Spin-wave turbulence

Nuclear Physics B (Prec. Suppl.) 2 (1987) 25-36 25 North-Holland, Amsterdam

SPIN-WAVE TURBULENCE

Paul BRYANT,* Carson JEFFRIES,* and K~tsuhiro NAKAMURA +

Physics Department, University of California, and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA

A high resolution study is made of spin-wave dynamics above the Suhl threshold in a sphere of yttrium iron garnet driven by microwave ferro- magnetic resonance. In di f ferent regions of parameter space observed behavior includes: excitation of a single spatial spin-wave mode at threshold; when two modes are excited, low frequency col lect ive osci l lat ions with a period doubling route to chaos; nonperiodic relaxation osc i l la t ions; when three modes are excited, quasiperiodic route to chaos; abrupt hystere- t i c onset of wide-band chaos. These results are accounted for in a unified way by numerical i terat ion of a model: coupled quantum osci l lators representing the photons of the cavity and the magnons, including four-magnon scattering processes.

I . INTRODUCTION

Spin-wave ins tab i l i t i es were f i r s t observed by Damon I and Bloembergen and

Wang2: microwave resonance in fe r r i tes , when strongly driven, showed onset of

anomalous absorption. Noisy low frequency osci l lat ions were also discovered. 3

To f i x ideas consider spin Sj on the crystal la t t ice of a fe r r i te sphere in an

external magnetic f ie ld Ho, with a hamiltonian

= -hy~ Sj.H o - 2J ~ Sj.Sj' + ~ Hdipole-dipole (1) j j , j '

where y is the gyromagnetic rat io and J (J>O) the Heisenberg nearest neighbor

exchange energy. The Zeeman interaction leads to a uniform precession of the

crystal magnetization M about H o at frequency ~o-YHo and to a narrow ferro-

magnetic resonance absorption at mpZ~ o when driven by a small ac f ie ld H 1 sin

(wpt), perpendicular to H o. The exchange term can give rise to spin-waves; the

overall dispersion relation for Eq. (1) is 4

2 ~M ~M ~k = (~o - ~ + yDk2)(~o ~ + Y Dk2 + ~Msin2Ok) (2)

for spin-waves of frequency m k and wave vector k in the direction 0 k relat ive

to H o. Here, M s = sample magnetization, mM~Y4~Ms, and the exchange constant

*Supported by Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-ACO3-76SFO0098. +Permanent address: Fukuoka Inst i tu te of Technology, Higashi-ku, Fukuoka, 811-02, Japan.

0920-5632/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

26 P. Bryant et al. / Spin-wave turbulence

D = 2JSa2/hy Gcm 2. Nonlinear coupling between the uniform and spin-wave modes

arises through the exchange and d ipolar terms, which, for certa in values of H o

and pump frequency Up, gives r ise to an i n s t a b i l i t y : when the dr ive f i e l d H I

is increased to a c r i t i c a l or " threshold" value Hlc , there is abrupt growth in

the spin-wave exc i ta t i on . A theory of th is was given in 1957 by Suhl, 5 who

remarked, " . . . th is s i tua t ion bears a cer ta in resemblance to the turbulent

state in f l u i d dynamics . . . " .

Suhl predicted several i n s t a b i l i t i e s : ( i ) of f i r s t order when Up = 2m k

2m o ( "subsid iary absorpt ion") ; ( i i ) of second order when Up z mk z mo ("pre-

mature sa tu ra t ion " ) ; ( i i i ) and f i r s t order at Up z 2m k with pump f i e l d H 1

para l le l to N o ( "pa ra l le l pumping"). An example of what can happen in case ( i )

is shown on the dispersion diagram, Fig. l ( a ) , computed from Eq. (2), with

parameters val id for y t t r ium iron garnet (YIG) spheres (~ = 17.7xi06 G-Zsec - I ,

D = 5.4 x 10 -9 Gcm 2, 4~M s = 1700 G) and for pump frequency Up = 2~ x 9.2 GHz,

and for mo = ~Ho = 0.5 Up, corresponding to H o = 1640 G; however, the exact

condit ion mp = 2~ o is not required. Spin-waves can ex is t in the manifold

shown, 0 5 Ok 5 7/2. How can they be excited? By a three-magnon scat ter ing

process6: a pump photon with up z 2m k can exci te a uniform magnon (mp,k=O)mag

which scatters into a pair of magnons, (mk,k) and (mk, -k ) , i . e . a standing

spin-wave mode, with locus along the dashed l i ne , Fig. l ( a ) . Since the maximum

wave vector, k = 10 5 cm -z , is much larger than the fundamental wave vector k o

= ~/d for a sphere of diameter d ~ 10 - I cm, a large number of spin-wave modes

are possible; they are damped at a rate y k ~ 106 sec-1, t y p i c a l l y , by magnon-

phonon processes. Cases ( i i ) and ( i i i ) are s im i l a r ; in a l l cases microwave

photons exci te spin-waves of frequency ~10 I° Hz.

Although these Suhl i n s t a b i l i t i e s were extensively studied 7 e a r l i e r , no

clear evidence of low dimensional chaotic motion was reported. Nakamura et

a__11.8 and Ohta and Nakamura 9 reexamined the theory for para l le l pumping,

numerically i terated the equations of motion assuming two modes, and found

onset of i n s t a b i l i t y , co l lec t i ve o s c i l l a t i o n s , and a period doubling cascade to

chaos, with a Hen6n-like return map. Gibson and Je f f r ies I0 observed a period

doubling route to chaos, per iodic windows, and a single-hump return map for the

second order i n s t a b i l i t y in YIG. Zhang and Suh111 i terated the or ig ina l

equations for th is i n s t a b i l i t y and found a period doubling cascade to chaos.

Simi lar theoret ica l conclusions were reported by Rezende et a l .12; de Aguiar

and Rezendel3 reported theory and experiments on para l le l pumping.

In th is paper we report a high resolut ion study of the f i r s t order

i n s t a b i l i t y in a YIG sphere. New f indings ind icate: a two-parameter (Ho,H~)

phase diagram; exc i ta t ion of single modes; co l lec t i ve osc i l l a t i ons with

quas iper iod ic i t y , locking and chaos; and abrupt onset of high-dimensional

P. Bryant et at / Spin-wave turbulence 27

6 m

i o o)

~ 4 3

> -

o z 3 LU

0 i i i

r r 2 i i

LU

(3. O9 0

~ ] ~ - - T - - - T ~

-4 -2 0 2 4 WAVEVECTOR k (10 5 cm -1)

I KLYSTRON I

+ i ~ I DETECTOR

LINEAR I WAVEGU'OE I COUPLING f +

I RESONATOR I DAMPING F (A) I ~" LINEAR co P,,NG

/ I UNIFORM I DAMPING 'P'o / I MODE (B) I \ "

NONLINEAR | + | COUPLING gk l,,C~':l~--~--I

\ ( ' I SPINWAVE II DAMPING 'Yk NONLINEAR~ MODES (bk) INTERACTIONS T, S YIG SPHERE

(b)

I

THERMAL RESERVOIR

FIGURE 1 (a) Dispersion diagram, ~k vs. k, from Eq. (2) for YIG sphere. experimental arrangement and model, Eq. (3).

(b) Diagram of

m "o cN,c. "1"

20

15-

10-

5 -

I I I I I I I I I

",, i / • I s S

~,.o/

",,, Very noisy | ", collective

",, oscillations ! /

\ Relaxation'-, / ~ , , [ Suhl ~osci l lat ions " . . oo/ threshold ~ . . . . . /

Period doubling, quasiperiodicity,

' ~ , locking, chaos Abrupt hysteretic onset -¢,~t~,~ / of wide-band noise - " ~ - / . ~ ; , / ~

Single Modes: ~, ~ ~ Threshold for ~, . . . . , . o,J collective oscillations J ~

I I I I I I I I I I I 1000 20O0

DC Magnetic field, gauss

200

100

E

so ~ O

E e~

20 m

O

10 ~

4

FIGURE 2 and boundaries of observed behavior in parameter space (Ho,H~). Regions

28 P. Bryant et aL / Spin-wave turbulence

turbulence with hysteresis. We present, in Sec. 3, a theoretical model to

account for these in a unified way.

2. EXPERIMENTS

Figure l(b) indicates schematically the experimental arrangement as well as

the theoretical model. Microwave power from a klystron (fp = 9.200 GHz) is

coupled via a precision attenuator into a waveguide, and into a resonator,

producing a field H I sin(2~fpt) on the YIG sphere, oriented with the [111] axis

parallel to a uniform and adjustable field H o. The sphere of diameter 0.066 cm

is spherical to within AR/R = 6 x 10 -s and highly polished, to within 0.15 ~m.

Incident power P is absorbed by the resonator (A), and the uniform mode (B),

and the spin-wave modes (bk). Power not absorbed is reflected onto a diode

detector giving a dc signal S O as well as an ac signal S(t) (100 Hz to I MHz).

This paper mainly reports on the subsidiary absorption instabi l i ty (H 1 ± Ho) ,

although some data are given for parallel pumping (H I II Ho), to be reported

separately. 14

Figure 2 shows regions and boundaries of observed behavior in the space of

two parameters: dc field H o and microwave pump power P ~ H~ . The Suhl

threshold boundary is found by monitoring signal So, which drops at onset of

any spin-wave excitation. This boundary is abrupt and reversible except near

the shaded area, where onset is abrupt and hysteretic, and accompanied by a

large signal S(t) of wideband character (~ MHz) with no resolvable spectral

peaks. For H ~ 600 G the boundary becomes increasingly less distinct.

For H o ~ 1600 G and P ~ 0.1 dB above the Suhl threshold we generally find

onset of sinusoidal "auto" oscillations [Fig. 3(a)] in the frequency range

104 ~ fco ~ 106 Hz, which are believed to arise from coupling between spin-

waves; we refer to them as "collective" oscillations. These oscillations show

period doubling [Figs. 3(b), 4(a)] and a transition to chaos [Figs. 3(e),

4(b)]. The frequency has a marked dependence on pump power, Fig. 5(a), which

is consistent with predictions, Fig. 5(b), of our model in Sec. 3, as well as

that of Zautkin et al. 15 Typically these single frequency oscillations display

a bifurcation to quasiperiodicity [Figs. 3(c), 3(d), 4(c)], along with fre-

quency locking and chaos; they may appear when at least three spin-wave modes

are excited. The spectral lines of these periodic oscillations are ~ 50 dB

above a broad spectral base believed to be of deterministic origin. In some

regions (H o ~ 1400 G, P ~ 50 mW) the base has a much higher intensity (+30 dB);

the rough boundary for these noisy oscillations is shown in Fig. 2.

The parallel pumped instabi l i ty is also observed, showing period doubling

[Fig. 4(d)] to at least period eight, onset of quasiperiodicity, and chaos,

Fig. 4(e). Another type of signal, nonperiodic "relaxation oscil lat ion," Fig.

P. Bryant et al. / Spin-wave turbulence 29

=S(t) (a)

t (ms)

(b)

0 0.5 0 0.5 0 I0

(f)

i i i i L i , i i l i i i i i i , = l i i i b , i i J

0 0.02 0 0.5 0 20

FIGURE 3 Observed signals, S(t) (relative units) vs. t , showing (a) periodic oscillations, fco = 16 kHz; (b) period doubled; (c) quasiperiodic; (d) quasiperiodic; (e) chaotic; (f) relaxation oscillation.

p(f) (=)

L i i ~ , , , i i

0 f (MHz)

0

(~

h , I = i i i i i

0

i i i i i i J J i

1 0

L L L i i J i , i

0

(f)

= i i i i i J i i

0 0.01

FIGURE 4 Observed power spectra, p(f) (10 dB/div) vs. f for: (a) period doubling; (b) onset of chaos; (c) quasiperiodicity; (d) parallel pumping, period doubling to period 8; (e) quasiperiodicity; (f) relaxation oscillations, Fig. 3(f).


3(f) , is found in a broad region, Fig. 2; they have essentially no resolved

spectral peaks, Fig. 4( f ) , and may be related to the "chaotic bursts" of the

model below.

The Suhl threshld in the region 1600 ~ H o ~ 1900 G, when examined at high

resolution, has a rich structure, Fig. 5(c). For constant pump power, the dc

signal S O vs. H o is recorded while slowly increasing Ho, showing a series of

peaks in the power absorption spaced by aH o = 0.157 G, which can be understood

as high order spatial resonances of single spin-wave modes within the sphere. 16

In a simplified picture assume that the spin-wave mode (mk,±k) has an integral

number n of half wavelengths that resonantly f i t in the diameter d of the

sphere i f n(~/d) = k; i f k is changed by Ak o = ~/d, the next spatial resonance

at n+l wi l l be excited. From Eq. (2) this small change ak o can be induced by a

f ie ld change AH o while holding constant ~k = ~p/2, i f AH o = 2DkAk o. I f we take

k ~ 3 x I0 s cm -I and 0 = 0 [Fig. l ( a ) ] , this expression yields AH o = 0.152 G in

agreement with the observed sp l i t t ing . Another sequence of modes wi l l have a

similar set of dips of s l ight ly different spacing with the irregular inter-

ferences most clearly seen to the right in Fig. 5(c). In a more refined

picture the dips in Fig. 5(c) may be viewed as the modes of a sphere Mjkl, say,

labelled by suitable indices. We find that for the f i r s t few dips in S O only

single modes are excited and are not accompanied by a collective osci l lat ion

signal S(t). Beyond this point simultaneous excitation of two or more modes

becomes possible owing to mode overlap. Signals at fco = 10s Hz may appear

when a second mode becomes simultaneously excited; a second frequency f 'co may

appear when a third mode is simultaneously excited.

3. THEORY

We model the system, Fig. 1(b), as a collection of coupled quantum osci l la-

t ions, with the hamiltonain7, 8

= h~rA+A + b~oB+B + z h~kb~b k k

n ~ (g~Bb~bt k + h.c.) (3) + h(G*A+B + h.c.) + ~ k

+ h z (Tkkb~bkb~,b k, + } Skk,bkb_kb~,b~k,) + h(FA+exp(-i~pt) + h.c.) k,k'

These terms are, respectively: (I) a single osci l lator with raising and lower-

ing operators A+,A representing a single electromagnetic mode of the resonator

of frequency mr; (2) a single osci l lator representing the uniform mode of fre-

quency mo = YHo; (3) a set of osci l lators representing spin-waves of frequency

mk, Eq. (2), for al l possible wave vectors k; (4) linear coupling G between

resonator and uniform mode; (5) nonlinear coupling between uniform and spin

P. Bryant et aL / Spin-wave turbulence 31

S - -

C" 0

4 - 0

~- 3 - o z uJ o 2 - r r LL

1-

0 0

,aJ I (b)~

1 2 3 P/Pc

Onean o,reouenc, , collective oscillations

tShUhl ~stabili~Jt A ~ ~ ~ I ,

L~ 8 gauss ~1 1650.5 13(3 Magnetic field, gauss

FIGURE 5 (a) Square of observed collective oscil lation frequency f~o vs. pump power P relative to threshold value Pc. The line is a f i t to the data. (b) lOx f~o vs. P/Pc, computed for two-mode model: the line has the same functional form as that of (a), fc~ = ((P/Pc)-1) • (c) Observed single mode resonances.

C k

I

Fc F

ck (b)

<-______ I

F c F

s(t ) (c) P(f) (e)

0 t (ms) 2 0 t (ms) 0.4 0 f (MHz)

FIGURE 6 (a) Subcritical, and (b) supercritical symmetry-breaking bifurcations; center line is t r i v i a l fixed point Ck=O ; dotted line represents unstable fixed points. (c) and (d) Time series of computed solution for two-mode model, showing "spiking" and "relaxation oscil lat ions;" compare to Fig. 3(f). (e) Power spectrum of (d); compare to Fig. 4(f).


wave modes; (6) nonl inear wave-wave coupl ing from the four magnon i n t e r a c t i o n ;

(7} microwave d r i v i ng of the resonator by the k lys t ron of frequency mp and

amplitude F. The pump power P ~ F 2. From Eq. (3) we obtain the equations of

motion fo r A, B, and bk, and then add phenomenological damping terms £, Yo, Yk,

respec t i ve l y .

Since Up z 1011 sec - I and we are in teres ted in signal f requencies ~0 to

10 7 sec - I , we transform to slow var iab les A, B, and C k using

= A e x p ( - i ~ p t ) ; B = B e x p ( - i ~ p t ) ; b k = C k exp( -~pt /2 ) (4)

Since waves are created in (k , -k ) pairs we take C k = C_k, and ICkl 2 becomes the

magnon occupation number.

parameters

This t ransformat ion

a£ o = m o - Up ~ 3 X I0 1° sec - I

A£k = (ink - rap/2) = 2~v k ~ 10 6 sec -z

A£ r = m r - Up, set to zero .

introduces the "detuning"

(5a)

(5b)

(5c)

We make the approximation that the resonator has the greatest damping and

thereby A a d i a b a t i c a l l y fo l lows B and C k. Since B may o s c i l l a t e rap id ly due to

large values of A~o, we replace i t by i t s average and regard the average of

as small . The only remaining dynamical var iables are the C k with equation

• * *

Ck = -YkCk " iA~kCk - 1QFgkCk

* 2,C~} (6) - i~,{2Tkk,ICk,12Ck + (Skk, ÷ Egk,gk)C k

where k = 1,2 . . . . N is an index for a spin-wave mode, k' = 1,2 . . . . N, and para-

meters Q and E are funct ions of the parameters prev ious ly in t roduced. Equation

(6) has the form of a set of N coupled damped dr iven nonl inear o s c i l l a t o r s ,

each represent ing one spin-wave mode; there is always one uniform mode, 2

somewhat hidden in these equations: B depends on C k as B = QF + E ~, gk,Ck,

We do not convert the var iab les C k into Cooper-pair var iab les 7 since th i s

obscures important symmetry proper t ies of the system. For su i tab le parameter

values 17 we numerical ly i t e r a t e Eq. (6) , f i r s t for N : I , then N = 2, e t c . ,

adding one mode at a t ime. Control parameters used include pump power P, and

the detuning frequency ~k [Eq. (5b)] corresponding to a change in the f i e l d H o.

In Eq. (6) we note that C k = 0 is always a so lu t ion or f i xed point of the

system. For small fo rc ing F th i s is a s table f i xed po in t , but i t becomes

unstable at the Suhl threshold where the fo rc ing is f i r s t able to overcome the

damping. Nonzero (or n o n t r i v i a l ) f i xed points may also ex is t and have

important consequences for the system's behavior. I f we assume one mode to be

exc i ted , then we may determine these n o n t r i v i a l f i xed points exac t l y . The

P. Bryant et al. / Spin-wave turbulence 33

equation Ck : 0 can be put in the form:

M + N l C k l 2 - (C~ )2 iCkl 2 - point on unit c irc le (7a)

where M = i (y k + iagk)/QFg ~ (7b)

and N = -(2TRR + Skk + Elgkl2)/QFg~ (7c)

The Suhl threshold corresponds to IMl = 1, and the solution C k = 0 is stable

for IMI > 1 (note that M varies inversely with the forcing F). There are two

dist inct possib i l i t ies for what may occur when the threshold is crossed. For

Re(M/N) ~ 0 one obtains an ordinary (or supercri t ical) symmetry breaking

bifurcation at the Suhl threshold. As the solution C k = 0 becomes unstable, a

complementary pair of stable nontrivial fixed points emerge from the origin,

Fig. 6(a). (These two solutions are actually equivalent: one represents a

sh i f t in phase by ~ of the other.) The second possib i l i ty is that Re(M/N) < 0

and a subcrit ical symmetry breaking bifurcation wi l l occur. In this case,

nontr ivial fixed points f i r s t appear below the Suhl threshold via saddle-node

bifurcation. As F crosses the Suhl threshold F c, C k jumps to one of these at a

f i n i t e value, and the system wi l l display hysteresis on reducing F, Fig. 6(b).

In our experiments we observe a region of hysteresis, Fig. 2, which we believe

is due to this effect.

Numerical i terat ion of Eq. (6) yields a wide variety of behavior, here

summarized. Wi th only one spin-wave mode allowed (in addition to the hidden

uniform mode) a Suhl threshold is found, but no osci l lat ions. For N = 2 modes

an example of behavior follows: choosing parameters [Eq. (5b)] ~1 = -60 kHz,

v 2= 40 kHz, yields Suhl thresholds at pump power P1 = 8.43 mW, P2 = 4.77 mW,

respectively, for C I and C 2. At P = 10 mW, C 2 has developed an asymmetric

orbi t at f2 = 97 kHz, Fig. 7(a); and C 1 a symmetric orbit at f l = ( I /2) f2.

Holding P = 10 mW but decreasing H o yields various computed behavior: for u I =

-77 kHz, v 2 = 23 kHz we find period doubling for C2, Fig. 7(c), and symmetry

breaking for C1, Fig. 7(d). For v I = -82 kHz, v 2 = 18 kHz, both orbits have

become chaotic: Figs. 7(e) and (f) show this for mode C 2. At other parameter

values, part icular ly when one mode is below and the other above Suhl threshold,

nonperiodic "relaxation osci l lat ions" are found, Fig. 6(c,d,e), rather similar

to those observed experimentally [Figs. 2, 3( f ) , and 4 ( f ) ] .

For N = 3 modes the computed behavior shows periodic orbits, then quasi ~

periodic as in the por t ra i t of Fig. 8(a) and power spectrum, Fig. 8(b). A

chaotic section is shown in Fig. 8(c), with power spectrum, Fig. 8(d); the

orbits are apparently near, but not locked, to a 1:5 winding number. Unusual

behavior is shown in the portrai t and time series, respectively, in Fig.


Im C2 (a)

Re C z

Im C1 (b)

Re Ct

Im C )

Re Cz

[m C~ p(f) (f}

Re C~

)

FIGURE 7 Computed behavior for two modes: (a) phase p o r t r a i t for per iodic o s c i l l a t i o n s , asymmetric mode; (b) symmetric mode; (c) period doubling; (d) symmetry breaking; (e) chaos; ( f ) power spectrum of above, fmax = 0.5 MHz.

8 (e , f ) . This o rb i t appears to be of the Si ln ikov type, and may be related to

that of Fig. 6(c) and (d), for N = 2 modes, but with addi t iona l high frequency

components. Behavior q u a l i t a t i v e l y l i ke th is is observed in the experiment.

4. SUMMARY AND CONCLUSIONS

Star t ing from the viewpoint of microscopic scat ter ing processes of spin

waves on a magnetic crysta l l a t t i c e , we make a connection to the theore t ica l

framework of nonl inear dynamics. We s p e c i f i c a l l y aim to elucidate the

dynamical behavior of the magnetization in a f i n i t e sphere of a real mater ia l ,

y t t r ium iron garnet, above the Suhl f i r s t order i n s t a b i l i t y , excited by

microwave pumping. The model used is a co l lec t ion of quantum osc i l l a t o r s

representing the photons and magnons; ef fects of the cavi ty mode are included;

the consequences of symmetry are invest igated. In a high reso lu t ion (I0 - s )

experimental explorat ion of parameter space we f i nd , on a global phase diagram,

the regions and boundaries of a wide var ie ty of behavior: ( I ) The Suhl

threshold is found when, in addi t ion Lo the uniform precession, a s ingle

P. Bryant et at /Spin-wave turbulence 35

(a) II m C1 (c)

[ Re Cl Re Cf

p(f) (b)

f

p(f) (d)

Im C 3 (e)

R e c3 (f)

FIGURE 8 Computed behavior for three modes: (a) phase po r t r a i t of quasiperiodic o rb i t and sect ion; (b) power spectrum of above; fmax : 0.25 MHz; (c) Poincar~ section of o rb i t beyond quasiperiodic t rans i t ion to chaos; (d) power spectrum of above; fmax = 0.25 MHz; (e) po r t ra i t of Si ln ikov-type o rb i t ; ( f ) time series of above.

spat ia l spin-wave mode is exci ted, f l ~ lOZU Hz. (2) When a second mode is

exci ted, at f2 ~ 101° Hz, there may occur onset of a low frequency co l lec t ive

osc i l l a t i on , fco ~ I04-i06 Hz, owing to the coupling between the two microwave

modes; the observed dependence of fco on pump power is in agreement with the

model. The osc i l l a t i on fco displays a period doubling route to chaos. (3)

When a th i rd mode is exci ted, an addit ional osc i l l a t i on f ' co may ar ise, and the

system then displays a quasiper iodi ic route, including frequency locking and

chaos. Regions (2) and (3) are closely inter twined. (4) There exists a small

region characterized by abrupt hysteret ic onset of wide-band chaos. (5) There

is an extended region characterized by nonperiodic " re laxat ion" osc i l l a t i ons .

The model predicts the observed behavior in regions ( I ) , (2), and (3), and

also predicts behavior qua l i ta t i ve ly l i ke (4) and (5). For four modes excited,

neither the experiment nor the model have yet yielded three-frequency quasi-

pe r iod ic i t y ; we expect th is only in a very small region of parameter space.

Further deta i ls w i l l be published elsewhere. 14


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6. See, e.g., F. Keffer, Spin Waves, in Encyclo. of Physics 18/2, chief editor S. Fl~gge (Springer-Verlag, 1966).

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13. F.M. de Aguiar and S.M. Rezende, Phys. Rev. Lett. 50 (1986) 1070.

14. P. Bryant, K. Nakamura, and C. Jeffr ies, to be published.

15. V.V. Zautkin, V.S. L'vov, and S.S. Starobinets, Soy. Phys. JETP 36 (1973) 96.

16. W. Jantz and J. Schneider, Phys. Stat. Solidi (a) 31 (1975) 595.

17. Typical values used for parameters: G = 1.2 x I0 II sec - l , gk = 14 sec -I r : i0 B sec " I , Yo : 107 sec-1, k 2 x 10 s sec " I , F = 4 x 10 14 sec "I , Tkk ~ ~2 x 10 -8 sec - I , Tkk, = O, Skk = 4 x 10 -8 sec "1, Skk, = 4.5 x 10 -8 sec-1,

(-8.5 x 10 -11 -i2 x 10 -12 ) sec, E = (1.6 x 10 -14 -i7.3 x 10 -13 ) sec. These values are preliminary, and only approximately correspond to the real physical system.

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Spin-wave turbulence