Two-dimensionally-modulated, magnetic structure of neodymium metal

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Two-dimensionally-modulated, magnetic structure of neodymium metal

B. Lebech, P. Bak

To cite this version:

B. Lebech, P. Bak. Two-dimensionally-modulated, magnetic structure of neodymium metal. Journal de Physique Colloques, 1979, 40 (C5), pp.C5-14-C5-15. �10.1051/jphyscol:1979503�. �jpa-00218875�

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JOURNAL DE PHYSIQUE Colloque C5, suppliment au no 5, Tome 40, Mai 1979, page C5-14

Two-dimensionally-modulated, magnetic structure of neodymium metal

B. Lebech

Physics Department, Risra National Laboratory, DK-4000 Roskilde, Denmark

and P. Bak

Nordita, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

RCsumk. - L'ordre magnCtique incipient dans Nd dhcp est caractCrisC par une structure bi-dimensionelle, modulCe d'une maniQe incommensurable (structure ⁽⁽triple-q D). L'ordre est accompagnk par une distorsion d e rkseau qui forme une configuration semblable.

Abstract. - The incipient magnetic order of dhcp N d is described by a two-dimensional, incommensurably modulated structure ((( triple-q ^)>structure). The ordering is accompanied by a lattice distortion that forms a similar pattern.

The crystal structure of Nd is hexagonal with a stacking sequence ABA'C along the c-axis (2). Moon et al. [l] reported the first neutron scattering data on Nd. They found satellites in the elastic spectrum displaced from reciprocal lattice points (z) _by vectors f q,, k ⁼1,2, 3 in the three equivalent 6 , direc- tions. They proposed a model in which only the spins in the B and C layers order at T,. This model predicts zero intensities of satellites on -the 8,-axes

TEMPERATURE (K)

Fig. 1. - A) Temperature dependences of the neutron intensities of the structural and the magnetic satellites. The solid lines are the results of fits to power laws. The hatched area corresponds to a 25 %variation of B'. B) The full width at half maximum (FWHM) of the satellites. The open circles show the (q, 0, 3) intensity and width in the critical regime [2].

and identical intensities of satellites at z _+ q,. Small, but finite intensities were observed on the 8,-axes, and different intensities were observed for satellites at z f qk' Our neutron diffraction measurements, performed at 10 K on single crystals of Nd using the steady-state reactor DR3 at Riso, agree with the results described by Moon et al. In addition, we studied the temperature dependence of selected satellites near TN.

From the temperature dependences of { q, 0, I } satellite intensities, we conclude that the transition at TN is of second order (figure 1). According to one of the Landau rules, the symmetry-breaking order parameter should then transform as an irreducible representation of the space group of the paramagnetic phase. The star of q, consists of the six equivalent basal plane vectors f q,, where q, varies from -- 0.144 at T, = 19.9 K to q, 1.118 at T -- 7.5 K.

The point group (C,,) that leaves q, invariant has four one-dimensional representations, so the order parameter has n = 6 x 1 components denoted

The most general structure described by the order parameters M, and a, is linear combinations of the terms

where the parameters p,, p,, p, and 0 must be determined experimentally from a neutron diffraction

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979503

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TWO-DIMENSIONALLY-MODULATED, MAGNETIC STRUCTURE OF NEODYMIUM METAL C5-15

study. In the Landau theory, the free energy, F, is expanded in terms of the components of the order parameter. By minimizing F, we find [2] that the phase diagram consists of three different regions, one region of first-order transition (111) and two regions (I and 11) of second-order transition with different ordered states. In region I, the ordered state has either MI, M2 or M, # 0 (cc single-q ⁾⁾structure).

By choosing p, = p, = 0 and 8 = 180°, we obtain the structure of Moon et al. In region I1 the ordered state has M I = M2 = M, # 0, i.e., the three different equivalent q-vectors are present simultaneously

(a triple-q ⁾⁾structure). In the critical region, Landau theory is insufficient, and near the phase transition the expansion coefficients of F should be replaced by the renormalized values that should converge to a stable fixed point at a second-order transition.

Since the only stable fixed point is in region 11, and we have found a second-order transition experimentally, we conclude [2] that the ordered state must have the cc triple-q ⁾⁾ structure. The most general cc tripleq)) structure is a superposition of the single-q)) components in (I), i.e.

which only depend on a, through a =

1

a,. The

k

sixth-order term in F fixes a to either 0° or 900.

The magnetic neutron diffraction cross sections for a pair of satellites at z

+

q, are identical for the cr single-q ⁾⁾and the ⁽⁽triple-q n structures. To dis- tinguish experimentally between the c( single-q )) and the cc triple-q )) structures, it is necessary to consider the coupling to the lattice [2]. The spin-lattice coupling introduces two additional terms in the free energy expansion ; firstly, the usual magnetostriction with wave vector 2 q,, and secondly lattice distortions that are 900 out of phase with the magnetization and with periodicities given by G. The latter term involves coupling to two different components $(q,) and t+b(q2) of the order parameter, thus it is active only in the cc triple-q ⁾⁾state. To visualize the resulting structure, we have plotted the basal plane A-site spins on top of an exaggerated distorted A-site lattice (figure 2).

The distortion gives rise to satellites in the diffraction spectrum which generally coincide with the magnetic satellites. However, while the magnetic satellites { h

+

^{q, 0, 0 )}vanish, the corresponding structural satellites are allowed. The existence of ( h f q, 0,O ) satellites is thus direct evidence of the ⁽⁽triple-q ⁾⁾ structure. As the lattice distortion (~(q,)) couples to second order in the primary order parameter, its temperature dependence in the critical

Fig. 2. -The magnetic structure of neodymium for a = 00. The basal plane components of the spins (7) ^{in the}A layers are centered at the atomic positions of the distorted lattice (thin lines).

region should be

I

u(q,)

I

a (TN - T)~', in contrast to that of the primary order parameter

I

^$('I,)

I

^(TN^-^T)'

^.

We compared the measured temperature dependences of the (1

-

q, 0,O) and the (q, 0, 3) satellites (figure 1A). The Ntel temperature was estimated to be (19.9

+

^0.1)^Kboth from the peak in the elastic scattering away from the (q, 0,3) Bragg point and from the divergence in the width of the satellites close to TN (figure 1B). The critical scattering, which was subtracted from the (q, 0, 3) intensity below TN, was estimated from the (q, 0,3) intensity above TN using the scaling laws. After a few iterations,

p

converged to

P

⁼(0.36 f 0.02), in agreement with theory [2]. The (1 - q, 0, 0) intensity is consis- tent with j?' N 1

+

^0.3

-

²

^p,

thus confirming that the peak arises from a second-order coupling to the order parameter that only exists for the ⁽⁽triple-q ⁾⁾ structure. The possibility of multiple scattering effects involving two magnetic satellites was ruled out by experimental tests and calculation.

The parameters giving the magnitude and the phase of the spins in the different layers and the corresponding lattice distortions were determined from the intensities at 10 K of

-

40 independent satellites. For the common phase cc = 0°, we found the maximum possible moments of the (c triple-q ⁾⁾ structure to be pMh 2.36 p,, pMc 0.4 p,, P M ~ 1.08 p,, and 8

-

1800. For a = 900, we found PMI,

-

2.54 PB, P M ~ 0.3 PB, pMz 0.78 p,, and 6 1800.

The associated lattice distortions are of the order of a few per cent. The data suggest that the distortions are confined to the basal planes and restricted to distortions of either the hexagonal or of the cubic sites.

References

[I] MOON, R. M., CABLE, J. W. and KOEHLER, W. C., J. Appl. [2] BAK, P. and LEBECH, B., Phys. Rev. Lett. 40 (1978) 800 and

Phys. 33 (1964) 1041. references therein.