi-ate vicinity of Tz =0.850 K.

q =0 gap in the IPC branch; further, the IPC gap ismanifest only at temperatures below T~. The ap-parently mysterious upper branch of excitations, pre-viously observed at high fields, ' is shown to be asso-ciated primarily with longitudinal fluctuations and ori-ginates from scattering involving two magnons, oneof which has wave vector q -0. This selection iscaused by singularities in the density of states at thegap energies and is a unique consequence of the 1Dcharacter of the dispersion. We also observe astrong coupling between the one-magnon and two-magnon resonances which also clearly demonstratesthat anharmonic effects play an important role in thespin dynamics.

The remaining part of this paper is divided into thefollowing main sections. In Sec. II we presenttheoretical calculations of the dynamical spin-correlation functions within the framework of con-ventional spin-wave theory. Section III contains theexperimental details. The data and discussion for thezero-field measurements are presented in Sec. IV,Sec. V contains the finite-field results, and analysis.Finally we give concluding remarks in Sec. VI.

II. THEORY

We take for the Hamiltonian of the system

with J=13.0 K; D =0.0086, and g =2.00. The z axisis along the chain axis. At this stage, we shall omitthe very small xy in-plane anisotropy. The above alsoomits the longer-range part of the dipolar interaction

X =J X(S; S, , DS;$~, ) gp, H —XS; (2.1)—I i

24 ONE- AND T%0-MAGNON EXCITATIONS IN A. . . 3941

S=S—aa"g

S = S"+iS"= 42Sat (2.2)

S =S"—iS»=v'2Sa

where z, the spin direction, is the axis of quantiza-tion. We have linearized the transformation, which,in order to make sense, requires that the spins arewell localized in a direction along the quantization

'"0' ' i'axis. We put cos8;=e 0 'cos8, where ko=qr/a(here a is the separation between nearest neighborsalong the chain) and sin8, =sin8, and by transform-

which is not quantitatively important here. In the ab-sence of a field, at low temperatures but above the3D ordering temperature, the rotational invarianceabout the z-axis may be exploited to obtain exactasymptotic results for spin-wave frequencies andlinewidths. ' If the field is sufficiently strong, thespins will be aligned nearly perpendicular to the fieldand one might expect to be able to use spin-wavetheory to describe the response. We will considerthis latter case, thus assuming that the configurationabout which we calculate small fluctuations has an in-finite coherence length along the chain. This is anapproximation in that there are quantum fluctuationseven at T =0 that will disorder the chain.

We choose for each site local axes (x y z) with zthe quantization direction (see Fig. 2). The angle 8;in Fig. 2 will be determined presently. TheHolstein-Primakoff prescription for the replacementof spin operators by boson operators is

Sl S'

FIG. 2. Orientation of the quantization axes for the spin-wave theory relative to the field H (assumed along y) andthe chain (assumed along z) axes.

ing to the system (xyz) we obtain

Si" e q '(S —ata;) cosH —i(S/2)' z(a;t —a;) sin8

S»=ie '(S/2)' (a; —a;) cos8

+ (S —a; a;) sin8

S,*= (S/2) 'i'(a t+ a, )

(2.3)

Substituting Eq. (2.3) into Eq. (2.1) we observe thatwe will obtain terms linear in a~ and a. The vanish-ing of these terms is assured when 8 is chosen as

sin8 =g p,sH/4JS =—l3 (2.4)

which value minimizes the sum of field energy andexchange energy. We note that this substitution doesnot eliminate terms which are cubic in a~ and a. TheHamiltonian then becomes

X=NJS2cos28 NgizsHS sin8—+2JS X(a, a;+-(a, —a, )(a;+~ —a, +~) cos28i

+—,'(1 D)(a, +a;)—(a,~t +a;~, )]+gpsH sine Xa; a;+i

2.5)

where the ellipsis indicates higher order terms. De-

fining aq=N '2 X, e ia, and a,t=N 'X,. e 'a

and setting the lattice spacing equal to unity. Wehave for the spin-wave part of the Hamiltonian

I

where a'+ P' = 1. We make the transformation tonormal modes by defining new boson operators b~~, b~

aq = pqbq—qiqb q, aq = pqbq

—qiqb q, (2.7)

t

X,=2JS X 1+ P' ——cosq aqaqD

e

r

+JS az ——cosq(aqtatq+aqa q)

(2.6)

where p~, q~ are real, symmetric in q, and satisfy

p~2 —v)~2 =1 (2.8)

The terms b, b ~ and b~b ~ can be eliminated fromthe Hamiltonian if we choose

2 pqqiq/( pq'+ qiq')'

=cosq(ot' —D/2)/[1+(P' D/2) cosq] . (2.9)—


in which case the Hamiltonian becomesr' t 1

—2JS X 1 + p2 cosq (pz+»12) 2 a cosqp»'g» 'b» b»D

q i i i

f 1

+2JS X' l. +. p2 cosq q a cosqp»q»'D 2 D

q i i i

Equation (2.8) is satisfied identically by p» =cosh8», »1» = sinh8» and 8» is determined from (2.9). We have

(2.10)

1+ (Pz D/2—) cosq[[1+(P'—D/2) cosql' —(n' D—/2)'cos'q]'t'

p» = [—(1+cosh28q)]' ', »t» =+[z

(cosh28q —1)]'(2.i i)

where the + sign is equal to sign (cosq). We havethen

I

have then

(S»(t)S"») =g» ((S—aita;))'+(5S»+ (t)8S»+ )s ~~q~q q +Xzero point

where the spin-wave energy is given by

ip» =2JS [[1+(P2—D/2) cosq]2

( 2 D/2) 2 Gos2i? }I/

(2.i2)

(2.i3)

(S»(t)S») = —( [a»+„(t)—a»(t)](a»+„—a») )

(S»( t) S*» ) = —([a t» ( t) + a, ( t) ](a t» + a, ) )

2where

or, equivalently

ip» =2JS ] [(a2 —i82) (1 cosq) +—2Pz]

x [1+(1 D) cosq]]'—tz (2.14)

For the maximum value of field, H —70 kOe for thepresent measurements, we have P =g p, &H/4 JS-0.08 and in calculating the correlation functions,

lq r.we can set 8=0. Defining S, =N 't' X, e 'S;, we

SS» = W 'tz Xe '(a ta —(a ta ) )?

=?V 't' $ A(q, —q2+q) (a»t, a», —(a»t a, ))(2.15)

(5 = 1 when the argument is equal to a reciprocal-lattice vector n2»r). The averages are readilyevaluated by converting to normal modes using (2.7).%e have

(S(t)S )= (p+ +-5+ ) [n+ e + +(n „+1)e»+ ]

(S,*(t)S'-,) = —(p, —»t»)'[n»e ' +(n, +1)e»] (2.16)

where

n=»( ~e' s —1) '

From Eq. (2.16) we can readily identify distinct IPCand OPC modes close to the zone center. For q = m

the yy response exhibits a sharp peak at

4, = ip( q = 2»r ) =g paH (1 ——,' D) 't' = g paH (2.17)

which is absent in the zz response since

p. ..= [-,' (1+P')/2P]'",

n -z.= [—'(1 /3')/2/3]'" = p»-2.—~

according to Eq. (2.11). Similarly, for q = »r the zz

response exhibits a sharp peak at

Az = »» (q = »r ) =tDi [2(2JS) ——(g?i, H ) ] )

'tz

(2.18)

which is almost absent in the yy response since

p„q = —qq . Thus, d~ =gp, &His the IPC mode,

the energy of which is proportional to the field H:the spins fluctuate "against" the field. For the rangeof H values in these measurements g p,&H (& 2JS=65 K and the energy of the OPC modeb,2=2JS42D, independent of the field: the spinsfluctuate in the plane perpendicular to the field. Therelative intensities of 6] and b 2 may easily be calcu-lated by means of Eqs. (2.16) and (2.11). For mag-non creation we obtain

I(h, ) a(Pp+»lp)'(np+1)

=4JS/g psH/(1 —e )

(2.19)?(h, )n(p —

vt )'(n +1)= (1/ J2D +1/4D)/(1 —e s )

ONE- AND TWO-MAGNON EXCITATIONS IN A. . . 3943

The relative intensities 1(bt) and 1(52) as a func-tion of H are plotted in Fig. 3 with T =1.8 K (experi-mental condition). D was put equal to D =0.0086,the value found from the measured energy of h2 (seeSec. IV). It is seen from Fig. 3 that for increasing Hthe IPC intensity I(b t) decreases and the OPC inten-sity is constant, the two intensities becoming equal atH =70 kOe.

When q moves away from q = m the IPC and OPCmodes gradually lose their distinct polarizations andthe two branches merge at the zone boundaryq = qr/2, where the spin-wave energy takes on thefield and anisotropy-independent value oftu(q = qr/2) =2JS. This zone-boundary degeneracyof the spin-wave spectrum at finite fields is charac-teristic for the spin-flopped configuration in whichthe two sublattices are equivalent.

The xx correlation in Eq. (2.15) contains, in addi-tion to the antiferromagnetic Bragg scattering, a termdescribing correlation in the length fluctuations of thespins, which we now want to evaluate. The averagedeviation from each spin being fully aligned along thequantization axis (practically, the x axis) is

(atra; ) = N t g (aqtaq ) = N t X [ ( pq2 + qtq2) nq +qt q ]

(2.20)P

and the fluctuation in the length of the spins

(SSq+ (t)8S q+ ) =N ' X A(qt —q2+q+qr)

IOO

80—

60—

(0La!

40—

20—

00

I

20I

40I

601 I

80

FIG. 3. Theoretical intensities of the in-plane componentl(ht) and the out-of-plane component, 1(52), at q = qr/a

from the noninteracting spin-wave theory.

x [(pq pq +qtq qtq )2nq (nq +1)cos(cuq taq )t-i (o) '+ so ) I (cat + rat ) f

+ 2(qtq, pq +qtq pq, ) [(nq, +1)(nq +1)e ' 2 +nq, nq e ' 2 ]] . (2.21)

These correspond to the various two-spin-wavescattering terms. " In this paper we shall focus onthe (nq, +1) (nq + I) term which represents the

two-spin-wave creation process. At zero field thespins are not at all localized along the x axis, and infact (2.20) diverges. To obtain some estimate of thevalidity of the spin-eave approximation, we have cal-culated Eq. (2.20) at T =0, for different values of H.The result is shown in Fig. 4. The spin-wave approx-imation will be reasonable when (a;ta;) « S =2.5.Actually, the validity of the theory depends on whatone calculates. For rotationally invariant quantitiessuch as SI SJ one may even. use it for H =0. Forquantities that we are calculating which are projec-tions onto a specific fixed axis, the aforementionedcriterion is appropriate. The theory evidently makesno sense for fields of the order of 2 kOe (H/4JS=0.002) but ought to be fairly good for fields above10 kOe, and improve with increasing field.

The experiments are done mostly at temperaturesof about 1.8 K. At the highest fields used, the max-imum value of the occupation number is

l

-gp, H/kT—e & —e~, and thus completely negligible.On the other hand, for fields of 30 kOe, the valuewould be 0.25, and not completely negligible. How-ever, the dominant contribution to Eq. (2.18) abovethe IPC spin-wave frequency is obtained by taking

I I I 1 1 I

OIII- 2.0—

O

I.O—

0O. I O.OI

H/4 JSO.OOI

FIG. 4. Zero-temperature deviation of the spins from fullalignment calculated using linear spin-wave theory.


80.0—(a)

60.0—

40.0—

Q= I.OO r.l.u.T= I.B K

H=30 kG

the T =0 limit. Note that there is also a contributionaround co =0, whose intensity would vary as

-gga H/kTe & for sufficiently small values of this parame-ter. At T =0, we obtain for the longitudinalresponse function

G, (o)) =(I/2qrtt) „7e' '(Sq(t)S" )dt,

20.0-Gq (~) = +('9q+t +q Pq +p +k y qt )

i

X, (Cq + . a+CO I)aq+kO+q q

(2.22)

pp0.0 0.4 0.8

I I I I I

I.2 I.6 2.0 2.4 2.8FREQUENCY where qi are the wave vectors in one zone for which

40 = OJq +k +q + Mq0 i i(2.23)

(b)50.0—

I-g 40.0—4JO

50.0—KI-.o 20.0-ILIA.CO

I 0.0—

H=50 kG

I20.0—(c)

80.0— ¹70kG

40.0—

pp I l i i i I I I

0.0 0.4 0.8 l.2 I.6 2.0 2.4 2.8FREQUENCY

This, of course, is just two-magnon creation scatter-ing which as we discuss below, is expected to have avery unusual form in one-dimensional magnets.There is a singularity in G, (co) for every maximumor minimum of co + +~ as a function of q',

q+ko+q q'

and the spectrum is in general quite complicated. Forq = qt/a = ko, the spectrum is easy to visualize sinceeo =2coq. The spectrum begins with a singularity at

co =2g p,&H, since our fields are such that g p,&H is thelowest spin-wave frequency. This is followed by asingularity at twice the frequency of the OPC mode2(2JSJ2D ) and the spectrum ends at twice themaximum spin-wave frequency 4JS with a muchweaker singularity. The strength of each peak isdetermined by the coefficients p, qt in (2.22), whichare largest where the frequency is least, so that therelative intensity of the second peak increases as thefield increases. We show in Fig. 5 the spectral func-tion (2.22) for q = qr calculated for H =30, 50, and70 kOe. The arrow indicates the position of thespin-wave peak at g p,&H. Once more we emphasizethat the very unusual form for the two-magnonscattering evident in Fig. 5 is a direct consequence ofthe singular density of states characteristic of a one-dimensional system.

III. EXPERIMENTAL TECHNIQUE

Oo I It I I I I I

~.0 0.4 0.8 l.2 l.6 2.0 2.4 2.8F RFQUENCY

FIG. 5. (a)-(c) Inelastic part of the longitudinal responsefunction G~(Q, ao) calculated using harmonic spin-wavetheory at Q =1 reciprocal-lattice unit (~/a), T =1.8 K andvarious fields. The energy is in units of meV. The arrowindicates the lower gap energy g p,~H. The lower and upperpeaks at 30 and 50 koe correspond to 2d i =2g p, ~H and

252 1.6 me& where LLi and LL2 are the one-magnon IPC-

and OPC gap energies, respectively.

The inelastic neutron scattering measurement re-ported here were greatly facilitated by the growth byS. L. Holt of a new 5 &8 &20 mm high-quality singlecrystal of deuterated TMMC. In order to reducestrains induced by the mounting, the sample crystalwas wrapped in Al foil and attached to an Al holderby means of glue. It was then mounted in a He-filledAl can.

The majority of the measurements were carried outon the H7 triple-axis spectrometer at the BrookhavenHigh-Flux-Beam Reactor; in addition some high-


resolution scans were carried out on the cold-sourcespectrometer TAS 7 at Ris National Laboratory,Denmark. The scattering geometry is shown in theupper part of Fig. 6. The Mn2+ chain direction, C, isoriented in the scattering plane defined by the wavevectors ki and kf of incident and scattered neutrons,respectively. The chain direction is equal to the caxis of the hexagonal high-temperature phase. Uponcooling below T, =128 K the crystal undergoes astructural phase transition which involves changes inbasal plane lattice parameters, but does not affect theinternal structure of the Mn2+ chains and thereby the1D magnetic properties. ~ It is convenient to maintainthe indexing of reciprocal space corresponding to thehigh-temperature hexagonal phase. The lo~er part ofFig. 6 shows the (h0!) zone of reciprocal space usedas the scattering plane. The hatched lines,(h, 0, 2n +1), indicate the intercept between thescattering plane and the planes (h k 2n +1) of dif-fuse magnetic scattering from the Mn + chains. 9 In

the 3D ordered phase below T~ the points(0,0,2n +1) become antiferromagnetic Braggpoints, "since there are two Mn'+ ions per latticeconstant c. In what follows we will refer to Cartesianaxes xyz in real space defined such that y is parallel tothe applied field and z is parallel to the chain direc-tion c. Thus, the scattering plane is the xz plane. Asis well known, '4 only components of the spin-spincorrelations perpendicular to the scattering vector Qcontribute to the magnetic coherent neutron scatter-ing cross section. Thus, in terms of the angle $between Q and the chain direction C(z) the crosssection can be written as

d QdE',ucos'd G (Q, ao) +G~(Q, &o)+sin'QG (Q, ~)

(3.1)where

Gae(Q ) dte '"'(S- (t)S - (0) )2mb Q —Q

Q= kf-k)

(200) ii

(000)

Jp

Jp

/y

Jp

c J is

QPrlrPrPr

(002)

j!Jy

Jp

/g

/y

ll-

FIG. 6. Scattering geometry in these experiments. Theupper diagram shows the scattering in real space, the lower

in reciprocal space.

Hence by varying $, one can control the relative con-tributions of the zz and xx fluctuations.

The measurements in an applied magnetic fieldwere carried out by means of a split-coil supercon-ducting magnet combined with a pumped 4He cryo-stat. The direction of the magnetic field H was per-pendicular to the scattering plany„ the highest obtain-able field strength was 70. kOe. The lowest obtain-able temperature with this configuration was 1.8 K.We also carried out zero-field measurements at tem-peratures down to 0.3 K by- means of a pumped 'Hecryostat combined with a conventional pumped 4He

cryostat.Standard techniques were applied in the operation

of the triple axis spectrometers. All inelastic scanswere carried out in the constant-Q mode; the neutronenergies as well as the horizontal collimations wereadjusted to meet the various requirements of resolu-tion and intensity. Details of the spectrometer con-figurations are given in the figure captions.

IV. ZERO-FIELD MEASUREMENTS. '

EXPERIMENTAL RESULTS. AND DISCUSSION

In the previous neutron studies of the magnetic ex-citations in TMMC, the inelastic scans were car-ried out for directions of the momentum transfer Qin the vicinity of the c" axis (@—0), and thus onlythe in-plane-correlation components were probed. Inthis study, we have extended the previous measure-ments at zero magnetic field, and special attentionhas been focused on the small-wave-vector regionwhere, as we have discussed in the previous section,an energy gap due to anistropy is expected.

Figure 7(a) shows inelastic scans performed atthree different momentum transfer vectors Q, all


TMMCI

I 60—E~ l20—COI-

o 80

40—

0.80O

~n 0.6

~ 0.4

0.2

(0)Q

0 (0,0, I)

0 (-0.8,0, I)

~ {-l.6,0, l)

I5 30

T= I. I KI I

45 60$ (deg)

75

H=O

I

l.2 {1TieV)

I

90

for the classical magnetic dipole-dipole couplingalone. This discrepancy may reflect single ion anis-tropy effects as well as 'quantum corrections to theclassical spin-wave theory for the spin-wave gap. Theformer explanation has been advanced by Lagendijk'5to explain the discrepancy between the theory andexperiment for the EPR linewidth in TMMC. ' If thepresent value of D =0.0086 is used, the theoreticallinewidth prediction is reduced by about a factor of 2,and would agree with the experiments to within thetheoretical uncertainty.

The measurements of the OPC excitations were ex-tended to finite values of reduced wave vector q, ; weshow in Fig. 8(a) the measured dispersion forq, ~0.1. Clearly, there are two branches of spin-wave excitations, an acoustic branch with co 0 forq, 0 as seen in the previous studies, ' associatedwith IPC fluctuations, and an optical branch with gapenergy A2 for q, =0, seen only in scans with Q form-ing a finite angle Q with the chain direction, and thuscorresponding to OPC fluctuations. The twobranches cannot be resolved for q, & 0.05.

The solid curves in Fig. 8(a) are the theoreticalspin-wave dispersion as given by Eq. (2.14) withD =0.0086 fitted to the observed gap 42 =0.80 meVand 2JS =6.1 meV fitted to the observed zone-

FIG. 7, (a) Upper panel: inelastic scans at q, =1 with

three different transverse momentum transfers. (b) Lowerpanel: intensity of the 0.8-meV spin-wave peak as a func-tion of $, the angle between the momentum transfer Q andthe chain axis. The outgoing neutron energy was fixed at5.5 meU and all collimators were 40'. 1.2

TMMC SPI N-WA VE D I SPER SION S H=O

having the same component Q, = 1 (c"units) alongc" but different components Q, (=0, —0.8, and —1.6,respectively, in a" units) along a". Excitations of re-duced wave vector q, equal to zero are thus probed.As the value of g, and thereby Q increases a sharpexcitation at 0.8 meV grows up. The integrated in-tensities of the peaks were determined approximatelyby subtraction of individual background levels andare plotted in Fig. 7 as a function of @ (open circles).The intensities are normalized to sin'Q at one valueof qh (P =47'); it is evident that the overall Q depen-dence is close to sin2$. Thus, from Eq. (3.1) we mayconclude that the excitation at 0.8 meV is associatedalmost entirely with zz fluctuations, and therefore canbe unambiguously identified as the OPC gap mode.As we have discussed in a previous publication, ' theresults of Fig. 7 give clear evidence for the dipolaranisotropy in TMMC and we can estimate directly thevalue of D in the spin Hamiltonian Eq. (2.1) fromthe measured gap energy. From Eq. (2.18) using62 =0.80 meV and 2JS =6.1 meV one findsD =0.0086. This may be compared with the value ofD =0.014 calculated on the basis of a full Ewald sua

0.8t

0.4)E

0.040 4 (b) T=030K

LLI

4J

0.08I

O. I 2

0.3

0.2

0 lt

0 I I I I

0.0 I 0.02REDUCED WAVE VECTOR(c" UNITS)

FIG. 8. Spin-wave dispersion relations in TMMC in zeromagnetic field at (a) T=1.8 K () T~) and at (b) T=0.3 K(( Tg =0.85 K).


TMMCI

(a) T=0.50K40—

H=OI

(0,0, I) —80C:

E

V)I-

O 20c)o

5=(0.7, 0, I).2, 0, I) E

FO

M

-40 o

0 0gO 0

I 60—E

V) I 20—I-

Oo 80—

I I

0.2 0.4(b)~ T = I. IK

o T =4.0K

o T=8.0K

I I

0.6 0.8I I

I.O l.2

Q = (-1.6,0, I )

40—

I I I

0.2 0.4 0.6 0.8, ENERGY(meV)

I

I.OI

I.2

FIG. 9. Inelastic scans in zero magnetic field at T =0.30K and as a function of temperature in the OPC energy re-gion. The solid lines are guides to the eye. The incomingneutron energy was 5 meV and the collimations (a) 40-'20'-40'-40' and (b) 20'-20'-20'-20'.

boundary excitation energy. We note that this valuefor 2JS includes a 9% quantum correction from theunrenormalized value of 5.6 meV. The spin-wavetheory is thus seen to account satisfactorily for theobserved dispersion of the magnetic excitations atzero field, but is, of course, unable to account for ef-fects due to the lack of long-range order, such as thebroadening of the acoustic modes in the long-wavelength limit. The latter is, however, correctlypredicted by the more elaborate theory due to Vil-lain. '

It is of interest to study the temperature depen-dence of the OPC mode. With a departure of only—1% from the isotropic Heisenberg interaction itmight at first glance seem counterintuitive that theanisotropy manifests itself at temperatures of T -2K and above. However, as pointed out by Villain,an enhancement is brought about by the well-

developed short-range order: that is, the conditionfor the XY behavior to manifest itself is that the ther-mal energy should be much less than the anisotropyenergy associated with the spins contained in a corre-lated block; algebraically this condition may be writ-

ten DJS2[2JS(S +1)]/ksT » ksT This giv.esks T « JS'J2D =12 K. In Fig. 9(b) we have plot-ted the results of scans at Q = (—1.6, 0, 1) taken at

three different temperatures. The sharp OPC com-ponent observed at T =1.1 K broadens as the tem-perature increases and is smeared out at T = 8 K,thus displaying behavior consistent with the above-mentioned condition.

The measurements presented above were carriedout for temperatures above the 3D-ordering tempera-ture T& =0.85 K. In a previous publication we re-ported some anomalous effects observed at highermagnetid fields which seemingly could be ascribed tothe field-induced 3D ordering occurring at tempera-tures above T~(H =0) (see Fig. 1). In order to elu-cidate further these effects we have studied the spec-trum of magnetic excitations in zero field at tempera-tures down to T =0.3 K. Figure 9(a) shows energyscans for various scattering vectors Q = ($, 0, 1), tak-en at T =0.3 K, that is, in the 3D-ordered phase.Comparison between Fig. 7(a) and the right-handside of Fig. 9(a) shows that the OPC excitationwithin experimental error is unaffected by the 3D or-dering. The left-hand side of Fig. 9(a) shows ahigh-resolution scan taken at Q = (—0.7, 0, 1), reveal-ing a sharp excitation at —0.1 meV. This newfeature was pursued in scans probing finite wave-vector excitations, the peak values of which are plot-ted in Fig. 8(b). Thus, the branch of IPC excitationsexhibits a small q, =0 energy gap, Attic =0.10(2)meV, reflecting the presence of in-plane anisotropy.The corresponding in-plane anisotropy field is 71 + 30Oe. This observation may be compared with very re-cent heat capacity measurements' of T~(H), with themagnetic field direction within the easy plane. In thisstudy an additional phase line to the left of the oneshown in Fig. 1 is found, the "new" and the "old"phase lines being interpreted as arising from domainswith an easy axis nearly coinciding with the fielddirection and from domains with an easy axis nearlyperpendicular to the field direction, respectively. Inthe first case, T~(H) exhibits bicritical behavior witha cusplike minimum at 0& =11.4 kOe, indicating a.spin-flop transition. The spin-wave energy gap corre-sponding to this field is 5 =g p, &H& =0.13 meV, andis thus seen to be in reasonable agreement with ourobservation, especially given that the phase diagrammeasurements extend down only to about 0.6T~(H =0). We should note that comparable valuesfor Hb have been obtained by Mangum and Utton 'and Magarino et al.

The present measurement of the in-plane anisotro-py is only possible below TN, since in the disorderedphase the excitations become overdamped in thesmall q, limit. However, it seems inherent in thephase diagram established in Ref. 7 that the in-planeanisotropy is present in the disordered phase as well;as noted previously such an anisotropy is allowed onpure symmetry grounds, since below T, = 126 K thecrystal symmetry is monoclinic. The physical originof this very small anisotropy presumably is a com-

243948 I. U. HEILMANN et al.

bination of the between-chain magnetic dipole-dipoleinteraction an a smad 11 crystal field term. In additionto the scans carried out for small q, 's as shown inFig. 8(b), we surveyed at T =0.3 K the entire spec-trum of magne icf tic excitations of wave vectors q,

in theh =0.5 (q =m/2). No new features in ethrough q, = . q=ared to thespin-wave spepectrum were detected as compare

thatprevious resu s olt btained in the disordered phase,is for T &0. . u,85 K. Thus, we are led to the conclusion

velen tht ataparh rt from converting the long-wavelengtmodes in o sa '

t harp elementary excitations, theif an,dering o t e spinf th in system in TMMC has little, i any,

effect on the spectrum of magnetic excitations.

T= I.SK

~ & '0 p

E~ 200

I-Z~ 100—O

H = 56kOeQ = (-l.4,0,

TMMCI

H = 70kOe~ Q = (-0.2

l20 p f=(-I.4E

,80—I-XO

40—

—240—E

CO—l60 I-

OO—80

v p ~

-0

V. FINITE FIELD MEASUREMENTS

In this section we report. a series of measurementsof the inelastic excitations at low temperatures inmagnetic fie s up o

'ld t 70 kOe. The measurements are

d'

antly at high fields, that is, at large g p,~k~ T so that the spins are well localized alongangle 8& to x c. igs."( f F' 2 and 4). In this regime thelongitudinal correlation length appears to be con-trolled by m-domain walls which are we11-described asclassical sohtons.1' ' Our studies, however, are cen-tere on e id th 'nelastic excitations as opposed to t equasie as ic s

'1 stic scattering which is determine y e

exten-domain-wall dynamics. The latter has been ex en-sively explored by Boucher and co-workers.

0..—H=Of= (-l.4, 0,

LA200—

CO

~ IOO-O

I

4ENERGY (meV)

FIG. 10. Inelastic scans at .T = 1.8 K in m gin ma netic fields of0.36 and 70 kOe. The solid lines are guides to the eye. The

14.8 meV and the collimationoutgoing neutron energy was20'-40'-40'-40'.

A. Experimental results

In Fig. we ave p' . 10 h lotted results of inelastic scans

f Q = (-1.4, 0, 1.25) at H =70, 36, andperformed for Q =—0 kOe and in addition for Q = (—0.2, 0, 1. atH =70 kOe. The former corresponds to I/I =42.3'

=7'. As the fieldw iehile the latter corresponds tothe single spin-wave peak observerved atincreases, e

H =0 splits into two: a strong peak shifting ow

energy and a weaker shifting up in energy. The reso-lution properties of the spectrometer are such thatoptimal focusing at this slope of dispersion is o-ai, ——1.4. Thus for Q, =0.2 the observedpeaks are broader and hardly resolved (closed cir-c es . One notices romthat the high-energy peak is relatively more intense

, = —0.2 th for Q =—1.4, thus suggestingpre ominan yd

"tl x polarization of the high-energy pea .

The scans of Fig. 10 were taken at T = . , ah' h t rature the 30 ordering occurs at magnet-

ic fields higher than H =35 kOe (see ig.Scans simiar o'1 t those of Fig. 10 were carried out

for a variety of wave vectors. ypt ical scans are shownin the upper part of Fig. 11. They all display a two-

peak structure with a splitting at-0.7S meV, nearly independent of the wave vector.Furthermore, the intensity ratio depends on q, such

TMMC

soo- (o)

c 400E

500—CAI-

200—O

I 00—

400—E

500—I-

200—OC3

100—

(b)

MAGNETIC EXCITATIONSI I I I I

T= I

5)

I.IO)

2 5 4 5 6I

7

—200

—I60-E

—l20 ~) (1)

—80O

0 O—40

200E

l50 ~

IOO

I

50

I I I

5 4 5ENERGY(meV)

FIG. 11. (a) Inelastic ~cans at T =1.8 K a=1.8 K and H =70 kOelu er panelj and at T=2.5 K and H =55 kOe. (b) The

14.8 meV and the collimationin neutron energy was . m20'-40'-40'-40' (lower panel).40'-20'-40'-40' (upper panel and

24 ONE- AND TWO-MAGNON EXCITATIONS IN A. . . 3949

that the high-energy peak is relatively weaker forsmall values of q, . The scans in Fig. 11(a) were tak-en in the 3D-ordered phase, as were the middle andupper scans in Fig. 10. Figure 11(b) shows two scansat q, =0.25 and 0.5, respectively, taken at T =2.5 Kand H =55 kOe. The system in this case is in thedisordered phase; these results thence give the im-portant result that the double-peak feature is in-

dependent of the 3D ordering and thus solely aconsequence of the applied field.

In Fig. 12 we have plotted the observed peak posi-tions for scans probing various values of the reducedwave vector q, between 0 and 0.5 and for appliedmagnetic fields of H =0 and H =70 kOe. The OPCbranch observed for small q, values at zero field hasbeen omitted for the sake of cl'arity. At H =70 kOewe observe two branches of excitations; the first isthe lower branch which has an intensity close to thatof the zero-field spin-eave branch. At the zonecenter, that is q, =0, an energy gap A~ =0.75 meV isobserved; we note, however, that the branch shiftsbelow the zero™fieldbranch at higher q, values: in

particular, at the zone boundary q, =0.5, the shift is-0.40 meV. We also observe an upper branch of re-latively weak intensity, especially in the q, ~0 limit,and separated in energy from the lower branch by-0.75 meV. The response is relatively more intensein scans with small values of Q„suggesting that theupper branch is associated with predominantly x fluc-tuations. We find, in addition, that at intermediatefields the two branches are closer to each other, the

T.MMC

(a)

160—

I

T 165K H 25koeQ= —80

separation being roughly proportional to the fieldstrength.

The energy of the lower q, =0 gap sho~n in Fig.12 is equal to 0.75{3)meV at H =70 kOe, and is

proportional to H. It thus agrees closely with thespin-wave-theory prediction of the IPC energy gap5& =g p,sH [Eq. (2.17)]. In contrast to this, the

q, =0 response observed at energies above -0.8meV is anomalous, as illustrated in Fig. 13. Here weshow scans performed at H =25 kOe, T =1.6S K (a)and at H = SO kOe, T =2.5 K (b), with an experi-mental configuration identical to that of Fig. 7(a).The energy resolution at 1 meV energy transfer in

this scan is 0.15 meV. The temperature was adjustedin the two cases so as to put the system equa11y closeto the 30-ordering line in the H-T phase diagram ofFig. 1. Figure 13(a) shows a sharp peak atE =0.85(3) meV with roughly the same polarizationproperty as that shown in Fig. 7(a). This may be as-

signed to the OPC excitation, although the energy ap-pears to be shifted up slightly relative to the zerofield peak. The strong increase in intensity below-0.6 meV arises from the IPC mode which occurs atthe energy 4~ =g p, &H =0.30 meV at H =25 kOe.The observed response at H =50 kOe shown in Fig.13(b) is qualitatively different from that of Fig.13(a). We observe a weak, broad band peaked at

TMMC T= I. 8Kl

Ecl

120—COI-

80—C3

o (0,0, 1.0)o (-0.6,0, 1.0)~ (-1.7, 0, 1.0)

E—60 ~

tAI-

—40 oC3

40— —20

E4C9CL4J

LLI

(b)

160—

EOJ~ 120—P-

o 80

I I

T =2.5K H =50koe

Q

0 (0,0, 1.0)o (-0.6,0, 1.0)

I.O)

40—

O. I 0.2 0.5 0.4REDUCED WAVE VECTOR (c"UNITS)

0.5

I I

0.4 0.8I I

1.2 I.6 2.0 2.4ENERGY (meV)

FIG. 12. IPC dispersion curve at H 0 and T =1.8 K, to-gether with the one- and two-magnon peaks at T =1.8 Kand H 70 kOe. The solid line is the sine-wave dispersioncurve; the dashed lines are guides to the eye.

FIG. 13. Inelastic scans at the zone center q, = (m/a)& (Q~ =1.0) for various transverse momentum transfers at(a) T -1.65 K and H =25 kOe and at (b) T =2.5 K andH 50 kOe. The outgoing neutron energy is 5.5 meV andthe collimation 40'-40'-40'-40'.


-1.2 meV with an asymmetrical line shape. The po-larization is predominantly xx, since the intensity ismaximum for / =0 and decreases as ($( increases.The OPC excitation of Fig. 13(a) is invisible in thescan of Fig. 13(b). Upon further increase of magnet-ic field H the peak energy shifts up, becoming 1.6meV at H = 70 kOe, as indicated by the upper branchat q, =0 in Fig. 12.

It is obvious from the results shown in Figs. 10—13that the observed magnetic neutron response fromTMMC in an applied magnetic field cannot be ex-plained in terms of one-magnon. scattering alone. Inparticular, as we have discussed in Sec. II, the one-magnon spin-wave spectrum is degenerate and in-

dependent of the field at the zone-boundary point

q = n7I'2 (q, =0.5), and thus distinctly different fromour observations. In addition, the unusual fielddependence of the OPC mode and the appearance ofa new band in going from H =0 to H =50 kOe sug-gests that one must go beyond the one-magnondescription to include two-magnon excitations as dis-

cussed in Sec. II.The observations presented in Fig. 13 clearly call

for a more detailed high-resolution study of the

behavior at intermediate fields. We have done this

by carrying out irielastic scans using Ef =3 meV asfixed outgoing neutron energy, and a spectrometerconfiguration such that the energy resolution is -0.1

meV. Figure 14 shows a series of such scans forq, =0 taken at different values of the magnetic fieldH. As H increases from zero the OPC excitation gra-

dually smears out and shifts up in energy. Then,above 30 kOe a new peak at -0.6 meV appears onthe high-energy flank of the IPC excitation (ht), andthe peak position shifts up as H is further increased.At H =41.2 koe (limit of this experimental setup)the two scans at g, =0.23 and 2.23 (black and opencircles, respectively) show that the polarization of thefluctuations giving rise to the peak at -0.7 meV ispredominantly along z that is, along the chain axis, asis the case with the peak observed at H =0. Conse-quently, the peak may be identified as the OPC exci-tation (4t).

In Fig. 15 we show inelastic scans taken with Ef =4meV and the same collimation as those of Fig. 14.Compared to Fig. 13(b) the energy resolution issomewhat better and additional details emerge. Thescans performed for Q = (0.23, 0, 1) (closed circles)

TMMC T = 1.8KI I I I I I

Ef =3meVI ~ I

T=1.8K Ef =4rneV

C

80 — ~ 0= (0.23, 0o Q=(2.23. 0

H=O

EC3

40—

0 I I I I I

H= 20 k0eo20

CE40

(3 0C:

E 20-LO

(3 0

.~80-EC) 40-

I I I I I I I I I I I

H 30 k0e

2.5 kOe

1.2 kOe

0 I I I . I I I I I . I I I I I I

0 0.2 OJ 0.6 0.8 1.0 1.2 1.4ENERGY(meV}

100—

80-

40-

20-

~ 5= (0.23, 0. t}o 6=(2.23, 0, I}—

II 0

2&~ ' H =28 kOe

II

0 I I I" I I i I I I I

E 40 I",tent '~, H= 32.5 kOe

-.2D-/ I, I

40 20

~ 1

7D k I( H=375kDs] Q ~

Ol, ( i' j+

DD- '~ l~, t P '~ H 7.i.7kD —=e~ ~40- 'x~ ~ ',

~ ~

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2ENERGY(meV}

FIG. 14. High-resolution (b,E =0.1 meV) inelastic scans

as a function of magnetic field at the zone center Q, =1 r.l.u.

{reciprocal lattice units) (q, = rr/a) and t ansverse mornen-

tum transfers Q„=0.23 and 2.23. The solid lines are guides

to the eye,

FIG. 15, Inelastic scans in the mode-crossing region as afunction of magnetic field. The arrows at 4~ identify theIPC magnon energy. The arrows at 2b, i and 2h~ simply lo-

cate the positions of twice the measured IPC (4~) and OPC

(b~) spin wave energies.

24 ONE- AND TWO-MAGNON EXCITATIONS IN A. . . 3951

TMMC ZONE-CENTER EXCITATIONSI I: I I I I I I I

2.2—polarization:

20 — 0 mostly zz~ most l y xx

18 — o mixedxx+ yy

o 14

~12E

QJ 1e0

~ 0.8

H

0.6

0.4

0.2

0.00 20 40 60

H (koe}80

FIG. 16. Zone-center peak excitation energies at T =1.8K as a function of magnetic field in the anticrossing region.The solid and dashed lines are guides to the eye at the indi-

cated energies. The polarizations are as indicated.

display two peaks, clearly present for H & 30 kOe.Peak positions as well as intensities are field depen-tent: As H increases, the separation of peak posi-tions decreases, mainly due to the shift in the posi-tion of the lower peak. Moreover, the intensitiesseem to increase somewhat with field. The open cir-cles at H =41.2 kOe show a scan taken atQ = (2.23, 0, 1), showing virtually no sign of struc-ture. Thus, at this field the polarization of the fluc-tuations giving rise to both peaks are clearly along x,in agreement with the observations at higher fields.That is, these are longitudinal fluctuations.

We have tried to compile in Fig. 16 the experimen-tal information obtained at the zone center (q, =0) asfunction of magnetic field H. The triangles denotethe peak positions of the lower energy gap b, ~ whichis observed in the entire range of magnetic field. Theresponse occurs in xx and yy polarization, characteris-tic of IPC fluctuations, and the peak positions con-form closely to the spin-wave result h~ =grM, &H. Atmoderate fields, 8 & 20 kOe the observed excitationat -0.8 meV conforms to the one-magnon OPC exci-

. tation. Upon further increase of H (H & 20 kOe)this peak shifts up in energy, the polarization changesgradually from zz to xx, and the intensity decreases. Inthe range 30 & H & 40 kOe a total of three excitationsare resolved, in addition to the IPC mode, The po-larizations are mixed around H —35 kOe but tend toseparate out for H & 40 kOe. Thus, at H =41.2 kOe

the polarization of the excitation at —0.7 meV ispredominantly zz (OPC) and the polarization of theexcitations at —1.0 and —1.3 meV are xx, as is thecase for the broad (unresolved) responses observedwith poorer resolution at higher fields. In Fig. 16 thesolid curves are guides to the eye only and thedashed line denotes the H dependence of the OPCgap excitations according to linear spin-wave theory.

It is evident from the q, =0 results shown in Fig.16, as well as from our observations at high fields forwave vectors through the entire Brillouin zone, dis-cussed earlier, that the observed neutron scatteringbehavior falls outside the framework of linear spin-wave theory. In particular, the OPC-mode mixeswith a new kind of excitation found at approximately2g p, &H instead of being independent of field aspredicted by linear spin-wave theory. In the crossingregion at -35 kOe the observed mode-mode cou-pling is reminiscent of the "anticrossing" of two pho-non branches of the same symmetry.

B. Discussion

Although the experimental results cannot be ex-plained in terms of the excitation of single spinwaves, the picture becomes clearer when we take intoaccount the possibility of exciting pairs of spin waves.Comparison of the spectral functions shown in Fig. 5

with the data of Fig. 15, together with the fact thatthe polarization of the high-frequency peaks is

predominantly xx, and increasingly so at higherfields, suggest strongly that what is being observed isthe two-spin-wave-excitation process described by Eq.(2.19).

We show in Fig. 17 the theoretical spectral functionappropriate to the experimental conditions describedin Fig. 15 at 41.2 kG, with the single spin-wave peakintensity reduced by a factor of 20 from its theoreti-cal value [see Eq. (2.16)] convolved with a Gaussianof variance 0.1 meV. Qualitatively, it is clear. that thesimple harmonic theory provides a g'ood descriptionof the data, and that the peaks in Fig. 15 occurring atroughly 2b, ~ and 242 with the lower peak tracking thefield, are indeed due to the excitation of pairs of spinwaves. There are, however, important quantitativediscrepancies between the data and the theorypresented in Sec. II. The theoretical intensity of thetwo-spin-wave band is too low by a factor of 2 or 3,relative to the spin waves. This is true also for theobservations at larger wave vectors. The relative in-

tensities of the two peaks in the band are incorrect,the theoretical prediction being that the lower peak is

more intense. The energies are also not quite wherethey should be. The lower peak is shifted up fromthe theoretical position, the upper peak down.

The origin of these discrepancies, and the fact thatthe OPC mode energy initially increases with increas-ing field, in contrast to the prediction of the nonin-


6.0—

Mz'4.0

Cl

metry is broken by the external field. The deviationsfrom the harmonic theory observed in the data havea plausible interpretation in terms of the trilinearanharmonic terms that Osano and Shiba pointed outin the Hamiltonian when the field is present. Moretheoretical calculation will be necessary to decidewhether these terms suffice to give a quantitativedescription of the observed mode intensities and fre-quency shifts.

0.0 i i

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8FREQUENCY (rneV)

FIG. 17. The theoretically predicted neutron scatteringspectrum (see Fig. 5) convoluted with a Gaussian of vari-

ance 0.1 me& to simulate the experimental conditions. Theone-magnon intensity has been reduced by a factor of 20 tocoincide with the presentation of the data in Fig, 15. Notethat the predicted intensity in the two-spin-wave band is

much less than that observed experimentally. The ordinatescale should be regarded as arbitrary.

teracting spin wave dispersion relation Eq. (2.22), arealmost certainly due to anharmonic terms neglectedin our derivation so far. In fact, taken as a whole,the dispersion relations for the OPC mode and thetwo-spin-wave branch look very much like the resultsone obtains for two coupled modes whose dispersionrelations anticross (see Fig. 16). Of course, theupper branch is not a single mode. What is requiredis an anharmonic term that couples the creation of apair of spin waves with the destruction of a'thirdmode. That such terms exist when the field isnonzero has been pointed out by Osano and Shiba. "The shift up of the lower of the two spin-wave peakscan be interpreted as the repulsion of the modes,while the upper peak appears at 242, with 52 shifteddown from the value predicted by the harmonictheory. The mixed nature of the polarization of themodes in the crossover region supports this interpre-tation.

We attribute the departure of the spin-wave energyfrom the prediction of the harmonic theory near thezone boundary (see Fig. 12) to this mechanism aswell. The additional intensity seen in the two spin-wave band is presumably the result of "borrowing"some of the single spin-wave intensity by means ofthis coupling, and one would not be surprised if thesame process reverses the relative intensity of thetwo peaks in the band.

To summarize, while it is clear from the harmonictheory presented here that the high energy "modes"we have observed are the two-spin-wave excitationband, it is equally clear that the large scale quantumfluctuations in this one-dimensional system lead tolarge anharmonic effects when the easy-plane sym-

VI. CONCLUDING REMARKS

It is evident that as this model one-dimensional an-tiferromagnet, TMMC, is looked at on a finer andfiner scale, one observes richer and richer behavior.This is particularly true for the static and dynamicfluctuations in a magnetic field. On the one hand,the longitudinal quasielastic scattering appears to bewell described using classical soliton concepts; on theother hand, the longitudinal inelastic scattering showsdramatic two magnon peaks which are presumablyentirely quantum mechanical in origin.

Our empirical characterization as well as ourtheoretical understanding of the spin-dynamicalbehavior of TMMC in zero field at low temperaturesnow seem to be rather complete. In particular, thereseems to be good accord between the neutron elasticand inelastic measurements, NMR, susceptibility,heat capacity, and other results; further, these mayall be understood using fairly straightforward theoret-ical concepts. It is clear, however, that substantialimprovement is required before we can claim as com-plete understanding of the behavior of the dyanmicsin a magnetic field. We have discovered two newbands of excitations which, outside the anticrossingregion, are longitudinal in character. We believe thatwe have established convincingly that they originatein two-magnon scattering processes which have anunusual spectral function in one dimension. Howev-er, we observe some discrepancies in the intensitiesand energies using standard harmonic theory. Mostimportantly, we observe a dramatic anticrossingbetween the transverse one-magnon spin-wavebranch and the longitudinal two-spin-wave modes. "Qualitatively, this may be readily understood as ananharmonic coupling between the one and two mag-non branches. It will be interesting to see if the de-tailed features we observe can indeed be understoodquantitatively using an appropriately elaborate spinwave theory incorporating anharmonic effects.

ACKNOWLEDGMENTS

We should like to thank M. Blume, K. Osano, andH. Shiba for helpful discussions. We should also liketo thank S. L. Holt for the growth of the TMMC

24 ONE- AND TWO-MAGNON BXCITATIONS IN A. . . 3953

crystal used in these experiments. Part of the workwas carried out by R. J. Birgeneau, I. U. Heilmann,and Y. Endoh while they were at Brookhaven Nation-al Laboratory and they would like to thank .their col-leagues at Brookhaven for their kind hospitality. Theresearch at Brookhaven was supported by the

Division of Basic Energy Sciences, U.S. Departmentof Energy, under Contract No. DE-AC02-76CH00016;the work at M.I.T. was supported by the NationalScience Foundation under Contract No. DMR-7923203.Research at Tohuku University was supported by aGrant for Fundamental Research in Chemistry.

'Present address: Department of Physics, Tohoku Universi-

ty, Sendai 980, Japan.For recent reviews see R. J. Birgeneau and G. Shirane, Phys.

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one- and two-magnon excitations in a one-dimensional antiferromagnet in a magnetic field

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