Combined three-dimensional polarization analysis and spin echo study of spin glass dynamics

Journal of Magnetism and Magnetic Matermls 14 (1979) 2 1 1 - 2 1 3 © North-Holland Publishing C o m p a n y

COMBINED THREE-DIMENSIONAL POLARIZATION ANALYSIS AND SPIN ECHO STUDY OF SPIN GLASS DYNAMICS F. MEZEI and A.P. MURANI Instttut Laue-Langevm, 156X, 38042 Grenoble Ckdex, France

We report dtrect m e a s u r e m e n t s of the time-dependent spin correlation function S(K, t) for a C_uu-Mn spin glass alloy over nearly 3 decades of time in a single scan in the range 10 - 1 2 < t < 10 - 9 s. This was achieved by using, for the first time, a combination of n e u t r o n spin echo and polarization analysis techmques. The results show m o s t clearly the particular dynamics of spin glasses, namely a behavlour that can be described by a spectrum of relaxation times spreading over a wide range

Neutron scattering techniques have been increasingly applied in recent years to study the dynamics of spin glass alloys [1 ]. With the conventional methods of measurement using unpolarized neutrons, the magnetic scattering law S(K, co) has been possible to determine because of its particular form - a broad quasielastic-like spectrum which is well separated from the phonon spectrum lying at high energies in classical spin glass system such as C_u-Mn. The elastic magnetic scattering effects are identified through their temperature dependence since non-magnetic elastic scattering is almost temperature independent in the temperature and K-range of interest in the spin glass problem. Direct measurements of magnetic scattering using polarized neutrons have also been carried out earlier on some C_u-Mn spin glasses by Ahmed and Hicks [2]. In these measurements, however, no energy analysis was employed with the result that information about spin dynamics is lacking. In the following we report the first measurements of the intermediate scattering law S(K, t) for a C u - 5 at% Mn spin glass where the magnetic scattering is directly observed by use of polarized neutrons. The measurements involve the neutron spin echo technique [3] and polarization analysis of the scattered beam in three dunensions with neutrons polarized sequentially along the three perpendicular axes of analysis. In the neutron spin echo technique, a neutron polarized perpendicular to the magnetic field direction

traverses a well-defined magnetic field region in which it carries out n Larmor precessions before impinging on the sample. The neutron is spin flipped at the sample and traverses another precession field with Identical strength times length product. There it undergoes a further m precesslons, which because of the spin flip appear to be in the opposite sense. Thus for elastic scattering from the sample n - rn = 0 for all neutrons and the beam is phase focused at the analyzer i.e., its polarization is a maximum. This is equivalent to the spin echo condition in an nmr experiment. Any inelasticity in the scattering will then result in a value o f m different from n and the measured polarization in the original direction x will be given by

:efs( ,

cos

~

d~o/fs(K, co) dco,

(1)

with CO

= 2zr(n - m) ~ 27rn - 2E0

(2)

and

S(K, CO)~NI1 - exp(w/kT)] X F2~ K) f eit°t(s~ (0)

S-K(t)) dt,

where P is the initial polarization, kT the thermal energy, K and co the neutron wave vector and energy 211

F. Mezet, A.P. Muram / Neutron spin echo study of spin glass

212

This way the P'I component of the polarization will produce a span echo with a maximum amplitude of

transfer, respectively, and Eo the incident energy. Hence,

IP'l I : I.

=Pf S(K,

cos(

co/Eo) d o/f s(K,

dco

= P Re S(K, t)/S(K, 0)

(3)

with t = 7rn/Eo. Thus the resultant polarization m the original direction gives a direct measure of the real part of the Intermediate scattering law, i.e., the time correlation functaon S(K, t) at various times t. A tame scan is then obtained by varying the product of the precession field times the path length. The above technique of measurement is apphcable to all types of scattering. In the special case of pardmagnetic as well as isotropic magnetic samples, such as spin glasses having no net magnetization along any preferred direction, the spin flip scattering from the sample can be directly used to achieve the spin echo. In other words the spin fhpper coil at the sample is ehmmated and only the magnetic scattering contributes to the measured spin echo signal. This can be seen by examining the scattered beam polarization given by [4,5]

V' = - ~ ( ~ " P ) ,

(4)

-

1 and C ' : -P:, = ~Py .

P.'~c= - P c o s 2 o ~

f o r P IIx,

!

Pz = - P sin2a

for P I1z,

Py = 0

f o r P Ily,

(5)

Px +P~ = - P ( cOs2a + sin2°0 = - P .

Thus the average polanzatlons (Px) and (Pz) add to give the total scattering intensity. Furthermore since the angle a is directly related to the inelasticity of the scattering, the measurements of (Px) and (Pz) can yield a measure of inelasticity as shown by Maleev [5 ] and Drabkm at al. [6]. In fact the difference (Px) (Pz) provides, to a good approximation, a measurement of the intermediate scattering law Re S(K, t) at t = ctg( { O)/2Eo. From eq. (5) we have (p~) = _ P ( c o s 2 a )

where P as the incident polarization. If ~ IIx axis (see fig. 1)

By = 0 = ~P),

In addatlon to the spin echo scans we have also carfled out simple polarization analysis with the incident neutron polarization sequentially along the three &rectaons x, y, z Since the scattering vector K is in the x, z plane as shown i n fig. 1, where a is the angle between the elastic scattering vector K0 and K, we have

~ p

-

1

~ P x

-

Ipx

,

(x, y ) being the Larmor precession plane. In this plane the vector P; = (½Py,- ½Px) corresponds to the n - n spin flap necessary for spin echo, while ( - ~1py, -½Px) gives only a 180 ° phase shift in the precession.

-

f

,.) cos do/f

,.) dw

½PfS(•, w) (l + cos 2a) d~/S(K, 0)

(6)

and

= -lefs(

,

60)(1 - cos 2a)dw/S(K, 0 ) ,

(7)

, el cos 2= d¢o/S(K, 0).

(8)

hence (Pz)

-

=efs(

For small energy transfers w, we find that a"-

Z

LO

4Eo

ctg(lO).

(9)

Thus Fig 1. The geometry of the scattering triangle for fixed scattermg angle O (nO and K are the elastic and inelastic momentum transfer vectors)

( P z ) - ( P x ) = P f s(•, oo) c o s ( ~ ° ctg(½0))dw/S(~, Ol = P Re S(~, t)/S(K, 0)

(10)

F. Mezei, A.P Muram / Neutron spin echo study o/spin glass

where t = ctg(½tg)/2E o. It turns out, however, that the requirement of small energy transfers (eq.(9)) is not very strict, basically because larger a's contribute little to the integrals (8) and (10) for typical quasi-elastic spectra. We note that for lsotroplc spin systems S(K, t) is purely real, thus in this case we measure directly the time correlation function. The measurements were made on the IN 11 spin echo spectrometer at the ILL using neutrons of incident wavelength 5.9 A with a wavelength spread of 35%, fwhm. Thus, the measured S(K, t) is in fact averaged over the K-range defined by the momentum resolution of the spectrometer. This r-averaging, however, has no influence on the results since we find in the present case that S(K, t)/S(K, 0) is independent of K within the accuracy of the measurement. Furthermore, it can also be shown that for relaxation type spectra eq. (2) still remains a good approximation with Eo corresponding to the average incoming neutron wavelength. The question of momentum resolution of the spectrometer aside, the K for the measured correlatmn function is given by the elastic scattering vector since only small energy transfers are involved in its measurement. This however is not so for the total scattering intensity (shown in fig. 2 for two scatterlng angles of 5 ° and 20 °) obtamed from + (eq. (5)), which represents the integral over energy transfers comparable with Eo. At high temperatures the broad oJ dmtrlbution results in an averaging

213

over a large range of scattering vectors whereas at low temperatures as the energy d~strlbution gets narrower the scattering is averaged over a narrower range of •. Since S(K) lS strongly peaked in the forward direction we have a relative increase of the total intensity with decreasing temperature from this effect alone. In fig. 3 we show the results for the intermediate scattering law S(K, t) measured at scattering angle t9 = 5 ° (= 0.092 A - l ) . In the abscissa we have used the logarithmic time scale in order to represent the data over a time range varying over several orders of magnitude. The data have been normalized to the total polarization efficiency of the spectrometer, and represented as the ratio S(K, t)/S(K, 0) where t = 0 corresponds in effect to times smaller than 10 - 1 4 S as seen from earlier neutron scattering experiments. In the diagram we have included a curve representing a simple relaxation process, given by the exponentml e -Tt for 3' = 0.5 meV. It should be noted that on such a diagram a rumple exponential function always maintams the same shape, being displaced laterally for different values of 7. The present practical hmlts of the time range over which S(K, t) can be measured directly by the spinecho techmque is 5 X 10 -11 ~< t ~< 5 X 10 - 9 S. In fig. 3 the data points for t = 3 X 10 -12 s for the two scattering vectors are obtained from polarization analysis along the x- and z-directions and using eq. (10). Th]s limited time range prevents the determination of

"S Z

{3

Cu - 5% Mn Z

Q

150

I--L) IaJ t/') i (/3 100 U")

k)

F-- 5o

}

{ <2)<.>=0093/~ -1

.{'} { ,t{ {{

{

t

}tSK

{ < x . > = 037 /~-1 {

20K

1

{ '}{{{{'},}

{ ,} {

0

{ {

W Z

,[ {

Ig

0

U 010-14

i

10-13

i

10-12

10-'11"

10-10

,I0-9

I0-8

Time (sec)

0

50

100

150

200

TEMPERATURE Fig. 2. Temperature dependence of the magnetic scattering cross-sectmn at scattering angles of ,9 = 5 ° and 20 °.

Fig. 3. The measured time dependent spin correlataon function for C u - 5 at% Mn at various temperatures The thick hne corresponds to the simple exponential decay. The thin lines are guides to the eye only

214

F Mezei, A.P. Mutant/Neutron spin echo stud)' of spin glass

the shape of S(~, t) at short times and especially its measurement at higher temperatures where S(K, t) decreases very rapidly with t. It is interesting however that within the limits o f the experimental error bars the shape of S(K, t) is found to be independent of K (over the momentum interval of 0.047 - 0.37 A -1 covered in the present measurements). It is emphasized that this observatmn holds for temperatures below 45 K where non-zero values of the correlation function can be measured in the time range 3 × 10 -12 < t ~ 2 X 10 - 9 s covered in the present experiment We therefore express S(K, t) as

S(K, t) =-S1(K )S(t),

(11)

where SI(K ) represents the spatial correlations between the spins and S(t) the time-dependent correlation function which in this case must coincide with the self correlation function (Sz(0) St (t)). Monte Carlo computer simulation calculations o f Ising spin glasses [7] yield self correlation functions which at first sight appear to be qualitatively similar to the present results. These, however, have the form

S(t) = const - in(t),

(12)

which would be represented by straight lines in fig. 3. The marked curvature in S(t) for t <~ 1 0 - I 1 s is therefore interesting and possibly reflects the influence o f the rapid Korrlnga relaxation mechanism (r ~< 1 0 - 11 s for T ~> 5 K) which is necessarily present in metallic spin glass systems with 3d magnetic atoms as suggested

by earlier measurements also [8]. The model calculations as well as computer simulations, of course, do not include the Korringa mechanism but treat s o l u t e solute exchange couphngs only. Since the latter must dominate the spectrum for t > 10-11 s the qualitative similarity of the measured S(t) with the model calculations for t > 10 -11 s can be regarded as reasonably supporting the models, or simply as showing that a distribution o f barrier heights may be present in real spin glasses [9]. In conclusion, we believe the present measurement of the lnterme&ate scattering law S(~, t) over a time Interval exceptionally large for a single experiment provide the most clear insight obtained up to now into the dynamics of spin glasses, and that the novel techniques we have used seem to be particularly well adapted for its measurement

References

[1] A.P. Muram, J. de Phys. C6 (1978) 1517. [2] N Ahmed and TJ Hicks, Solid State Commun 15 (1974) 415 [3] F Mezel, Z. Phys. 255 (1972) 146 [4] O Halpern and H.R Johnson, Phys. Rev 55 (1939) 898 [5] S.V. Maleev, Soy. Phys. JETP Lett. 2 (1966) 338. [6] G.M Drabkm, A.I. Okorokov, E I Zabldarov and Ya A Kasman, Soy. Phys. JETP 29 (1969) 261. [7] K. BlnderandK Schroder, Phys Rev B14(1976) 2142 [8] A P Muranl, Phys. Rev Lett 41 (1978) 1406 [9] Juho F Fernandez and Rodngo Medina, Phys Rev BI9 (1979) 3561