179 SPIN REORIENTATION
G.J. BOWDEN, P. MILES, J. P O P E and K.N.R. TAYLOR School o[ Physics, University o[ New South Wales, P.O. Box 1. Kensington, N S W 2033, Australia
It is s h o w n that a n u m b e r of previously unexplained features of the 2~AI N M R spectrum in the cubic L a v e s c o m p o u n d GdAI2 can be u n d e r s t o o d if it is a s s u m e d that the direction of preferred magnetization changes from a  to a [11 I] axis as the temperature is raised from 4.2 to 77 K.
In recent years Kaplan and his co-workers have made an exhaustive study of the 27A1 NMR spectrum of the Laves phase compound GdA12 [1-4]. In particular, these authors believe that this compound magnetizes along a  axes of the cubic structure. However, a close inspection of the experimental results reveals a number of outstanding problems. (a) For a (111) easy direction, theory predicts  that the spectrum should consist of two peaks with an intensity ratio of 3 : 1. However, workers [1-3] invariably report a third peak in the 4.2 K spectrum just to the high frequency side of the most intense resonance line. (b) The experimentally observed [2-4] ratio of the quadrupole parameters for the AI nuclei at the two "" resonance peaks is only 2.17_+ 0.22 whereas the theoretical value should be close to 4 for a  preferred direction. (c) Degani and Kaplan  have reported that the quadrupole hyperfine parameter decreases as the temperature is lowered, and suggested that there is a magnetically induced contribution to the electric field gradient at the AI sites, which augments the lattice contribution. This has been interpreted by Zevin and Kaplan  in terms of a magon-induced pseudo-nuclear quadrupole interaction. In the following we show that all the above phenomena can be understood if we assume that the direction of magnetization in GdAl 2 rotates from a  to a [l I l] axis as the temperature is raised from 4.2 to 77 K.
certain directions of magnetization this latter term has the effect of making the AI atoms, within a given tetrahedral group, inequivalent. Thus, for a general direction of magnetization it is necessary to employ four nuclear hamiitonians  to describe the A!27 hyperfine interactions. These are of the form H = y h I ( H s + H~a) + ½(3 c o s ~ Oa -- 1)P
[3I~ - I ( I + 1)], where (i)
Hs is the magnetic field due to conduction electrons, etc., (ii) H i is a local dipolar field for the AI site " a " in question (fig. 1), (iii) P is the electric quadruple parameter due to the lattice, and (iv) 0a is the angle between Hs and /'/d and the electric symmetry axis characterizing the " a " Al site.
2. Hyperfine fields at the A! sites
It is well known that the spectral details of the 27A1 resonance in the cubic rare earth-Al2 compounds are determined not only by the core and conduction electron polarization hyperfine field contributions but also by the dipolar fields. For Physica 86-88B (1977) 179-180 (~) North-Holland
Fig. 1. Calculated magnetic hyperfine fields at the AI 2~ nuclei as M rotates in the  plane.
180 Experimentally H~= 52 MHz ( - 4.8 T), Hd is -=--8.5 MHz ( - 0.8 T) and P is = 0.5 MHz. Thus the dipolar fields play the major role in determining the character of the A127 N M R spectra. The expression for the dipolar fields in the Laves compounds take on a particularly simple form if the matrix methods of Arif et ai.  are employed. This shows that if the atom in question possesses a threefold rotational axis, then the local dipolar fields may be written in the simple matrix form
direction of magnetization. Moreover if we take H~ = 4.8 T and (Dz:)6d = 0.75 T then excellent agreement is obtained between the experimental data and the theoretical predictions embodied in fig. 2, provided we assume that the preferred direction at 4.2 K lies along a  cubic axis. Turning now to the electric quadrupole interaction we have computed the values of Pen[ = ½(3 c o s 2 0 - 1 ) P ] for a spin rotation in the  plane, to allow a comparison between theory and experiment. This reveals that the ratio of the I I l l l / o/~eff [III] - - 2.22, for a quadrupole parameters = D'-eft  direction of magnetization which is very close to the experimentally determined value of 2.17  at 4.2 K. In this regard we note that the magnon induced nuclear quadrupole interaction  vanishes identically at T = 0 K and is therefore fundamentally incapable of accounting for the experimental value at T = 4.2 K. Finally we comment on the observed decrease in the effective quadrupole parameter at the  site as the temperature is raised . This behaviour can be understood if it is assumed that the direction of preferred magnetization rotates from a  to a  cubic axis as the temperature is raised. This would lead to a decrease in the angle 0 between the magnetic hyperfine field and the  axis and hence give rise to an increase in the quadrupole parameter Per at the  site. -
where o D[Illl
o Dtml = Dzz
o -½ -½
-5 o :
(3) D [fIll
D Ilft] = Dzz 1
Eq. (2) may now be used to examine the effects of the local dipolar fields on the magnetic hyperfine fields on the magnetic hyperfine fields at the Ai sites, as the magnetization is rotated with respect to the crystal axes. The results of such a calculation for a rotation in the plane containing the [I l m] axes may be seen in fig. 1. From an examination of this diagram it will be observed that the N M R spectrum consists of three peaks, in the ratio of 2 : 1 : 1 , for a 
 N. Shamir, N. Kaplan and J.H. Wernick, J. Phys. Paris 32 (1971) CI, 902.  E. Dormann, K.H.J. Buschow, K.N.R. Taylor, G, Brown and M.A.A. lssa, J. Phys. F. 3 (1973) 220.  N. Kaplan, E. Dormann, K.H.J. Buschow and D. Lebenbaum, Phys. Rev. B7 (1973) 40.  J. Degani and N. Kaplan, Phys. Rev. B7 (1973) 2132.  V. Zevin and N. Kaplan, Phys. Rev. B12 (1975) 4604.  S.K. Arif, D.St.P. Bunbury, G.J. Bowden and R.K. Day, J. Phys. F. 5 (1975) 1048.