ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
Re-entrant magneto-elastic transition in HoFe4Ge2 a neutron diffraction study P. Schobinger-Papamantellosa,*, J. Rodr!ıguez-Carvajalb, G. Andre! b, C. Ritterc, K.H.J. Buschowd a Laboratorium fur CH-8092, Switzerland . Kristallographie, ETHZ, Zurich . Laboratoire Leon Brillouin, (CEA-CNRS) Saclay, Gif sur Yvette Cedex 91191, France c Institut Laue-Langevin, 156X, Grenoble C!edex 38042, France d Van der Waals-Zeeman Institute, University of Amsterdam, NL-1018 XE Amsterdam, Netherlands b
Received 24 October 2003
Abstract The re-entrant magneto-elastic transition of the antiferromagnetic HoFe4Ge2 compound has been studied by neutron powder diffraction as a function of temperature. The magnetic phase diagram comprises the wave vectors: (q1o ; q2o ; q1t ) and three magnetic transitions, two of them occurring simultaneously with the structural changes at Tc ; TN ¼ 52 and Tc0 ; Tic1 ¼ 15 K, the third being purely magnetic at Tic2 ¼ 40 K. The first transition is of second order while the latter two of first order. The sequence of phases follows the path: P42/mnm (HT), Tc ; TN ¼ 52 K-Cmmm (IT): ðq1o ¼ ð0; 1=2; 0Þ; Tic2 ¼ 40 K ) q2o ¼ ð0; qy ; 0ÞÞ; Tc0 ; Tic1 ¼ 15 K ) P42/mnm (LT): q1t ¼ ð0; 1=2; 0Þ: The magnetic structures described by the wave vectors (q1o ; q2o and q1t ), where the components are referred to the reciprocal basis of the conventional Cmmm cell, correspond to canted multi-axial arrangements. The q2o wave vector length of the amplitude modulated phase varies non-monotonously, decreasing fast just below Tic2 ;—slowly between 36 K—Tc0 ; Tic1 and jumping to the q1t ¼ ð0; 1=2; 0Þ lock-in value at Tc0 ; Tic1 simultaneously with the first order re-entrant transition to the (LT) tetragonal phase. In the coexisting meta-stable orthorhombic phase from Tc0 ; Tic1 down to 1.5 K the length of the wave vector q2o continues to decrease. To solve the magnetic structures of all the phases appearing in this complex situation, arising from competing ordering mechanisms and anisotropies of the underlying sublattices, we have used the simulated annealing method of global optimisation on high-resolution neutron powder diffraction data. r 2004 Published by Elsevier B.V. PACS: 75.25.+z; 61.12.Ld; 71.20.Eh; 75.80.+q Keywords: Magnetic structure; Neutron diffraction; Rare earth alloys; Re-entrant magneto-elastic transition; Simulated annealing
1. Introduction
*Corresponding author. Tel.: +41-632-3773; fax: +41-1632-1133. E-mail address:
[email protected] (P. Schobinger-Papamantellos). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2004.02.028
Ternary rare earth compounds of the formula RFe4Ge2 where (R=Y, Dy, Er, Lu) crystallise with the ZrFe4Si2 type of structure [1,2] (space ( c ¼ 3:755 A, ( Z ¼ 2: group, P42/mnm, a ¼ 7:004 A R at 2b site: (0,0,1/2), Fe at 8i: (0.092, 0.346, 0)
ARTICLE IN PRESS 120
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
and Ge at 4g: (0.2201, 0.7799, 0). These compounds were originally reported to order ferromagnetically [2] at very elevated temperatures (963643 K for R ¼ Y and Er respectively). More . recently they were found by 57Fe Mossbauer spectroscopy, high field magnetic measurements [3–5] and our actual investigation on the magnetic properties of the RFe4Ge2 compounds with R ¼ Y ; Dy, Ho, and Er [6–9], to order antiferromagnetically at considerably lower Ne! el temperatures, TN : 43.5 [6], 65(1) [3,4], 56(2) [5] and 47(2) K [3,4,6,7] respectively. Our neutron diffraction studies on the magnetic ordering of the YFe4Ge2 and ErFe4Ge2 compounds [6,7] indicate that the magnetic ordering is accompanied by simultaneously occurring structural changes. In ErFe4Ge2 the magnetic transition is associated with a double symmetry breaking. Just below TN the high temperature (HT) tetragonal phase, P42/mnm, disproportionates, by a first-order transition, into two distinct phases of orthorhombic symmetry. The two phases coexist, with varying proportions, in a large temperature range. The transition paths and magnetic wave vectors are: (a) P42/mnm-Cmmm (majority phase) q ¼ ð0; 1=2; 0Þ; (b) P42/mnm-Pnnm (minority phase) qa0 (unknown). A high-resolution X-ray (XRPD) study [8] has shown that the Cmmm phase is dominating the low temperature (LT) range 1.5–20 K while the Pnnm phase reaches its highest percentage of 30% in the range 20–35 K (see also Section 2.3). The transition to the Cmmm phase is associated with a distortion of the tetragonal angle g leading to the enlarged orthorhombic phase Cmmm with ao=atbt and bo=at+bt. The thermal variation of the orthorhombic distortion (with ao/bo>1) could be brought into connection with the dominant R–R magnetic interaction which is ferromagnetic along ao and antiferromagnetic along bo leading to a cell enlargement 2bo : The underlying magnetic structure associated with the wave vector q ¼ ð0; 1=2; 0Þ corresponds to a canted arrangement with 10 sublattices (8 for the Fe moments and 2 for Er). The Er moments are
confined to the plane ð1 1 0Þt or ð1 0 0Þo : The structural transition to the Pnnm space group is associated to a at =bt a1 distortion as explained below. The magnetic ordering of this phase is still not fully understood. The double symmetry breaking in this compound most likely originates from the existence of different competing coupling mechanisms between the R–R, the Fe–Fe and the R–Fe magnetic moments and their coupling with the lattice strains. YFe4Ge2 was found to order antiferromagnetically below TN ¼ 43:5 K. The magnetic ordering occurs simultaneously with a structural transition following path (b) of ErFe4Ge2 from tetragonal to orthorhombic symmetry: P42/mnm-Pnnm ðq ¼ 0Þ: For the non-magnetic Y this transition can exclusively be induced by magneto-elastic strains of the compact tetrahedral Fe arrangement with antiferromagnetic interactions. Both transitions (structural and magnetic) are of first order. The symmetry breaking is connected with primary displacive order parameters involving shifts of the Fe atoms. The Fe site splits into two sites in Pnnm. The shifts of the Fe atoms with respect to the tetragonal phase have opposite signs along the a (dilatation) and b (contraction) axes and therefore to an at =bt > 1 distortion. The planar canted antiferromagnetic arrangement, with eight Fe sublattices, realised at lower temperature results from a considerable anisotropy within the basal plane and non-exclusively antiferromagnetic interactions. The moments of the two Fe orbits are perpendicular and have the same value of 0:63ð4Þ mB /Fe atom at 1.5 K. On the one hand, these results triggered the interest to enlarge the experimental basis and to extend this study to isomorphic RFe4Ge2 and RFe4Si2 compounds with and without magnetic rare earths (R). On the other hand, they stressed the importance of studying the magnetic and structural transitions with temperature, by a simultaneous analysis of high-quality X-ray powder (XRPD, Synchrotron radiation) and neutron data for each distinct case because of the complexity of the situation. This strategy has been adopted to study the sequence of magneto-elastic phase transitions in HoFe4Ge2. In a first stage a set of high-flux neutron data traced the strategy for the
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
XRPD data collection in order to cover the relevant critical points with sufficient experimental information. The analysis of XRPD data led to the phase diagram as a function of temperature given in Ref. [9] and reproduced in Section 2.3. Of particular interest is the occurrence of a re-entrant structural transition discovered in HoFe4Ge2: P42 =mnmðTc ; TN Þ ¼ 52 K -CmmmðTc0 ; Tic1 ¼ 15 KÞ ) P42 =mnm: The first symmetry breaking transition in HoFe4Ge2 is of second order and occurs below Tc ; TN ¼ 52 K [9], which within error agrees with the reported magnetic ordering temperature TN ¼ 56ð2Þ K in Ref. [5] from magnetic measurements. The second transition is of first order and occurs at lower temperatures Tc0 ; Tic1 ðE15 KÞ; restoring the tetragonal symmetry. Both transitions were connected with the presence of anisotropic micro-strain making the modelling of the line shape parameters as important as the structural parameters for the refinements described in detail in Ref. [9]. In the present study we will focus, by using neutron diffraction, on the magnetic ordering of HoFe4Ge2 associated with the structural transitions and its relation to the lattice strains. Our results will be summarised in a magnetic phase diagram, incorporating the results of our XRPD study [9] and the relevant information concerning the magnetic ordering found in the present study, which will be compared to that of the ErFe4Ge2 compound. For the sake of simplicity the description of all magnetic structures will refer to the frame of the space group Cmmm. The subscript used throughout the text in the vector notation q1t and q1o ; q2o denotes the crystal symmetry, but the components are always referred to the reciprocal basis of the conventional direct cell of Cmmm. In the appendix we present extensively the use of the simulated annealing technique, which enabled the present data analysis.
2. Neutron diffraction Neutron diffraction experiments were carried out in the temperature range 1.5–150 K. The
121
HoFe4Ge2 sample used for neutron diffraction, prepared by arc melting from starting materials of at least 99% purity, was the same as used in the XRPD structural study [9] and was found to contain some weak contributions of the HoFe2Ge2 (ThCr2Si2 type I4/mmm) phase. Two data sets were collected with the D1B (double axis multi-counter diffractometer) and the high resolution D2B spectrometer at the facilities of the ILL in ( Grenoble using a wavelength of 2.52 and 2.39 A respectively. The step increment of the diffraction angle 2y was 0.2 and 0.05 , respectively. The D1B data were collected in the 2y region 3–83 for a full set of temperatures in the range 1.5–54 K in steps of 1, 2 K. The D2B data, which extend from 3 to 160 , were collected for some selected temperatures in the range 1.5–73 K for structural refinements. The data were analysed with the program FullProf [10]. Two further data sets were collected at the facilities of the Orphe! e reactor (LLB-Saclay) using the G4.1 (800-cell Position Sensitive Detector: PSD) and the G42 (high resolution: 70 detectors with Soller collimators) diffractometers with the ( respectively. wavelengths of 2.426 and 2.3433 A, The step increment in 2y was 0.1 . The outline of the data analysis in Section 2 is as follows. Section 2.1 refers to structural refinements of high-resolution D2B data based on the phase diagram and parameters from the XRPD refinements given in Ref. [9]. Section 2.2 gives a general view of the interplay between structure and magnetic ordering based on D1B data over the entire magnetically ordered regime. Sections 2.2.1 and 2.2.2 show in detail the sequence of magnetic transitions, the critical points and the corresponding magnetic wave vectors on heating, which lead to the magnetic phase diagram in Section 2.3. The refinement of the magnetic structures from highresolution D2B data in Section 2.4 are carried out for a few characteristic temperatures around the critical points found in the phase diagram and are preceded by symmetry analysis (Section 2.4.1) and simulated annealing (see Appendix A). The refinements of commensurate and incommensurate magnetic phases are given in Sections 2.4.2 and 2.4.3, respectively. The two types of structures are then compared in the last Section 2.4.4.
ARTICLE IN PRESS 122
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
2.1. Nuclear structure, HR (D2B) data The HR neutron diffraction pattern (D2B) collected in the paramagnetic state at 65 K is shown in Fig. 1. The refined parameters and Rfactors ðRB ¼ 2:5%; Rwp ¼ 11%) given in Table 1 confirm the ZrFe4Si2 type of structure [1]. In agreement with the X-ray findings the powder pattern contains small amounts (2–3%) of HoFe2Ge2 and a-Fe impurity phases that have been included in the refinement. According to Ref. [11], the HoFe2Ge2 compound orders antiferromagnetically below 1.5 K but no coherent magnetic reflections could be detected at this temperature. More recently we found [12] that this compound orders magnetically below TN ¼ 20 K with the wave vector q ¼ ð1=2; 1=2; 0Þ: However, its strongest magnetic reflections at 2y ¼ 24:2 and 27.5 would be too weak to be observed. The structural parameters obtained at lower temperatures 42 and 18 K in the magnetically ordered state included also in Table 1 are those obtained from the XRPD data [9]. For the 1.5 K data (not available from XRPD) the structural parameters were refined simultaneously for the coexisting tetragonal and orthorhombic phases. The later phase exists as a metastable phase down
to 1.5 K due to the first order transition at Tc0 ; Tic1 : The refined magnetic parameters for the same temperatures are discussed in Section 2.4.2 after the symmetry analysis (Table 2) and are summarised in Table 3. As already pointed out in Ref. [9] a major problem in the refinements of the data collected for the HoFe4Ge2 compound at any temperature was the anisotropic broadening of the Bragg peaks. All reflections are broadened if compared with the experimental resolution. In particular the full width at half-maximum (FWHM) of the general ðh k lÞt tetragonal reflections is larger than that of the corresponding ðh h lÞt reflections due to the sequence of magneto-structural transitions associated with micro-strain effects that were modelled similarly to the XRPD data. The formulation used in Ref. [9] of micro-strain is independent of the used wavelength and since the D2B instrumental parameters are known the XRPD refined micro-strain parameters were directly implemented in the calculations for each distinct temperature. The peak shape corrections used in the refinements led to a 5% decrease of the Rwp weighted profile factor. The largest anisotropy is along the tetragonal axes a and b and is observed in the tetragonal HT (52 K) phase (see
Fig. 1. Observed and calculated HR (D2B instrument) neutron intensities of HoFe4Ge2 in the paramagnetic state at 65 K. The indexing refers to the tetragonal phase, P42/mnm.
ARTICLE IN PRESS
0.1331 0.27949 0.21576 0 10.30186(8) 10.27086(8), 1.00301 3.85367(3) 407.752(6) 3, 13 7 3.85641(3) 407.918(4) 2.5, 11 7
3.85641(3)
( c (A) ( 3) Vol (A RB ; Rwp (%) Rexp (%)
The refined parameters of the 65 K tetragonal (t) phase P42/mnm are also given in the orthorhombic setting (o) Cmmm for comparison reasons with the 1.5 K coexisting ( 2) is fixed to the 0.04. RB ; Rwp and Rexp (%) are: the Bragg, the weighted profile and the tetragonal (t) and orthorhombic (o) phases. The overall temperature factor B (A experimental reliability factors.
3.85274(4), 407.769(5) 2, 11 7
0 0.0 0.22299 0.13371 0 0.28343 0 0.5 0.13277 0.27957 0.21471 0 10.3003(4) 10.2701(5), 1.00294 3.8549(1) 407.79(3) 2.5 0 0 0.2220(1) 0.1324(1) 0 0.2845(2) 1 2
1/2 0 0 1/2 0 1/2 0 1/2 0.3546(2) 0.4117(1) 0.2154(2) 0.7154
y
Ho1:2d Ho2:2b Fe1:8p Fe2:8q Ge1:4g Ge2:4j 10.28478(6) 10.28478(6), 1 0 1/2 0.0883(1) 0.8546(2) 0.2154(2) 0.7154 7.27244(4) 7.27244(4), 1 Ho:2b 1 2 Fe:8i (x; y; 0) 1=2 þ y; 1=2 x; 1=2 Ge:4g (x; x; 0) x þ 1=2; x þ 1=2; 1=2 ( a (A), ( a=b b (A),
0 1/2 0.1331 0.2786 0.2154 0
0 0 0.2214 0.1331 0 0.2846
0 1/2 0.1331 0.2794 0.2157 0 10.29518(7) 10.27626(6), 1.0018 3.85442(2) 407.783(4) 2.1, 11 7.8
0 0 0.2224 0.1333 0 0.2839(3)
0
0 0 0.2224 0.13334 0 0.28398
0 1/2 0.1324(1) 0.2791(1) 0.2153(3) 0 10.28781(5) 10.28787(5), 1
y
Cmmm 1.5 K (o)
x y x
Cmmm 1.5 K (t)
y x
Cmmm 18 K
y x
Cmmm 42 K
y Atom/site
x Cmmm 65 K (o)
Atom/site
z P42/mnm 65 K (t)
x
P42/mnm
Table 1 Refined structural parameters of HoFe4Ge2 from high-resolution neutron diffraction data (D2B) in the paramagnetic state at 65 K and in the magnetically ordered state at 42, 18 and 1.5 K
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
123
Ref. [9, Fig. 6]). By contrast, in the LT tetragonal phase below the re-entrant transition also shown in Ref. [9, Fig. 6], the micro-strain anisotropy is less pronounced.
2.2. The interplay of structural and magnetic transitions, D1B data The interplay of magnetic ordering and structural transitions can be best followed in the low 2y angle part (13–53 ) of the neutron thermodiffractograms (see Fig. 2) obtained in the 1.5–54 K range. Long-range magnetic order becomes visible below about 52 K simultaneously with the structural transition, in good agreement with our XRPD study [9] and within error with the magnetic measurements [5]. The results displayed in Fig. 2 involve at least three regions of distinct magnetic order with regard to the phase diagram as a function of temperature [9], comprising three stability regions, of two distinct crystal structures and two transitions among them. The magnetic contributions occur at reciprocal lattice positions other than the nuclear reflections ðqa0Þ and, as stated above, the indexing refers to the Cmmm cell for all considered temperatures. Ferromagnetism can be ruled out from the unchanged intensity of the (2 0 0)/(0 2 0) nuclear reflections with temperature. Figs. 3 and 4, display the very different thermal behaviour of selected well-resolved magnetic reflections in more detail. The strongest magnetic contribution (2 0 0)7q at 2yE28 becomes already visible on the right part of the (2 0 0) nuclear reflection below 52 K cf. Fig. 2 and persists over the entire magnetically ordered regime (see Fig. 3, top part) for this reason its behaviour will be described more extensively in the next section as it offers the most direct experimental observation. By contrast, the reflections shown in the middle part of Fig. 3 display quite a different behaviour and appear at lower temperatures (4036 and 15 K). The presence of the (2,0,0)7q satellite is compatible with the presence of wave vector(s) along the orthorhombic b- or c-axis. From the similarity in peak topology found in the Cmmm magnetic phase with q ¼ ð0; 1=2; 0Þ in ErFe4Ge2, it can be assumed that the
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
124 Table 2
(a) The split of the Ho 2b site into the Ho1 and Ho2 sites and of the Fe 8i site into the sites IFe, IIFe, IIIFe, IVFe caused by the action of the wave vector (x; x; 0) in the parameter space of the P42/mnm space group at 1.5 K. The labelling of the atoms corresponds to that of Fig. 9a. In the C-cell the labelling of the atoms of the Fe orbits are denoted by dotted numbers 10i ; 20i ; 30i ; 40i where i ¼ 1; 4 thus four Fe atoms with the same i belong to the same Fe tetrahedron Site/atom
Nr.
x
y
z
Nr.
x
y
z
IHo IIHo IFe IIFe IIIFe IVFe
1 2 1 2 3 4
0 1/2 x x y þ 1=2 y þ 1=2
0
1/2 0 0 0 1/2 1/2
7 8 6 5
y y x þ 1=2 x þ 1=2
x x y þ 1=2 y þ 1=2
0 0 1/2 1/2
1 2
y y x þ 1=2 x þ 1=2
(b) Irep’s for q¼ ð0;qy ; 0Þ in Cmmm
G1 G2 G3 G4
E
2y
mz
mx
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
(c) Fourier components Sq ¼ ðu; v; wÞ of the 6 magnetic sublattices (orbits), without centring translation (1/2,1/2,0), of HoFe4Ge2 for the four representations associated with the wave vectors q1o ¼ ð0; 1=2; 0Þ; q2o ¼ ð0; qy ; 0Þ referring to the C-cell (Figs. 9a–c). ðuo ; vo ; wo Þ are the corresponding refined parameters at 1.5 K for q1t for G2 þ G3 : The positional coordinates of the Fe atoms of the Fe1i 0 and Fe2i 0 orbits, i ¼ 1; 2 are ðxi ; yi ; 0Þ; ð1 xi ; yi ; 0Þ respectively and those of Fe3i 0 and Fe4i 0 ðxi ; yi ; 1=2Þ; ð1 xi ; yi ; 1=2Þ Site
1 2 3 4 5 6
Atom
Ho1 Ho2 Fe101 Fe102 Fe201 Fe202 Fe301 Fe302 Fe401 Fe402
x
y
0 0.5 0.1324 0.8676 0.3676 0.6324 0.2791 0.7209 0.2209 0.7791
z
0 0 0.2220 0.2220 0.2780 0.2780 0.1324 0.1324 0.3676 0.3626
0.5 0 0 0 0 0 0.5 0.5 0.5 0.5
G1
G2
w
u
u3 u3 u4 u4 u5 u5 u6 u6
W3 w3 W4 w4 w5 w5 w6 w6
wave vector(s) are also confined to the b-axis in the HoFe4Ge2 compound. 2.2.1. First-order magneto-structural transition at Tc0 ; Tic1 : P42 =mnmðLTÞ: qlt ¼ ð0; 1=2; 0ÞTc0 ; Ticl ¼ 15 K )ÞCmmmðITÞ; q2o ¼ ð0; qy ; 0Þ: The intensity and position of the (2,0,0)7q1t magnetic reflection see Fig. 3 (top part) remain stable in the low temperature range 1.5 K—Tc0 ; Tic1 ¼ 15 K (on heating) where the LT tetragonal structure prevails. The refined wave vector value (using profile matching) has the commensurate
G3
G4
v
w
u
v1 v2 v3 v3 v4 v4 v5 v5 v6 v6
w1 w2 w3 w3 w4 w4 w5 w5 w6 w6
u1 u2 u3 u3 u4 u4 u5 u5 u6 u6
G2 þ G3 v
v3 v3 v4 v4 v5 v5 v6 v6
uo
vo
0.5(1) 0.5(1) 0.2(1) 0.2(1) 0.1(1) 0.1(1) 0.3(1) 0.3(1)
2.36(7) 5.4(1) 0.4(1) 0.4(1) 0.93(5) 0.93(5) 0.95(4) 0.95(4) 0.13(3) 0.13(3)
wo 0.3(15) 0.70(1) 0.70(1) 0.2(1) 0.2(1) 0.3(1) 0.3(1) 0.93(5) 0.93(5)
components q1t ¼ ð0; 1=2; 0Þ with respect to the reciprocal basis of the orthorhombic C-cell. (As stated in the introduction the subscript 1t denotes the crystal symmetry, which is tetragonal). At Tc0 ; Tic1 ; the temperature of the first-order structural transition P42/mnm (LT) (Tc0 ; Tic1 ¼ 15 K) ) Cmmm (IT), the intensity and 2y position of this reflection decrease abruptly as the amount of the orthorhombic phase increases at the cost of the now metastable tetragonal phase with increasing temperature. The fast decrease of the 2y position of the (2, 0, 0)7q satellite just above Tc0 ;
1.5 K, q1t ¼ ð0; 0:5; 0Þ
1.5 K, q2o ¼ ð0; qy ; 0Þ
18 K, q2o ¼ ð0; qy ; 0Þ
42 K, q1o ¼ ð0; 0:5; 0Þ
Ho1:2d (0,0,1/2) my ; mz ðmB Þ mT ðmB Þ; F; y ðdegÞ; j ð2pÞ
2.36(7), 0.30(15) 2.38(7), 90, 83(4), 0.125
4.1(5), 0.9(4) 4.1(5), 90, 77(6), 0
4.75(9), 0.6(1) 4.8(1), 90, 98(1), 0
1.52(7), — 1.52(7), 90, 90, 0.125
Ho2:2b (1/2,0,0) my ; mz ðmB Þ mT ðmB Þ; F; yðdegÞ; jð2pÞ
5.4 (1), — 5.34(2), 90, 90, 0.125
5.8(6), 1.3(6) 5.9(6), 90, 77(6), 0
5.37(9), 0.2(1) 5.38(9), 90, 87(1), 0
2.43(7), — 2.43(7), 90, 90, 0.125
Fe10 :ðx; y; 0Þ at 8p mx ; my ; mz ðmB Þ mT ðmB Þ; F; yðdegÞ; j ð2pÞ
0.5(1), 0.4(1), 0.70(1) 0.97(5), 141(6), 138(6), 0.125
2.3(2), 1.9(3), 0.7(4) 3.1(2), 321(6), 103(8), 0.4(2)
0.8(1)1.2(1), 1.47(3) 2.08(4), 303(4), 45(1), 0.19(1)
0.34(3), 0.26(3),— 0.43(3), 38(6), 90, 0.125
Fe20 :8p ð1=2 x; 1=2 y; 0Þ mx ; my ; mz ðmB Þ mT ðmB Þ; F; y ðdegÞ; j ð2pÞ
0.2(1), 0.93(5), 0.2(1) 0.97(5), 284(5), 100(7), 0.125
1.0(3), 2.4(4), 1.6(4) 3.1(2), 293(6), 122(9), 0.2(1)
1.43(6), 0.4(1), 1.48(3) 2.08(4), 166(4), 45(1), 0.18(1)
0.003(3), 0.43(3), — 0.43(3), 270(12), 90, 0.125
Fe30 :ðx; y; 1=2Þ at 8q mx ; my ; mz ðmB Þ mT ðmB Þ; F; yðdegÞ; jð2pÞ
0.1(1), 0.95(4), 0.3(1) 0.99(5), 276(4), 75(5), 0.125
2.9(2), 0.5(3), 0.9(4) 3.1(2), 191(6), 107(8), 0.5(1)
1.94(5), 0.2(1), 1.37(69) 2.39(5), 173(2), 55(1), 0.17(1)
0.08(2), 0.41(3), — 0.42(3), 281(12), 90, 0.125
Fe40 :8q ð1=2 x; 1=2 y; 1=2Þ mx ; my ; mz ðmB Þ mT ðmB Þ; F; yðdegÞ j ð2pÞ qy (r.l.u) RB ð%Þ; Rm1 ; Rm2 ð%Þ Rwp ð%Þ; Rexp ð%Þ
0.3(1), 0.13(3), 0.93(5) 0.99(5), 45(8), 116(4), 0.125 0.5 2.8, 6, 12, 4
1.2(3), 1.4(4), 2.4(3) 3.1(2), 230(10), 142(9), 0.3(1) 0.4346(2) 2.8, —, 8
1.68(6), 1.0(1), 1.37(6) 2.38(5), 149(2), 55(1), 0.75(1) 0.44164(8), 3.4, —, 7 13, 7
0.38(3), 0.17(3), — 0.42(3), 24(6), 90, 0.125 0.5 2, 9, 11 12, 8
F and y are the angles of the magnetic moments with the ao ; co axes, j is the phase of the wave with the origin. RB ; Rm1 ; Rm2 ; Rwp and Rexp (%) are: the Bragg, the magnetic ðq1t or q1o ; q2o Þ; the weighted profile and the experimental R-factors for integrated intensities.
ARTICLE IN PRESS
HoFe4Ge2 Cmmm
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
Table 3 Refined magnetic parameters of HoFe4Ge2 at various temperatures from HR neutron data. mx ; my ; mz ; mT are the moment components for qy ¼ 1=2 (fixing the origin of the wave to 7p=4) and the Fourier coefficients for qy a1=2 of the Ho and Fe sublattices
125
ARTICLE IN PRESS 126
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
HoFe 4 Ge2 1.5-54K 2,1/2,0
D1B, λ =2.52Å
TN=55K 200/020
T1=15K
111-q y 310 130
020-qy 200±q y020+qy 1,3/2,0
1,1/2,0
1,1/2,1
0,3/2,0
0,5/2,0 2,3/2,0
220 220-qy 1,5/2,0
111 3,1/2,0
220+q y 111+q y 3,3/2,0
Fig. 2. The low 2y angle part of the neutron thermodiffractograms of HoFe4Ge2, collected (on heating) for various temperatures with the D1B instrument above and below the ordering temperature TN ¼ 56ð2Þ K indicating the interplay of re-entrant magnetic and structural transitions. The indexing of the nuclear reflections refers to the Cmmm ðat O2; at O2; ct Þ cell. The indexing of the magnetic satellites requires three wave vectors q1t ¼ ð0; 1=2; 0Þ; q1o ¼ ð0; 1=2; 0Þ; q2o ¼ ð0; qy ; 0Þ: However for simplicity only two of them are given q1o ; q2o :
Tic1 ; corresponds to a decrease of the wave vector length qy o1=2 indicating the occurrence of a simultaneous transition to an incommensurate magnetic phase at Tc0 : The refined qy value just above the transition at 16 K equals 0.4419(1) r.l.u. see bottom part of Fig. 3. Here we would like to stress that, because the transition is of first order, the proportions of the coexisting—P42/mnm (LT) and Cmmm (IT)— crystal structures depend on the experimental conditions, such as the heating speed and the thermal history of the sample. In the XRPD study the amount of the Cmmm phase at the lowest measured temperature 4.2 K was 26%, but in some of the neutron experiments it was less at the same temperature. It is expected that for sufficiently long cooling time the tetragonal phase will prevail at the cost of the metastable orthorhombic phase with incommensurate magnetic structure. Profile matching as well as full Rietveld LineProfile Analysis of the high-resolution D2B data (see next section) led to the important result that the magnetic ordering of each of the two coexisting phases is associated with a distinct
wave vector: P42 =mnm ðLTÞ : qlt ¼ ð0; 1=2; 0Þ and Cmmm ðITÞ : q2o ¼ ð0; qy ; 0Þ: The incommensurate wave vector length q2o ¼ ð0; qy ; 0Þ increases linearly with temperature from qy ¼ 0:434 r.l.u at 1.5 K where it exists as a metastable phase to qy ¼ 0:448 r.l.u. at 40 K and then jumps to the lock-in value q1o ¼ ð0; 1=2; 0Þ (see Fig. 3, bottom part). At 42 K the amount of the incommensurate phase is less than 5%. 2.2.2. First-order magnetic transition at Tic2 : Cmmm: q2o ¼ ð0; qy ; 0Þ; Tic2 ¼ 40 K ) q1o ¼ ð0; 1=2; 0Þ In the low-temperature part Tc0 ; Tic1 ¼ 15 Ko To36 K of the stability range of the Cmmm (IT) structure type, the magnetic ordering is associated with the wave vector q2o ¼ ð0; qy ; 0Þ: In this range the 2y position of the ð2; 0; 0Þ7q2o reflection shows a linear displacement with increasing temperature. A change in the slope sets in above 36 K in a very narrow range 36–40 K where a first-order lock-in magnetic transition takes place to the
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
0
P42 /mnm
q2o=(0,qy,0)
q1o=(0,1/2,0)
Cmmm
0
20
40
3
0,3/2,0 000±q
y
10
T
ic1
2
ic2
T ,T c
Intensity(a.u)x10
t
y
(1,1/2,0)
Intensity(a.u)x10
3
y
020-q
c'
0 20
40
14K
40K
16K
42K
N
0 0
2,2,0-q2o
27.8
60
20
T ,T
34K
28.2
110-q
4
1.5K
2,2,0-q2o
2θ ( °) P42 /mnm
6
(b)
0,2,0+q2o
28.6
8
(2,3/2,0)o
(2,3/2,0)t
TN,TC T2
4
HoFe4Ge2
0,2,0+q2o
T1
Neutron Intensity (a.u)
(a)
29.0
D1B, data
Io(200)±qy
12
q1t=(0,1/2,0)
Integrated Intensity (a.u)
HoFe4Ge2
127
60
34
36
34
36
2θ deg.
0.460
Fig. 4. Deconvolution of characteristic magnetic neutron peaks showing a succession of magnetic phase transitions vs. T around the transition temperatures Tc0 ; Tic1 ¼ 15 K and Tic2 ¼ 40 K. The intensity of the central peak(s) at 2yE35 ð2; 3=2; 0Þt (tetragonal) or ð2; 3=2; 0Þo orthorhombic displays a re-entrant behaviour competing with that of the adjacent peaks ð2; 2 qy ; 0Þ and ð0; 2 þ qy ; 0Þ: The left part of the figure has a doubled y-scale.
HoFe Ge
0.455
4
2
D2B, data
0.445 0.440 c'
0.420
(c)
0
N
y 2o
20
C
P4 /mnm
q =(0,q ,0)
1t
0.425
T ,T
ic2
Cmmm
q =(0,1/2,0)
0.430
T
ic1
P42 /mnm
2
1o
T ,T 0.435
q =(0,1/2,0)
2o
q (r.l.u)
0.450
40
60
T[K]
Fig. 3. Temperature dependence of the HoFe4Ge2 magnetic neutron intensity of (a) the strongest magnetic peak ð2; 7qy ; 0Þ: Also shown are the changes of the 2y angle and the stability regions of the corresponding crystal and magnetic structures with the wave vectors q1t ; q2o and q1o resulting from the refinements (top part). (b) The well-resolved satellites at low 2y angles of the commensurate and incommensurate phases: ð0; 7qy ; 0Þ; ð1; 1 qy ; 0Þ; ð1; 1=2; 0Þ; ð0; 2 qy ; 0Þ; ð0; 3=2; 0Þ showing the competition between the wave vectors (middle part). (c) Temperature dependence of the q2o ¼ ð0; qy ; 0Þ wave vector length. The dotted line is a guide for the eye.
commensurate value q1o ¼ ð0; 1=2; 0Þ at Tic2 ¼ 40 K. Contrary to the situation found below Tc0 ; Tic1 ¼ 15 K, the two wave vectors pertain here to two distinct magnetic domains whose relative portions change with temperature but are of the same crystal structure (Cmmm). This hypothesis is supported by the XRPD results [9] showing only a small anomaly of the orthorhombic a=b ratio at Tic2 ¼ 40 K, which lies well below the secondorder transition to the tetragonal high-temperature phase occurring at TN ; Tc ¼ 52 K. In the hightemperature part of the Cmmm stability range Tic2 2TN ; Tc the position of the ð2; 0; 0Þ7q1o reflection remains unchanged according to the
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
HoFe Ge
P4 /mnm
2
Cmmm
2
y 2o
1t
0
P4 /mnm
q =(0,q ,0)
q =(0,1/2,0)
2
1o
40
4
q =(0,1/2,0)
80
Phase content %
commensurate wave vector value q1o ¼ ð0; 1=2; 0Þ while its intensity decreases monotonously. The thermal variation of the intensity of the reflections shown in Fig. 3 (middle part) provides additional information. An interesting observation is that the zero point satellite ð0; 0; 0Þ7q of both commensurate structures q1t ¼ ð0; 1=2; 0Þ and q1o ¼ ð0; 1=2; 0Þ is almost absent in the corresponding stability range of each phase. By contrast, the ð0; 0; 0Þ7q2o satellite (dotted line) is visible in the entire ordered regime below Tic2 ¼ 40 K. However, its intensity with decreasing temperature undergoes an abrupt change below Tc0 ; Tic1 ¼ 15 K, decreasing by 70%. A similar behaviour is displayed by the 020-q2o reflection. This indicates that the magnetic transition might be related to a spin reorientation towards the baxis for the rare earth in the commensurate phase as only in-plane mx;z moment components may contribute to the magnetic intensity of both reflections ð0; 7qy ; 0Þ and ð0; 2 qy ; 0Þ or the sum of all Mx and Mz contributions to the magnetic structure factor are zero (see also Section 2.4.4). The opposite is observed for the {1 1 0}-qy reflection, whose intensity increases below the lock-in transition at Tc0 ; Tic1 ¼ 15 K. The intensity of this reflection may comprise mxyz components.
T ,T c'
0
T
ic1
4
T ,T
ic2
20
c
40
ErFe Ge
80
Phase content %
128
N
60
2
Cmmm
P4 /mnm 2
q =(0,1/2,0) 1o
40
T ,T c
N
Pnnm q
2.3. The magnetic phase diagram
2
The D1B data and the high-resolution neutron D2B data, which will be discussed in the next section, together with the information obtained from our XRPD study led to the magnetic phase diagram shown in the top panel of Fig. 5. The phase diagram comprises three transition temperatures one of second order at Tc ; TN ¼ 52 K and two of first order at Tc0 ; Tic1 ¼ 15 K and Tic2 ¼ 40 K. The first two transitions occur simultaneously with structural changes. The third is a purely magnetic transition at Tic2 ¼ 40 K. The magnetic phase diagram comprises the wave vectors q1o ¼ ð0; 1=2; 0Þ; q2o ¼ ð0; qy ; 0Þ in the (IT)-orthorhombic range, q1t ¼ ð0; 1=2; 0Þ in the (LT)-tetragonal range following the path: P42/mnm (HT), Tc ; TN ¼ 52 K-Cmmm (IT): (q1o ¼ ð0; 1=2; 0Þ; Tic2 ¼ 40 K ) q2o ¼ ð0; qy ; 0Þ), Tc0 ; Tic1 ¼ 15 K ) P42/mnm (LT): q1t ¼ ð0; 1=2; 0Þ:
0
λ1
XRD: 0
λ2
20
40
60
T[K] Fig. 5. Schematic representation of the structural and magnetic phase diagram of HoFe4Ge2 (top part). Also shown is the magnetic phase diagram of ErFe4Ge2 (bottom part). Full and dashed lines are first and second order transitions. Dash dotted lines denote the lower and upper limit of coexistence regions around the first-order transitions (Tc0 ; Tic1 ; and Tic2 ) of the adjacent magnetic and/or crystal structures.
There are three stability ranges of two distinct crystal structures and four ranges of magnetic order (including the paramagnetic state). This behaviour is very different to that observed for
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
the ErFe4Ge2 compound shown in the bottom part of the figure as will be discussed in Section 3. The interplay of the three magnetic propagation vectors relevant to the re-entrant structural transition is shown in Fig. 4 in the vicinity of the firstorder transition temperatures Tc0 ; Tic1 ¼ 15 K and Tic2 ¼ 40 K. One could say that the re-entrant structural transition is accompanied by a reentrant magnetic transition with regard the peak topology associated with the two commensurate wave vectors i.e. the aspect of the 1.5 K diffraction pattern is very similar to that shown for 42 K in the figure. The indexing and de-convolution of the magnetic satellites around the magnetic satellite (2, 3/2, 0) at 2y ¼ 35 follows from the data analysis. At 42 K the magnetic order is described by q1o=(0,1/2,0) and one observes a single
129
magnetic peak of the orthorhombic phase: (2, 3/ 2, 0)o (Fig. 4 right part). The q2o ¼ ð0; qy ; 0Þ magnetic domain is just evolving and the ð2; 2 qy ; 0Þ satellite becomes visible on the right side of the central peak (2, 3/2, 0)o. At 40 K its intensity has increased and has become well resolved and slightly shifted to higher 2y angles showing a decrease of qy at 34 K, while the central peak is not distinguishable from the background. The opposite shift direction is observed for the ð0; 2 þ qy ; 0Þ satellite on the left part of the (2, 3/2, 0)o peak. In the temperature range 14–16 K around the Tc0 ; Tic1 transition the (2, 2, 0)-q2o satellite splits into two each one corresponding to a distinct domain. In a part of the sample with orthorhombic symmetry the q2o wave vector length continues
8300
521
HoFe4Ge2
7500
412
52K
440
6700
49K Intensity (a.u.)
5900
332
5100 4300 3500
36K
2700 1900 1100 300 17.83
17.89
17.95
18.01
18.07
18.13
18.19
18.25
18.31
18.37
18.43
18.37
18.43
8300
521
7500
HoFe4Ge2
412
4.2K
Intensity (a.u.)
6700
14K
5900
440
5100
332
4300 3500
21K
2700 1900 1100 300 17.83
17.89
17.95
18.01
18.07
18.13
18.19
18.25
18.31
2θ (deg.)
Fig. 6. Detail of the XRPD patterns for various temperatures displaying the pronounced peak splitting of ðh h lÞ tetragonal reflections towards ðh k lÞ orthorhombic reflections in the re-entrant transition: tetragonal (HT)-orthorhombic (IT)-tetragonal (LT) observed in HoFe4Ge2.
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
2.4. Magnetic refinements, D2B data
1,1/2,1
1,3/2,1,2,5/2,0
0,5/2,0 2,3/2,0
110 0,3/2,0 020110+ 1,3/2,0
20
40K
I I
200
30
40
50
60
70
80
30
40
50
60
70
80
60
70
80
obs cal
I -I
obs cal
131-
111111
220
110+
020- 110
110-
200
0,q ,0 y
36K
020+ 220-
200, 020
20
200+-
0
10
Intensity (a.u.)
Symmetry analysis allows in cases of no strong symmetry reduction upon magnetic ordering a considerable reduction in the number of free parameters in the refinements. In our case the star of the wave vector k10 ¼ ðx; x; 0Þ found for To15 K referring to the tetragonal P4/mmm (Q) Bravais lattice has four arms [13]. The symmetry operations of the group P42/mnm that leave the wave vector arm (1/4, 1/4, 0) invariant are Gk ¼ fE; mz ; mxxz ; 2xx0 g: Symmetry analysis leads to a splitting of the Ho site 2b into two sites and a splitting of the Fe site 8i into 4 sites, as listed in Table 2a. The moments of these six sites (orbits) may have different values and directions. Atoms belonging to the same orbit may have the same
0
10
Intensity (a.u)
200,020
λ=2.398Å 1,1/2,0 110-
200
000+-q y 0,1/2,0
Intensity (a.u)
42 K HoFe4Ge2
2,1/2,0
to decrease on cooling and the (2, 2, 0)-q2o satellite moves to lower angles see Fig. 3 (bottom part). However for the major part of the sample undergoing the re-entrant transition to the tetragonal phase the q2o wave vector length jumps from qy ¼ 0:4419ð1Þ or E11/25 r.l.u to the lock-in value q1t ¼ ð0; 1=2; 0Þt : The intensity of the (2, 3/2, 0)t peak continues to increase at the cost of the q2o structure. At 1.5 K the amount of the tetragonal phase reaches 80% see Fig. 4 (left part). For comparison Fig. 6 shows the de-convolution of well-resolved X-ray reflections for characteristic temperatures of the re-entrant structure transitions reproduced from Ref. [9] allowing a direct comparison with the simultaneously occurring reentrant behaviour of the magnetic ordering as observed by neutron diffraction.
130,310 111+
130
0
10
20
30
40
50 2θ deg
Fig. 7. Observed calculated and difference diagrams of HoFe4Ge2 below and above the first order magnetic transition at Tic2 ¼ 40 K. For TN > T ¼ 42 K > Tic2 the magnetic ordering of the Cmmm phase is associated with the wave vector q1o ¼ ð0; 1=2; 0Þ (top part) traces of the q2o ¼ ð0; qy ; 0Þ become visible. For Tc0 ; Tic1 oT ¼ 36 KoTic2 the magnetic ordering is described by q2o ¼ ð0; qy ; 0Þ (bottom part) while in the intermediate range the two phases coexist in variable portions (middle part).
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
131
1500 2000
HoFe Ge , 1.5K 4
2
λ=2.398A
q =(0,q ,0)
I
q =0.4346(2)
I -I
1,1/2,1
1,3/2,1,2,5/2,0
1,1/2,0 110110 0,3/2,0 020110+ 1,3/2,0 200,020 2,1/2,0 0,5/2,0 2,3/2,0
y
000+-q 0,1/2,0
Intensity (a.u.)
I
1t
1000
500
q =(0,0.5,0) 2o
y
obs cal obs
y
cal
0
-500
20
40
60
80
100
120
140
1950
18K 1200 y
0,q ,0
q =0.4416(1)
I
y
y
obs cal
110+
110-
obs
cal
131-
200,020 200+020+ 220220 111111 130,310 111+
I -I
110 020-
Intensity (a.u.)
I
2o
800
400
q =(0,q ,0)
0
20
40
60
80
100
120
140
2θ deg Fig. 8. Observed calculated and difference diagrams of HoFe4Ge2 below and above the first-order structural transition occurring at Tc0 ; Tic1 ¼ 15 K. For Tic2 > T ¼ 18 K > Tc0 , Tic1 the magnetic ordering of the Cmmm phase is associated with the wave vector q2o ¼ ð0; qy ; 0Þ (top part). For T ¼ 1:5 KoTc0 ; Tic1 the magnetic ordering is described by q1t ¼ ð0; 1=2; 0Þ for the tetragonal LT phase in the orthorhombic setting and by q2o ¼ ð0; qy ; 0Þ for the orthorhombic part of the sample (bottom part).
moment value and their moment direction is related by symmetry. However due to the g-angle deformation found for T > 15 K one has to use in the description of the magnetic structure the four times larger orthorhombic cell ðat O2; 2at O2; ct Þ where (0, 1/2, 0) is an anti-translation (Pb lattice).
In view of the presence of incommensurate regions the magnetic calculations were carried out referring to the same Cmmm cell and using the wave vector description. The coordinates for the 65 K in the Cmmm cell are xo ¼ ðx yÞt =2; yo ¼ ðx þ yÞt =2; zo ¼ zt (see Table 1). Refined magnetic
ARTICLE IN PRESS 132
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
(a)
(b)
(c) Fig. 9. (a) The lock-in (LT) tetragonal phase, q1t ¼ ð0; 1=2; 0Þ; at 1.5 K. (b) The lock-in (HT) orthorhombic, q1o ¼ ð0; 1=2; 0Þ; phase at 42 K. (c). The incommensurate (IT) magnetic structure at 18 K, q2o ¼ ð0; qy ; 0Þ:
parameters obtained from high-resolution D2B data for a few characteristic temperatures are summarised in Table 3. Fig. 7 displays refinements above (42 K) and below (36 K) the first-order incommensurate to commensurate (on heating transition at Tic2 ¼ 40 K and Fig. 8 displays
refinements above (18 K) and below (1.5 K) the first order magnetostructural transition at Tc0 ; Tic1 ¼ 15 K. In Table 2 a, the notation of the orbits is with roman numbers when one refers to the tetragonal cell i.e. IFe is used for orbit one, which comprises
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
the atoms 1 and 7. When one refers to the orthorhombic conventional cell (see Fig. 9a) the orbits are labelled by primed Arabic numbers Fe10i ; Fe20i ; Fe30i ; Fe40i i.e. Fe10i ; where i ¼ 1; 4; denotes the four atoms of the orbit that are symmetry related. In this way Fe atoms situated at the corners of the same compact tetrahedron belong to different orbits but have the same i subscript. We summarise below the results of symmetry analysis in the conventional Cmmm cell. 2.4.1. Irreducible representations of the propagation vector group Gq for q ¼ ð0; qy ; 0Þ in Cmmm The propagation vector q ¼ ð0; qy ; 0Þ is at the interior of the Brillouin Zone. The star of the propagation vector is constituted by the two arms {q, q}. The present analysis is common for both the commensurate qy ¼ 1=2 and the incommensurate qy ¼ 0:43 magnetic structures. The propagation vector group is Gq ¼ Cm2m ¼ fE; 2y ; mz ; mx g: There are four real irreducible representations (Irreps) of dimension 1 as given in Table 2b. *
*
The Fe atom sites 8p and 8q (Table 1): Fe1 and Fe2 are split in 2 orbits. The total number of sites has consequently increased from 4 to 6 as found in the tetragonal system.
The data about atoms are given in Table 2c (only the content of a primitive cell is considered, the moment of atoms related by the centering translation t ¼ ð1=2; 1=2; 0Þ is obtained from the phase factor expf2piq:tg: The symmetry analysis for all sites has shown that the decomposition of the global magnetic representation Gm in Irreps is the same for the two Ho sites and the same for all Fe sites: Gm ¼ G2 þ G3 þ G4 (Ho sites) and Gm ¼ G1 þ 2G2 þ G3 þ 2G4 (Fe sites). The Fourier coefficients deduced from the basis functions are given for each Irrep. The coefficients ðu; v; wÞ are, in principle different for each site but the magnetic modes are the same for all Fe orbits for the corresponding Irreps. Ho may have only uniaxial structures. For the Fe sites Table 2c shows that the G and G3 representations allow only w Fourier components, while the G2 and G4 allow u and v components.
133
2.4.2. The commensurate phases: q1o ¼ ð0; 1=2; 0Þ or q1t ¼ ð0; 1=2; 0Þ As shown in Table 2c due to their highsymmetry position, the direction of the Ho moments is constrained to one of the orthorhombic mirror planes mx ; my ; mz : The Ho atoms may have a single-moment component, which is perpendicular to any of the mirror planes. Each of the four Fe orbits comprises two atoms related by the mirror operation across the plane mx : The moment components of the atom pairs belonging to the same orbit are symmetry related. For instance, atoms Fe101 and Fe102 of the first orbit have the same moment components parallel to the antimirror plane m0x but opposite signs for the component perpendicular to that plane. These relations were used in the parameter space throughout the refinements. However the refinements indicated that the magnetic structure could not be described by a single representation. In fact the observed magnetic structures for all temperature ranges are well described by a mixture of the representations G2 and G3 for all magnetic sites, so that the magnetic symmetry (in the case of a commensurate structure qy ¼ 1=2) is lowered to monoclinic and the number of free parameters increased. The mirror plane mx has negative character in both representations: it becomes an anti-mirror plane m0x : The refined moment values at 1.5 K ðuo ; vo ; wo Þ for all atoms, are included in Table 2c see also Fig. 9a. The other refined parameters are given in Table 3 and in Fig. 8 (top part). The results given in Table 3 for q1o ¼ ð0; 1=2; 0Þ at 42 K (Fig. 9b) and the tetragonal phase at 1.5 K q1t ¼ ð0; 1=2; 0Þ refer to the same C-cell with tetragonal constraints for the structural parameters for the latter phase. Additional constraints were added after analysing the results found by simulated annealing runs for the magnetic structures. This is illustrated also in Figs. 9a and b. Translationally equivalent atoms in the enlarged cell have the same moment value but may have different signs. For instance, the sign change (+ + ) of four translationally equivalent Ho atoms along the orthorhombic diagonals (1,71,0) or at or bt can be achieved by choosing the phase p=4 (or (+ +) for p=4) for the origin Ho
ARTICLE IN PRESS 134
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
atom at (0, 0, 1/2) and by applying Eq. (3) to Rn ðn ¼ 0; 3Þ see next section. Atoms in planes perpendicular to the wave vector have the same phase. These relationships can also be verified in Figs. 9a and b. The simulated annealing runs, as discussed in the appendix, showed that the data were not sensitive to some of the parameters and resulted to large errors in the moment value and the angle y with the c-axis for the Fe atoms. As a consequence the moment values and the y angles of the atom pair (Fe10i ; Fe20i ) located at z ¼ 0 belonging to different orbits, and those of the atom pair (Fe30i ; Fe40i ) at z ¼ 1=2 were set to the same values. Here we would like to note that powder diffraction cannot distinguish between y and y in order to derive the absolute configuration. In the refinement of the 42 K data, Fig. 7 (top part) all moments were constrained to the (0 0 1) plane (y ¼ 90 in Table 3 and Fig. 9b). The refined magnetic moments of the Fe sublattices at 42 K are 0.43(3) mB. The magnetic moments of the Fe atoms located at the corners of a compact tetrahedron, have a planar canted arrangement but a different one to that found in the orthorhombic (Pnnm) YFe4Ge2 at 1.5 K [6] where the mutual moment direction of the atom pairs belonging to the same orbit (Fe1, Fe2) and (Fe3, Fe4) are anti-parallel but point to perpendicular directions restoring the local tetragonal symmetry. In the Cmmm symmetry found in HoFe4Ge2 atom pairs (Fe101 ; Fe201 ) and (Fe301 ; Fe401 ) belong to different orbits and their angles are 52 and 77 , respectively. The moment of Fe101 is almost parallel to Fe401 and that of Fe201 parallel to Fe301 : The rare earth ordered moment values are 1.52(7) and 2.43(7) mB for Ho1 and Ho2 respectively and point along the b-axis. The local surrounding of both Ho sites within the plane ðx y 0Þ is a rectangular oblong with 4 nearest Fe atom neighbours located at its corners and Ho in the centre. The Ho1 atom is surrounded by the Fe30 and Fe40 orbits (full circles), the moments of the two Fe atoms of the former orbit are almost parallel to the b-axis and anti-parallel to the Ho1 moments while the moments of the two Fe atoms of the Fe40 orbit are anti-parallel and almost perpendicular to the Ho1 moment. A similar situation is found for the moment of the
Ho2 atom surrounded by the Fe10 and Fe20 orbits (open circles). The moment of Ho2 has an opposite direction to that of the Fe moments of orbit Fe20 and is canted with respect to the moments of Fe10 . The refined magnetic canted model with the wave vector q1o has only monoclinic symmetry Pbm0 11. The reliability factors RB ¼ 2%; Rm ¼ 9% indicate a satisfactory agreement between the model calculations and observations. The 1.5 K refined parameters led to a threedimensional canted arrangement Fig. 9a of the Fe moments with eight sublattices similar to the arrangement found in the ErFe4Ge2 [7] compound at low temperature. The projection of this structure in the tetragonal plane, shown in Fig. 9a, is not very different from that of the 42 K structure shown in Fig. 9b. In spite the tetragonal lattice symmetry, the magnetic configuration remains orthorhombic as found in the Er compound. The ordered moment values of the Ho1 and Ho2 sites of 2.38(7) and 5.34(2) mB, respectively, are very different and strongly reduced relatively to the free ion value for Ho3+, gJ(mB)=10 mB presumably due to crystal field effects, but this fact could also have a contribution from frustration in the interactions Ho–Fe. More direct information illustrating the difference in crystal field interaction at the two rare earth sites . was obtained by 161Dy and 166Er Mossbauer spectroscopy on the corresponding RFe4Ge2 compounds [4]. The Ho moments are almost confined to the tetragonal plane y ¼ 83ð4Þ for Ho1 and 90 for Ho2 in variation with the angles found in ErFe4Ge2 of 55 and 42 , respectively. The moment values of the four Fe sublattices are within error equal 0.97(5) mB for Fe10 and Fe20 and 0.99(5) mB for Fe30 and Fe40 and are inferior to the corresponding value of 2.4 and 1.4 mB found in ErFe4Ge2. 2.4.3. The refinement of the incommensurate phase: q2o ¼ ð0; qy ; 0Þ For the incommensurate phase the number of orbits is the same but the number of free parameters increases depending on the model. Once the moment components (Fourier coefficients) within the orthorhombic cell are defined the
ARTICLE IN PRESS P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
magnetic moment direction of any translationally equivalent atom may be calculated. The magnetic moment value lnj of the jth atom in the nth cell at position Rn ; is given by the relation: X S expf2piqRn g lnj ¼ fqg qj X ¼ ½R cosf2pðqRn þ uqj Þg 1=2fqg qj þ Iqj sinf2pðqRn þ uqj Þg ;
ð1Þ
where the general Fourier coefficients are: Sqj ¼ 1=2fRqj þ iIqj gexpf2pijqj g ¼ Sqj
ð2Þ
for a sinusoidal modulated structure (single pair q,q and Iqj ¼ 0; Rqj ¼ moj =2z) this expression is lnj ¼ l0j z cosð2pq Rn þ jj Þ;
ð3Þ
where loj is the amplitude of the moment, jqj ðjj Þ is the phase relative to the origin, and z is a unit vector along the moment direction. The results given in Table 3 for the incommensurate orthorhombic phase with q2o ¼ ð0; qy ; 0Þ with qy ¼ 0:44164ð8Þ at 18 K refer to the C-cell. The simplest possible model is that of a multi-axial sine wave modulated structure displayed in Fig. 9c. The starting parameters for the sine wave model were those obtained by the simulating annealing method, see Fig. 10 in the appendix for the 18 K data. Results given in Table 3 and Fig. 8 (bottom part) for the same temperature are not very different to the starting parameters, which greatly support the use of this method in refinements. The reliability factors RB ¼ 3%; Rwp ¼ 13% and Rm ¼ 7% show a satisfactory agreement between calculated and observed data. In this model the moment of each sublattice has a distinct orientation but different from that found in the commensurate phases as can be seen in Fig. 9c for a few cells. 2.4.4. Comparing the commensurate and incommensurate phases The differences between the two structures consist of the ordered moment values and the moment direction along the c-axis. The local ordered moments of each sublattice point always to the same direction. This means that the incommensurate phase is also canted but the moment value varies as a sine wave function in
135
the direction of the wave vector. For the Ho1 and Ho2 sites the amplitude of the wave (Fourier coefficient) is 4.8(1) and 5.38(9) mB, respectively at 18 K. For the Fe10 and Fe20 sublattices the amplitude is 2.08(4) mB and for the Fe30 and Fe40 sublattices it is slightly larger, 2.38(5) mB. Comparing Figs. 9a and c one observes that in the incommensurate phase the Mz components of the four Fe atoms at the corners of a tetrahedron have the same sign. In the 42 K commensurate phase shown in Fig. 9b, Mz is zero for all atoms in the cell, as the moments were a priori constrained to lie within the (0 0 1) plane because of large errors in the z-components. In the 1.5 K structure (Fig. 9a) one finds that the Mz moment components of the atom pairs at z ¼ 0 (Fe10 and Fe20 ) belonging to different orbits have the same sign but opposite to that of the atom pairs (Fe30 and Fe40 ). Here it should be noted that the ordered moment value of the commensurate tetragonal phase given in Table 3 is obtained from the Fourier coefficient by fixing the origin of the wave to 7p=4 in order to obtain (via Eq. (3)) equal moments for atoms related by the non-primitive translation. Contrary for an incommensurate amplitude modulated phase the structure depends on the choice of origin, which cannot be defined by diffraction. In our case it was set to zero therefore for comparison of the amplitude of the wave (Fourier coefficient) of the coexisting incommensurate and commensurate phases at 1.5 K, the moment values given for the q1t phase have to be scaled by O2 (set phase to zero). This leads to 3.36(9) mB for Ho1 and 7.55(3) mB for Ho2. Disregarding the different crystal symmetry, these values are within 3s as expected larger to those of the coexisting incommensurate phase. Here we would like to show in detail that it is the arrangement along the c-direction, which explains the behaviour of the zero point satellite of the commensurate structures. Many times when dealing with complex problems it appears useful to analyse more carefully the singularities found in the experiment. The main difference in the aspect of the diffraction pattern of the commensurate (q1o and q1t) and incommensurate q2o phases, as discussed in previous sections, was the intensity of the zero point satellite, which is non-zero for the
ARTICLE IN PRESS 136
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
COMM HoFe4Ge2 P42 /mnm
D2B
18K Integrated intensities Nre Cry Uni Cor Opt Aut
!Job Npr Nph Nba Nex Nsc Nor Dum Iwg Ilo Ias Res Ste
1 0 1 0 0 0 0 0 0 0 0 0 0 16 3 0 0 0 1 16 !Number of refined parameters !--------------------------------------------------------------------! Data for PHASE number: 1 ==> Current R_Bragg for Pattern# 1: 15.35 !--------------------------------------------------------------------HoFe4Ge2 magnetic 18K !Nat Dis Mom Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ Nvk Npr More 6 0 0 0.0 0.0 1.0 -1 4 -1 0 0 0.00 -1 0 0 C -1 20% but become well defined at lower temperatures.The solution found from the 237 reflections after 150 cycles from the SAnn refinement was implemented in the refinement of the 18 K entire profile and led to the results given in Fig. 9c and the parameters given in Table 3. References [1] Ya.P. Yarmoluk, L.A. Lysenko, E.I. Gladyshevski, Dopovidi Akad. Nauk Ukr. RSR Ser. A 37 (1975) 279. [2] O.Ya. Oleksyn, Yu.K. Gorelenko, O.I. Bodak, 10th International. Conference on Solid Compounds of Transition Elements, Munster . May 21–25, 1991. P-254-FR, SA. [3] A.M. Mulders, P.C.M. Gubbens, Q.A. Li, F.R. de Boer, K.H.J. Buschow, J. Alloys Compd. 221 (1995) 197. [4] P.C.M. Gubbens, B.D. van Dijk, A.M. Mulders, S.J. Harker, K.H.J. Buschow, J. Alloys Compd. 319 (2001) 1. [5] P.C.M. Gubbens, et al., HoFe4Ge2 unpublished results. [6] P. Schobinger-Papamantellos, J. Rodr!ıguez-Carvajal, K.H.J. Buschow, J. Magn. Magn. Mater. 236 (2001) 14. [7] P. Schobinger-Papamantellos, J. Rodr!ıguez-Carvajal, G. Andr!e, C.H. de Groot, F.R. de Boer, K.H.J. Buschow, J. Magn. Magn. Mater. 191 (1999) 261. [8] P. Schobinger-Papamantellos, J. Rodr!ıguez-Carvajal, K.H.J. Buschow, E. Dooryhee, A.N. Fitch, J. Magn. Magn. Mater. 210 (2000) 121. [9] P. Schobinger-Papamantellos, J. Rodr!ıguez-Carvajal, K.H.J. Buschow, E. Dooryhee, A.N. Fitch, J. Magn. Magn. Mater. 250 (2002) 225;
ARTICLE IN PRESS 142
P. Schobinger-Papamantellos et al. / Journal of Magnetism and Magnetic Materials 280 (2004) 119–142
P. Schobinger-Papamantellos, J. Rodr!ıguez-Carvajal, K.H.J. Buschow, E. Dooryhee, A.N. Fitch ESRF Highlights, 2003, p. 41. [10] J. Rodr!ıguez-Carvajal. Physica B 192 (1993) 55 (The manual of FullProf can be obtained from a Web browser looking at ftp://ftp.cea.fr/pub/llb/divers/ fullprof.2k/). [11] A. Szytula, S. Baran, J. Leciejewics, B. Penc, N. Stusser, . A. Yon fan Ding, J. Zygmunt, Z. Zukrowski, J. Phys.: Condens. Matter 9 (1997) 6781. [12] P. Schobinger Papamantellos, J. Rodr!ıguez-Carvajal, K.H.J. Buschow, J. Alloys Compd. 284 (1999) 42.
[13] O.V. Kovalev, Representations of the crystallographic space groups, in: H.T. Stokes, D.M. Hatch (Eds.), Gordon and Breach Science Publishers, Switzerland, Australia, Belgium, France, 1993. [14] M. Hidaka, M. Yoshimura, N. Tokiwa, J. Akimitsu, Yong Jun Park, Jae Hyun Park, sung dae Ji, Ki Bong Lee. Phys. Stat. Sol. (b) 236 (3) (2003) 570. [15] J. Rodr!ıguez-Carvajal, Mater. Sci. Forum 378–381 (2001) 268. [16] S. Kirkpatrick, C.D. Gellat Jr., M.P. Vecchi, Science 220 (4598) (1983) 671. [17] A. Corana, M. Marchesi, C. Martini, S. Ridella, ACM Trans. Math. Software 13 (1987) 262.
