Pulsed magnetic resonance as a probe of longitudinal spin waves and magnetic second sound

Volume 100A, number 2

PHYSICS LETTERS

9 January 1984

PULSED MAGNETIC RESONANCE AS A PROBE OF LONGITUDINAL SPIN WAVES AND MAGNETIC SECOND SOUND L.R. CORRUCCINI Physics Department, Universityof California, Davis, CA 95616, USA Received 25 July 1983 Revised manuscript received 19 September 1983

Under appropriate conditions, longitudinal spin waves and spin wave second sound, a magnetic temperature wave, can be observed in quantum liquids and solids using pulsed magnetic resonance. A "hole-burning" experiment yields directly the spectrum w (k) of the propagating magnetic wave.

Under special conditions, pulsed magnetic resonance can be used to observe non-resonant longitudinal spin waves and spin wave second sound in quantum liquids and solids. The principle of this technique is to heat the nuclear spin system locally, creating a localized disturbance in Mz, and then monitor the subsequent propagation of this magnetic disturbance with time. In addition to the static external field H, a uniform gradient G = dH[dz is applied in the same direction, to inhomogeneously broaden the nuclear magnetic resonance. The spatial dependence of the sample magnetization can then be monitored, since ~ e magnetic resonance frequency is spatially dependent. The spin system is heated in a narrow region of the sample by applying a long burst of rf power at a fixed frequency. This "bums a hole" in the center of the broad resonance line. The resulting magnetic disturbance then propagates away in one of three ways: (1) Diffusively; (2) As spin wave second sound, a magnetic temperature wave, under certain conditions; or (3) As r/on-interacting longitudinal spin waves. The spin system must be decoupled as much as possible from the lattice to prevent loss of energy by relaxation. The changing shape of the magnetic resonance line with time is most easily observed with short sampling pulses, applied at a time t o after the rfpower is turned off. Each pulse is followed by a free induction decay which is equal to the Fourier transform of the resonance envelope. The broad overall line envelope yields

a free induction signal which decays very rapidly. The narrower magnetic disturbance evolving from the central hole produces a free induction signal with a longer decay time, and a characteristically different shape as a function of time. This Fourier transform is shown to directly yield the dispersion relation and damping of the propagating magnetic disturbance. In the limit of a delta-function disturbance in Mz(X), i.e., an inf'mitely narrow hole burned in the NMR line, the propagating disturbance is given by a one-dimensional Green's function

G(x, t) = f

dk exp (i[kx - co(k)t]),

where co(k) is the (unknown) dispersion relation of the wave. In the actual hole-burning experiment, the hole will not be a delta-function in x (and co). It will have a profile H(co) which can be taken to be gaussian, H(co) = exp[-(co - co0)2/2a 2] . In the simplest, or local, approximation, the presence of the field gradient maps spatial position x into the local NMR frequency domain according to the linear relation co -

coo = ~G(x

-

Xo) ,

where G is the gradient and 3' is the appropriate gyromagnetic ratio. 113

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Therefore, the propagating wave as a function o f x and t will be described by the wave function

AMz(X , t) = f f H ( x ' ) G ( x - x', t)dx' . - -

o o

In the frequency domain, this is

9 January 1984

Here co(k) is the wave frequency in the frame rotating at coO. This result is essentially the inverse Fourier transform of G(x, t) with respect to x, with x mapped into co. Thus it is not necessary to know the form of G(x, t) or AMz(x, t) (although they must apparently exist). The result

V(t) = exp [ - ( t - %)2o2/2]

AMz(co - coO, t) × exp[-icok(TG(t - t0))t0] , = f f H(co' - co0)G(co - co', t)d(co' - coo). The application of a short sampling pulse at time t o after the r f p o w e r is turned offproduces a free induction decay V(t) which is the Fourier transform with respect to co of AMz(co - coO, to). In the rotating frame, this will be given by

if

e~

V(t) = ~

exp [-i(co - co0)(t - to) ]

X AMz(co - coO, to)d(co - coO) = V(H)V(G), by the convolution theorem, where oo

v(m = f d(co - coo) exp [-i(co

- co0)(t - to)]

--oo

X H(co - coO) = exp [ - ( t - t0)202/2 ] , and o~

V(G) = 1

f

d(co - coO) e x p [ - i ( c o - c o 0 ) ( t - t0) l

×f

f

dk exp[-ico(k)t0]

d(co - coO) exp(-i(co - coO) [(t - to) - k/TG] }

dk e x p [ - i c o ( k ) t 0 ] 6(k - 3,G(t - to)),

or

V(G) = exp {-ico k [TG(t - to)] t0}. 114

k = 7G(t-

to).

The type of wave excited in this experiment, and hence the co(k) which enters eq. (1), depend on the type o f scattering processes the spin waves undergo. if they propagate diffusively, due to scattering which does not conserve k (impurity, m a g n o n - p h o n o n , umklapp), the hole burned in the resonance line broadens diffusively and

V(t) o~ exp { - D [ T G ( t - to) ] 2to}. This case has been analyzed for paramagnetic system by Hunt and Thompson [ 1 ]. If scattering is elastic and dominated by magnon-magnonN-processes, a density wave in the gas of spin waves may propagate [2,3]. This is termed spin wave second sound after the analogous mode observed in phonons [4]. In this regime the hole in the line propagates outward as a temperature wave with a dispersion relation which is predicted to be linear. The free induction decay follows the relation

where c 2 is the spin wave second sound velocity. Finally, if scattering processes are negligible, and the spin waves are longitudinal, the spin waves propagate ballistically and the hole spreads in a manner determined by their dispersion. In this regime

V(t) o¢ exp [--icok(Ta(t - t0))t0] ,

oo

= f

yields the sougth-after dispersion relation co(k) directly, where k is mapped into the time domain by the relation

V(t) ~ exp[ic27G(t - to)t0] ,

--oo

× c(co - coo, to) = 1

(1)

and V(t) determines the spin wave spectrum co(k) itself. This technique is not sensitive to transverse spin waves, which will lead to a diffuse propagation of A M z because all phases are excited. This technique is limited to materials where the

Volume 100A, number 2

PHYSICS LETTERS

9 January 1984

longitudinal and transverse spin relaxation times T 1 and T 2 are long enough that

6Ok(TGT2)T 1 > 2rr.

N

!

One candidate is the nuclear antiferromagnet solid 3He, where spin wave second sound may exist as a well-defined mode below the ordering temperature o f approximately 1 inK. Fig. 1 shows the propagating magnetic temperature wave AMz(~O - ~ 0 , to) superimposed on the broadened resonance, along with the detected free-induction decay expected. Another possible candidate system, using NMR, is longitudinal spin waves [5] and/or solitons [6] in the A and B phases o f superfluid 3He. Helpful discussions with M.C. Cross are gratefully acknowledged.

v

References

~-t o

Fig. 1. Wavelike disturbance AMz(to - too, to) in the magnetization of antiferromagnetic solid 3He, due to spin wave second sound, and the corresponding free induction decay V(t) predicted by eq. (1), inthe frame rotating at to o. AMz(to too, to) evolves from a narrow "hole" saturated in the center o f the gradient-broadened resonance. The free induction signal follows a sampling pulse, applied a time t o after the saturating power is turned off. -

[1] E.R. Hunt and J.R. Thompson, Phys. Rev. Lett. 20 (1968) 249. [2] K.H. Michel and F. Schwabl, Z. Phys. 238 (1970) 264. [3] J.J. Forney and J. J~ickle, Phys. Kondens. Mater. 16 (1973) 147. [4] R.A. Guyer and J.A. Kxumhansl, Phys. Rev. 148 (1966) 778. [5] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [6] C.M. Gould, T.J. Bartolac and H.M. Bozler, J. Low Temp. Phys. 39 (1980) 291.

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