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Spontaneous magnetostriction and thermal expansibility of TmFeO3 and LuFeO3 rare earth orthoferrites . b, Joachim Kuszc, Andrzej W. Pacynad,* Andrzej Bombika, Horst Bohm a

Faculty of Physics and Nuclear Techniques, Stanisław Staszic Academy of Mining and Metallurgy, ul. Władysława Reymonta 19, ! Poland 30-059 Krakow, b Institut fur . Mainz, Johann Joachim Becher Weg 21, 55099 Mainz, Germany . Geowissenschaften, Johannes Gutenberg-Universitat c Institute of Physics, University of Silesia, ul. Uniwersytecka 4, 40-007 Katowice, Poland d ! ! Poland Henryk Niewodniczanski Institute of Nuclear Physics, ul. Walerego Eljasza Radzikowskiego 152, 31-342 Krakow, Received 8 November 2000; received in revised form 12 December 2001

Abstract At low temperatures the TmFeO3 orthoferrite shows a strong spontaneous magnetostriction generated by rare earth Tm atoms. This eﬀect essentially depends on the temperature and orientation of the weak ferromagnetism vector of the Fe sublattice and reaches largest values for the directions [1 0 0] and [0 0 1]. The spin reorientation process in TmFeO3 marks itself mainly in the change of the signs of the linear thermal expansion coeﬃcients in the directions mentioned above. Some additional singularities of thermal dependencies of expansion coeﬃcients are observed below 20 K, which may be due to a magnetic compensation eﬀect in the investigated orthoferrite. r 2001 Elsevier Science B.V. All rights reserved. PACS: 61.10.Nz; 75.30.Gw; 75.40.Cx; 75.80.+q Keywords: Rare earth orthoferrites; Single crystal; Magnetostriction; Thermal expansibility

1. Introduction The atomic structure of the rare earth orthoferrites REFeO3 (REFrare earth element) is described by the Pbnm (D16 2h ) space group and referred to as orthorhombic-distorted perovskite [1,2]. Their unit cell contains Z ¼ 4 chemical (REFeO3) formulae whose atom’s space distribution is shown in Table 1. An important feature, occurring in numerous compounds of this family *Corresponding author. Tel.: +48-12-370222; fax: +48-12375441. E-mail address: [email protected] (A.W. Pacyna).

and following from their symmetry, is the presence of a magnetic order called weak ferromagnetism [3,4]. This term describes such a type of magnetic ordering in which a small declination (B0.51) from collinear antiferromagnetic arrangement of spins generates a nonzero net magnetic moment of the elementary cell. In turn, this magnetic moment is a source of a small magnetisation that is registered in compounds of the orthoferrite family [1,2]. In rare earth orthoferrites the exchange antiferromagnetic interaction between iron magnetic moments (Fe–Fe) is exceptionally strong, which results in high temperatures (about 620–760 K) of transition to paramagnetic phases TN1 : On the

0304-8853/01/$ -see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 1 3 6 - 6

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Table 1 Atom positions in elementary cell of REeO3 orthoferrites (space group Pbnm) Fe+3 (4(b)) RE+3 (4(c)) O2 (4(c)) O2 (8(d))

(0, 1/2, 0) (1=2 x; 1=2 þ y; 1/4) (1=2 x; y þ 1=2; 1/4) 7(1=2 þ x; 1=2 y; z)

(0, 1/2, 1/2) (1=2 þ x; 1=2 y; 3/4) (1=2 þ x; 1=2 y; 3/4) 7(x; y; 1=2 þ z)

other hand, the mutual exchange interaction of rare earth magnetic moments (RE–RE) is about three orders of magnitude smaller than the (Fe– Fe) interaction, which is consistent with the Ne! el temperatures for that sublattice (TN2 ). These temperatures do not exceed 9 K in these compounds [1,2]. Various magnetic ordering temperatures, diﬀerent for Fe sublattice (TN1 ) and rare earth magnetic moments (TN2 ) point to a rather weak magnetic mutual bonding (Fe–RE) of the mentioned subsystems. Rare earth orthoferrites distinguish themselves by complex phase diagrams with compensation temperatures, Morin’s type phase transitions, spin reorientation process, critical ﬁelds and other observed singularities [1,2,5]. The above mentioned spin reorientation process (strictly speaking spontaneous spin reorientation process) can be considered as a continuous rotation of the Fe sublattice antiferromagnetism vector between the [1 0 0] and [0 0 1] crystal directions; of course the small magnetisation vector (weak ferromagnetism vector) rotates simultaneously from [0 0 1] to [1 0 0]. Spontaneous spin reorientation occurs only in those orthoferrites whose rare earth atoms carry nonzero magnetic moments. In the opposite case, i.e. when rare earth atoms do not have any localised magnetic moments, as happens for LaFeO3, EuFeO3, LuFeO3 and for the isostructural orthoferrites YFeO3, spin reorientation can be caused by magnetic ﬁeld [1,2]. It is then called the induced spin reorientation process. The orthoferrite properties have been the subject of intensive studies on single crystal and polycrystalline substances, carried out by numerous research groups using various experimental techniques and theoretical concepts. Some of these research projects have been very successful; however, many issues still remain unsolved and need

(1/2, 0, 1/2) (x; y; 3/4) (x; y; 3/4) 7(1=2 x; 1=2 y; 1=2 z)

(1/2, 0, 0) (x; y; 1/4) (x; y; 1/4) 7(x; y; z)

further investigation. State-of-the-art measurement methods and perfected experimental devices allow for a more accurate determination of the physical properties of orthoferrites and a more precise investigation of the dependencies between these properties. As a consequence, analyses of these more exact data sometimes make it possible to ﬁnd new unexpected correlations between examined parameters, which leads to a deeper understanding of physical processes in matter. The thermal study of ErFeO3 and TmFeO3 orthoferrites, performed recently [6], is a case in point. In this investigation, it was shown that the modern microcalorimetry technique permits the direction of very small changes in speciﬁc heat, such as the subtle changes, which accompany the spin reorientation process. In this context, it is interesting to ask to what extent magnetic phase transitions observed in the rate earth orthoferrites correlate with their lattice parameter. As the lattice parameters can be determined with high precision by modern X-ray methods, both for single crystal and powder specimens of a substance, it should be possible to ﬁnd some correlations between structural and magnetic properties in orthoferrites. This idea stimulated us to undertake the study of the cell parameter temperature dependencies for two rare earth orthoferrites: TmFeO3 and LuFeO3. The ﬁrst chosen compound (TmFeO3) is characterised by a plurality of magnetic phases and transitions between them. Neutron diﬀraction investigations have shown that the Fe sublattice is ordered magnetically with a main antiferromagnetic component and another, considerably weaker, ferromagnetic component. At temperatures below the lower limit of the spin reorientation range (Tt1 E80 K), the anti-ferromagnetism vector of this structure is oriented according to the c parameter of the primitive cell, while the ferro-

A. Bombik et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 443–453

magnetism vector is oriented along the a parameter. For temperatures exceeding the upper limit (Tt2 E95 K) of the spin reorientation range mentioned above, an opposite conﬁguration is observed: antiferromagnetism and ferromagnetism vectors are oriented along a and c parameters, respectively. This magnetic order of the Fe sublattice remains stable up to the transition temperature to the paramagnetic phase (TN1 ¼ 6302633 K) [7,8]. In the temperature range Tt1 2Tt2 ; which separates the two above mentioned magnetic phases, the spontaneous spin reorientation process takes place. It is worth remembering that over this temperature range a change of the sign of the intersublattice interaction (Fe–RE) occurs (from ‘‘antiferro’’ at ToTt1 temperatures, to ‘‘ferromagnetic’’ at T > Tt2 temperatures) [1,2,5,7]. However, down to 1.6 K, no magnetic order of the Tm subsystem was observed either in a neutron diﬀraction experiment [9] or in the recent calorimetric measurements (even down to 0.8 K) [6]. In the neighbourhood of 20 K disappearance of the net magnetic moment is observed. This eﬀect is interpreted as evidence of magnetic compensation of the weak ferromagnetic Fe sublattice moment and partial magnetic polarisation of paramagnetic Tm subsystem, the latter one being induced by the former [7,8]. At low temperatures, the phase diagram of LuFeO3 is considerably simpler than for the TmFeO3 orthoferrite. As Lu3+ does not carry any localised magnetic moment, this orthoferrite shows neither magnetic compensation nor spin reorientation eﬀects. The Fe subsystem magnetic order, with the weak ferromagnetic vector of iron sublattice oriented according to the [0 0 1] direction, is stable in this orthoferrite from the lowest temperatures to its Ne! el temperature (TN ¼ 6232625 K) [1,2,5]. Thus the magnetic structure of LuFeO3 is the same as the magnetic order of TmFeO3 iron sublattice for temperatures above the spin reorientation temperature. For this reason, LuFeO3 can be considered as a kind of a standard in the comparative analysis of cell parameter temperature dependencies of both examined orthoferrites. Thus the diﬀerences in corresponding cell parameters of both compounds,

445

expected to show in low temperatures (ToTt2 ), should express the inﬂuence of the magnetic character of rare earth ions (Tm) on structural properties, that is to say, the magneto-elastic eﬀect in TmFeO3.

2. Experimental details Lattice parameter measurements of TmFeO3 and LuFeO3 orthoferrites have been carried out on single crystal specimens. The samples were sent to Cracow several years ago by S. Shtrikman from the Weizmann Institute of Science, Rehovoth, Israel. The single crystal TmFeO3 samples were also used in our earlier investigations [8]. Measurements were performed using the graphite-monochromatised Cu Ka radiation with Enraf-Nonius rotating anode and a four-circle Huber diﬀractometer with a 250 mm w-circle in front of the rotating anode. The diﬀractometer was controlled by a PC with the STOE STADI4 programme [10]. The measurement device was equipped with a two-stage closed-cycle helium low-temperature attachment (CTI-Cryogenics). Cell parameter temperature dependencies were investigated in the 10–290 K interval with temperature stability within 0.1 K. The reﬁnement of the cell parameters were carried out using about 50 reﬂections with 2y values, including Friedel pairs at both sides of the primary beam. An o scan was carried out at 72y and o: The centre of gravity was determined for both scans and 2y value was taken as the diﬀerence of the two o centres. The above procedure is capable of ensuring that the results are free of zero-point errors, absorption eﬀects and systematic errors resulting from a miscentring of the crystal.

3. Results and discussion The measured values of cell parameters are shown in Tables 2 and 3, while their temperature dependencies are presented in Figs. 1–3. Additionally, the unit cell volume temperature dependence is shown in Fig. 4. As it could be expected, temperature plots are more complex for the

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Table 2 Cell parameter and cell volume data of TmFeO3 orthoferrite T (K)

( a (A)

( b (A)

( c (A)

( 3) V (A

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 180 200 220 240 260 280 300

5.2445(25) 5.2440(30) 5.2436(24) 5.2437(23) 5.2434(23) 5.2433(23) 5.2432(23) 5.2430(22) 5.2430(21) 5.2426(22) 5.2427(21) 5.2428(21) 5.2429(21) 5.2430(21) 5.2431(20) 5.2434(19) 5.2439(19) 5.2444(18) 5.2453(17) 5.2459(19) 5.2468(18) 5.2476(13) 5.2486(13)

5.5646(19) 5.5649(19) 5.5653(19) 5.5656(18) 5.5659(17) 5.5664(18) 5.5668(17) 5.5673(17) 5.5680(16) 5.5681(17) 5.5689(16) 5.5691(16) 5.5698(16) 5.5699(16) 5.5707(16) 5.5710(15) 5.5718(15) 5.5723(14) 5.5732(13) 5.5739(13) 5.5746(14) 5.5755(10) 5.5763(10)

7.5769(23) 7.5772(24) 7.5770(23) 7.5766(22) 7.5760(21) 7.5760(22) 7.5757(21) 7.5756(21) 7.5759(20) 7.5757(20) 7.5760(20) 7.5760(20) 7.5764(20) 7.5764(20) 7.5769(19) 7.5772(18) 7.5779(18) 7.5786(17) 7.5798(16) 7.5808(17) 7.5820(17) 7.5832(12) 7.5846(12)

221.12(10) 221.12(11) 221.11(10) 221.11(10) 221.10(10) 221.12(10) 221.12(10) 221.13(9) 221.16(9) 221.15(9) 221.19(9) 221.20(9) 221.25(9) 221.25(9) 221.30(9) 221.34(8) 221.41(8) 221.47(8) 221.58(7) 221.66(7) 221.76(8) 221.87(6) 221.99(6)

Fig. 1. LuFeO3 and TmFeO3 orthoferrite a parameters.

Table 3 Cell parameter and cell volume data of LuFeO3 orthoferrite T (K)

( a (A)

( b (A)

( c (A)

( 3) V (A

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 180 200 220 240 260 280 300

5.2103(8) 5.2102(8) 5.2102(8) 5.2103(8) 5.2105(8) 5.2107(8) 5.2107(8) 5.2111(8) 5.2113(8) 5.2115(8) 5.2118(8) 5.2122(8) 5.2196(8) 5.2130(9) 5.2133(8) 5.2136(8) 5.2147(8) 5.2157(9) 5.2168(9) 5.2180(9) 5.2192(9) 5.2206(8) 5.2215(8)

5.5559(14) 5.5562(13) 5.5561(13) 5.5561(13) 5.5563(13) 5.5566(13) 5.5567(14) 5.5567(14) 5.5567(14) 5.5570(14) 5.5571(14) 5.5573(14) 5.5576(14) 5.5578(14) 5.5581(14) 5.5580(14) 5.5585(14) 5.5592(19) 5.5600(19) 5.5607(15) 5.5614(16) 5.5621(13) 5.5632(13)

7.5583(9) 7.5582(8) 7.5583(8) 7.5584(9) 7.5585(9) 7.5585(8) 7.5588(9) 7.5588(9) 7.5593(9) 7.5596(9) 7.5600(9) 7.5603(9) 7.5607(9) 7.5612(9) 7.5615(9) 7.5620(9) 7.5632(9) 7.5643(9) 7.5655(10) 7.5669(10) 7.5683(10) 7.5693(8) 7.5711(9)

218.79(5) 218.80(5) 218.80(5) 218.81(5) 218.83(5) 218.85(5) 218.86(5) 218.88(5) 218.90(5) 218.93(5) 218.96(5) 218.99(5) 219.03(5) 219.07(5) 219.10(5) 219.13(5) 219.22(5) 219.33(5) 219.44(6) 219.56(6) 219.68(6) 219.79(5) 219.93(5)

Fig. 2. LuFeO3 and TmFeO3 orthoferrite b parameters.

Fig. 3. LuFeO3 and TmFeO3 orthoferrite c parameters.

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Fig. 4. LuFeO3 and TmFeO3 orthoferrite unit cell volumes.

TmFeO3 compound than for the LuFeO3 one, especially below the spin reorientation temperature region. In the LuFeO3 orthoferrite, the cell parameter values, as well as elementary cell volume, increase monotonically with temperature. Such a dependence of cell parameters on temperature is consistent with a simple anharmonic model of thermal expansion of nonmagnetic materials. A diﬀerent situation is encountered for TmFeO3, where only parameter b shows a similarly regular run. The a and c parameters of this compound behave quite diﬀerently. At low temperatures (ToTt1 ), they decrease in spite of the increasing temperature and not until the spin reorientation region is surpassed does the monotonic increase occur (Figs. 1 and 3). A more regular increase of both parameters take place only above spin reorientation temperatures (T > Tt2 ). In order to compare the cell parameters of both compounds as well as to emphasise their diﬀering temperature behaviour, the diﬀerences between corresponding parameters have been calculated and are shown in Figs. 5–7. As can be seen from the presented ﬁgures, the diﬀerences depend on temperature and reach the highest absolute values for the a parameter. Of course, the temperature dependence plots have a similar character only for the a and c parameters, and a diﬀerent one for parameter b: For the ﬁrst two parameters, the corresponding diﬀerences have their greatest values for low temperatures, and then they decrease (but not linearly) with the temperature. The b

Fig. 5. TmFeO3 and LuFeO3 orthoferrite a parameter diﬀerences.

Fig. 6. TmFeO3 diﬀerences.

and

LuFeO3

orthoferrite

b

parameter

parameter diﬀerence behaves in an opposite way: it reaches the minimal value in the lowest temperature and grows as temperature increases, but about 150 K its value stabilises at an almost ( This means that for constant level (1.3 102 A). temperature exceeding 150 K the coeﬃcients of thermal expansion in the [0 1 0] direction should be approximately the same for both orthoferrites. The diﬀerent cell parameter temperature dependencies of both the orthoferrites studied, particularly visible in low temperatures, are interpreted as a consequence of the inﬂuence of magnetic moments of Tm atoms on the elastic properties

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of Fe sublattice antiferromagnetic order on lattice parameters a general formula [12] VðTÞCVð0Þ þ IV TFðyD =TÞ; where IV ¼ 3kB rgK; 3 FðtÞ ¼ 3 t

Fig. 7. TmFeO3 diﬀerences.

and

LuFeO3

orthoferrite

c

parameter

of TmFeO3. To put it more precisely, they express a mutual relationship between exchange and elastic interactions in rare earth orthoferrites. The inﬂuence of rare earth elements on the properties of orthoferrites is signiﬁcant at low temperatures and, as it is well known, leads to such eﬀects as the spin reorientation process mentioned above, the magnetic compensation eﬀect, some singularities in elastic, electric and optical properties, as well as other eﬀects [1,2,5]. No doubt the diﬀerent behaviour of the a; c and b parameters is connected with the fact that spin reorientation (Tt1 Tt2 ) in TmFeO3 occurs in the (x; z) plane, which is perpendicular to the b parameter. As it is commonly assumed, the last process results from the mutual relations between exchange interaction and temperature dependent anisotropy. Continuous lines in Figs. 1–4 denote the unit cell parameter temperature dependencies calculated on the basis of Debye–Gruneisen . approximation, without making allowance for magnetic interactions (only the phonon contribution to thermal expansion). Very often for such calculations one exploits the high-temperature part of experimental cell parameter temperature dependencies and the Debye temperature value (yD ) of the investigated compound (cf. Ref. [11]). Because of the lack of the correct value of the Debye temperature we calculated the required parameters for LuFeO3 orthoferrite in a slightly diﬀerent way. Assuming a small and weakly temperature-dependent inﬂuence

Z 0

t

x3 dx; ex 1

t ¼ yD =T

has been adapted. In this expression Vð0Þ is the volume of the elementary lattice cell at 0 K and IV is a constant value determined by the number r of atoms in primitive cell, Boltzmann’s constant kB ; Gruneisen’s parameter g; and the isothermal compressibility K ¼ 1ð1=VÞðqV=qpÞT : The above quoted formula gives the simplest theoretical description of the lattice thermal expansion of cubic crystals that contain only a single kind of atom. However, as it has been shown [12], it can also be used for single compounds and for even crystals of lower symmetry. Therefore, we used an analogous expression XðTÞ ¼ Xð0Þ þ IX TFðyD =TÞ to ﬁt the experimental data X (where X denotes a; b; c and V parameters) with least-squares reﬁnement of the three relevant parameters Xð0Þ; IX and yD : Obviously, in the last formula only yD and Xð0Þ (að0Þ; bð0Þ; cð0Þ; Vð0Þ) have their physical meanings. In the case of LuFeO3, the ﬁtting procedure was carried out in the 10–280 K temperature range and the best results were obtained with Xð0Þ ¼ ( 5.5562 A, ( 7.5583 A ( and 218.815 A ( 3 as 5:2104 A, well as yD ¼ 432; 419, 451.5 and 426.5 K for a; b; c and V parameters respectively. In our opinion (see Figs. 1–4) the use of this formula gives a rather good approximation to the experimental date on the crystal parameters of the LuFeO3 orthoferrite. Moreover, taking into account the polyatomic character of the unit cell and its appreciable X parameter anisotropy, the calculated yD temperature divergence does not seem surprising. The average value of cited temperatures is about 432 K and for this particular value (yD ¼ 432 K) the Xð0Þ ad IX parameters as well as the ﬁtting correlation coeﬃcients were recalculated (Table 4).

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A. Bombik et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 443–453 Table 4 Fitted parameters of XðTÞ ¼ Xð0Þ þ IX TFðyD =TÞ formula yD ¼ 432 K

( að0Þ (A)

( Ia (A/K)

( bð0Þ (A)

( Ib (A/K)

( cð0Þ (A)

( Ic (A/K)

( 3) Vð0Þ (A

( 3/K) IV (A

LuFeO3 (10–280 K) TmFeO3 (150–300 K)

5.2104 5.2410

6.6539 105 4.4037 105

5.5562 5.5687

3.8252 105 4.5115 105

7.5583 7.5738

7.2743 105 6.2297 105

218.815 221.041

6.4289 103 5.5026 103

In the case of the TmFeO3 orthoferrite the above procedure could not be applied in such a simple way because the magnetic contribution of Tm ions to the thermal expansion process cannot be neglected in this compound. Direct attempts to adapt the above expression lead to absurdly divergent yD temperature values (814, 434, 892 and 766 K), which obviously cannot be identiﬁed with the Debye temperatures. However, it is worth noting that the value related to parameter b (434 K) is close to the value (yD ¼ 419 K) obtained for the same direction in LuFeO3. Undoubtedly, it results from the already mentioned fact that in TmFeO3 the inﬂuence of magnetic rare earth sublattice on the b parameter is the smallest. Therefore, as the crystal structure of both investigated orthoferrites is the same, for the determination of X we have assumed the Debye temperature (yD ¼ 432 K), the average value yD temperatures obtained for LuFeO3. Besides, the ﬁtting calculations were made for the upper parts (150–300 K) of XðTÞ dependencies, as in this temperature range XðTÞ the dependencies are least disturbed by Tm magnetism. The Xð0Þ and IX parameters are also presented in Table 4. As it can be seen from Figs. 1–4, in TmFeO3 orthoferrite, unlike in LuFeO3, there are considerable diﬀerences between experimental Xexp ðTÞ and calculated Xcal ðTÞ values (presented by the continuous line). In principle, knowledge of calculated cell parameters makes it possible to extract a phonon contribution from a total thermal expansion eﬀect with the purpose of obtaining some information about the participation of other factors (magnetic, electron) in the considered process. Corresponding diﬀerences of measured and calculated unit cell parameters and cell volume, normalised to the calculated values ðXexp Xcal Þ=Xcal are shown in Fig. 8 for the TmFeO3 orthoferrite. A qualitative correspondence with the values from Figs. 5 to 7

can be easily noticesFtemperature plots of deﬁned expression are similar for a; c and V parameters; the maximal positive values (B103) for a and c are reached at low temperatures; they are negative for b parameter. The ðXexp Xcal Þ=Xcal negative sign for b indicates, of course, the fact that at low temperatures the experimental b values are smaller than the calculated ones. It is also worth noting that the corresponding quotients ðXexp Xcal Þ=Xcal vanish at higher temperatures (above B150 K). The evident similarities between temperature dependencies, presented in Figs. 5–9, as well as their correlation with the magnetic properties of the TmFeO3 orthoferrite, strongly suggest that the investigated diﬀerences in measured and calculated cell parameters (Fig. 8) are a manifestation of a spontaneous magnetostriction eﬀect. Spontaneous magnetostriction is deﬁned as the relative change of cell parameters induced by inner (spontaneous) magnetisation of a substance. The ratio oA ¼ ðA Ac Þ=Ac may be taken as its measure, where A is the parameter value in magnetically ordered (ferromagnetic, anitferromagnetic, etc.) state and Ac is its calculated value for hypothetical paramagnetic state. Similarly, the quotient oV ¼ ðV Vc Þ=Vc is taken as a measure of volume spontaneous magnetostriction, where V stands for unit cell volume in magnetically ordered state, and Vc its calculated cell volume [11]. It follows from the above deﬁnitions that ðXexp Xcal Þ=Xcal diﬀers from oA : However, as the magnetic structure of the Fe sublattice structure is stable and the weak magnetic moment value does not change over the whole temperature range (up to TN1 ), the ﬁrst relation can be considered as some measure of a spontaneous magnetostriction eﬀect, generated only by the Tm sublattice. The magnetostriction eﬀect in the TmFeO3 orthoferrite was studied in the external magnetic

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A. Bombik et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 443–453

Fig. 8. Normalised experimental and calculated cell parameter diﬀerences of TmFeO3 orthoferrite.

ﬁeld H ¼ 1:4 T and temperatures T > 67 K [13]. These investigations show that the value and sign of the magnetostriction eﬀect is determined by the orientation of H ﬁeld according to crystal directions as well as the mutual arrangement of ﬁeld and Fe sublattice antiferromagnetic vector. A magnetic ﬁeld H oriented parallel to the [0 0 1] direction causes a signiﬁcant eﬀect (Dl=lE105 ) for temperatures in which the antiferromagnetic vector of the Fe subsystem is parallel to the c parameter (To80 K), and a weaker eﬀect (Dl=lE107 ) when H and the antiferromagnetic vector are perpendicular (T > 95 K). External magnetic ﬁeld oriented parallel to the a parameter causes contraction of crystal lattice in the direction b; while magnetic ﬁeld parallel to b does not induce any measured dimensional eﬀect. If one assumes that in TmFeO3 the deviation of measured cell parameters a; b; c; and the cell volume from their calculated values (Figs. 1–4) is determined mainly by the spontaneous magnetostriction eﬀect, then one has to conclude that there is some diﬀerence between the magnetostriction induced by the external ﬁeld and the spontaneous magnetostriction. Fig. 8 shows that the marked

spontaneous magnetostriciton eﬀect appears only below B160 K and it is positive in the spin rotation plane (x; z), whereas in the direction perpendicular to this plane (direction b) it is negative. If the ratio ðXexp Xcal Þ=Xcal is taken as its measure, then for the a direction the eﬀect reaches its maximal value (B7 104) in the lowest measured temperature T ¼ 10 K. In the c direction its maximal value (4.7 104) is reached at T ¼ 20 K; at temperature T ¼ 10 K the magnetostriction reaches a slightly lower (B4.1 104) value. A negative magnetosriction eﬀect is observed in the b direction; its maximal absolute value (B7.4 104) is reached at T ¼ 10 K. A more precise numerical analysis of aðTÞ; cðTÞ and even bðTÞ and VðTÞ in the spin reorientation temperature region (80–95 K) shows existence of some small disturbances of monotonousness in these dependencies. The spin reorientation process manifests itself much more explicitly in the linear coeﬃcients of thermal expansion deﬁned as aX ¼ ð1=XÞðqX=qTÞ (Figs. 9–12). From Figs. 9–12 it follows that in TmFeO3 the coeﬃcient ab is positive and keeps an

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451

Fig. 9. Thermal expansion aa coeﬃcients of TmFeO3 and LuFeO3 orthoferrites.

Fig. 11. Thermal expansion ac coeﬃcients of TmFeO3 and LuFeO3 orthoferrites.

Fig. 10. Thermal expansion ab coeﬃcients of TmFeO3 and LuFeO3 orthoferrites.

Fig. 12. Volume thermal expansion coeﬃcients of TmFeO3 and LuFeO3 orthoferrites.

almost constant value (6–8 106 K1) over the whole measured temperature range (10–280 K). Unlike ab ; coeﬃcients aa and ac are a complex function of temperature especially in their lower

range. The coeﬃcient aa takes its minimal value (B9 106 K1) in the lowest measured temperature (10 K) and changes its sign from negative to positive at B110 K. In the lowest temperature

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the value of the ac coeﬃcient is B4 106 K1; it changes its sign to negative at 20 K and reaches its minimal value (6 106 K1) at 40 K. Starting from this temperature, the ac value grows monotonically with temperature and at TB80 K changes its sign again, but this time from negative to positive. The continuous lines in Figs. 9–12 denote a hypothetical (calculated from the Debye– Gruneisen . approximation) behaviour of the thermal expansion coeﬃcients. Two features of the presented behaviour seem to be worth emphasising. The ﬁrst one is the fact that the aa and ac coeﬃcients change their sign in the neighbourhood of spin reorientation temperatures, and the second one is the stabilisation of thermal expansion coeﬃcients on almost the same asymptotic value in higher temperatures. The second fact points to a lack of magnetic contribution to the thermal expansion process at higher temperatures. To compare, the coeﬃcients of thermal expansion for LuFeO3 are also shown in Figs. 9–12 (continuous lines). As can be seen, in this orthoferrite the temperature dependencies of the investigated coeﬃcients are much more regular, and ﬁt the calculated ones better than in the previous case. In our opinion, the considerable deviation of the measured structural parameters from their calculated values observed in the TmFeO3 orthoferrite, as well as diﬀerent low temperature behaviours of these parameters in TmFeO3 and LuFeO3, clearly prove the existence of a strong magneto-elastic interaction in TmFeO3. It manifests itself as the spontaneous magnetostriction eﬀect generated by the magnetic nature of the rare earth atom (Tm) in the lower temperature region.

4. Conclusions Studies carried out on LuFeO3 and TmFeO3 reveal a close correlation between the magnetic and the elastic properties of rare earth orthoferrites. Among other things, these correlations manifest themselves by the spontaneous magnetostriction eﬀect observed in TmFeO3. Generally, this eﬀect reaches maximal values at low temperatures and depends on the orientation of the

antiferromagnetic vector of the Fe sublattice versus the basic vectors of the crystal lattice. The largest positive values of spontaneous magnetostriction are observed along the [1 0 0] and [0 0 1] directionsFin the rotation plane of the Fe spins. Along the [0 1 0] direction, which is perpendicular to the rotation plane, the eﬀect is slightly weaker and negative. High spontaneous magnetostriction values and their temperature dependencies suggest that they are induced mainly by the magnetism of the rare earth element. The spin reorientation process in TmFeO3 is accompanied by the change of the signs of the thermal expansion coeﬃcients in the [1 0 0] and [0 0 1] directions. Additionally, at To20 K the temperature dependencies of thermal expansion coeﬃcients show some singularities. These are probably caused by the compensation eﬀect in TmFeO3, but this suggestion needs to be veriﬁed by further research.

Acknowledgements The authors remain in gratitude to Miss Justyna Les!niewska for her revision and proof-reading of the English version of the manuscript. This work was partly supported by research grant No. 2 P03B 035 14 from the State Committee for Scientiﬁc Research.

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