Spin dynamics of a binary alloy (spin glass)

Solid State Communications,

Vol. 22, pp. 25-28,

SPIN DYNAMICS

1977.

Pergamon

OF A BINARY ALLOY

Printed

Press.

in Great Britain

(SPIN GLASS)

A.P. Murani Institut

Laue-Langevin,

156 X, 38042 Grenoble-Cedex,

France

and J.L. Tholence Centre de Recherches sur les Trss Basses TempGratures, 166 X, 38042 Grenoble-Cedex, France (Received

31 January

1977 by E.F. Bertaut)

Neutron scattering measurements on a & 8 % Mn alloy fail to show the normal critical slowing down of spins associated with a sharp magnetic scattering width is found phase transition. However, the quasi-elastic to approach an approximately constant q-independent value at temperatures around that of the maximum in the susceptibility, following a continuous process of freezing of spins. An important observation from the measurements is that the temperature of the maximum in x depends on the time constant of the probe:being 39 2 1 K in the a.c. susceptibility measurements (T z 10m2 S) against 52 r 3 K obtained from the neutron scattering results (T % lo-l1 s).

of the data with respect to the relative efficiency of the counters and their calibration was achieved by4measurements on a standard 1 mm A thin gold plate (s 0,5 mm) vanadium plate. with approximatly the same transmission as the sample was used as a "dummy" for the background measurement in order to make proper allowance for the scattering from the cryostat. The sample was prepared by induction melting and rolled into an approximately rectangular plate (5 x 20 x 30 mm) which was annealed under argon at 800°C for 12 hours, and quenched into water. The a.c. susceptibility measurements were made on the same large sample by winding a special set of mutual induction coils ; an a.c. bridge working in the frequency range 10-2000 Hz was used for the measurements. The magnetic neutron scattering cross-

An interesting aspect of the study of spin glasses at the present time is the question whether a sharp magnetic phase transition occurs in these random substitutional solid solution systems as suggested by some recent theories. 1 Calculations of the dynamics2 as well as computer simulations of the Edwards-Anderson model, in the case where slow relaxation is neglected? indicate critical slowing down of the spin system at a well defined temperature T,. We report neutron scattering measurements on a Cu-8 at. % Mn alloy using the time-of-flight technique which, however, fail to show the critical behaviour of the type normally associated with a sharp magnetic phase transition, although the temperature variation of the quasi-elastic line-width is found to approach an approximatly constant q-independent value close to the temperature of the maximum in X. In addition, the results appear to demonstrate a continuous process of freezing of spins over a finite temperature range beginning well above the temperatire of the maximum in the susceptibility. Finally, the results show also that the latter temperature itself is not unique but depends on the measurement time constant of the probes used : two such probes in the present experiments being the neutron scattering measurements and the low frequency a.c. susceptibility measurenents made on the same sample.

section

is written d2c? -= dRdw

as 5

N(&2 mc 2

:

!&(q k0

W)

’

1

(1)

where N is the number of spins, (52 the interIncident action constant, k, and k' the and the final wave-vectors and the function S(q,o) describes the wave-vector q and the energy w dependence of the scattering. We have evaluated S(q,w) from the data first for each group of counters at a fixed scattering angle 2 8. From the resultant closely spaced mesh of S(q,w). constant-q cuts were made by an interpolation method using a cubic spline fitting routine. The S(q,w) with q constant thus obtained are plotted against w in fig. 1. The of a strong elastic line superspectra consist posed over a broad quasi-elastic structure as shown by the continuous curves plotted on the reduced scale. Although the elss;ic q-range in the lowest the experiment is 0.08 + 0.64 A q-value for which sufficient w range is covered

The neutrons scattering measurements were made on a polycrystalline Cu-8 at. X Mn alloy close to the (O,O,O) reciprocal lattice point using the IN5 TOF spectrometer at I.L.L. Nettrons of incident energy 3.0 meV (X = 5.14 A) were selected by a multichopper system with the overall spectrometer resolution of 0.2 meV (fwhm).3 He counters were placed mostly at the low scattering angles to cover the elastic qrange between 0.08 and 0.64 w-1. Normalisation 25

SPIN DYNMICS OF A BINARY ALLOY (SPIN GLASS)

Vol.

22,

No.

1

Fig. 2 : The half-widths r of the Lorentzian fits to the quasielastic scattering for three q values as a function a the temperature T, for the Cu-8 at % Mn alloy.

-1.0

0

1.0

-1.0

0

I.0

,w (meV) 1 : S(q,w) vs w for the Cu-8 at. % Mn alloy Fig. for several temperatures and qO-values. The solid lines through the data points represent the Lorentzian fits to the quasi-elastic scattering. tg_Tllow meaningful subby the constant-q cuts, sequent analysis, is 0.2 A . The S(q,w) arising from paramagnetic sc&5terin the isotropic case, as : ing may be written,

s(q*w) =

k2(q)++f(q,w)(2) 2’1_exp(:,,,,kT) g uB

where static f(q,o) tem.

F(q) is the magnetic form factor, x(q) the wave-vector dependent susceptibility and describes the dynamics of the spin sys-

In the paramagnetic regime, i.e. at high it is reasonable to assume a temperatures, Lorentzian form for the quasi-elastic scattering, i.e. : f(q,w)

r = 1. Ti r2+02

(3)

where P the half-width of the scattering is in general a function of q. Results of least-squares fit of the observed quasi-elastic spectra using the Lorentzian convoluted with the instrumental resolution are shown by the solid iines in fig. 1. It is interesting that the quasi-elastic scattering may be described so closely by the simple Lorentzian form even at low temperatures. The computed values of the half-width r are shown plotted against T in fig. 2 for three different q-values P is seen to increase with q at higher temperatures (although there is considerable uncertainly in the results for the 200 K data due to moorer statistics and broader widths). Both the

magnitude of r and its q-dependence decrease with decreasing temperature but somewhere below about 50 K, r becomes approximately constant and also roughly q-independent (I’20.55+0.15 meV) possibly suggesting the freezing of the spin system as a whole such that the spins giving rise to the broad quasi-elastic spectrum are bound, or correlated to some time average spatial orientations about which they oscillate. The susceptibility x(q), given by eq. 2, has been evaluated by simple numerical integration over w using the results of the Lorentzian fits to S(q,w) (for constant q). The computed results are shown in fig. 3 which includes also the measured a.c. susceptibility of the same large sample. In the diagram we have also included, for comparison, the susceptibility computedOfor the lowest counter angle (qelastic = 0.08 A-l) using the original.non-constant-q data points everywhere except in the central region around the elastic peak where a fit was utilised to make a separation between the elastic and the quasi-elastic components. The susceptibility thus obtained shows the same general features as the other x(q) curves, in particular a maximum in x(q) at 52 -+ 3 K, which is considerably higher than the peak in the low-field a.c. susceptibility at 39 + 1 K. This difference is real and due to the difference in the time constants of measurement of the two methods (see below), suggesting a freezing phenomenon rather than a 6 sharp phase transition. Using the results of the Lorentzian fits to the spectra the integrated elastic and quasielastic cross-sections as well as their sum giving the total cross-section have been determined. The results are shown in fig. 4. We note that the elastic scattering begins to increase markedly with decreasing temperature below about 80 K and simultaneously the quasi-elastic cross-section begins to diminish (the systematic deviation of the data points at 80 K from the curves drawn is thought to be due to some anomaly in the experimental conditions for that measurement compared with the rest of the data, it being the first temperature point studied). The diffuse magnetic elastic scattering

Vol.

22,

No.

cross-section for a random substitutional glass is proportional to the square of tic, or frozen-in, local magnetization do _=__ dQ

2 N(s)’ 3 mc2

27

SPIN DYNA?iICS OF A BINARY ALLOY (SPIN GL_qSS)

1

F’(q)

temperature of the maximum in the susceptibility’, the present neutron scattering data, because of its much shorter time constant, show that the “frozen-in” magnetization remains finite up to much higher temperatures. This “frozen-in” magincludes the contribution netization, however, from slowly relaxing spins or groups of correlated spins. Whose observation is supported by both the specific heat8 and indirectly, the susceptibility measurements 9 which show that the spins are correlated into magnetic clusters at temperatures well above that of the maximum in the susceptibility.

spin the sta i.e.

‘.

(4)

is pro orFor wave-vector q=O, the scattering tional to2the spin glass order parameter q P, the long time average at since represents each site. However, because of the limited instrumental energy resolution of 0.2 meV in the present experiment the scattering due to any slowly fluctuating paramagnetic spins, or groups wit-n relaxation times of correlated spins,

I

0.5 I

T 2 lo-11 s will appear within the elastic line and be indistinguishable from truly frozen spins with T + m. Hence this limit of resolution sets the time constant of the present neutron scattering experiment at % lo-l1 s which is much shorter compared with the time constant of % 10-Z s in the a.c. susceptibility measurement.

I ,

I--, 07 3

0

I =

I

xl

I

100

I

I

I50

200

T(K) Fig.

4

: The scattering

cross-sections

$

per

average atom vs T for (a) the total, (b) the elastic and (c) the quasielastic scattering from the Cu-8 at. % Mn alloy. The vertical arrows indicate the temperature where marked changes in the cross-sections become evident.

I

I

I

50

100

I

150

I

200

T(K) x(q) vs T for the Cu-8 Fig. 3 : The susceptibility at. % Mn alloy. The solid dots represent the result obtained for the lowest elastic q-value of 0.08 w-l which, because of the variation of q with w in the is only accurate to within a slowly integration, varying factor ; the dashed curves give the measured a.c. magnetic susceptibility of the same large sample as used for the neutron scattering experiment. The strong temperature variation of the “elastic” scattering in the present measurements reflects the build-up of the “frozen-in” magnetization which can be observed by conventional magnetization measurements also. However, although in the latter measurements the isothermal remanence may be found to disappear close to the

the above results fail to In conclusion, show the normal critical slowing down of the spin system associated with a sharp magnetic although the approximate consphase transition, tancy with q and temperature of the quasielectric scattering width at low temperature suggests that the whole spin system appears frozen below some finite, even if not a sharp, temperature. In addition the marked continuous increase of the elastic scattering accompanied by the continuous decrease of the quasi-elastic cross-section suggests that the continuous process of freezing of spins begins well above the temperature of the maximum in the susceptibility. the results show also that the latter Finally, temperature itself is not unique but depends on the time-constant of measurement. ACKNOWLEDGEMENT - It is a pleasure to ackrowledge helpful discussions with many colleagues and including some of the helpful comments visitors, of referees of another journal.

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(SPIN GLASS)

Vol. 22,

No.

REFERENCES -1. EDWARDS, S.F. and ANDERSON, P.W., J. Phys. F 5, 965 (1975). FISCHER, K., Phys. Rev. Letters 2, 1438 (1977). SHERRINCTON, D. and SOUTHERN, B.W., J. Phys. F 2, L49 (1975). SHERRINGTON, D. and KIRKPATRICK, S., Phys. Rev. Letters 35, 1792 (1975). 415 (1976). HARRIS, A.B., LUBINSKY, T.C. and JING-HUE1 CHEN, Phys. Rev. Letters 2, KOSTERLITZ. J.M.. THOULESS. D.J. and JONES Ravmond C. Phvs. Rev. Letters 36. 1217 (1976). GRINSTEI&'G., B&R, A.N: CHALUPA, J., and Michael WORTIS,Phys. Rev. Lezers 2, 1508 (1976). 2. EDWARDS, S.F. and ANDERSON, P.W., J. Phys. F 5, 1927 (1976). 3. BINDER, K. and STATFER, D. Phys. Lett. j7A, 177 (1976). 1361 (1976). BINDER, K. and SCHRODER, K. Solid State Corma. 2, 4. A check for the absolute ca.i ibration is provided by the incoherent nuclear scattering crosssection of the alloy._The f!lastic scattering from the sample at high _temperatures is found _ to be 180 2 A; mb.st-l at-' against a calculated value of 151 mb.st-' at-l for the quantity : (1-c)a~~,+coin,+c(l-c)(bcu-bMn)2 U sing the standard cross-section for Cu and Mn taken from Bacon, Neutron diffraction, Oxford 1972. 705 (1968). 5. MARSHALL, W. and LOWDE, R.D., Rep. Prog. Phys. 1, A.P.,Phys. Rev. Letters 37, 450 (1976) and references therein. 6. See for exampleMURAN1, L4-229 (1974). 7. THOLENCE, J.L. and TOURNIER, R., J. Phys. (Paris) 2, GUY, C.N., to be published. 3497 (1975). a. WENGER, W.E. and KEESOM, P.H., Phys. Rev. E, 9. CANNELLA, V. in Amorphous Magnetism edited by H.O. Hooper and A.M. de Graaf (Plenum N.Y. 1973) pp. 195-206.

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