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Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

E-m

JRC@ill.

Pleas

journal homepage: www.elsevier.com/locate/jmmm

Low-temperature magnetic structures in the DySi FeB-type compound27.03.09

P. Schobinger-Papamantellos a,�, K.H.J. Buschow b, J. Rodrıguez-Carvajal c, C. Ritter c

a Laboratory of Crystallography, ETH-Zurich, 8093 Zurich, Switzerlandb Van der Waals-Zeeman Institute, University of Amsterdam, NL-1018 XE, The Netherlandsc Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, France

a r t i c l e i n f o

Article history:

Received 27 March 2009

PACS:

75.25.+z

75.50.Ee

75.30.Gw

Keywords:

Long-period magnetic structure

Incommensurate magnetic structure

Neutron diffraction

Dysprosium germanium alloys

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.04.032

esponding author. Tel.: +4144 632 3773, fax:

ail addresses: Schobinger@mat.ethz.ch (P.

fr (J. Rodrıguez-Carvajal).

e cite this article as: P. Schobinger-P

a b s t r a c t

The low-temperature magnetic ordering of the dimorphic DySi compound has been studied at 1.5 K by

neutron diffraction on two polycrystalline samples. The samples comprise various amounts of the two

orthorhombic modifications: CrB-type (Cmcm Nr. 63, all atoms at 4c site: (0, y, 14)) and FeB-type (Pnma

Nr. 62, all atoms at 4c site: (x, 14, z)), both order antiferromagnetically (TNE38 K). The CrB-type phase

orders with a uniaxial structure with the wave vector q1 ¼ (0, 0, 12) requiring a doubling of the c-axis. The

Dy moments point along the linear chain with the shortest distance c. At 1.5 K, the ordered moment

value is 8.57(1)mB/Dy atom.

Two symmetry independent wave vectors describe the 1.5 K magnetic ordering of the FeB-type

phase: q2 ¼ (0, 12, 1

6) and q3 ¼ (0, 0.484(1), 0.0892(1)), coexisting in form of domains. In both structures

the magnetic moments are confined to the (0 0 1) plane at an angle of 2(2)1 and 22(3)1 from the shortest

axis b, respectively. Both structures correspond to sine wave modulations. The amplitude of the q2 wave

is mo ¼ 7.5(1)mB/Dy atom and that of q3 8.2(1)mB/Dy atom. The wave vector q2 when referring to the (a,

2b, c) cell and the wave vector q ¼ (0, 0, 16) corresponds to a transversal modulation, which by a proper

origin choice can be also described as an antiphase domain structure with two amplitudes. The

moments point to the b-axis and are stacked in the sequence (+mo/2, �mo/2, �mo, �mo/2, +mo/2, +mo,

y) along the c-direction, while tb acts as an antitranslation. For the q3 phase, the local moment value

depends on the atom position in the wave. We also discuss the case where q3 and q2 act simultaneously

in physical space.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

DySi like all of the heavy rare-earth compounds except TmSi,occurs with two orthorhombic modifications: the CrB (Cmcm) andthe FeB (Pnma) types of structure. According to Hohnke andParthe [1], they have a common structural unit: the rare-earthtrigonal prism, centered by zigzag silicon chains that are stackeddifferently in the two structures. One may obtain correlationsbetween their lattice constants [1]. The CrB phase stabilized byaddition of a small amount of Ge (20%) is reported to orderantiferromagnetically [2] below TN ¼ 43 K with a uniaxial anti-ferromagnetic arrangement associated with a cell doubling in thec-direction as found by neutron diffraction. The Dy magneticmoments point along the c-axis and the ordered moment value is8.8(2)mB/Dy atom similarly to the isomorphic DyGe compound[3]. Magnetic measurements of the FeB-type DySi compoundreveal an ordering temperature of 41 K [4]. More recent magne-tisation data reveal an ordering temperature of 38 K for both DySi

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Schobinger-Papamantellos),

apamantellos, et al., J. Mag

modifications [5] and the magnetic isotherm at 5 K shows a broadmetamagnetic transition in a field of E65 kOe with hysteresiseffects above 20 kOe.

The present investigation focuses on the low-temperaturemagnetic structure at 1.5 K of the high-temperature FeB-typemodification of the DySi compound that has not been reportedpreviously. We report on the crystal and the magnetic structure inthe paramagnetic and the magnetically ordered state, respec-tively, based on neutron powder data. We report results obtainedon two distinct samples having different contents of the twophases. The thermal evolution of the magnetic ordering and thederivation of the individual ordering temperatures of the twomodifications are on the way and will be reported in a futurestudy.

2. Experimental procedures and results

2.1. Sample preparation

The DySi samples were prepared by arc melting of theappropriate amounts of the elements in an atmosphere of purified

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argon gas. The purity of the starting materials was 99.9% fordysprosium and 99.99% for silicon. Standard X-ray diffractionusing CuKa radiation shows that both samples are dimorphic andthat one of the samples (sample II) contains mainly the FeBmodification.

2.2. Neutron diffraction and results

Neutron data are collected on both DySi powder samples in theparamagnetic state at 42 K (sample I), 293 K (sample II) and in themagnetically ordered state at 1.5 K. The data were collected at theILL in Grenoble with the high-flux D1B diffractometer (l ¼ 2.52 A,2y: 0-801, step increment 0.201) for sample I and the high-resolution D1A diffractometer (l ¼ 1.91 A, 2y: 0-1601, stepincrement 0.101) for sample II. In view of the high-absorptioncross-section, a sample holder of 0.5 cm outer diameter was usedand the DySi samples were diluted by polycrystalline Al metalthat was included in the refinements. The data analysis is madewith the Fullprof Suite of programs [6]. The structure plots aremade with the program Fullprof Studio [7] incorporated in Ref. [6].

2.3. The paramagnetic state

The refinements of the data in the paramagnetic state showthat sample I contains equal amounts of the two modifications,while sample II contains 74% of the FeB-type and 26% of the CrB-type.

For sample I, the structural parameters were not refined due tothe limited sin y/l range of the available D1B data. At 42 K:RB ¼ 14.5%, for the CrB modification and 7% for the FeB modifica-tion Rwp ¼ 15%. For the 1.5 K data, Rwp ¼ 13.4% and the magneticreliability factors are 9–12% for both types of structures. Resultsare given in Table 1 and Fig. 1.

For sample II, the refinements of the D1A data also comprisethe structural parameters. At 293 K, the reliability factors areRB ¼ 6%, for the CrB and 5% for the FeB modifications andRwp ¼ 15.8%. For the 1.5 K data, Rwp ¼ 12.7% and the magneticreliability factors are 6.9% for CrB and 7.9% and 10.9% for the twophases found in FeB as given in Sections 2.6.1 and 2.6.2. Resultsare given in Table 2 and in Fig. 2.

2.4. Magnetic refinements

The 1.5 K data shown in Fig. 1 (sample I) and Fig. 2 (sample II)compared to those of the paramagnetic state indicate that themagnetic reflections of both modifications occur at reciprocal

Table 1Refined structural and magnetic parameters from the neutron data of the CrB

modification of samples I and II of the dimorphic DySi compound.

CrB-type content Sample I (50%) D1B data Sample II (26%) D1A data

Parameter 42 K 1.5 K 293 K 1.5 K

a (A) 4.234(2) 4.235(2) 4.2551(3) 4.2472(2)

b (A) 10.424(6) 10.442(5) 10.5019(5) 10.4755(4)

c (A) 3.783(3) 3.787(2) 3.8220(2) 3.8008(1)

yDy 0.1412 0.1414 0.1412(6) 0.1414(7)

ySi 0.428 0.4331 0.428(2) 0.4331(2)

mz (mB) – 7.8(2) – 8.58(2)

RB %, Rm% 14.5, – 9.2, 12.3 6, – 7.3, 6.9

Rwp %, Rexp %, x2 15, 6.6 13.4, 1 15.8, 12 12.7, 4.8

Space group Cmcm (Nr. 63), all atoms at sites 4c: 7(0, y, 14).

mz is the ordered moment value of dysprosium parallel to the c-axis. RB, Rwp, Rm %

and Rexp are the Bragg factor, the reliability factor for the magnetic intensities, the

weighted profile intensities, and the expected value related to the statistical

accuracy of the data.

Please cite this article as: P. Schobinger-Papamantellos, et al., J. Mag

lattice positions different from those of the chemical cell. As willbe shown below, in spite of the different compositions of the twosamples and different experimental conditions (as instrumentalresolution, sin y/l range, etc.) of the two data sets, the refinedmagnetic structures are identical for each modification in the twosamples within experimental error. We will first discuss the CrBordering and then the FeB ordering that is more complex.

2.5. The magnetic structure of the CrB-type modification

The magnetic ordering for the CrB phase is associated with thecommensurate wave vector q1 ¼ (0, 0, 1

2) or alternatively with adoubling of the c-axis in agreement with the results of Ref. [2]. InFig. 1, the CrB calculated magnetic intensity is in red colour. Theabsence of the first magnetic reflection (0, 0, 1

2) suggests that theDy magnetic moments point along the c-axis. The refinements ledto a parallel arrangement of the magnetic moments within thechemical cell and a sign change going from one cell to the next inthe direction of the wave vector. The difference in the refinedmoment value 7.9(2)mB/Dy atom for sample I and 8.58(2)mB/Dyatom for sample II might be connected to the limited 2y range ofthe data available for sample I and the higher absorption due tothe larger wavelength. The latter moment value comparesfavorably with the results of the DySi0.8Ge0.2 compound [2]. Thesevalues fall below the free-ion value (gJJ mB ¼ 10mB for Dy3+),presumably due to crystal field effects.

2.6. The magnetic satellites of the FeB-type DySi phase

The low-temperature magnetic pattern in Fig. 1 is collectedwith a three times larger counting time than the 42 K data forsample I. It comprises next to the dominant magnetic reflectionsof the CrB-type phase ((0, 2, 1

2) at 341 (1, 1, 12) at 42.51 and (1, 3, 1

2) at601) (red colour) a strong double peak around 271 and a largenumber of weak peaks. These peaks are then attributed to the FeBmodification. The first and most dominant reflection at 2y ¼ 27.3(1) is located close to the reciprocal lattice position (1, 1

2, 0) of theFeB structure (see Fig. 1), which suggests a wave vector close tothe (0, 1

2, 0) value. Because of the similarity with the isomorphicTbSi and CeSi [8–11] compounds with two-dimensional wavevectors (0, 1

2, qz), the indexing of the FeB modification isstraightforward.

2.6.1. The FeB wave vector q2 ¼ (0, 12, 1

6)

The best results are obtained using a two-component wavevector q ¼ (0, qy, qz) close to the commensurate value q2 ¼ (0, 1

2, 16).

This wave vector explains the majority of FeB magnetic peaks.However, to our surprise it became evident that the use of a singlecommensurate wave vector q2 ¼ (0, 1

2, 16) does not explain all

magnetic peaks. The indexing of the FeB-type reflections in Fig. 1is given as satellites of the main nuclear reflection (h k l)-q. As asimple example, we refer to the first double satellite (at 27.31 inFig. 1) indexed as (110)-q2, (10 0)-q2, which is not explained bythe qy ¼

12 value as this leads to a single peak. The same holds for

several weak reflections that cannot be explained by q2 and alsofor the corresponding observations of sample II that display thesame peak topology. This can be easily verified by a carefulinspection of the low angle 1.5 K data in Figs. 1 and 2 (in Fig. 2 thefirst FeB peak is located at 181).

2.6.2. The FeB wave vector q3 ¼ (0, qy, qz)

A single wave vector is not sufficient to explain the largenumber of FeB magnetic peaks, which exist in both samples. As anext step, we first checked for the presence of the third harmonic3q2 ¼ (0, 1

2, 12) of the wave vector, q2 ¼ (0, 1

2, 16), and for the wave

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Table 2Refined structural and magnetic parameters from the neutron data of the dimorphic DySi compound.

FeB-type content Sample I (50%), D1B data Sample II (74%), D1A data

Parameter 42 K 1.5 K 293 K 1.5 K

Atom x z x z x z x z

Dy 0.1782 0.1150 0.1776 0.1150 0.1789(4) 0.1147(5) 0.1776(4) 0.1150(4)

Si 0.041 0.628 0.0425 0.629 0.0378(12) 0.639(2) 0.0435(1) 0.629(2)

q2 ¼ (0, 1/2, qz) – 0.5, 0.1647(9) – – 0.5, 0.16666

mox, moy (mB) – 1.5(8), 9.3(5) – – 0.2(3), 7.5(1)

j12, j21, j22 (2p) 0.1699, �0.37(2), �0.54(2) 0.0833, �0.401(7), �0.484(7)

moT(mB), jb deg – 9.5(5), 10(5) – – 7.5(1), 2(2)

q3 ¼ (0, q3y, q3z) – 0.4837(7), 0.089(2) 0.4839(2), 0.0892(3)

mox, moy (mB) – 2.1(1.1), 9.0(7) 3.0(4), 7.7(2)

j12, j21, j22 (2p) 0.089, �0.421(3), �0.51(3) 0.089, �0.433(8), �0.522(8)

moT(mB), jb deg – 9.3(5), 13(7) 8.2(1), 22(3)

a (A), b (A) 7.825(5) 3.788(3) 7.838(3) 3.790(2) 7.8712(2) 3.8205(1) 7.8585(1) 3.7994(1)

c (A) 5.638(4) 5.643(3) 5.6744(1) 5.6614(1)

RB, Rm2, Rm3 % 7, – 3.7, 10.1, 11.1 5.6, – 6.0, 7.9, 10.9

Rwp %, Rexp % 15, 6 13.4, 1 15.8, 12 12.7, 4.7

Space group Pnma (Nr. 62), all atoms at sites 4c: (x, 14, z).

mox, moy are the components of the refined Fourier coefficient parallel to the a, b axes. moT and jb are the amplitude of the wave and its angle with the b-axis. j12, j21 and

j22 are the phases for the sites Dy12, (�x+12, y+1

2, z+12) Dy21(�x,�y,�z) and Dy22 (x+1

2,�y+12�z+1

2) in fractions of 2p. RB, Rwp, Rm2, Rm3% and Rexp are the Bragg factor, the reliability

factors, the weighted profile intensities, for the q2, q3 magnetic intensities and the expected value related to the statistical accuracy of the data.

Fig. 1. Observed and calculated neutron diffraction patterns of the dimorphic DySi compound sample I: (a) in the paramagnetic state at 42 K (b) in the magnetically ordered

state at 1.5 K with q ¼ (0, 0, 12) for the CrB-type (Cmcm) red colour, and q2 ¼ (0, 0.5037(8), 0.1649(9)) blue colour and q3 ¼ (0, 0.4837(7), 0.089(2)) cyan colour for the FeB-

type (Pnma). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]] 3

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Fig. 2. A part of the observed and calculated neutron diffraction patterns of the dimorphic DySi compound sample II: (a) in the paramagnetic state at 293 K (b) in the

magnetically ordered state at 1.5 K with q ¼ (0, 0, 12) green colour for the CrB-type (Cmcm), and q2 ¼ (0, 1

2, 16) blue colour, q3 ¼ (0, 4839(2), 0.0892(3)) in red colour for the

FeB-type (Pnma). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]]4

vector (0, 12, 0). Several efforts to include these sets of reflections in

our calculations failed and led to rather unstable refinements.Finally, using the profile match program tool for the 1.5 K data ofboth samples we detected a second wave vector deviating fromthe commensurate value in both the y- and z-directions q3 ¼ (0,q3y, q3z). The best fit was achieved for the refined valuesq3y ¼ 0.484(1)E12

25 and q3z ¼ 0.0892(1)E 111 at 1.5 K that are within

error the same for both samples.The wave vectors q2, q3 are not symmetry-related. They

describe two independent magnetic systems as they give rise totwo distinct reflection sets. The various models developed for theq2 and q3 structures and used in the magnetic structurerefinements are based on the restrictions provided by symmetryanalysis in terms of irreducible representations and their linearcombinations for the wave vector q ¼ (0, qy, qz) and space groupPnma as given below (see Table 3).

The low-temperature refinements consist of at least sixoverlapping patterns comprising the structural and the magneticcontributions of the CrB, and FeB DySi modifications with thewave vectors: q1 ¼ (0, 0, 1

2), and (q2 ¼ (0, 12, 1

6) and q3E (0, 1125, 1

11)),respectively, and of the Al powder (added to reduce the sampleabsorption). For this reason in the low-temperature plots, weinclude next to the standard Iobs and Icalc profiles the calculated

Please cite this article as: P. Schobinger-Papamantellos, et al., J. Mag

magnetic profile intensities of the CrB magnetic phase q1 (redcurve). The profiles of the FeB phases for q2 (blue dotted curve)and q3 (cyan dotted curve) are shifted vertically in Fig. 1 (sampleI). Similar presentation is used for the magnetic data of sample IIin Fig. 2. The refined parameters corresponding to the bestmodel(s) are summarised in Tables 1 and 2, respectively.

2.6.3. Symmetry analysis

The magnetic ordering of DySi described by the propagationvector

q ¼ ð0;qy;qzÞwith qy �12 and qz ¼

16 for q2 and qy �

12qz � 0:09 for q3

The components refer to the reciprocal frame of the Pnma spacegroup. The star of the wave vector has four arms and q is notequivalent to �q

q1 ¼ ð0;qy;qzÞ; q2 ¼ ð0;�qy;�qzÞ; q3 ¼ ð0;�qy; qzÞ; q4 ¼ ð0; qy;�qzÞ.

The magnetic atoms (Dy3+ ions) in the Pnma space group occupythe positions 4c of the low-symmetry site m

Dy1 : ðx; 14; zÞ; Dy2 : ð�xþ 1

2;34; zþ

12Þ; Dy3 : ð�x; 3

4;�zÞ;

Dy4 : ðxþ 12;

14;�zþ 1

2Þ.

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Table 3Irreducible representations and magnetic modes for q ¼ (0, qy, qz) in Pnma.

Irrep (1|0) (mx|12,12,12) x y z

G1 1 o Fx Ay Az o ¼ exp(�pi(qy+qz))

G2 1 �o Ax Fy Fz

Atom

1 1 2 o ¼ Permutation of atoms and ‘‘returning’’ translations

2 2 1[0 11]

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Under the action of the wave vectors q2 ¼ (0, 12, 1

6) and q3 ¼

(0, qy, qz), the 4c site splits into two orbits. Orbit I comprises atoms1 and 2 and orbit II the atoms 3 and 4. The Fourier coefficients andphases of atoms of the same orbit are symmetry–related, whileatoms of different orbits may have various values and orienta-tions.

The point group of q is G0¼ m11 ¼ {1, mx}, which has two one-

dimensional representations. The propagation vector q belongs tothe interior of the Brillouin Zone, thus the small representationsare easily obtained from the representations of the point group ofthe propagation vector. Symmetry analysis has to be done for allarms of the wave vector star separately as these are nonequivalent.

The representations of the space group of q, G ¼ Pn11, areobtained from those of G0 by the expression

DgðhjthÞ ¼ e�2piq th D0g

ðhÞ (1)

The co-set representatives of G are: ~1 ¼ ð1j0;0;0Þ; ~nx ¼

ðmxj12;

12;

12Þ. Due to the particular form of the translations the

products qth are non zero and the matrices Dg are not identical toD0g.

Table 3 gives the small representations for the propagationvector q and the space group Pnma and provides the basisfunctions and signs of the Fourier components of the magneticatoms located at the mirror plane my (Dy1 at the Wyckoff site4c7(x, 1

4, z)), that were used to develop various models in ourrefinements. The symbols F and A given in Table 3 for the basisfunctions are explained below. To obtain the basis functions, wemay apply the projection operator formula (1) for all representa-tions. For the representation G1 this results to two distinctmagnetic modes

PS1x ¼ ð1 � ~1þo � ~nxÞS1x ¼ S1x þo � ~nxS1x ¼ S1x þoðS2xe�piðqyþqzÞÞ

¼ S1x þ S2xe�2piðqyþqzÞ ¼ S1x þ bS2x

¼ Fx ðwith b ¼ e�2piðqyþqzÞÞ (2)

PS1y ¼ ð1 � ~1þo � ~nxÞS1y ¼ S1y þo � ~nxS1y ¼ S1y þoð�S2ye�piðqyþqzÞÞ

¼ S1y � S2ye�2piðqyþqzÞ ¼ S1y � bS2y ¼ Ay (3)

PS1z ¼ ð1 � ~1þo � ~nxÞS1z ¼ S1z þo � ~nxS1z ¼ S1z þoð�S2ze�piðqyþqzÞÞ

¼ S1z � S2ze�2piðqyþqzÞ ¼ S1z � bS2z ¼ Az (4)

The same procedure is applied for obtaining the basis functionsof the representation G2. The Fourier components, for the atoms oforbit I, corresponding to the representation G1 are therefore

S1ðqÞ ¼ ðu;v;wÞ; S2ðqÞ ¼ ðu;�v;�wÞe�2piðqyþqzÞ

For the representation G2 we have

S1ðqÞ ¼ ðu;v;wÞ; S2ðqÞ ¼ ð�u;v;�wÞe�2piðqyþqzÞ

If the coefficients u, v and w are taken as real numbers, theseFourier coefficients, together with the arm q2, define sinusoidalmodulated magnetic structures. The different orientations of thevectors S1(q) and S2(q) means that both representations would

Please cite this article as: P. Schobinger-Papamantellos, et al., J. Mag

allow three-dimensional canted models. For the commensuratecase (0, 1

2, qz) things are in fact more complex as there is only onetwo-dimensional complex representation that allows six para-meters for each orbit. This does not provide parameter restrictionsand limitations to the number of possible models.

2.6.4. The q2 ¼ (0, 12, 1

6) magnetic structure of the FeB-type DySi

phase

In the following, we will discuss in full detail the commensu-rate model. The best fit of the 1.5 K neutron data for the q2 ¼ (0, 1

2,16) phase was obtained for a collinear amplitude-modulatedstructure with the moments within the plane (0 0 1) at an angleof 2(2)1 from the b-axis choosing a global origin shift of 2p/6 oneobtains

S1ðqÞ ¼ ðu;v;0Þ; S2ðqÞ ¼ ðu;v;0Þ

This corresponds to a mixture of the representations G1+G2 or afurther symmetry reduction. Furthermore, the refined amplitudesof the two orbits were restricted to the same value as the datawere not sensitive to these parameters. In the refined structure,the average value of a single magnetic moment component variessinusoidally along the direction of the wave vector (Fig. 3). Themoment of jth atom in the nth unit cell is given by expression

Sjn ¼ zm0j cosð2pq:Rn þjjÞ (5)

where z is a unit vector in the direction of the varying spincomponent, mj

0 the amplitude of the sinusoidal variation (Fouriercoefficient), Rn the position of the origin of the nth cell and jj thephase angle of the jth spin.

At 1.5 K, for sample II where the FeB phase is in majority (D1Ahigh-resolution data), the refined amplitude of the wave (Fouriercoefficient) mj

0¼ 7.5(1)mB/Dy ion is close to the free-ion value

(gJJ mB ¼ 10mB for Dy3+) if one considers that this value is scaled tothe nuclear intensities for the entire sample volume thatcomprises the q3 phase as well. Assuming that the two FeB-typemagnetic phases have equal volumes, the amplitude has to bescaled by O2 that corresponds to 10.6(2)mB/Dy. The reason of thissimplification is that not knowing the relative amount of the twophases the calculations are carried out using the scale factor of theFeB structural refinement.

Fig. 3 shows that the magnetic moment lies within the (a, b)plane at an angle of 2(2)1 with the b-axis. The refined wave vectorvalue at 1.5 K for the two DySi samples is close to thecommensurate value q ¼ (0, 1

2, 16). This suggests alternatively, that

the magnetic structure of DySi can be approximated by a 12 timeslarger commensurate cell (ta, 2tb, 6tc) with 48 Dy atoms, where tb

and 3tc are antitranslations. This can be easily followed in Fig. 3for four cells along the c-direction and two cells along b. Therefinement has converged for the same Dy moment values for thetwo orbits shown by different colours magenta for orbit I (Dy1and Dy2) and cyan for orbit II (Dy3 and Dy4). Furthermore, themoments of orbits II and I are antiferromagneticaly coupled overthe center of symmetry transforming atoms Dy1–Dy3 andDy2–Dy4 (anti-inversion). The labeling of the atoms is the same

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Fig. 3. The FeB-type antiphase domain with two amplitudes magnetic structure of DySi with the wave vector q ¼ (0, 12, 1

6) at 1.5 K for (a, 2b, 4c) cells. The magnetic moments

are confined to the the b-axis, tb and 3tc are antitranslations. The Dy nearest- and next nearest neighbours are located at the corners of a trigonal Dy prism i.e. the six atoms

around Dy10 (Dy2, Dy3, Dy20 and those translated by �tb are antiferromagnetically coupled). This arrangement results to a zero molecular field at Dy10.

P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]]6

as in Fig. 4, where we show another aspect of the structure withregard the Dy coordination polyhedra.

Until now, the existence of the third harmonic 3q ¼ (0, 12, 1

2) inour neutron powder data could not be unambiguously confirmed,because its weak contributions are not discernable from thebackground, or because they overlap with strong nuclear reflec-tions and possibly with other unidentified magnetic peaks. Theantiphase domain structure with two amplitudes shown in Figs. 3and 4 is derived from the refined Fourier coefficients of only thefirst harmonic of the wave vector q2 ¼ (0, 1

2, 16). The collinear

magnetic moment arrangement of the four Dy atoms shown inFigs. 3 and 4 in three successive cells along c and two cells along b,changes sign collectively in the next three cells along c resulting inthis way to a six-fold enlargement of this axis (6tc). The momentlength and sign change can be calculated by expression (5) fromthe refined Fourier coefficient mj

0 and the phases jj that are givenin Table 2. Table 4 gives the local moment values along the c-linear chains for the atoms of orbit I (Dy1 and Dy2). Choosing aglobal origin phase FG ¼ 2p/6, one obtains the distribution for thetotal phase FT

j¼ jj+FG for three translation equivalent atoms

along c. For the linear chain starting at Dy1 (atoms 1, 10 at tc and 100

at 2tc), this corresponds to the arrangement (+m01,�m1

1,�m21) see

Figs. 3 and 4. In the next three cells, the signs are inverted.For Dy1 (n ¼ 1) and jj ¼ 0

m10 ¼ S1ðqÞ cosð2p=6Þ ¼ S1ðqÞ=2 ¼ 3:75mBz (6a)

m11 ¼ S1ðqÞ cosð2p=6þ 2p=6Þ ¼ �S1ðqÞ=2 ¼ �3:75mBz (6b)

m12 ¼ S1ðqÞ cosð4p=6þ 2p=6Þ ¼ �S1ðqÞ ¼ �7:5mBz. (6c)

For Dy2 (n ¼ 2) with jj ¼ 2p/6

m20 ¼ S2ðqÞ cosð2p=6þ 2p=6Þ ¼ �S2ðqÞ=2 ¼ �3:75mBz (6d)

m21 ¼ S2ðqÞ cosðpÞ ¼ �S2ðqÞ ¼ �7:5mBz (6e)

m22 ¼ S2ðqÞ cosð4p=3Þ ¼ �S2ðq2Þ=2 ¼ �3:75mBz (6f)

The chains starting at the atoms of orbit II Dy3 and Dy4 along c

are coupled antiferromagnetically to those of Dy1 and Dy2,respectively (over the center of symmetry). The sign change andmoment value along the c-linear chains starting from any of theDy atoms in the 0th cell is the same (+mo/2, �mo/2, �mo, �mo/2,+mo/2, +mo, y) with mo ¼ 7.5mB/Dy but the chains are shiftedrelatively to each other according to their phase. This arrangementof antiphase domain with two amplitudes might be revisedtowards an equal-moment long-period superstructure once thepresence of the third harmonic would be confirmed by single

Please cite this article as: P. Schobinger-Papamantellos, et al., J. Mag

crystal data. Assuming a 50% content of this phase at 1.5 K, theaverage moment value in this arrangement is 4O2�7.5mB/6 ¼ 7.0mB/Dy3+ atom. As expected, this is lower (due to thepresence of q3), than the value of 8.0mB/Dy3+ at 5 K found bymagnetic measurements in a field of 90 kOe in Ref. [5] for adimorphic sample with a different content of the CrB and FeBphases. The saturation magnetisation, reported in Ref. [5] of9.6mB/Dy3+ is close to the full moment value.

2.6.5. The q3 ¼ (0, 0.48, 0.089) magnetic structure of the FeB-type

DySi phase

The refinement of the q3 FeB magnetic structure led to a sinewave modulated structure as no higher harmonics were observedfor this wave vector. The best fit was obtained for a collinearmodel with Fx and Fy modes within the basis cell which isdescribed by a mixture of representations G1+G2 or a furthersymmetry reduction as can be seen in Table 3.

The main difference to the q2 structure is, (i) that the wavevector q3 is incommensurate with the crystal lattice in twodirections (0, 0.4839(2), 0.0892(3)), which is close to the value (0,1225, 1

11), and (ii) that the magnetic moments deviate from the b-axisby 22(2)1, the refined amplitude of the wave being 8.2mB/Dy3+.Here, we would like to note that the moment deviation of 21 fromthe b-axis found in the q2 structure is negligible within error.

In this structure, the local moment orientation is fixed withinthe plane (0 0 1), presumably through the crystal field anisotropy,but its value derived by expression (5) depends on the position ofthe atom in two directions and may vary between 0 and moj in thedirection of the wave vector. This can be seen in Fig. 5a for a singleDy1, (x, 1

4, z) atom (in yellow colour) for a large number of cells:a�4b�14c. This means that within certain ranges in the crystal,the Dy1 moments can be considered as quasi-disordered, even atlow temperatures. For comparative reasons, we show in Fig. 5b forthe same Dy atom and the same number of cells the momentvariation of the uniaxial commensurate (q2) magnetic structurethat can be described as a two amplitude antiphase domainarrangement in the direction of c. The horizontal axis is a and thevertical axis is b, while c runs out of the plane. The refinedparameters are given in Table 2 together with the parameters ofthe q2 results. As already mentioned the refined parametersobtained for both samples are in good agreement.

2.6.6. The magnetic structure of coherent superposition of q2 and q3

In refinements of multiple q-vectors magnetic structures onemay assume that the various structures coexist in form ofdomains. There is no way to distinguish this case by diffraction

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Fig. 4. The FeB-type modulated magnetic structure of DySi with the wave vector q ¼ (0, 12, 1

6) at 1.5 K shown for (a, b, 4c) cells. The magnetic moments are confined to the (a,

b, 0) plane at an angle of 251 away from the b-axis. The Dy atoms are surrounded by 10 Dy and 7 Si atoms. The molecular field at the central atom is zero.

P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]] 7

from the case where all vectors act in the same physical space(coherent superposition). For this reason as a last point we wouldlike to show the structure resulting in DySi under the assumptionthat the q2 and q3 vectors act in the same physical space.

Both q2 and q3 magnetic structures correspond to sine wavestructures. The refined Fourier coefficients at 1.5 K are: 7.5mB forq2 and 8.2mB for q3 see Table 2. The real structure for eachindependent system may be derived by Fourier expansion fromthe refined Fourier coefficients Sn(qj) of the magnetic moments inthe basic unit cell by (7). The Fourier expansion of any periodicmagnetic moment arrangement with the moments at positions

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Rnn ¼ rn+Rn, where Rn ¼ n1a+n2b+n3c is a lattice translation vectorwith (n1, n2, n3) integer for a primitive lattice, rv is the vectorposition of the atom n (n ¼ 1, 4 in our case) in a primitive unit cellcan be expressed by the formula

mnðRnÞ ¼X

fjg

SnðqjÞ expð�2piqj � RnÞ ¼X

j

SRn ðqjÞ cosf2pqj � Rn þjjng

(7)

The first sum is extended for all pairs (qj, �qj) of propagationvectors; in the second one, we have assumed that the Fouriercoefficients have the form: SnðqjÞ ¼ 1=2SR

n ðqjÞe�ijjn , so that only

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cosine terms appear after grouping the pairs (qj, �qj). Only twosets of magnetic satellites (H7qj) of the wave vectors q2 and q3

are observed and no higher harmonics could be detected. Thephases jn(qj) ¼ jjn between atoms of the same orbit are deducedby symmetry analysis from the basis functions of the representa-tion. In commensurate phases these are fixed. The phases betweendifferent orbits are refined. Fig. 6 displays the resulting calculatedFeB-type DySi structure for a�4b�14c cells under theassumption that q2 and q3 act in the same physical space. Inthis structure, the magnetic moments change both in length anddirection within the plane (0 0 1). The Dy3+ local moment values

Table 4

The magnetic moment distribution along the linear chains c starting at the atoms Dy1

Dy1, jj ¼ 0

Cell z FT(2p) m m

0 z 0 mo cos(0) mo

1tc z+1 1/6 mo cos(p/3) mo/2

2tc z+2 1/3 mo cos(2p/3) �mo/2

3tc z+3 1/2 mo cos(p) �mo

4tc z+4 2/3 mo cos(4p/3) �mo/2

5tc z+5 5/6 mo cos(5p/3) mo/2

FT ¼ q �Rj+jj is the total phase depending on the z coordinate, m the local moment valu

been added to the wave (origin shift).

Fig. 5. (a) The variation of the Dy1 (x, 14, z) ordered magnetic moment in the incommen

type) for a large number of cells (a�4b�14c). The moment points always along the sam

horizontal axis is a, the vertical axis is b while c runs out of the plane. (b) The variation

magnetic structure of DySi (FeB-type). Described also as a antiphase domain arrangem

comparison with Fig. 5a, a large number of cells (a�4b�14c) is shown. The horizonta

Please cite this article as: P. Schobinger-Papamantellos, et al., J. Mag

were found to vary between �0–15.2mB/Dy3+. The fact that themaximum moment value is larger to that of the Dy3+ free-ionsuggests that this structure type is unrealistic.

3. Discussion and concluding remarks

The low-temperature FeB-type magnetic structures of DySi aredescribed by two symmetry independent wave vectors q2 ¼ (0, 1

2,qz) with qzE1

6 and q3 ¼ (0, q3y, q3z) with qyE1225 and qzE 1

11.Neglecting the moment deviation of 2(2)1 from the b-axis, the q2

at (x, 14, z) and Dy2 at (�x+1

2, 34, z+1

2) (orbit I) for six cells along c.

Dy2, jj ¼16

Cell z FT(2p) m m

0 z 1/6 mo cos(p/3) mo/2

1tc z+1 1/6+1/6 mo cos(2p/3) �mo/2

2tc z+2 2/6+1/6 mo cos(2p/3) �mo

3tc z+3 3/6+1/6 mo cos(4p/3) �mo/2

4tc z+4 4/6+1/6 mo cos(5p/3) mo/2

5tc z+5 5/6+1/6 mo cos(6p/3) mo

e. The phase jj is defined by symmetry. In Figs. 3 and 4, a global phase of 16(2p) has

surate q3 ¼ (0, 0.48, 0.089) amplitude-modulated magnetic structure of DySi (FeB-

e direction but the ordered moment value varies 0–8.2 lB/Dy3+ (formula (5)). The

of the Dy1 (x, 14, z) magnetic moment in the uniaxial commensurate q2 ¼ (0, 1

2, 16)

ent with two amplitudes (mo and mo/2) in a 12 times larger cell (a, 2b, 6c). For

l axis is a, the vertical axis is b while c runs out of the plane.

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Fig. 6. The variation of the Dy1 (x, 14, z) magnetic moment assuming that q2+q3 act in the same physical space for a large number of cells (a�4b�14c). In this structure,

both the value and orientation change with distance. The ordered moment value varies between 0 and 15.7 lB/Dy3+ within the plane (0 0 1). The horizontal axis is a, the

vertical axis is b while c runs out of the plane.

P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]–]]] 9

structure can be described by a single Fy mode corresponding tothe representation G2 while the q3 phase is described by a mixtureof G1+G2 representations. According to Izyumov [12], magneticstructures described by a mixture of two representations of thespace group of the crystal of the same wave vector star, becomestabilized by predominant isotropic exchange forces in theHamiltonian (exchange multiplets). This situation may occurwhen the energies of two distinct states described by differentrepresentations are very similar or equal. In this case, the twomodes can condense together in the paramagnetic to magneticordered phase transition giving rise to the mixing of representa-tions.

Please cite this article as: P. Schobinger-Papamantellos, et al., J. Mag

The two structures (q2 and q3) may coexist in form of domainsor create one single complex incommensurate structure, whereboth the amplitude and the direction change with the position.This cannot be directly decided only by diffraction methods.However, the model where the two wave vectors act in the samephysical space can be rejected as it led (see previous section) tounrealistic maximal moment amplitude for Dy3+ of 15.7mB.

Both DySi FeB magnetic structures bear some similarity withthe high-temperature TbSi [9] structure with the wave vector (0, 1

2,qz) qzE1

8 stable in a limited temperature range 39 K-TN ¼ 57 K.This phase is incommensurate with the crystal lattice andundergoes a first-order transition at lower temperatures to a

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commensurate phase with q ¼ (0, 12, 0), and displays also

coexistence regions. Incommensurate magnetic structures occurfrequently in magnetic systems due to competing interactionsusually in the high-temperature range and lock-in at lowertemperatures to a constant moment structure. Both DySi struc-tures correspond to collinear antiferromagnetic arrangements ofthe magnetic moments within the (0 0 1) plane close to theshortest axis b most probably fixed by the important role of the ofthe crystal field induced anisotropy. The difference with TbSi isthat in DySi the two q2, q3 structures coexist at low temperatures.As the thermal behaviour of the two phases is not yet available, wedo not know whether q2 is the lock-in phase of q3 (or of anotherwave vector) and what is the temperature of the transition. For thisreason, we restrict ourselves to discussing the resulting models ingeometric terms and focus on the commensurate structure.

Although it is difficult to describe in simple terms a magneticstructure with atoms located at such a low-symmetry position, wewould like to comment on the relation between magnetic orderand the underlying crystal structure for the q2 case. Thecharacteristic building blocks of the FeB and the CrB structuresare elongated trigonal rare-earth prisms that are stacked side-by-side to infinite blocks centered by zigzag Si chains. These blocksspan along the b-direction in the FeB structure (see Fig. 4).

In this structure, the Dy atoms are surrounded by 10 Dy atomsat distances (d) ranging from 3.5 to 4.3 A and 7 Si atoms atdistances 2.91–3.1 A (Dy coordination number N ¼ 17). In ourmodel, the moments of Dy atoms at distances ranging from 3.802to 4.219 A are found to be coupled antiferromagnetically (anti-parallel). These comprise the two Dy–Dy bonds along the b-axislinear chains tb ¼ 3.802 A and those in the zigzag chains along a

with d ¼ 4.219 A (elongated trigonal prism side between Dy1–D4or Dy2–Dy3). Therefore, ta ¼ 7.87 A remains a lattice translation(see atoms Dy1 and Dy1+ta in Fig. 4 with opposite signs) while theb-axis is twice enlarged 2tb ¼ 7.65 A.

The remaining six Dy–Dy bonds comprise the four nearest- andthe two next nearest neighbours. One sees in Figs. 3 and 4 that theDy atoms are located in the center of a trigonal prism with thefour nearest neighbours at d ¼ 3.594 A along the zigzag chains inthe c-direction and two next nearest neighbours at d ¼ 3.628 A.For Dy1, this corresponds to the four bonds between the atomsDy1–Dy2 (3.594 A) and the two bonds between Dy1–Dy3(3.628 A). More precisely, Dy1 is at the top of a flat pyramid withan orthorhombic basis (b� c) defined by the four Dy2 atoms. As b

is an antitranslation, two Dy2 moments of the (b� c) pyramidebasis point in one direction and two have opposite signs (see inFig. 3 Dy2, Dy20 and Dy2-tb, Dy20-tb). The same holds for the twonext nearest neighbours of Dy1 to the two Dy3 and Dy3-tb atomsat the corners of an isosceles triangle with basis b and site lengthd ¼ 3.628 A across the center of symmetry at the origin (0, 0, 0)and at (0, 1

2, 0)). Again, because b is an antitranslation the two Dy3atoms have also opposite moment signs.

Thus, for the Dy1 atom located in the center of a trigonal prism(4 Dy2 atoms and 2 Dy3 atoms) the molecular field is always zeroas three atoms have their moments in parallel and three points inopposite directions. This might be the reason why the arrange-ments between Dy1–Dy2 and Dy1–Dy3 along the c-directionmight be ferro- or antiferromagnetic. Apparently, the twoarrangements are energetically equivalent. The same situation isfound for Dy1–Dy10 at distance tc ¼ 5.667 A or 2tc Dy1–Dy100,

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while at d ¼ 3tc ¼ 16.97 A the moment arrangement is exclusivelyantiferromagnetic and this finally leads to the six-fold enlarge-ment of the c-axis.

The above presented analysis gives a hint why in such a systemmore than one incommensurate phases with similar values in freeenergy can coexist, resulting from a large number of competingexchange interactions, at least along the c-axis.

Comparing the q2 and q3 structures in Fig. 5a and b, onerealises the very different arrangement along the c-axis relatedwith the strong decrease of the qz component from 1

6 to � 111 in the

two structures and the increase of the number of ferromagneti-cally ordered moments along c for q3. More precisely, the numberof atoms with the same moment orientation along the c-axisincreases from 3 in the q2 phase to almost �6 in the q3 phase. Thisindicates a tendency in this compound to build larger ferromag-netic chains along c, which are stacked antiferromagneticallyalong b. A rough approximation of the average local moment valuefor the q3 structure estimated for 12 atoms along c (Fig. 5a) resultsin 5.46mB/Dy3+. Scaling this value by O2�5.46mB leads to theaverage value of 7.73mB/Dy3+ that is higher than that found in q2

(7.0mB/Dy3+ see at the end of Sections 2.6.4). The average of themoment values of the two FeB phases calculated according to Ref.[13] for a two magnetic phase compound O(52+5.462)mBE7.4mB

Dy3+ is strongly reduced if compared to the full Dy3+ momentvalue of (gJJ mB ¼ 10mB for Dy3+) and the reported 9.5mB/Dy3+

value of the saturation magnetisation from the field-dependentmeasurements in Ref. [5]. This can be explained by the presence ofpartial disorder in the neutron experiments (at zero fields) as thefield action may induce ferromagnetic alignment of the moments.

One can figure out that the portions of the competing q2 and q3

magnetic structures found in the FeB-type DySi compound at 1.5 Kwill vary with temperature, pressure or magnetic field. The field-dependent first-order metamagnetic transition reported in Ref. [5]indicates that these structures undergo easy changes. Because thefield favours ferromagnetic interactions it could easily influencethe arrangement along the c-axis which is very different alreadyin the two observed structures.

References

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139 (1988) 221.[4] R. Iraldi, V.N. Nguyen, J. Rossat-Mignod, F. Tcheou, Solid State Commun. 15

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[8] V.N. Nguyen, J. Laforest, J. Sivardiere, Solid State Commun. 8 (1970) 23.[9] P. Schobinger-Papamantellos, T. Janssen, K.H.J. Buschow, J. Magn. Magn.

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