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Experimental study of the magnetic phase diagrams in NdZn
ELSEVIER
Physica B 210 (1995) 157 170
Experimental study of the magnetic phase diagrams in NdZn M. A m a r a a' *, P. M o r i n h, P. Burlet c aphysikalisches Institut der Universitiit, Robert Mayer Str. 2-4, D-60054 Frankfurt, Germany b Laboratoire Louis-Nbel 1, C.N.R.S., BP 166X,, 38042 Grenoble C~dex, France CMagn~tisme et Diffraction Neutronique, D.R.F./M. C., Centre d'Etudes Nuclbaires B.P. 853(,, 38041 Grenoble c~dex, France Received 7 November 1994
Abstract
The NdZn compound (CsCl-type) orders antiferromagnetically at TN = 70 K. In the magnetic state, a first-order transition at TR = 18.1 K separates two different spontaneous magnetic phases. These two phases have been associated with two multiaxial structures on the basis of powder neutron diffraction and crystal-field analysis. Present work confirms these results by means of magnetization and neutron diffraction measurements on single crystals and completes the H - T phase diagrams for magnetic fields applied along the [0 0 1], [1 1 0] and [1 1 1] directions. The complementary analysis of magnetization and neutron diffraction measurements allows us to associate a magnetic structure with each of the phases seen. All these structures are multiaxial and confirm the existence of strong quadrupolar interactions in NdZn.
1. Introduction Rare earth intermetallic systems present a large variety of magnetic structures, which result from the complex coexistence of two-ion and one-ion interactions I-1]. In low-symmetry antiferromagnetic compounds, the crystalline electric field (CEF) anisotropy and the RKKY-type bilinear interactions generally determine a unique magnetic structure. On the contrary, in high-symmetry lattices, the magnetic ground state would be highly degenerate if only these two couplings are considered and the actual structure is determined by higher-order
* Corresponding author; on leave from the Laboratories LouisNbel (C.N.R.S.) and Magn6tisme et Diffraction Neutronique (D.R.F./M.C., CEN-G) of Grenoble. 1Associated with the University Joseph-Fourier of Grenoble.
pair interactions. In numerous rare earth cubic compounds, the most important higher-order terms have been established to be of quadrupolar type I-2]: quadrupolar interactions are large for the cubic rare earth compounds RAg, RCu, RCd, RZn having the CsCl-type structure. For TmZn the quadrupolar interactions are strong enough to drive a quadrupolar ordering in the paramagnetic phase [3], although, in most cases, the quadrupolar couplings are dominated by the bilinear ones and only the magnetic order occurs. Nevertheless, quadrupolar couplings are active in selecting the most energetically favourable structure with respect to the quadrupolar energy: Ferroquadrupolar interactions give rise to collinear antiferromagnetic structures described by a unique propagation vector k, whereas antiferroquadrupolar couplings lead to multiaxial arrangements. The latter are characterised by the presence, within a given magnetic domain, of several propagation vectors k belonging
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 1 0 1 - X
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M. Amara et al./Physica B 210 (1995) 157 170
to the same crystallographic family, which leaves the bilinear energy invariant. The magnetic moments of constant modulus point from site to site along different equivalent axes, with the same CEF energy. Such structures have been observed in DyCu [4], DyAg [5] and TmGa3 [6], where a triple-k structure (k e ( 1 / 2 1 / 2 0 ) ) is stabilized at low temperature with the moment directions along the threefold axes. Note that, from an experimental point of view, these multiaxial structures are indistinguishable from the collinear ones when using powder neutron diffraction and studies on a single crystal under an applied magnetic field are necessary [7, 8]. The 21 different high-symmetry structures associated with each of the ( 1 / 2 1 / 2 0 ) and ( 1 / 2 0 0 ) families of propagation vectors possible for the spontaneous magnetic state have been indexed in previous papers [9, 10]. The latter paper, called I in the following, also studies the 24 fieldinduced arrangements with moments along directions belonging to one of the three principal crystallographic families. The degeneracy of these magnetic models is removed by the quadrupolar energy in accord with the quadrupolar dispersion curves for the two quadrupolar symmetry modes, tetragonal and trigonal. The CsCl-type compound NdZn antiferromagnetically orders at TN = 70 K [11]. Although the powder neutron diffraction patterns could be indexed with the hypothesis of an unique ( 1 / 2 0 0 ) propagation vector and with a moment directed along the c axis of the magnetic cell at any temperature, two multiaxial structures could also give the same patterns: a double-k structure and a triple-k one with magnetic moments along twofold and threefold directions, respectively. A clear anomaly in the temperature variation of the magnetic susceptibility indicates a change of the magnetic structure at TR = 18.1 K, the temperature at which specific heat measurements reveal a first-order transition [12]. On the basis of CEF calculations, this transition was interpreted as a change of easy magnetization direction from the threefold axes at high temperature to twofold ones at low temperature. This was confirmed by the measured thermal variation of the [101/2] reflection in a neutron diffraction experiment, which revealed a discontinuity (about 0.25/t~) of the magnetic moment at
T R [13]. The ordered moment is larger in the low temperature phase, which agrees with an anisotropy of the calculated magnetic moment with a maximum along the twofold axes. The two successive structures are both multiaxial, triple-k and double-k at high and low temperatures, respectively, and give evidence for the strong influence and nontrivial character of the quadrupolar dispersion curves K,Aq ) and K~(q) and in this compound. In order to confirm and clarify the complex interplay of bilinear, CEF and quadrupolar interactions in NdZn, we have investigated by means of magnetization and neutron diffraction measurements the ordered range for magnetic fields applied along the [001], [-110] and [ 1 1 1 ] axes. We present first the magnetic phase diagrams thus established and then the determination of the magnetic structure associated with each phase, following the methods proposed in I. The quantitative description of both the phase diagrams and the magnetization processes will be given in a forthcoming paper (Ill).
2. Magnetization measurements Most of the magnetization measurements have been performed at the Laboratoire Louis-N6el by the extraction method in two cryomagnets with maximum applied fields of 7.6 and 16 T. Additional measurements up to 27T, were done at the High Magnetic Fields Laboratory in Grenoble. The temperature range of the measurements extends from 1.5 K to room temperature. Isothermal and isofield measurements have been carried out for the three main crystallographic directions. The same spherical sample with a diameter of 2 mm was used in all the experiments. This single crystal was cut from an ingot prepared by the Bridgman technique.
2.1. Magnetization along the [1 1 Of axis For temperatures lower than TR = 18.1 K, the magnetization curves show only one metamagnetic transition at about 3.5 T, with a step of approximately 1 #B (Fig. 1). This feature remains unchanged above TR, except for an increase of the critical field, which reaches a maximum value of 6.5 T at about
159
M. Amara et al./Physica B 210 (1995) 157-170 1.6
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Fig. 1. Magnetization processes along the [1 101 direction at T = 1.5,40 and 55 K. Provided that the ferromagneticcomponent of the structure is parallel to the field, its value is determined by the intersection of the magnetization curve with the M = H/AJ line (see text). The inset shows the derivative of the T = 55 K curve.
40 K. At higher temperatures, this anomaly becomes smooth and a second transition is visible as a change in the slope of the magnetization curve. With increasing temperature, this second critical field rapidly decreases towards the lower critical field. This first determination of the phase diagram is completed by results from isofield measurements, which better define the transition lines almost parallel to the field axis in the H - T plane. The firstorder transition lines are defined at the point where the temperature-derivative is m a x i m u m and the paramagnetic-antiferromagnetic line as the temperature at mid point of the j u m p of the temperature-derivative (Fig. 2, upper part). This definition has the advantage that it can be applied for the whole magnetic field range; experimentally, the magnetization curve always displays a change of slope at the transition temperature, whereas there is a m a x i m u m of M only for low values of the magnetic field. The outline of the phase diagram is already apparent from the isofield measurements (Fig. 2, lower part): in fields lower than 5 T, two transitions can be observed, one corresponding to TN and the other to TR in perfect agreement with the specific heat results for H = 2 T. Above 5 T, the thermal evolution of the magnetization becomes more
0.5
0
0
20
40
60
80
T (K) Fig. 2. Upper part: thermal variation of the magnetization (dots and left scale) and the corresponding derivative (solid line and fight scale) for a H = 14T field applied along the I1 10] direction. The arrows indicate the magnetic transition and a change of magnetic structures. Lower part: Thermal variations of the magnetization for the field values indicated.
subtle: the analysis of its temperature derivative gives evidence for two transition temperatures, the higher one corresponds to the antiferromagnetic ordering, the lower one to a change of magnetic structure (see for example the curve of the temperature derivative at 14 T in the upper part of Fig. 2). At 6.5 T, three transitions are observed. The resulting phase diagram is plotted in Fig. 3. Note that the subscripts b, t and q correspond to binary, ternary and quaternary directions for the field. If we consider the inflection point of the isofield curves for H larger than 7 T (see Fig. 2, lower part) as a transition between phases IIIb and VIb, one would conclude that there are five different phases.
2.2. Magnetization along the [1 1 1 ] axis
The isothermal magnetization processes (Fig. 4) present two steps at low temperature: the first one
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M. Amara et al. / Physica B 210 (1995) 157-170 20
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T (g) Fig. 3. Magnetic phase diagrams of NdZn for the three field directions.
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Fig. 4. Magnetization processes along the [1 I 1] direction at T = 1+5,35 and 60K. The inset gives the field derivative of the 60 K curve.
Fig. 5. Thermal evolution of magnetization for different values of the field applied along the [111] direction. The arrows correspond to magnetic transitions at 16 T.
at H --- 4 T with a 1 #r+ amplitude and the second at H=13.5T with a 0.5/JB amplitude. Above T = 35 K, only the first step is observed at a critical field of about 7 T and is almost temperature independent up tO 65 K. For temperatures ranging from 55 to 65 K, a second weak anomaly can be detected in the magnetization derivative (inset of Fig. 4) at a temperature which decreases rapidly with field, similar to the behaviour along the [1 1 0] field direction.
The isofield measurements (Fig. 5) are similar, at low field, to those obtained along the twofold direction. At low temperature, the differences in magnetization between the 4 and the 6 T curves and between the 13 and 14 T curves reflect the metamagnetic transitions observed in the isothermal results. Above H = 6 T and T = 50 K, the changes of slope seen in the thermal derivative of the magnetization reveal two transitions, as for the [1 10] direction: the higher temperature corresponds to the
161
M Amara et al./Physica B 210 (1995) 157 170
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Fig. 6. Magnetization processes along the [001] direction at 2 K (open triangles) and 25 K (filled squares). Provided that the ferromagnetic component of the structure is parallel to the field, its value is determined by the intersection of the magnetization curve with the M = H/AJ line (see text). The inset shows the magnetic behaviour in the thermal range corresponding to the change of the easy magnetization axes. Note the large value of the susceptibility close to TR in phase IIIq.
magnetic-paramagnetic transition and the lower one to a change of structure. Here also, we attribute the inflection point of the isofield magnetization for H larger than 7T in the 30-40K range to an additional transition. Thus the H - T phase diagram exhibits six different phases (Fig. 3) and, except at low temperatures and large fields (i.e. ignoring phase VIIt), the phase diagram for the [1 1 1] direction resembles closely that found for the [1 10] axis.
2.3. Magnetization along the [0 01] axis At low temperature (T < TR) the isothermal magnetization curves (Fig. 6) display two sharp metamagnetic steps at about 4.5 and 9 T. Measurements up to 27T do not give evidence of any additional metamagnetic transition for this temperature range. Above TR, only one metamagnetic transition can be observed in a field which monotonously decreases from 5.5 to 0 T between TR and TN. The inset of Fig. 6 shows that close to TR, at say 22K, the susceptibility is larger than at higher temperatures. This is due to the fact that at the crossing point TR of the free energies for the (1 1 0) and (1 1 1) magnetization directions, the anisot-
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0.2 ~ , p 3 T ~ m _ 0
20
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T (K)
60
80
100
Fig. 7. Thermal dependence of the magnetization for different values of the field applied along the [001] direction.
ropy is smaller and thus the structure is easier to distort than at high temperature. From the isofield measurements (Fig. 7), the firstorder transition temperature, TR, between the two spontaneous phases, exhibits only a small field dependence. At high temperatures, above H = 3 T, the temperature derivative of the isofield magnetization indicates an additional transition close to the antiferromagnetic one. When increasing the field, this transition is shifted toward lower temperatures and corresponds to the metamagnetic transition observed above TR in the isothermal processes. For fields higher than 6 T, only one sharp transition in the vicinity of TR is observed within the ordered range. The magnetic phase diagram shows five different phases (Fig. 3). It is reminiscent of the [1 1 1] diagram, except that there exists only one phase at high fields and temperatures. 3. Neutron diffraction experiments
All the neutron diffraction experiments were carried out on the DN3 spectrometer (double axis with
162
M. Amara et al./Physica B 210 (1995) 157-170
a moving up counter), at the C.E.N. Grenoble reactor Silo6. Depending on the cryomagnet, magnetic fields below 6 or 10T were available at variable temperature. Two samples were used for the measurements: the previous spherical one and a wedge-shaped sample coming from the same Bridgman ingot. Two different neutron wavelengths were used: 2 = 1.29 and 0.993 A, the respective 2/2 harmonic contributions to the intensities were evaluated from the nuclear [m/2 n/2 p/2] reflections to be of 1% and 0.33% and included in the computed intensities. In order to determine the magnetic structure, numerous integrated intensities were collected in each of the accessible phases. The structures were then determined with some specially developed software. As no other propagation vectors except the ( 1 / 2 0 0 ) branches, associated for the field-induced structures to the 1-000] ferromagnetic propagation vector, have been observed, the procedure starts from one of the high-symmetry structures indexed in I (i.e. a model structure with moments of constant amplitude frozen along only one family of high symmetry directions). For fieldinduced structures, the number of possible models can, however, be reduced by comparing the ferromagnetic component of the model to the magnetization measured along the field. In practice, this comparison is difficult to perform because the real magnetic structure has a nonzero susceptibility and is then distorted with regard to the rigid model. However, as shown in I, when considering CEF anisotropy and isotropic bilinear exchange within the molecular field approximation, the real structure is identical to the model at a particular value of the field: H* = (J(k) - J(0))Mo -- AJ M0,
(1)
where J(q) is the Fourier transform of the exchange interaction, Mo the ferromagnetic component of the structure and k belongs to the star of the antiferromagnetic propagation vectors. Provided that the magnetization, i.e. the ferromagnetic Fourier component, is parallel to the field, this relation determines Mo from the intersection of the line H / A J and the isothermal magnetization curve (see Fig. 1, for example). In the case of a second order transition toward the antiferromagnetic state, AJ
may be directly determined from the measurement of the reciprocal susceptibility at TN: for NdZn, AJ = 105 +__3 kOe//~B [11]. As the neutron diffraction measurements are also carried out on a distorted structure, a distortion-parameter describing the change in the ferromagnetic component is introduced in the refinements. For each tested model, the procedure determines the different magnetic domains which must be considered with their relative proportions as additional parameters for minimizing the reliability factor 1,14]: l
[-/talc
--
I ....
7
where Pi is the inverse of the statistical error of the intensity (lmeas) measured for the ith reflection.
3.1. Field along [ 0 01
]
In order to confirm the proposed spontaneous structures, we have measured the thermal evolution of diffraction peaks associated with the three independent branches of the ( 1 / 2 0 0 ) star in a field of 1.8 T (Fig. 8). This field value was chosen in order to induce possible magnetic domain motions without distorting the magnetic structure. In the 20-60 K range, the three intensities are equal and prove the
200
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oo o
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40
i
NdZn H / / [ 0 0 1] H = 1.8T
,~. 160
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I
A
A
t II
0
i
0
TR
I
I
10
20
I I
I
30 40 T (K)
50
60
70
Fig. 8. Thermal evolution in a [001] field of 1.8 T of the intensities of three antiferromagneticreflections associated to each branch of the (1/200) propagation star. The two structures correspond to the spontaneous phases I and II.
163
M. Amara et al./Physica B 210 (1995) 157-170
single domain character of phase I, which is, under the constraint of the magnetic field, only compatible with a cubic structure. Thus, among the three structures allowed by the powder neutron diffraction results [12], the triple-k (Fig. 8) with moments along threefold axes, labelled tl in I, is confirmed. The fit of the diffracted intensities collected at 25 K and H = 0 to this triple-k model gives an ordered moment of 2.16 p~ with an excellent reliability factor equal to 5.2%. At TR, the three intensities become different revealing the partition of the sample into domains: the structure associated with phase II is then no longer cubic. The sum of the three intensities and thus the magnetic moment increase at TR, which confirms that the easy magnetization axes have changed to the twofold axes in agreement with previous neutron diffraction measurements [13] and the CEF calculations [12]. The structure of phase II is then confirmed to be double-k (dl in I) and the refinement using the intensities collected at T = 3.7K and H = 1.8T gives a reliability factor of 5.4% and an ordered moment of 2.47 Pa- The analysis of the quadrupolar energy of the spontaneous structures may provide us with information on the quadrupolar dispersion curves. From I, these energies are
for phase I'I; S is the amplitude of the magnetic moment, p, and pr describe the coupling of the quadrupolar components to the molecular exchange field. To agree with the determined structures, the quadrupolar dispersion curves should reproduce the following relations:
Kr(0) > K~(½½0),
for phase I and EQ(II) = -
1 4S4( 2 _2 K~(½½0)] ~J(k) ~p, Kr(O) + p~ -~ ;
Table 1 Fourier components of the model structures with twofold axes of easy magnetization. The ferromagnetic and M1 and M2 antiferromagnetic components correspond to a moment of unit modulus on each site. Note that there is no model with M0 parallel to a threefold axis. The last three columns give the magnetization (#B/at.) parallel to the fourfold, twofold and threefold direction, for the 2.5/~B moment observed at low temperature. /140
Ml
1
[0 0 l/x/'2 ]
[ l/x/2 0 0]
2 3 4 5
[1/2~f2 1/2x/'2 l/x/2 ] I-0 1/2,~f2 1/2,~f2] [0 0 l/x/2 ] [0 1/2x/2 l/2xf2 ]
[1/2x/'2 - 1/2x/2 0] [l/x/2 -- 1/2v/2 1/2,J2] [ - 1/2x/2 1/2x/'2 0] [1/x~ 0 0 ]
(3)
for the tetragonal and trigonal symmetries. When increasing the magnetic field up to 5.7 T at T = 2K, the intensity of the [1/200] reflection suddenly increases from zero at the same critical field as the jump in magnetization. Note that several Fourier components belonging to different domains can be propagated by the same [1/200] vector and contribute to the [1/200] intensity. Thus, in phase IIIq, at least one of them has rotated away from the [100] axis. In order to determine the structure of phase IIIq, we use the field-induced models with twofold axes of easy magnetization shown in Table 1 (Tables 7 and 8 of I). The indicated magnetization might be expected to be parallel to the applied field along the main crystallographic directions. From the [001] isothermal magnetization curves (Fig. 6), the value corresponding to a rigid model is 1.1 #B. There is no structure in Table 1 that gives a magnetization in agreement with this, which indicates that the ferromagnetic component is in fact not parallel to the field applied along the [001] axis. We thus conclude that there is no significant rotation or closure of the structure toward the applied field direction due to a huge anisotropy. The closest values in Table 1 to the observed value are for structures
11 1 4 p~2s4Ke(LiLiO) ~J(k) 3
EQ(I) = --
K , ( 1: 1: 0 ) > K~(0)
M2
M[oo 11
M[o I 11
Mr1 ~ ~1
[1/2x/2 1/2x/'2 0] [0 - 1/2x/2 l/2xf2]
1.77 1.77 0.8g 1.77 0.88
1.25 1.98 1.25 1.25 1.25
1.02 2.04 1.02 1.02 1.02
164
M. Amara et al./Physica B 210 (1995) 157 170
with a ferromagnetic component directed along a twofold axis (lines 3 and 5). According to the nomenclature of I, there are two double-k models, H d l 0 and H d l 1, and five triple-k ones, Ht7, Ht8, Ht9, H t l 0 and H t l l , to test. The best agreement between the intensities measured at H = 5.7 T and T = 2 K and those calculated for each model is obtained for the structure defined by: Mo = [0 - 0.86 0.94]/~B/at, M1 = [1.73 00] ktB/at (propagated by k~ = [1/2 0 0]), M2 = [0 0.86 0.86] ~B/at (propagated by k2 = [0 1/20]), which is derived from the Ht7 rigid model shown in Fig. 9. The refinement was carried out taking account the partition of the sample into height domains. The reliability factor is equal to 7.7% and the field-component of Mo to 0.94/~B, which is in close agreement with the magnetization measured at 5.7 T (see Fig. 6). The diffracted intensities as functions of temperature at H = 4 T have confirmed the sharp thermal range of phase IVq in low field just below TN.
•
160 L .
However, the set of intensities available for this phase is not sufficiently large to do a meaningful numerical refinement of its structure. Fortunately, the structure can be determined from qualitative considerations alone. Firstly, the intensities measured for [h k 1/2] reflections are null. Second, the fact that the intensities of [1/200] and [0 1/20] reflections are negligible indicates that the antiferromagnetic components are parallel to their ( 1 / 2 0 0 ) propagating vectors: among the models with threefold easy magnetization axes, the only possible structures are related to lines 2 (Hd8 and Hd9) and 3 (Htl, Ht2, Ht3 and Ht4) in Table 2. The magnetization measured at T = 25 K has a value of about 1.6 #B larger than the values in Table 2: as in phase IIIq, the ferromagnetic component of the structure is not collinear to the field in phase IVq. Thus models from line 3 have to be rejected and the retained model, derived from line 2, Hd8, is drawn in Fig. 10(a) and is defined by: M0 = [0 l/x/3 l/x/3 ] #n/at, M~ = [ 1/x/3 0 0] #B/at (parallel to kl = [ 1/2 0 0]). The unfavourable orientation of the ferromagnetic component, perpendicular to the field direction in
N d Z n , phase lllq T = 2 K
. . . . . . .
•
Measured
[]
Com uted
120
o
Fig. 9. The field-inducedmodel with moments along twofoldaxes of easy magnetizationin phase IIIq. The diagram shows both the computed and measured diffractedintensities at T = 2 K and H = 5.7 T, the reliabilityfactor being equal to 7.7%.
165
M. Amara et al./Physica B 210 (1995) 157-170
Table 2 Fourier components of the model structures with threefold axes of easy magnetization. The M0 ferromagnetic and M~ and M2 antiferromagnetic components correspond to a moment of unit modulus on each site. Note there is no model with Mo parallel to a threefold axis. The last three columns give the magnetization (gB/at) parallel to the fourfold, twofold and threefold direction for the moment values observed at 25 K (2.25 #B/at) and at 40 K (2.1/in/at).
l 2 3
M0
M1
[00 l/x/3 ] [0 l/x/3 l/x/'3 ] [00 l/x/3 ]
[ 1 / ~ 1/x~0 ] [l/x~00 ] [ l/x/3 0 0]
M2
M[001](25 K)
M[o 111(40K)
MI~ 1q(40 K)
[0 l/xfl30]
1.3 1.3 1.3
0.86 1.71 0.86
0.7 1.34 0.7
that the ferromagnetic component is already parallel to the fourfold direction in 10 T. As explained in I, the quadrupolar energies associated with lines 1 and 4 would respectively read as E Q ( V I ) = _ ~ j ( k ) 4 S 4 pZK,(O ) + p2 a
b
c
Fig. 10. (a) Field-induced model with moments along threefold axes proposed for phase IVq (b) (c) Field-induced structure with twofold easy magnetization axes in phases Vq and VIIt.
the domain associated with the [001/2] propagation vector, explains why this domain does not exist under the field. The actual structure has a large magnetization at 25 K because, in the vicinity of TR, it is easy to distort with respect to the high symmetry structure. This agrees with the magnetization measurements close to the point where there is a change of easy axes (inset of Fig. 6). No neutron diffraction experiment could be carried out in phase Vq since the sample fractured at 8 T, in phase IIIq, when increasing the field to enter phase Vq. (Note that the fracture due to torque effects is a direct consequence of the non collinearity of the ferromagnetic component and the field.) The only available information comes from the magnetization measurements. The determination of the ferromagnetic component of the model structure (Fig. 6) gives a value of 1.85 #B close to the 1.77 #a obtained for lines 1, 2 and 4 of Table 1. Such a good agreement would be surprising for a structure based on line 2, as the ferromagnetic component is 35 ° out of the field direction and then, cannot strictly obey Eq. 1. Owing also to the absence of any additional transition up to 27 T, it is then likely
1
Eo(V4) = - 2 J(k)4 $4
fp X,(o) + 3K,(1½0) 4 k
As K ~ (11 ~ 0 ) < gr(0)(Eq. 3), the set of Fourier components of line 1 is the most likely. The only difference between models Hd5, Hd6 and Hal7 of line 1 is the way in which the ( 1 / 2 0 0 ) propagation vector is associated with the antiferromagnetic Fourier component (see Table 8-third line-in I). To be consistent with the structures determined in phases II and IIIq, which are described with antiferromagnetic Fourier components parallel to the [,1/200] and [,0 1/20] propagation vectors, it is logical to choose the model Hd5 shown in Fig. 10(b) and defined by Mo = [00 l/x/2 ]/aB/at,
MI = [-l / x / 2 0 0 ] kta/at (parallel to kl = [-1/200]). 3.2. Field along El-1 O] We now report on the integrated intensities collected in Phase IIIb at T = 4 K and H = 7.9 T. At low temperatures the measured magnetization, 1.4#n (Fig. 1), allows us to exclude solutions
166
M. Amara et al./Physica B 210 (1995) 157 170
I 1200
T=4K H = 7.9 T / / [ 1 - 1 0
I
~
t
I
[]
Computed I
800
•~
400
0
Fig. 11. The field-induced model with moments along twofold axes in phase lllb. Figure shows both the computed and measured diffracted intensities at T = 1.5 K and H = 7.9T. the reliability factor being equal to 10.1%. associated with line 2 in Table 1. After testing the 12 remaining models by the same technique as in Section 3.2, the retained structure is the triple-k model Ht7 in line 5, drawn in Fig. 11 and defined by Mo = [1.02 - 1.02 0] #B/at, M1 = [0 0 1.74] #B/at (propagated by kl = [-0 0 1/2]), M2 = [ - 0.87 - 0.87 0] #B/at
(propagated by k2 = [0 1/2 0]). The refinement (Fig. 11) is carried out over two domains and yields a reliability factor equal to 10.1%. The deduced magnetization is 1.38 #B, in agreement with the 1.3 #B value measured at T = 1.5 K and H = 7.9 T. The value obtained for the ordered moment is 2.57 #B, a value slightly larger than the 2.47/~B spontaneous value. This difference can be understood in terms of a more favourable quadrupolar arrangement in phase IIIb than in phase II, which is supported by the low value of the critical field separating these two phases. The Ht7 model for phase IIIb is then identical to the one determined in phase IIIq (the Fourier components and the associated propagation vec-
tors in Sections 3.1 and 3.2 are related to two different domains of this Ht7 model). In addition to the fact that the magnetization is nonparallel to the field, the magnetic structure in phase III is insensitive to the field direction. Beyond the large anisotropy in favour of the twofold axes, this behaviour also indicates two-ion effects which cannot be related to the isotropic bilinear exchange and which are likely of quadrupolar origin. Integrated intensities have also been collected in phase VIb at T = 4 0 K in a field of 7.9 T. From Fig. 1, the ferromagnetic component of the associated model has (within the hypothesis of collinearity to the field) a value of 1.3/~B which permits us to exclude solutions from line 2 of Table 2. The refinement starting from the 6 remaining models of lines 1 and 3 is not very selective, but seems to determine the model H t l defined by Mo = [1.1400] pB/at, M1 = [0 1.140] pB/at (propagated by kl = [0 0 1/2]), M2 = [00 1.14]/~B/at (propagated by k2 = [00 1/2]), as the most likely (Fig. 12).
M. Amara et al./Physica B 210 (1995) 157-170
167
12oo
•
]
600 "~ 300
!
o
i
.
i"i "!! i i
-2' °
"2
i
Fig. 12. The field-induced model with moments along threefold axes in phase VIb. The diagram shows both computed and measured diffracted intensities at T = 40 K and H = 7.9 T, the reliability factor being equal to 15.3%.
Here also the ferromagnetic component does not point along the field and the deduced [1 0 1] magnetization is 0.81 #B, comparable with the 0.9 #n value measured in 8 T (Fig. 1). However, the reliability factor is too high (15.3%) to be completely confident in this result. Measurement of intensities versus temperature for the same field have confirmed the existence of phase IVb. Integrated intensities have been collected for T = 61 K and H = 7.9 T. Owing to the weak anisotropy of the isothermal magnetization curves at this temperature, the structure may be strongly distorted and the value of the ferromagnetic component a debatable criterion. The best agreement (R -- 9.2%) between measured and collected intensities (Fig. 13) is obtained for the model Hd8 from line 2 in Table 2, defined by
3.3. Field along f l 1 1J
Mo = [0.46 - 0.46 0] #B/at,
Mo = [0.05 0.92 0.92] #B/at,
M1 = [0 0 0.75] #B/at
M1 = [1.7400] #B/at
(propagated by kt = [00 1/2]).
For these measurements, the field has been increased up to H = 8.2 T and two sets of integrated intensities collected at T = 5 K (Phase IIIt) and at T = 40 K, (Phase VIi). Since there exists no model structures with a ferromagnetic component along a threefold direction (see Tables 7 and 8 in I), the isothermal magnetization curves are only of qualitative use in the selection of the models for each phase. At 5 K, the magnetization processes show a first magnetization jump slightly larger than 1 #B. From Table 1, this value is reminiscent of the ones in lines 1, 3, 4 and 5. The refinement (Fig. 14) leads to a reliability factor minimum (11.1%) for the same model Ht7 as in Phases IIIq and IIIb with
(propagated by kl = [1/2 0 0]),
Me = [0 - 0.87 0.87] #B/at This model is then the same as the one proposed for the phase IVq (Section 3.1).
(propagated by k2 = [0 1/2 0]).
168
M. Amara et al./Physica B 210 (1995) 157-170
[ NdZn,phaseIVb 450 IH = 7.9 T=61K T / l [ 1 - 1 0]
~
~
I~/i/[
300 150 "=
o
Fig. 13. The field-induced model with moments along threefold axes in phase IVb. The diagram shows both computed and measured diffracted intensities at T = 61 K and H = 7,9 T, the reliability factor being equal to 9.2%.
800 41,
= 600 t--
L..
400 200
),,,q
- -
on
Fig. 14. The field-induced model with twofold easy magnetization axes in phase lilt. The diagram shows both computed and measured diffracted intensities at T = 5 K and H = 8.2 T, the reliability factor being equal to 11,1%. T h e r e f i n e m e n t is c a r r i e d o u t by t a k i n g a c c o u n t of six d o m a i n s , of w h i c h t w o are f o u n d to r e p r e s e n t 8 7 % of the sample. T h e d e d u c e d m a g n e t i z a t i o n , 1.09/~B, is close to the m e a s u r e d value, 1.15/tB ( T = 1 . 5 K a n d H = 8.2 T). A t T = 40 K, the best a g r e e m e n t b e t w e e n m e a s u r e d a n d c a l c u l a t e d intensities is o b t a i n e d for
m o d e l H t l in line 3 of T a b l e 2 d e f i n e d by Mo = [0 1.08 0] #B/at, M1 = [ 1.08 0 0]/~B/at ( p r o p a g a t e d by kl = [ 1 / 2 0 0 ] ) ,
169
M. Amara et al./Physica B 210 (1995) 157 170
shown in Fig. 10 and defined by
Mz : [-001.08]/aB/at (propagated by k2 = [001/2]).
Alto = [1/2x//2 1/2x/~ 1/x/~]/aB/at,
This model, which leads to three domains in the refinement (Fig. 15), is the same as in phase VI b. The reliability factor (15.7%) is not fully satisfactory and is reminiscent of that found for phase VIb. AS we do not have enough integrated intensities for phase IVt, we are only able, using the argument of an unique structure observed for phases IIIq/b/t and for phases VIwt, to suggest tentatively that the Hd8 model determined in phases IVq and IVb is again applicable for the present case of phase IVt. Phase VIIt is not accessible by any of the available neutron diffraction cryomagnets. However, its structure is identified from Table 1 owing to the low temperature twofold axes of easy magnetization. The ferromagnetic Fourier component, not parallel to the [111] applied field, has to be large in order to give an important projection on the [-111] field direction as experimentally observed in Fig. 4 (about 2 #B). A model with a [1 1 2] direction for this ferromagnetic component, as in line 2, is then likely. Among the two corresponding double-k models, Hdl and Hd2, the former appears more in light of the field-sequence of phases 1I and lilt. It is
/ 4so1~
M, = [1/2x/~ - 1/2w/2 0] #,/at (propagated by ka = [01/20]).
4. Conclusion The present work has determined the H - T phase diagrams of the compound NdZn for the main crystallographic directions of the applied field below H = 16 T. At low temperature the isothermal magnetization curves show multistep processes that are rather common for uniaxial systems, but quite unusual for cubic ones. The corresponding lines of the phase diagrams are then well defined. Equivalent is the analysis of the temperature dependence of the magnetization, although some smooth thermal evolutions are less easy to interpret. For the three field directions, the change of easy magnetization axes is maintained over all the accessible range of the field. This is particularly pronounced for the fourfold direction for which this
t T=4o
= .
[] 7---__
I
ompo, o
I I
00
"~'~ 150
0 i
i
i
~
t
Fig. 15. The field-inducedmodel with moments along threefold axes in phase VIt. Figure shows both the computed and measured diffractedintensitiesat T = 40 K and H = 8.2T, the reliabilityfactor being equal to 15.7%.
170
M. Amara et al./ Physica B 210 (1995) 157-170
change occurs at a nearly constant temperature, equal to TR at zero field. The spontaneous multiaxial structures are here confirmed. For the fieldinduce d phases, the combined use of magnetization processes and neutron diffraction results has permitted the determination of the structures associated with most of the phases revealed by the magnetization measurements. Consideration of the multidomain state in the refinement is essential to the determination of the magnetic structures arising in a cubic lattice. In the absence of neutron diffraction results, the magnetization processes and the expression for the quadrupolar energy of the possible high-symmetry models has allowed us to propose the most likely structures for phases Vq and VIIt. An important result of this study is that most of the magnetic structures are stabilized whatever the field direction is. This is particularly true for the structures associated with the phases IIIq, IIIb and IIIt. Since C E F anisotropy and bilinear isotropic interactions are compatible with numerous magnetic arrangements, such a behaviour has to be related to other couplings. The strength of these quadrupolar couplings appears through the lowtemperature sequence of magnetic structures versus the magnetic field, especially for the fourfold direction. In the sequence (II-IIIq-Vq), the spins always point along different twofold axes, which minimize the C E F energy: phase II results from fourfold planes consisting of a square of four spins pointing to the center, i.e. along ( 1 1 0 ) and stacked along the [ 0 0 1 ] axis (Fig. 8). At 4.5 K, a pair of spins along one side of the square rotates as a block towards the [ 1 0 1 ] and [ - 101] axes to give the structure in phase III, which is not symmetric with respect of the field. At 9.5 T, the second pair of spins undergoes an identical j u m p to give a symmetric structure again. These aspects relate to the multiaxial character of all the observed structures and
give evidence for strong antiferroquadrupolar couplings in NdZn. Although there is no direct way to experimentally determine the dispersion curves for quadrupolar interactions away from the center, of the Brillouin zone, the present study has allowed us to fix the relations: K~(0) > K~(1/2 1/2 0) and g~(1/2 1/2 0) > g~(0) which, by themselves, determine many features of the magnetic order. In a forthcoming paper, we will confirm, by calculations within the periodic field method, that the quadrupolar interactions are indeed essential for the description of the ordered states of NdZn.
References [1] B. Coqblin, The Electronic Structure of Rare-earth Metals and Alloys: The Magnetic Heavy Rare Earths (Academic Press, New York, 1977). [2] P. Morin and D. Schmitt, in: Ferromagnetic Materials, Vol. 5, eds. K.H.J. Buschow and E.P. Wohlfarth (NorthHolland, Amsterdam, 1990) p. 1. [3] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B 17 (1978) 3684. [4] R. A16onard, P. Morin and J. Rouchy, J. Magn. Magn. Mater. 46 (1984) 233. [5] P. Morin, J. Rouchy,K. Yonenobu, A. Yamagishiand M. Date, J. Magn. Magn. Mater. 81 (1989) 247. [6] P. Morin, M. Giraud, P. Burlet and A. Czopnik, J. Magn. Magn. Mater. 68 (1987) 107. [7] M. Wintembergerand R. Chamard-Bois,Acta Crystallogr. A 28 (1972) 341. I-8] J. Rossat-Mignod, in: Neutron Scattering in Condensed Matter Research, eds. K. Sk61d and D.L. Price (Academic Press, New York, 1986). [9] P. Morin and D. Schmitt, J. Magn. Magn. Mater. 21 (1980) 243. [10] M. Amara and P. Morin, Physica B 205 (1995) 379. [11] P. Morin and J. Pierre, Phys. Stat. Sol. (A) 30 (1975) 549. [12] P. Morin and A. de Combarieu, Solid State Commun. 17 (1975) 975. [13] H. Fujii, Y. Uwatoko, K. Motoya, Y. lto and T. Okamoto, J. Magn. Magn. Mater. 63-64 (1987) 114. [14] M. Amara, R.M. Gal/~ra, P. Morin, J. Voiron and P. Burlet, J. Magn. Magn. Mater. 131 (1994) 402.
Physica B 210 (1995) 157 170
Experimental study of the magnetic phase diagrams in NdZn M. A m a r a a' *, P. M o r i n h, P. Burlet c aphysikalisches Institut der Universitiit, Robert Mayer Str. 2-4, D-60054 Frankfurt, Germany b Laboratoire Louis-Nbel 1, C.N.R.S., BP 166X,, 38042 Grenoble C~dex, France CMagn~tisme et Diffraction Neutronique, D.R.F./M. C., Centre d'Etudes Nuclbaires B.P. 853(,, 38041 Grenoble c~dex, France Received 7 November 1994
Abstract
The NdZn compound (CsCl-type) orders antiferromagnetically at TN = 70 K. In the magnetic state, a first-order transition at TR = 18.1 K separates two different spontaneous magnetic phases. These two phases have been associated with two multiaxial structures on the basis of powder neutron diffraction and crystal-field analysis. Present work confirms these results by means of magnetization and neutron diffraction measurements on single crystals and completes the H - T phase diagrams for magnetic fields applied along the [0 0 1], [1 1 0] and [1 1 1] directions. The complementary analysis of magnetization and neutron diffraction measurements allows us to associate a magnetic structure with each of the phases seen. All these structures are multiaxial and confirm the existence of strong quadrupolar interactions in NdZn.
1. Introduction Rare earth intermetallic systems present a large variety of magnetic structures, which result from the complex coexistence of two-ion and one-ion interactions I-1]. In low-symmetry antiferromagnetic compounds, the crystalline electric field (CEF) anisotropy and the RKKY-type bilinear interactions generally determine a unique magnetic structure. On the contrary, in high-symmetry lattices, the magnetic ground state would be highly degenerate if only these two couplings are considered and the actual structure is determined by higher-order
* Corresponding author; on leave from the Laboratories LouisNbel (C.N.R.S.) and Magn6tisme et Diffraction Neutronique (D.R.F./M.C., CEN-G) of Grenoble. 1Associated with the University Joseph-Fourier of Grenoble.
pair interactions. In numerous rare earth cubic compounds, the most important higher-order terms have been established to be of quadrupolar type I-2]: quadrupolar interactions are large for the cubic rare earth compounds RAg, RCu, RCd, RZn having the CsCl-type structure. For TmZn the quadrupolar interactions are strong enough to drive a quadrupolar ordering in the paramagnetic phase [3], although, in most cases, the quadrupolar couplings are dominated by the bilinear ones and only the magnetic order occurs. Nevertheless, quadrupolar couplings are active in selecting the most energetically favourable structure with respect to the quadrupolar energy: Ferroquadrupolar interactions give rise to collinear antiferromagnetic structures described by a unique propagation vector k, whereas antiferroquadrupolar couplings lead to multiaxial arrangements. The latter are characterised by the presence, within a given magnetic domain, of several propagation vectors k belonging
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 1 0 1 - X
158
M. Amara et al./Physica B 210 (1995) 157 170
to the same crystallographic family, which leaves the bilinear energy invariant. The magnetic moments of constant modulus point from site to site along different equivalent axes, with the same CEF energy. Such structures have been observed in DyCu [4], DyAg [5] and TmGa3 [6], where a triple-k structure (k e ( 1 / 2 1 / 2 0 ) ) is stabilized at low temperature with the moment directions along the threefold axes. Note that, from an experimental point of view, these multiaxial structures are indistinguishable from the collinear ones when using powder neutron diffraction and studies on a single crystal under an applied magnetic field are necessary [7, 8]. The 21 different high-symmetry structures associated with each of the ( 1 / 2 1 / 2 0 ) and ( 1 / 2 0 0 ) families of propagation vectors possible for the spontaneous magnetic state have been indexed in previous papers [9, 10]. The latter paper, called I in the following, also studies the 24 fieldinduced arrangements with moments along directions belonging to one of the three principal crystallographic families. The degeneracy of these magnetic models is removed by the quadrupolar energy in accord with the quadrupolar dispersion curves for the two quadrupolar symmetry modes, tetragonal and trigonal. The CsCl-type compound NdZn antiferromagnetically orders at TN = 70 K [11]. Although the powder neutron diffraction patterns could be indexed with the hypothesis of an unique ( 1 / 2 0 0 ) propagation vector and with a moment directed along the c axis of the magnetic cell at any temperature, two multiaxial structures could also give the same patterns: a double-k structure and a triple-k one with magnetic moments along twofold and threefold directions, respectively. A clear anomaly in the temperature variation of the magnetic susceptibility indicates a change of the magnetic structure at TR = 18.1 K, the temperature at which specific heat measurements reveal a first-order transition [12]. On the basis of CEF calculations, this transition was interpreted as a change of easy magnetization direction from the threefold axes at high temperature to twofold ones at low temperature. This was confirmed by the measured thermal variation of the [101/2] reflection in a neutron diffraction experiment, which revealed a discontinuity (about 0.25/t~) of the magnetic moment at
T R [13]. The ordered moment is larger in the low temperature phase, which agrees with an anisotropy of the calculated magnetic moment with a maximum along the twofold axes. The two successive structures are both multiaxial, triple-k and double-k at high and low temperatures, respectively, and give evidence for the strong influence and nontrivial character of the quadrupolar dispersion curves K,Aq ) and K~(q) and in this compound. In order to confirm and clarify the complex interplay of bilinear, CEF and quadrupolar interactions in NdZn, we have investigated by means of magnetization and neutron diffraction measurements the ordered range for magnetic fields applied along the [001], [-110] and [ 1 1 1 ] axes. We present first the magnetic phase diagrams thus established and then the determination of the magnetic structure associated with each phase, following the methods proposed in I. The quantitative description of both the phase diagrams and the magnetization processes will be given in a forthcoming paper (Ill).
2. Magnetization measurements Most of the magnetization measurements have been performed at the Laboratoire Louis-N6el by the extraction method in two cryomagnets with maximum applied fields of 7.6 and 16 T. Additional measurements up to 27T, were done at the High Magnetic Fields Laboratory in Grenoble. The temperature range of the measurements extends from 1.5 K to room temperature. Isothermal and isofield measurements have been carried out for the three main crystallographic directions. The same spherical sample with a diameter of 2 mm was used in all the experiments. This single crystal was cut from an ingot prepared by the Bridgman technique.
2.1. Magnetization along the [1 1 Of axis For temperatures lower than TR = 18.1 K, the magnetization curves show only one metamagnetic transition at about 3.5 T, with a step of approximately 1 #B (Fig. 1). This feature remains unchanged above TR, except for an increase of the critical field, which reaches a maximum value of 6.5 T at about
159
M. Amara et al./Physica B 210 (1995) 157-170 1.6
].5
M
14it ----
.
NdZnH//[I B~
M = 1.3 P'a
.
i
,
i
.
i
,
i
.
i
.
i
,
i
.
i
2O
m******'*****"~~
--
1.2 f
.
0
1.1
..
-10
i
-20
"~
0.9
"~ 0 . 8 *
~° " ~ -
II
.
.
..
..
.
.
.
dM)dH .
:3-
0.4
0
•
._.~
10]
0
4
8 H (T)
12
16
Fig. 1. Magnetization processes along the [1 101 direction at T = 1.5,40 and 55 K. Provided that the ferromagneticcomponent of the structure is parallel to the field, its value is determined by the intersection of the magnetization curve with the M = H/AJ line (see text). The inset shows the derivative of the T = 55 K curve.
40 K. At higher temperatures, this anomaly becomes smooth and a second transition is visible as a change in the slope of the magnetization curve. With increasing temperature, this second critical field rapidly decreases towards the lower critical field. This first determination of the phase diagram is completed by results from isofield measurements, which better define the transition lines almost parallel to the field axis in the H - T plane. The firstorder transition lines are defined at the point where the temperature-derivative is m a x i m u m and the paramagnetic-antiferromagnetic line as the temperature at mid point of the j u m p of the temperature-derivative (Fig. 2, upper part). This definition has the advantage that it can be applied for the whole magnetic field range; experimentally, the magnetization curve always displays a change of slope at the transition temperature, whereas there is a m a x i m u m of M only for low values of the magnetic field. The outline of the phase diagram is already apparent from the isofield measurements (Fig. 2, lower part): in fields lower than 5 T, two transitions can be observed, one corresponding to TN and the other to TR in perfect agreement with the specific heat results for H = 2 T. Above 5 T, the thermal evolution of the magnetization becomes more
0.5
0
0
20
40
60
80
T (K) Fig. 2. Upper part: thermal variation of the magnetization (dots and left scale) and the corresponding derivative (solid line and fight scale) for a H = 14T field applied along the I1 10] direction. The arrows indicate the magnetic transition and a change of magnetic structures. Lower part: Thermal variations of the magnetization for the field values indicated.
subtle: the analysis of its temperature derivative gives evidence for two transition temperatures, the higher one corresponds to the antiferromagnetic ordering, the lower one to a change of magnetic structure (see for example the curve of the temperature derivative at 14 T in the upper part of Fig. 2). At 6.5 T, three transitions are observed. The resulting phase diagram is plotted in Fig. 3. Note that the subscripts b, t and q correspond to binary, ternary and quaternary directions for the field. If we consider the inflection point of the isofield curves for H larger than 7 T (see Fig. 2, lower part) as a transition between phases IIIb and VIb, one would conclude that there are five different phases.
2.2. Magnetization along the [1 1 1 ] axis
The isothermal magnetization processes (Fig. 4) present two steps at low temperature: the first one
160
M. Amara et al. / Physica B 210 (1995) 157-170 20
'
'
'
I
•
'
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+
'
'
M(T)
•
'
I
'
"'
,
'
,
,
l
j
r
l
l
l
l
I + I I
NdZn
o M(H)
( I L l l t ' " l l l J i l ' | l l
U#ltlZl
H//[O01]
H#[llO] o
•
VIIt
•
15
•
o o a~ °~°
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~.
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+lo
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+•
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•
~ o°
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,
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i
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o
!1 •
•
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i
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20
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11
i
60
|
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q0~i
oo~
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o OOoi
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•
•
o o'm~=oommooo~ o~ o
o
• o
IVb
60
•
o~ o0 0
I
; 20
40
o
60
80
T (g) Fig. 3. Magnetic phase diagrams of NdZn for the three field directions.
NdZn H//[I
~ ,
1 1]
1.5
T=i.$ K 0.$
00
°°
=3S~ •
°°-'..J 4
~
.
~
1.5
A-"--'-~--m~nx : ~ °o~/
8 H (T)
'
12
A
4"
0.5
1,,,, t-
16
0
20
40
T (K)
60
80
Fig. 4. Magnetization processes along the [1 I 1] direction at T = 1+5,35 and 60K. The inset gives the field derivative of the 60 K curve.
Fig. 5. Thermal evolution of magnetization for different values of the field applied along the [111] direction. The arrows correspond to magnetic transitions at 16 T.
at H --- 4 T with a 1 #r+ amplitude and the second at H=13.5T with a 0.5/JB amplitude. Above T = 35 K, only the first step is observed at a critical field of about 7 T and is almost temperature independent up tO 65 K. For temperatures ranging from 55 to 65 K, a second weak anomaly can be detected in the magnetization derivative (inset of Fig. 4) at a temperature which decreases rapidly with field, similar to the behaviour along the [1 1 0] field direction.
The isofield measurements (Fig. 5) are similar, at low field, to those obtained along the twofold direction. At low temperature, the differences in magnetization between the 4 and the 6 T curves and between the 13 and 14 T curves reflect the metamagnetic transitions observed in the isothermal results. Above H = 6 T and T = 50 K, the changes of slope seen in the thermal derivative of the magnetization reveal two transitions, as for the [1 10] direction: the higher temperature corresponds to the
161
M Amara et al./Physica B 210 (1995) 157 170
2
,
'
'
i
,
,
,
I
,
,
,
i
,
,
l
i
,
,
,
i
'
'
'
"l
M= 1.85 IXa ~ 1.5 [ [M
=
t-
1.5
~, ~ II
~t f
1.8
/
NdZn
/
q
H//[IO0]
= ~;---'+++~-'+ -+L+~+
. . . .
.~
1.2 0.9 0.6
0.5 ~ 2K~/,~/~
0.5~
42.1K.-" ~+22.1K
+'"
I ~z~/ICM~H]AJo 0
5
10
, 15 H (T)
20
4~
7 25
II
"~'Cfl 0.8 - S T
30
Fig. 6. Magnetization processes along the [001] direction at 2 K (open triangles) and 25 K (filled squares). Provided that the ferromagnetic component of the structure is parallel to the field, its value is determined by the intersection of the magnetization curve with the M = H/AJ line (see text). The inset shows the magnetic behaviour in the thermal range corresponding to the change of the easy magnetization axes. Note the large value of the susceptibility close to TR in phase IIIq.
magnetic-paramagnetic transition and the lower one to a change of structure. Here also, we attribute the inflection point of the isofield magnetization for H larger than 7T in the 30-40K range to an additional transition. Thus the H - T phase diagram exhibits six different phases (Fig. 3) and, except at low temperatures and large fields (i.e. ignoring phase VIIt), the phase diagram for the [1 1 1] direction resembles closely that found for the [1 10] axis.
2.3. Magnetization along the [0 01] axis At low temperature (T < TR) the isothermal magnetization curves (Fig. 6) display two sharp metamagnetic steps at about 4.5 and 9 T. Measurements up to 27T do not give evidence of any additional metamagnetic transition for this temperature range. Above TR, only one metamagnetic transition can be observed in a field which monotonously decreases from 5.5 to 0 T between TR and TN. The inset of Fig. 6 shows that close to TR, at say 22K, the susceptibility is larger than at higher temperatures. This is due to the fact that at the crossing point TR of the free energies for the (1 1 0) and (1 1 1) magnetization directions, the anisot-
0.6 0.4
..
•
~4T
0.2 ~ , p 3 T ~ m _ 0
20
.
~
_ -+'~ 40
T (K)
60
80
100
Fig. 7. Thermal dependence of the magnetization for different values of the field applied along the [001] direction.
ropy is smaller and thus the structure is easier to distort than at high temperature. From the isofield measurements (Fig. 7), the firstorder transition temperature, TR, between the two spontaneous phases, exhibits only a small field dependence. At high temperatures, above H = 3 T, the temperature derivative of the isofield magnetization indicates an additional transition close to the antiferromagnetic one. When increasing the field, this transition is shifted toward lower temperatures and corresponds to the metamagnetic transition observed above TR in the isothermal processes. For fields higher than 6 T, only one sharp transition in the vicinity of TR is observed within the ordered range. The magnetic phase diagram shows five different phases (Fig. 3). It is reminiscent of the [1 1 1] diagram, except that there exists only one phase at high fields and temperatures. 3. Neutron diffraction experiments
All the neutron diffraction experiments were carried out on the DN3 spectrometer (double axis with
162
M. Amara et al./Physica B 210 (1995) 157-170
a moving up counter), at the C.E.N. Grenoble reactor Silo6. Depending on the cryomagnet, magnetic fields below 6 or 10T were available at variable temperature. Two samples were used for the measurements: the previous spherical one and a wedge-shaped sample coming from the same Bridgman ingot. Two different neutron wavelengths were used: 2 = 1.29 and 0.993 A, the respective 2/2 harmonic contributions to the intensities were evaluated from the nuclear [m/2 n/2 p/2] reflections to be of 1% and 0.33% and included in the computed intensities. In order to determine the magnetic structure, numerous integrated intensities were collected in each of the accessible phases. The structures were then determined with some specially developed software. As no other propagation vectors except the ( 1 / 2 0 0 ) branches, associated for the field-induced structures to the 1-000] ferromagnetic propagation vector, have been observed, the procedure starts from one of the high-symmetry structures indexed in I (i.e. a model structure with moments of constant amplitude frozen along only one family of high symmetry directions). For fieldinduced structures, the number of possible models can, however, be reduced by comparing the ferromagnetic component of the model to the magnetization measured along the field. In practice, this comparison is difficult to perform because the real magnetic structure has a nonzero susceptibility and is then distorted with regard to the rigid model. However, as shown in I, when considering CEF anisotropy and isotropic bilinear exchange within the molecular field approximation, the real structure is identical to the model at a particular value of the field: H* = (J(k) - J(0))Mo -- AJ M0,
(1)
where J(q) is the Fourier transform of the exchange interaction, Mo the ferromagnetic component of the structure and k belongs to the star of the antiferromagnetic propagation vectors. Provided that the magnetization, i.e. the ferromagnetic Fourier component, is parallel to the field, this relation determines Mo from the intersection of the line H / A J and the isothermal magnetization curve (see Fig. 1, for example). In the case of a second order transition toward the antiferromagnetic state, AJ
may be directly determined from the measurement of the reciprocal susceptibility at TN: for NdZn, AJ = 105 +__3 kOe//~B [11]. As the neutron diffraction measurements are also carried out on a distorted structure, a distortion-parameter describing the change in the ferromagnetic component is introduced in the refinements. For each tested model, the procedure determines the different magnetic domains which must be considered with their relative proportions as additional parameters for minimizing the reliability factor 1,14]: l
[-/talc
--
I ....
7
where Pi is the inverse of the statistical error of the intensity (lmeas) measured for the ith reflection.
3.1. Field along [ 0 01
]
In order to confirm the proposed spontaneous structures, we have measured the thermal evolution of diffraction peaks associated with the three independent branches of the ( 1 / 2 0 0 ) star in a field of 1.8 T (Fig. 8). This field value was chosen in order to induce possible magnetic domain motions without distorting the magnetic structure. In the 20-60 K range, the three intensities are equal and prove the
200
I
i
i
I
I
I
I
120
.o o
o
I1/2 -1 O] 11 -1/2 01 11 0 1/21
oo o
t_
~
80
~
40
i
NdZn H / / [ 0 0 1] H = 1.8T
,~. 160
~
I
A
A
t II
0
i
0
TR
I
I
10
20
I I
I
30 40 T (K)
50
60
70
Fig. 8. Thermal evolution in a [001] field of 1.8 T of the intensities of three antiferromagneticreflections associated to each branch of the (1/200) propagation star. The two structures correspond to the spontaneous phases I and II.
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M. Amara et al./Physica B 210 (1995) 157-170
single domain character of phase I, which is, under the constraint of the magnetic field, only compatible with a cubic structure. Thus, among the three structures allowed by the powder neutron diffraction results [12], the triple-k (Fig. 8) with moments along threefold axes, labelled tl in I, is confirmed. The fit of the diffracted intensities collected at 25 K and H = 0 to this triple-k model gives an ordered moment of 2.16 p~ with an excellent reliability factor equal to 5.2%. At TR, the three intensities become different revealing the partition of the sample into domains: the structure associated with phase II is then no longer cubic. The sum of the three intensities and thus the magnetic moment increase at TR, which confirms that the easy magnetization axes have changed to the twofold axes in agreement with previous neutron diffraction measurements [13] and the CEF calculations [12]. The structure of phase II is then confirmed to be double-k (dl in I) and the refinement using the intensities collected at T = 3.7K and H = 1.8T gives a reliability factor of 5.4% and an ordered moment of 2.47 Pa- The analysis of the quadrupolar energy of the spontaneous structures may provide us with information on the quadrupolar dispersion curves. From I, these energies are
for phase I'I; S is the amplitude of the magnetic moment, p, and pr describe the coupling of the quadrupolar components to the molecular exchange field. To agree with the determined structures, the quadrupolar dispersion curves should reproduce the following relations:
Kr(0) > K~(½½0),
for phase I and EQ(II) = -
1 4S4( 2 _2 K~(½½0)] ~J(k) ~p, Kr(O) + p~ -~ ;
Table 1 Fourier components of the model structures with twofold axes of easy magnetization. The ferromagnetic and M1 and M2 antiferromagnetic components correspond to a moment of unit modulus on each site. Note that there is no model with M0 parallel to a threefold axis. The last three columns give the magnetization (#B/at.) parallel to the fourfold, twofold and threefold direction, for the 2.5/~B moment observed at low temperature. /140
Ml
1
[0 0 l/x/'2 ]
[ l/x/2 0 0]
2 3 4 5
[1/2~f2 1/2x/'2 l/x/2 ] I-0 1/2,~f2 1/2,~f2] [0 0 l/x/2 ] [0 1/2x/2 l/2xf2 ]
[1/2x/'2 - 1/2x/2 0] [l/x/2 -- 1/2v/2 1/2,J2] [ - 1/2x/2 1/2x/'2 0] [1/x~ 0 0 ]
(3)
for the tetragonal and trigonal symmetries. When increasing the magnetic field up to 5.7 T at T = 2K, the intensity of the [1/200] reflection suddenly increases from zero at the same critical field as the jump in magnetization. Note that several Fourier components belonging to different domains can be propagated by the same [1/200] vector and contribute to the [1/200] intensity. Thus, in phase IIIq, at least one of them has rotated away from the [100] axis. In order to determine the structure of phase IIIq, we use the field-induced models with twofold axes of easy magnetization shown in Table 1 (Tables 7 and 8 of I). The indicated magnetization might be expected to be parallel to the applied field along the main crystallographic directions. From the [001] isothermal magnetization curves (Fig. 6), the value corresponding to a rigid model is 1.1 #B. There is no structure in Table 1 that gives a magnetization in agreement with this, which indicates that the ferromagnetic component is in fact not parallel to the field applied along the [001] axis. We thus conclude that there is no significant rotation or closure of the structure toward the applied field direction due to a huge anisotropy. The closest values in Table 1 to the observed value are for structures
11 1 4 p~2s4Ke(LiLiO) ~J(k) 3
EQ(I) = --
K , ( 1: 1: 0 ) > K~(0)
M2
M[oo 11
M[o I 11
Mr1 ~ ~1
[1/2x/2 1/2x/'2 0] [0 - 1/2x/2 l/2xf2]
1.77 1.77 0.8g 1.77 0.88
1.25 1.98 1.25 1.25 1.25
1.02 2.04 1.02 1.02 1.02
164
M. Amara et al./Physica B 210 (1995) 157 170
with a ferromagnetic component directed along a twofold axis (lines 3 and 5). According to the nomenclature of I, there are two double-k models, H d l 0 and H d l 1, and five triple-k ones, Ht7, Ht8, Ht9, H t l 0 and H t l l , to test. The best agreement between the intensities measured at H = 5.7 T and T = 2 K and those calculated for each model is obtained for the structure defined by: Mo = [0 - 0.86 0.94]/~B/at, M1 = [1.73 00] ktB/at (propagated by k~ = [1/2 0 0]), M2 = [0 0.86 0.86] ~B/at (propagated by k2 = [0 1/20]), which is derived from the Ht7 rigid model shown in Fig. 9. The refinement was carried out taking account the partition of the sample into height domains. The reliability factor is equal to 7.7% and the field-component of Mo to 0.94/~B, which is in close agreement with the magnetization measured at 5.7 T (see Fig. 6). The diffracted intensities as functions of temperature at H = 4 T have confirmed the sharp thermal range of phase IVq in low field just below TN.
•
160 L .
However, the set of intensities available for this phase is not sufficiently large to do a meaningful numerical refinement of its structure. Fortunately, the structure can be determined from qualitative considerations alone. Firstly, the intensities measured for [h k 1/2] reflections are null. Second, the fact that the intensities of [1/200] and [0 1/20] reflections are negligible indicates that the antiferromagnetic components are parallel to their ( 1 / 2 0 0 ) propagating vectors: among the models with threefold easy magnetization axes, the only possible structures are related to lines 2 (Hd8 and Hd9) and 3 (Htl, Ht2, Ht3 and Ht4) in Table 2. The magnetization measured at T = 25 K has a value of about 1.6 #B larger than the values in Table 2: as in phase IIIq, the ferromagnetic component of the structure is not collinear to the field in phase IVq. Thus models from line 3 have to be rejected and the retained model, derived from line 2, Hd8, is drawn in Fig. 10(a) and is defined by: M0 = [0 l/x/3 l/x/3 ] #n/at, M~ = [ 1/x/3 0 0] #B/at (parallel to kl = [ 1/2 0 0]). The unfavourable orientation of the ferromagnetic component, perpendicular to the field direction in
N d Z n , phase lllq T = 2 K
. . . . . . .
•
Measured
[]
Com uted
120
o
Fig. 9. The field-inducedmodel with moments along twofoldaxes of easy magnetizationin phase IIIq. The diagram shows both the computed and measured diffractedintensities at T = 2 K and H = 5.7 T, the reliabilityfactor being equal to 7.7%.
165
M. Amara et al./Physica B 210 (1995) 157-170
Table 2 Fourier components of the model structures with threefold axes of easy magnetization. The M0 ferromagnetic and M~ and M2 antiferromagnetic components correspond to a moment of unit modulus on each site. Note there is no model with Mo parallel to a threefold axis. The last three columns give the magnetization (gB/at) parallel to the fourfold, twofold and threefold direction for the moment values observed at 25 K (2.25 #B/at) and at 40 K (2.1/in/at).
l 2 3
M0
M1
[00 l/x/3 ] [0 l/x/3 l/x/'3 ] [00 l/x/3 ]
[ 1 / ~ 1/x~0 ] [l/x~00 ] [ l/x/3 0 0]
M2
M[001](25 K)
M[o 111(40K)
MI~ 1q(40 K)
[0 l/xfl30]
1.3 1.3 1.3
0.86 1.71 0.86
0.7 1.34 0.7
that the ferromagnetic component is already parallel to the fourfold direction in 10 T. As explained in I, the quadrupolar energies associated with lines 1 and 4 would respectively read as E Q ( V I ) = _ ~ j ( k ) 4 S 4 pZK,(O ) + p2 a
b
c
Fig. 10. (a) Field-induced model with moments along threefold axes proposed for phase IVq (b) (c) Field-induced structure with twofold easy magnetization axes in phases Vq and VIIt.
the domain associated with the [001/2] propagation vector, explains why this domain does not exist under the field. The actual structure has a large magnetization at 25 K because, in the vicinity of TR, it is easy to distort with respect to the high symmetry structure. This agrees with the magnetization measurements close to the point where there is a change of easy axes (inset of Fig. 6). No neutron diffraction experiment could be carried out in phase Vq since the sample fractured at 8 T, in phase IIIq, when increasing the field to enter phase Vq. (Note that the fracture due to torque effects is a direct consequence of the non collinearity of the ferromagnetic component and the field.) The only available information comes from the magnetization measurements. The determination of the ferromagnetic component of the model structure (Fig. 6) gives a value of 1.85 #B close to the 1.77 #a obtained for lines 1, 2 and 4 of Table 1. Such a good agreement would be surprising for a structure based on line 2, as the ferromagnetic component is 35 ° out of the field direction and then, cannot strictly obey Eq. 1. Owing also to the absence of any additional transition up to 27 T, it is then likely
1
Eo(V4) = - 2 J(k)4 $4
fp X,(o) + 3K,(1½0) 4 k
As K ~ (11 ~ 0 ) < gr(0)(Eq. 3), the set of Fourier components of line 1 is the most likely. The only difference between models Hd5, Hd6 and Hal7 of line 1 is the way in which the ( 1 / 2 0 0 ) propagation vector is associated with the antiferromagnetic Fourier component (see Table 8-third line-in I). To be consistent with the structures determined in phases II and IIIq, which are described with antiferromagnetic Fourier components parallel to the [,1/200] and [,0 1/20] propagation vectors, it is logical to choose the model Hd5 shown in Fig. 10(b) and defined by Mo = [00 l/x/2 ]/aB/at,
MI = [-l / x / 2 0 0 ] kta/at (parallel to kl = [-1/200]). 3.2. Field along El-1 O] We now report on the integrated intensities collected in Phase IIIb at T = 4 K and H = 7.9 T. At low temperatures the measured magnetization, 1.4#n (Fig. 1), allows us to exclude solutions
166
M. Amara et al./Physica B 210 (1995) 157 170
I 1200
T=4K H = 7.9 T / / [ 1 - 1 0
I
~
t
I
[]
Computed I
800
•~
400
0
Fig. 11. The field-induced model with moments along twofold axes in phase lllb. Figure shows both the computed and measured diffracted intensities at T = 1.5 K and H = 7.9T. the reliability factor being equal to 10.1%. associated with line 2 in Table 1. After testing the 12 remaining models by the same technique as in Section 3.2, the retained structure is the triple-k model Ht7 in line 5, drawn in Fig. 11 and defined by Mo = [1.02 - 1.02 0] #B/at, M1 = [0 0 1.74] #B/at (propagated by kl = [-0 0 1/2]), M2 = [ - 0.87 - 0.87 0] #B/at
(propagated by k2 = [0 1/2 0]). The refinement (Fig. 11) is carried out over two domains and yields a reliability factor equal to 10.1%. The deduced magnetization is 1.38 #B, in agreement with the 1.3 #B value measured at T = 1.5 K and H = 7.9 T. The value obtained for the ordered moment is 2.57 #B, a value slightly larger than the 2.47/~B spontaneous value. This difference can be understood in terms of a more favourable quadrupolar arrangement in phase IIIb than in phase II, which is supported by the low value of the critical field separating these two phases. The Ht7 model for phase IIIb is then identical to the one determined in phase IIIq (the Fourier components and the associated propagation vec-
tors in Sections 3.1 and 3.2 are related to two different domains of this Ht7 model). In addition to the fact that the magnetization is nonparallel to the field, the magnetic structure in phase III is insensitive to the field direction. Beyond the large anisotropy in favour of the twofold axes, this behaviour also indicates two-ion effects which cannot be related to the isotropic bilinear exchange and which are likely of quadrupolar origin. Integrated intensities have also been collected in phase VIb at T = 4 0 K in a field of 7.9 T. From Fig. 1, the ferromagnetic component of the associated model has (within the hypothesis of collinearity to the field) a value of 1.3/~B which permits us to exclude solutions from line 2 of Table 2. The refinement starting from the 6 remaining models of lines 1 and 3 is not very selective, but seems to determine the model H t l defined by Mo = [1.1400] pB/at, M1 = [0 1.140] pB/at (propagated by kl = [0 0 1/2]), M2 = [00 1.14]/~B/at (propagated by k2 = [00 1/2]), as the most likely (Fig. 12).
M. Amara et al./Physica B 210 (1995) 157-170
167
12oo
•
]
600 "~ 300
!
o
i
.
i"i "!! i i
-2' °
"2
i
Fig. 12. The field-induced model with moments along threefold axes in phase VIb. The diagram shows both computed and measured diffracted intensities at T = 40 K and H = 7.9 T, the reliability factor being equal to 15.3%.
Here also the ferromagnetic component does not point along the field and the deduced [1 0 1] magnetization is 0.81 #B, comparable with the 0.9 #n value measured in 8 T (Fig. 1). However, the reliability factor is too high (15.3%) to be completely confident in this result. Measurement of intensities versus temperature for the same field have confirmed the existence of phase IVb. Integrated intensities have been collected for T = 61 K and H = 7.9 T. Owing to the weak anisotropy of the isothermal magnetization curves at this temperature, the structure may be strongly distorted and the value of the ferromagnetic component a debatable criterion. The best agreement (R -- 9.2%) between measured and collected intensities (Fig. 13) is obtained for the model Hd8 from line 2 in Table 2, defined by
3.3. Field along f l 1 1J
Mo = [0.46 - 0.46 0] #B/at,
Mo = [0.05 0.92 0.92] #B/at,
M1 = [0 0 0.75] #B/at
M1 = [1.7400] #B/at
(propagated by kt = [00 1/2]).
For these measurements, the field has been increased up to H = 8.2 T and two sets of integrated intensities collected at T = 5 K (Phase IIIt) and at T = 40 K, (Phase VIi). Since there exists no model structures with a ferromagnetic component along a threefold direction (see Tables 7 and 8 in I), the isothermal magnetization curves are only of qualitative use in the selection of the models for each phase. At 5 K, the magnetization processes show a first magnetization jump slightly larger than 1 #B. From Table 1, this value is reminiscent of the ones in lines 1, 3, 4 and 5. The refinement (Fig. 14) leads to a reliability factor minimum (11.1%) for the same model Ht7 as in Phases IIIq and IIIb with
(propagated by kl = [1/2 0 0]),
Me = [0 - 0.87 0.87] #B/at This model is then the same as the one proposed for the phase IVq (Section 3.1).
(propagated by k2 = [0 1/2 0]).
168
M. Amara et al./Physica B 210 (1995) 157-170
[ NdZn,phaseIVb 450 IH = 7.9 T=61K T / l [ 1 - 1 0]
~
~
I~/i/[
300 150 "=
o
Fig. 13. The field-induced model with moments along threefold axes in phase IVb. The diagram shows both computed and measured diffracted intensities at T = 61 K and H = 7,9 T, the reliability factor being equal to 9.2%.
800 41,
= 600 t--
L..
400 200
),,,q
- -
on
Fig. 14. The field-induced model with twofold easy magnetization axes in phase lilt. The diagram shows both computed and measured diffracted intensities at T = 5 K and H = 8.2 T, the reliability factor being equal to 11,1%. T h e r e f i n e m e n t is c a r r i e d o u t by t a k i n g a c c o u n t of six d o m a i n s , of w h i c h t w o are f o u n d to r e p r e s e n t 8 7 % of the sample. T h e d e d u c e d m a g n e t i z a t i o n , 1.09/~B, is close to the m e a s u r e d value, 1.15/tB ( T = 1 . 5 K a n d H = 8.2 T). A t T = 40 K, the best a g r e e m e n t b e t w e e n m e a s u r e d a n d c a l c u l a t e d intensities is o b t a i n e d for
m o d e l H t l in line 3 of T a b l e 2 d e f i n e d by Mo = [0 1.08 0] #B/at, M1 = [ 1.08 0 0]/~B/at ( p r o p a g a t e d by kl = [ 1 / 2 0 0 ] ) ,
169
M. Amara et al./Physica B 210 (1995) 157 170
shown in Fig. 10 and defined by
Mz : [-001.08]/aB/at (propagated by k2 = [001/2]).
Alto = [1/2x//2 1/2x/~ 1/x/~]/aB/at,
This model, which leads to three domains in the refinement (Fig. 15), is the same as in phase VI b. The reliability factor (15.7%) is not fully satisfactory and is reminiscent of that found for phase VIb. AS we do not have enough integrated intensities for phase IVt, we are only able, using the argument of an unique structure observed for phases IIIq/b/t and for phases VIwt, to suggest tentatively that the Hd8 model determined in phases IVq and IVb is again applicable for the present case of phase IVt. Phase VIIt is not accessible by any of the available neutron diffraction cryomagnets. However, its structure is identified from Table 1 owing to the low temperature twofold axes of easy magnetization. The ferromagnetic Fourier component, not parallel to the [111] applied field, has to be large in order to give an important projection on the [-111] field direction as experimentally observed in Fig. 4 (about 2 #B). A model with a [1 1 2] direction for this ferromagnetic component, as in line 2, is then likely. Among the two corresponding double-k models, Hdl and Hd2, the former appears more in light of the field-sequence of phases 1I and lilt. It is
/ 4so1~
M, = [1/2x/~ - 1/2w/2 0] #,/at (propagated by ka = [01/20]).
4. Conclusion The present work has determined the H - T phase diagrams of the compound NdZn for the main crystallographic directions of the applied field below H = 16 T. At low temperature the isothermal magnetization curves show multistep processes that are rather common for uniaxial systems, but quite unusual for cubic ones. The corresponding lines of the phase diagrams are then well defined. Equivalent is the analysis of the temperature dependence of the magnetization, although some smooth thermal evolutions are less easy to interpret. For the three field directions, the change of easy magnetization axes is maintained over all the accessible range of the field. This is particularly pronounced for the fourfold direction for which this
t T=4o
= .
[] 7---__
I
ompo, o
I I
00
"~'~ 150
0 i
i
i
~
t
Fig. 15. The field-inducedmodel with moments along threefold axes in phase VIt. Figure shows both the computed and measured diffractedintensitiesat T = 40 K and H = 8.2T, the reliabilityfactor being equal to 15.7%.
170
M. Amara et al./ Physica B 210 (1995) 157-170
change occurs at a nearly constant temperature, equal to TR at zero field. The spontaneous multiaxial structures are here confirmed. For the fieldinduce d phases, the combined use of magnetization processes and neutron diffraction results has permitted the determination of the structures associated with most of the phases revealed by the magnetization measurements. Consideration of the multidomain state in the refinement is essential to the determination of the magnetic structures arising in a cubic lattice. In the absence of neutron diffraction results, the magnetization processes and the expression for the quadrupolar energy of the possible high-symmetry models has allowed us to propose the most likely structures for phases Vq and VIIt. An important result of this study is that most of the magnetic structures are stabilized whatever the field direction is. This is particularly true for the structures associated with the phases IIIq, IIIb and IIIt. Since C E F anisotropy and bilinear isotropic interactions are compatible with numerous magnetic arrangements, such a behaviour has to be related to other couplings. The strength of these quadrupolar couplings appears through the lowtemperature sequence of magnetic structures versus the magnetic field, especially for the fourfold direction. In the sequence (II-IIIq-Vq), the spins always point along different twofold axes, which minimize the C E F energy: phase II results from fourfold planes consisting of a square of four spins pointing to the center, i.e. along ( 1 1 0 ) and stacked along the [ 0 0 1 ] axis (Fig. 8). At 4.5 K, a pair of spins along one side of the square rotates as a block towards the [ 1 0 1 ] and [ - 101] axes to give the structure in phase III, which is not symmetric with respect of the field. At 9.5 T, the second pair of spins undergoes an identical j u m p to give a symmetric structure again. These aspects relate to the multiaxial character of all the observed structures and
give evidence for strong antiferroquadrupolar couplings in NdZn. Although there is no direct way to experimentally determine the dispersion curves for quadrupolar interactions away from the center, of the Brillouin zone, the present study has allowed us to fix the relations: K~(0) > K~(1/2 1/2 0) and g~(1/2 1/2 0) > g~(0) which, by themselves, determine many features of the magnetic order. In a forthcoming paper, we will confirm, by calculations within the periodic field method, that the quadrupolar interactions are indeed essential for the description of the ordered states of NdZn.
References [1] B. Coqblin, The Electronic Structure of Rare-earth Metals and Alloys: The Magnetic Heavy Rare Earths (Academic Press, New York, 1977). [2] P. Morin and D. Schmitt, in: Ferromagnetic Materials, Vol. 5, eds. K.H.J. Buschow and E.P. Wohlfarth (NorthHolland, Amsterdam, 1990) p. 1. [3] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B 17 (1978) 3684. [4] R. A16onard, P. Morin and J. Rouchy, J. Magn. Magn. Mater. 46 (1984) 233. [5] P. Morin, J. Rouchy,K. Yonenobu, A. Yamagishiand M. Date, J. Magn. Magn. Mater. 81 (1989) 247. [6] P. Morin, M. Giraud, P. Burlet and A. Czopnik, J. Magn. Magn. Mater. 68 (1987) 107. [7] M. Wintembergerand R. Chamard-Bois,Acta Crystallogr. A 28 (1972) 341. I-8] J. Rossat-Mignod, in: Neutron Scattering in Condensed Matter Research, eds. K. Sk61d and D.L. Price (Academic Press, New York, 1986). [9] P. Morin and D. Schmitt, J. Magn. Magn. Mater. 21 (1980) 243. [10] M. Amara and P. Morin, Physica B 205 (1995) 379. [11] P. Morin and J. Pierre, Phys. Stat. Sol. (A) 30 (1975) 549. [12] P. Morin and A. de Combarieu, Solid State Commun. 17 (1975) 975. [13] H. Fujii, Y. Uwatoko, K. Motoya, Y. lto and T. Okamoto, J. Magn. Magn. Mater. 63-64 (1987) 114. [14] M. Amara, R.M. Gal/~ra, P. Morin, J. Voiron and P. Burlet, J. Magn. Magn. Mater. 131 (1994) 402.