Saw-tooth peaks in elastic neutron scattering by rhombohedral Heisenberg antiferromagnets

Physica B 156 & 157 (1989) 115-117 North-Holland, Amsterdam

SAW-TOOTH PEAKS IN ELASTIC NEUTRON SCATTERING BY RHOMBOHEDRAL HEISENBERG ANTIFERROMAGNETS

E. RASTELLI

and A. TASS1

Dipartimento di Fisica dell’

lJniversitaa' , 43100 Parma, Italy

The phenomenology of the Heisenberg rhombohedral antiferromagnet (RAF) in classical approximation is characterized by the frustration of the minimum energy configuration consisting of infinite inequivalent helices. In absence of anisotropy long range order (LRO) is lost but a weak single ion anisotropy is sufficient to restore LRO and to produce unusual elastic neutron scattering profiles as Bragg lines for monocrystalline samples and saw-tooth peaks for policrystalline samples. The effect of quantum and thermal fluctuations as well as of a magnetic field is also considered.

The Heisenberg rhombohedral antiferromagnet (RAF) was studied in view of explaining the magnetic properties of solid oxygen in P-phase, where the 0, molecules are localized on the sites of a rhombohedral lattice and interact via direct nearest neighbour (NN) antiferromagnetic exchange [l]. An in-plane 120” three-sublattice short range order (SRO) was extensively assumed to explain the broad peak observed in neutron scattering experiments [2-41 on policrystalline samples at low energy transfer. The current opinion is that no long range order (LRO) is present even if the peak intensity is of the same order of magnitude as the elastic peaks observed in the low-temperature ordered CYphase [2]. We have studied the properties of the RAF model in classical approximation and we have found that previous hypothesis about the minimum energy configuration as an in-plane 120” three-sublattice configuration is wrong. Such a configuration indeed does not minimize the energy of the model and it is not able to support well-defined spin waves [5]. The Hamiltonian we consider reads

NN, respectively, D, is the planar single ion anisotropy strength. The x-axis of our reference frame is along an in-plane NNN row and z-axis is along the c-axis. We have looked for helical configurations and we have found that the classical energy of Hamiltonian (1) for 1j[ < 3, where j = J*/J,, is minimized by infinite inequivalent helices given by sin z0 = 2 sin(x,/3)[cos

(24

cos z. = -[2 cos (x,/3) cos y, + cos (2x,/3)] lj , (2b)

where x0 = fiaQJ2,

(3a)

aQJ2,

(3b)

Y, =

z. = cQJ3,

(3c)

a is the in-plane

lattice constant and c is three times the interplane distance, respectively. In the limit of vanishing interplane coupling j the solutions of eqs. (2) reads x0 = -j sin z. + ( j2/6) sin 22, -(j3/9)sinz0 y, = 2n/3 + (j/a)

where cy runs over 1,2, J, is the in-plane NN antiferromagnetic coupling, J2 is the out-of-plane NN coupling. Si is the spin of the site i, S, and S, joins the site i with its in-plane and out-of-plane

y, - cos(x,/3)] lj ,

+ [ j”/(Sti)]

+. . . , cos z. + [ j2/(6fi)] cos z. + . . . ,

z. arbitrary .

We refer to this frustrated

0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(4a) cos 22, (4b) (44

helix as a degenerate

116

E. Rastelli

and A.

Tassi

I Saw-tooth

helix (DH). The locus JZQ in the reciprocal space, solution of eqs. (2), that we call degeneration lines, represents also a set of soft lines in absence of anisotropy, because the magnon energy spectrum E, vanishes for any k E .TQ. This causes a low energy catastrophe in thermal average of the occupation number that destroys LRO at any finite temperature. If D, # 0, the degeneration lines are still present but the soft lines are lifted, so that LRO is restored. Let us consider the elastic neutron scattering profile of a system represented by the classical RAF model. For a monocrystalline sample instead of the customary a-shaped Bragg peaks one should observe Bragg lines, so that the elastic neutron scattering profile should appear very similar to the diffuse neutron scattering profile of a 2D compound with SRO. We have also accounted for the response of a polycrystalline sample [6] and we have found a saw-tooth peak profile of the elastic neutron scattering that reads

x

(,+Qz-QL 1

6(k-Q),

Q'

where Q, = 4~/(3a), Q, = 2Q,, Q3 = fiQ,, , and g, = 6, g, = 6, g, = 12, . . . . The exponential factor approximates the Debye-Waller factor. As one can see the neutron scattering profile given by eq. (5) is saw-tooth like with peaks in correspondence of Qi. The peak heights become finite if one accounts for the limited instrumental resolution [6]. We stress that this is the first example of broad peaks in elastic neutron scattering. We remember that a polycrystalline sample of a “normal” helimagnet with LRO should give S-peaks in correspondence of a magnitude of the scattering wave vector equal to ]Q + ~1 where Q is the helix wave vector and 7 is a reciprocal lattice wave vector [7]. The difference between the elastic response of a degenerate helix and of a normal helix is dramatic even if no monocrystalline samples are available and one could easily distinguish between them. The situa-

peaks

in neutron

scattering

by RAF

tion is much less promising if LRO is not present and one has to examine the diffuse neutron scattering: we have found [8], indeed, that only a careful investigation of the peak profile could distinguish between a DH configuration and a stacking of layers without interlayer correlation and with 120” three-sublattice in-plane SRO. It is an interesting fact that the P-phase of solid oxygen, stable in the temperature range 24-44 K [ 11, corresponds to a rhombohedral Heisenberg antiferromagnet. Inelastic neutron scattering at zero energy transfer by a polycrystalline sample of p-oxygen [2] shows a broad peak around Q = 1.3 A-’ which, if elastic, should be an evidence of the DH configuration. We notice that the intensity of this peak is of the order of the elastic peak observed in the lowtemperature monoclinic a-phase where two sublattice antiferromagnetic LRO exists. Unfortunately LRO seems excluded in P-phase on the basis of triple-axis polarized neutron scattering experiment [9], so that it would be very difficult to establish whether the SRO is DH-like or in-plane 120“ three-sublattice-like. On the other hand we notice that the diffuse scattering peak profile is temperature independent [2] over the whole range 24-44 K where p-phase is stable, what seems a little surprising in a paramagnetic phase. Anyway our theoretical results are able to explain possible deviations from a 120” order [4] as one can see from eqs. (2)-(4), at least from a qualitative point of view. In this connection we remember that our model accounts for magnetic degrees of freedom but neglects any coupling with elastic degrees of freedom and it is well known that the magneto-elastic interaction is surely important in solid oxygen because it is responsible for the cy-j? phase transition [8]. In default of a direct experimental test to assess the existence of the DH phenomenology in solid oxygen, we have studied the influence of quantum and thermal fluctuations on the DH phase that we have found in classical approximation. For small interplane coupling the ground state energy reads

EdQ) = -K,(Q)(l where

+ l/S> + A >

(6)

E. Rastelli and A. Tassi ! Saw-tooth peaks in neutron scattering by RAF

&r(Q)

= -6]J,]S2N(1/2

+j*/6)

(7)

is the classical energy of the RAF model with Q E .Y& and the zero point motion energy A reads A = 2]J, ]NS[a, + a2j2 - (a3 + b, cos 3z,)j3 + . . .] .

(8)

We have found [lo] that b, is positive so that the zero point motion chooses .zO= 2n1~/3 with n integer: order is produced by quantum disorder. It is interesting that thermal fluctuations near k = 0 and k = Q E Ze, provide a contribution to the free energy whose leading term is proportional to j2t3, where t = kBT/(4]J1]S), and has the same modulation as A but with opposite sign. At intermediate temperature the classical DH scenario could be recovered. The coefficient of cos 32, in the free energy vanishes at T = 14 K for parameters suitable for P-oxygen. For this set of parameters the simple spin wave approximation provides a transition temperature to a paramagnetic phase of about 60 K. Consider now the effect of a magnetic field H. If the field is directed along the c-axis, the response of the RAF model is similar to that of a normal helix, even if the infinite degeneration is not solved. On the contrary, if the field is applied perpendicular to the c-axis, a particular helix is selected: z,, = 2nn/3 or z0 = (2n + 1)1r/3 depending on whether h = g&H/( 1215, IS) is greater or lesser than h, [ll] where h, = 3/2{0.034j/[2S(l-

j/3)]}1’2 .

(9)

117

In conclusion we stress that the DH scenario found in classical approximation for the RAF model should have an actual chance to occur in real systems and it should not be a mere artefact of the approximation. Although zero point motion destroys the infinite degeneration, thermal fluctuations and (or) a magnetic field perpendicular to the c-axis could restore the classical phenomenology. We stress the opportunity of performing neutron scattering experiment of P-oxygen in presence of a magnetic field to test the sudden shift of the peak location that a magnetic field should tune, as we expect for the critical field (9) corresponding to H,, = 6 T for parameters proper to P-oxygen.

References [l] G.C. De Fotis, Phys. Rev. B 23 (1981) 4714. [2] P.W. Stephens, R.J. Birgeneau, C.F. Majkrzak and G. Shirane, Phys. Rev. B 28 (1983) 452. [3] R.J. Meier and R.B. Helmholdt, Phys. Rev. B 29 (1984) 1387. [4] F. Dunstetter, V.P. Plakhti and J. Schweizer, J Magn. Magn. Mat. 72 (1988) 258. [5] E. Rastelli and A. Tassi, J. Phys. C 19 (1986) L423. (61 E. Rastelli and A. Tassi, J. Appl. Phys. 61 (1987) 4117. [7] S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, vol. 2 (Clarendon, Oxford, 1984). [8] E. Rastelli and A. Tassi, J. Phys. C 21 (1988) 1003. [9] P.W. Stephens and CF. Majkrzak, Phys. Rev. B 33 (1986) 1. [lo] E. Rastelli and A. Tassi, J. Phys. C 21 (1988) L35. [ll] E. Rastelli and A. Tassi, J. Appl. Phys. 63 (1988) 3823.