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Critical magnetic behaviour of a ferromagnet with antiferromagnetic fourth-order exchange interactions: GdMg
Journal of Magnetism and Magnetic Materials 188 (1998) 333—345
Critical magnetic behaviour of a ferromagnet with antiferromagnetic fourth-order exchange interactions: GdMg U. Ko¨bler!,*, R.M. Mueller!, W. Schnelle", K. Fischer! ! Institut fu( r Festko( rperforschung, Forschungszentrum Ju( lich, D-52425 Ju( lich, Germany " Max-Planck Institut fu( r Festko( rperforschung, D-70506 Stuttgart, Germany Received 11 September 1997; received in revised form 17 May 1998
Abstract We report on investigations of the critical magnetic behaviour of GdMg. In this intermetallic ferromagnet the sum of all fourth-order interactions (i.e. biquadratic, three-spin and four-spin interactions) is 50 K and antiferromagnetic at the Curie temperature of ¹,"110 K. This has been shown with measurements of the third-order susceptibility s . Although C 3 the Curie temperature can be assumed to be defined mainly by bilinear (Heisenberg) interactions, the critical indices for the linear susceptibility s and the spontaneous magnetization are molecular field like c"c@"1, b"0.5. The 1 third-order susceptibility s is discontinuous at ¹,: for ¹'¹,, s is finite but for ¹(¹, it diverges with a critical 3 C C 3 C exponent of one. The ferromagnetic order—disorder transition is therefore weakly first order. At a second critical temperature of ¹M"91 K the rise of an ordered antiferromagnetic component oriented perpendicular to the ferromagN netic one has been reported two decades ago based on neutron scattering investigations. Here we will show that the third order susceptibility s diverges at ¹M but that the linear susceptibility s stays finite. This is characteristic for a phase 3 N 1 transition driven by the fourth-order interactions. We must therefore assume that the transverse components of the ferromagnetically ordered moments order antiferromagnetically at ¹M. This view is in accordance with measurements of N the susceptibility s perpendicular to an applied static magnetic field. As a result, a second-order parameter perpendicuM lar to the spontaneous magnetization is generated by the fourth-order interactions. As a further consequence of the perpendicular moment configuration, the spontaneous longitudinal magnetization reaches only 0.75 of the theoretical saturation value for ¹P0 and decreases like ¹2 with temperature as has been observed for other ferromagnets with fourth-order interactions, such as EuS and CrBr . ( 1998 Elsevier Science B.V. All rights reserved. 3 PACS: 75.30.Et; 75.40.Cx Keywords: Biquadratic exchange; Critical behaviour; Ferromagnets
1. Introduction
* Corresponding author. Tel: #49 2461 613324; fax: #49 2461 612016.
GdMg is an unconventional metallic ferromagnet [1]. The Gd moments occupy a simple cubic lattice and are not aligned strictly parallel for
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 2 1 6 - 9
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¹P0 [2]. The resulting ferromagnetic moment is only 5.24 l instead of 7 l expected for Gd. FerB B romagnetic saturation is not reached even in fields of 35 T [3]. Since any paramagnet would be magnetically saturated for ¹"4.2 K and B "35 T, 0 strong antiferromagnetic interactions must exist in GdMg in addition to the ferromagnetic ones which are responsible for the high Curie temperature of 110 K. In fact, half-indexed magnetic scattering lines could be observed for ¹(90 K in neutron scattering investigations and a magnetic structure model including an ordered antiferromagnetic component perpendicular to the ferromagnetic one has been proposed [2]. This biaxial magnetic ordering type is also named the flopside spin structure [4]. There is evidently a discrepancy between the high Curie temperature of 110 K and the strong antiferromagnetic interactions at low temperatures which prevent ferromagnetic saturation even in very large magnetic fields. If these antiferromagnetic interactions were of the conventional bilinear (Heisenberg) exchange type (S ) S ) they would add 1 2 to the ferromagnetic interactions and lower the Curie temperature accordingly. Here we will show that the transverse, antiferromagnetically ordered magnetization component is the result of antiferromagnetic fourth-order exchange interactions. There have been two different theoretical approaches to explain the unusual observation of mutually perpendicular ferromagnetic and antiferromagnetic sublattices in a material with pure spin magnetism and therefore with a vanishing magnetic anisotropy [4,5]. Both models require competing interaction mechanisms which destabilize the ferromagnetic order with decreasing temperatures leading eventually to the observed canted spin structure. Here we propose to consider the perpendicular configuration of two order parameters as a consequence of the general principle that an additional class of magnetic interactions will tend to increase the magnetic order further. Though the individual fourth-order interaction processes are sizeable they are prevented from increasing the Heisenberg transition temperature accordingly. Therefore, the magnetic order is increased by inducing an additional order parameter whereby the entropy of the
system can be lowered more efficiently. In the present state of our knowledge this explanation must be viewed as a working hypothesis which awaits a rigorous theoretical justification but which is able to explain the experimental facts from a general point of view without the necessity of making special assumptions on the interaction parameters. Apparently, the fourth-order exchange interactions (i.e. biquadratic, three-spin and four-spin interactions) are such a particular class of magnetic interactions able to generate a new order parameter. Since the transverse moment components are the only degrees of freedom left for an additional ordering process once the longitudinal moment components are ordered it is natural to assume that the fourthorder exchange interactions govern the order of these transverse moment components meaning that the rotational symmetry of the uniaxial magnetic order will be broken. This view is intuitively supported by the fact that an antiferromagnetic biquadratic interaction requires a perpendicular moment orientation but two orthogonal order parameters seem to occur also if other members of the fourth-order interaction class dominate as in the Eu-chalcogenides, in which the ferromagnetic three-spin interactions dominate [6,7]. As a consequence of our hypothesis, we consider the experimental value for the canting angle between the ferromagnetic and antiferromagnetic magnetization component [2] to be essentially 90° in GdMg. In Refs. [6,7] it was shown that in the hightemperature limit the fourth-order exchange interactions give rise to a Curie—Weiss law for the thirdorder susceptibility s which is defined according 3 to 1 1 B " m# m3#2, * s s 1 3
(1)
where B is the applied magnetic field converted to * its value inside the sample and m is the reduced magnetization. The Curie—Weiss temperature H 3 associated with s was shown to be a weighted 3 average over all fourth-order interaction processes [7]. The Curie—Weiss law of s shows that there 3 exists, even in the molecular field approximation, a second ordering transition at H in addition to 3
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the conventional one at H . Both transitions are of 1 the order—disorder type. Since a collinear spin system can order only once it is natural to assume that the ordering process at H affects the disordered 3 transverse moment components. These components order at a finite temperature whenever the fourth-order interaction sum as given by H has 3 a finite value [8]. This transverse order is called ferromagnetic for H '0 and antiferromagnetic for 3 H (0 [8]. 3 The longitudinal and transverse ordering phenomena are mutually independent and occur at clearly different temperatures in GdMg. At ¹,"110 K the longitudinal moment components C order ferromagnetically while at a lower critical temperature of ¹M"91 K the transverse moment N components order antiferromagnetically [2]. This independence of the ordering processes becomes evident from the fact that the spontaneous magnetization of the longitudinal (Heisenberg) order parameter does not exhibit any visible anomaly at the Ne´el temperature of the transverse order parameter. Measurements of the AC susceptibility s M perpendicular to a longitudinal static magnetic field performed in this work further confirm the perpendicular orientation of the antiferromagnetic component. In this communication, we will show by an investigation of the third-order susceptibility s that 3 there are strong antiferromagnetic fourth-order interactions for ¹*¹, in GdMg. These interactions C must be considered as the origin of the antiferromagnetically ordered transverse magnetization component found earlier with neutron scattering studies for ¹)¹M"91 K [2]. The different N observed transition temperatures can clearly be attributed either to second-order (Heisenberg) interactions or to fourth-order interactions. This was demonstrated in the diamagnetically diluted antiferromagnets Eu Sr Te. In this system both x 1~x order parameters are antiferromagnetic for x( 0.85 with definitely different Ne´el temperatures and clearly different critical field values [8—10]. The critical field lines do not touch and no multicritical behaviour occurs. This behaviour allows a clear distinction between second-order phase boundaries and fourth-order phase boundaries. As a further consequence of the perpendicular configuration of
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two ordered magnetization components, magnetic saturation will not be reached in finite magnetic fields. Both components will not reach the full magnetic moment for ¹P0. The antiferromagnetic total fourth-order interaction strength as identified by means of the thirdorder susceptibility s for ¹*¹, must be conC 3 sidered an average over all individual biquadratic, three-spin and four-spin interaction processes. Comparison with other investigated systems, such as the insulating solid solutions Eu Sr Te and x 1~x Eu Sr S [7], shows that these individual interacx 1~x tion processes are much larger than the Curie— Weiss temperature H which is a measure for the 3 high-temperature average over all fourth-order interactions [7]. In metals, however, no such Curie—Weiss behaviour is observed for s even at 3 temperatures several times larger than the ordering temperature. It appears that in metals s exhibits 3 a strong temperature dependence only near ¹ but C approaches a rather weak temperature dependence in the high temperature limit [11]. Only the observed shift of the experimental s (¹) data relative 3 to the calculated Curie law of s provides some 3 information for the strength of the fourth-order interactions which for metals change inherently with temperature such that s is only weakly tem3 perature dependent and may even cross the calculated Curie line. We must therefore assume there are much larger individual antiferromagnetic and ferromagnetic fourth-order interaction processes responsible for the observed mean value near ¹,. C Though both perpendicular ordering processes are mutually independent and the associated order parameters clearly differ, they show common properties and tend to order at similar temperatures. For instance, it was observed in Refs. [8,12] that both order parameters of EuS are ferromagnetic and both decrease according to a ¹2 law instead with a ¹3@2 law for increasing temperatures for ¹P0. This could be concluded from the fact that the zero-field 153Eu NMR signal, which averages over an isotropic distribution of both order parameters on account of an isotropic orientation of magnetic domains, shows a ¹2 law with high precision [8]. The ¹2 law must therefore be common to both order parameters. This law seems to be
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exact until the critical temperature range near the ordering temperature. The fourth-order interactions also dominate the spin dynamics near ¹, and are found to be the C relevant interactions for defining the asymptotical critical magnetic behaviour of both order parameters, just as they define their behaviour for ¹P0, where they give rise to a ¹2 law instead of a ¹3@2 law. The asymptotical temperature range in which the mean field critical behaviour is clearly observed in GdMg is as large as 10 K. This we attribute to the fact that the resulting total fourthorder interaction is as large as 50 K. Additionally there seems to be no strong competition between the different fourth-order interaction processes. Since the total interaction is antiferromagnetic the ordering of the transverse order parameter is also antiferromagnetic. It is therefore easily possible to orient the longitudinal magnetization component by means of a small magnetic field and to measure its critical behaviour with field-parallel magnetization measurements. In other ferromagnetic compounds such as EuO and EuS the total fourthorder interaction is ferromagnetic [6,12]; therefore, the order of the transverse moment components is ferromagnetic. In such a situation it is virtually impossible to distinguish both order parameters because neither of them will be oriented perfectly parallel to an applied magnetic field. As a consequence, in conventional magnetization measurements a mixture of both order parameters, depending on the strength of the applied magnetic field, will be measured. One may therefore speculate whether or not the same asymptotical critical exponents as observed here for GdMg may apply also for EuO and EuS but only if care is taken to properly discriminate between the two order parameters. In fact, in EuO, the experimental critical exponent for the magnetic specific heat is clearly not consistent with the Heisenberg model prediction [13], compare however Ref. [14]. Of course, the specific heat is a scalar quantity which averages over all space directions. The critical magnetic behaviour of the longitudinal order parameter of GdMg is essentially molecular field like: the linear susceptibility v diverges 1 with the mean field exponent c"c@"1 at ¹,. The C spontaneous magnetization also exhibits the mo-
lecular field exponent of b"0.5. The third-order susceptibility s shows more complicated behav3 iour. It should be recalled that s is the quantity 3 responding most strongly to the fourth-order interactions. Approaching ¹, from higher temperatures C s is finite at ¹, in agreement with the molecular C 3 field approximation but in disagreement with the Heisenberg model according to which s should 3 diverge with an exponent of 2b!c"!0.66. Approaching ¹, from lower temperatures, s diverges C 3 with an exponent of unity. We believe this discontinuous behaviour of s at ¹, shows different C 3 fourth-order interaction types are differently weighted by the rise of the bilinear order parameter. As a consequence of the discontinuous behaviour of s the phase transition at ¹, has to be conC 3 sidered weakly first order. The associated latent heat may be small and in fact no obvious latent heat could be seen in the magnetic specific heat [5]. Evaluation of the AC-susceptibility s measured M perpendicular to a longitudinal static magnetic field reveals clearly the ordering processes of the longitudinal and transverse moment components occur at different temperatures: at ¹,"110 K the C longitudinal components order ferromagnetically which is ascertained from a diverging longitudinal susceptibility s . The perpendicular susceptibility , continues increasing monotonicly at ¹, without C showing any definite anomaly. At ¹M"91 K, on N the other hand, s exhibits a rounded but clear M maximum characteristic for a Ne´el transition but here the longitudinal susceptibility s shows vir, tually no anomaly. The ordering transition at ¹M is N also distinguished by an absolute maximum of the magnetic specific heat [5], whereas at ¹, there is C a rounded shoulder in the magnetic specific heat.
2. Sample preparation The single crystal of GdMg was grown from a nominal composition of GdMg by directional 1.05 freezing. Gd and Mg were sealed in a molybdenum crucible. The crucible, surrounded by a molybdenum block and standing on a cold pin, was RF-heated in vacuum. The upper temperature of the crucible was about 200°C above the melting point of 868°C before slow cooling.
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The crystal did not adhere to the crucible wall and was covered by the excess Mg. It was rather easily cleaved along (1 0 0) planes and showed sharp Laue spots. For the magnetic measurements a nearly perfect sphere of about 1 mm diameter was fabricated.
3. Longitudinal magnetic measurements As a basis for the evaluation of the linear (longitudinal) susceptibility s and the third-order sus1 ceptibility s we make use of the definition 3 according to Eq. (1). For the evaluation of s and s magnetic iso1 3 therms are measured and plotted as m2 versus B /m * (Arrott plot). Fig. 1 shows an Arrott plot for some selected magnetic isotherms with ¹'¹,. In this C representation the slopes of the straight lines give the third-order susceptibility s and the intersec3 tions with the abscissa give the reciprocal linear susceptibility s~1"B /m. 1 * The s -data obtained by extrapolation of the 1 magnetic isotherms in the Arrott plot from the high-field range down to m2P0 (B P0) character* ise the behaviour of the longitudinal magnetization component which is easily turned into the field
Fig. 1. Magnetic isotherms of GdMg for ¹'¹, plotted as C squared reduced magnetization m2 versus B /m (Arrott plot). * The slopes of these lines give the third-order susceptibility s and the extrapolated intersections with the abscissa the recip3 rocal linear susceptibility s~1"B /m of the longitudinal (fer1 * romagnetic) magnetization component.
337
direction in a system in which the transverse magnetization exhibits antiferromagnetic interactions. In small fields with B (0.05 T (not resolved in 0 Fig. 1), orientational processes between longitudinal and transverse magnetization components take place and the susceptibility decreases again with decreasing field (see discussion of Fig. 11). In genuine zero-field measurements no orientation of the longitudinal and transverse magnetization components is ascertained and a mixture of both might be measured. The slopes of the Arrott lines in Fig. 1 change only little with temperature. In particular it is evident that the critical isotherm, i.e. the fictive line passing exactly through the origin, will have a finite slope. s is therefore finite at ¹,. According to the C 3 Heisenberg model s should diverge with an expo3 nent of 2b!c"!0.66 at ¹ . 2b!c results in C zero if the molecular field values of b"0.5 and c"1 are inserted. Moreover, a finite slope of the critical isotherm in the Arrott plot means that the critical exponent d equals 3. A finite value of s at 3 ¹, conforms to the molecular field prediction for C the longitudinal order parameter. Fig. 2 shows the temperature dependence of the reciprocal third-order susceptibility s~1 for tem3 peratures ¹'¹,. In this diagram the calculated C Curie law for s has been added for comparison. 3 The Curie law applies for vanishing fourth-order
Fig. 2. Reciprocal third-order susceptibility v~1 of GdMg ver3 sus temperature for ¹'¹,. The shift of the experimental C s~1 data to the theoretical Curie law s~1"C~1. ¹ is an 3 3 3 indicator for the existence of fourth-order interactions. At ¹,, s is finite coming from ¹'¹,. C 3 C
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interactions. As was shown in Refs. [6,7] for finite fourth-order interactions, s obeys a Curie—Weiss 3 law in the high-temperature limit according to C 3 s " 3 ¹!h 3 with
(2)
10g(S#1)3k B . C " (3) 3 9[(S#1)2#S2]k B The Curie—Weiss temperature H in Eq. (2) is 3 a measure for the high-temperature average of all fourth-order interactions. For metals, however, no such Curie—Weiss behaviour results even at temperatures several times larger than the ordering temperature [11]. Fig. 2 also shows the experimental s data assume nearly 3 temperature independent behaviour at larger temperatures and even cross the calculated Curie law for s at about 210 K instead of approaching a line 3 parallel to C~1 ) ¹ asymptotically for large temper3 atures. The total fourth-order interactions, namely correlations, can therefore change sign as a function of temperature in metallic systems [11]. The only conclusion we can draw from Fig. 2 is that there are appreciable fourth-order interactions in GdMg and these are antiferromagnetic for ¹(210 K but ferromagnetic for ¹'210 K. At the Curie temperature of ¹,"110 K the horizonC tal shift of the experimental s -data to the cal3 culated Curie law for s is 50 K. Therefore, the 3 order—disorder transition occurs in the presence of strong antiferromagnetic fourth-order interactions in GdMg. The critical behaviour of the linear susceptibility s is shown in Fig. 3 where the intersections of 1 the Arrott isotherms with the abscissa B /m" * s~1 from Fig. 1 are plotted versus temperature. As 1 can be seen, the data for ¹'¹, follow with high C precision a (¹!¹,)~1 law until ¹+120 K meanC ing that the critical exponent of s is c"1. Fig. 3 1 includes the Curie slope C~1 calculated using the 1 theoretical value for the Curie constant of the linear susceptibility which following the nomenclature of Eq. (1) is g(S#1)k B. C " 1 3k B
(4)
Fig. 3. Reciprocal linear susceptibility s~1 in the critical tem1 perature range around ¹, showing that c"c@"1. For ¹'¹, C C the theoretical Curie—Weiss line s~1"C~1(¹!¹,) has been 1 1 C added using g"2 and S"7/2 for the calculation of C accord1 ing to Eq. (4).
With g"2 and S"7/2, Eq. (4) gives C " 1 2.015 K/T. The experimentally observed coefficient of the asymptotical (¹!¹,)~1 law is, however, C C!4:.1"2.72 K/T. On the other hand, the ob1 served parameters of the high-temperature Curie—Weiss law are H "114 K and C=" 1 1 2.37 K/T. This value for C= is still much larger than 1 the free ion value and even larger than most reported values for metallic systems. As is well known [15], the effective Bohr magneton number of Gd is always increased in metallic systems. This applies also for alloys in which the gadolinium is strongly magnetically diluted by other diamagnetic metals [3]. It is therefore difficult to relate the conduction band polarization with the strength of the secondorder and fourth-order exchange interactions. Because the existence of two order parameters has been observed also in insulating materials [8,9] the biaxial spin structure of GdMg seems not to be a specifically metallic effect caused by particular properties of the conduction band in GdMg as was proposed in Ref. [5]. In spite of the fact that the critical exponent of the molecular field model of c"1 is observed for ¹P¹, for the linear susceptibility s , H is clearly C 1 1 larger than ¹, and the slope of the asymptotical C (¹!¹,)~1 law is larger than the observed highC temperature Curie constant. It should be noted that H "114 K is unusually close to ¹,"110 K but C 1 that the true molecular field conditions H "¹, C 1
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and C!4:.1."C= are not quite fulfilled. In other 1 1 words, spin fluctuations are unusually small which is confirmed also with the specific heat measurements. In order to evaluate s and s for ¹(¹, in C 1 3 a way consistent with the evaluation for ¹'¹, we C have to replace the reduced magnetization m in Eq. (1) by m!m S. mP (5) 1!m S Eq. (5) means that we subtract from the experimental magnetization value m the spontaneous magnetization m and normalize by 1!m . In this S S way, our new reduced magnetization tends to zero for mPm but approaches unity for mP1 irreS spective of the value of m . The s -data for ¹(¹, C S 1 can then be compared in magnitude with those for ¹'¹,. C Prior to evaluating s and s for ¹(¹, we C 1 3 must know the spontaneous magnetization m (¹). S This quantity was obtained as usual [5] by an extrapolation of the magnetization curves m(B ) to * B P0. Fig. 4 shows the obtained spontaneous * magnetization data as function of temperature. We have included the fitted ¹2 function describing the behaviour of m at low temperatures as well as the S (¹,!¹)0.5 function fitted to the m (¹) data near C S ¹,. The quality of both fits will be shown more C
Fig. 4. Temperature dependence of the reduced spontaneous magnetization m (¹). At low temperatures a ¹2 law has been S fitted to the experimental data. In the critical temperature range m (¹) is described by a (¹,!¹)0.5 law. S C
339
clearly in two consecutive diagrams. Both fitted functions in Fig. 4 reproduce the experimental m (¹) data rather completely. The gap region where S neither of the functions fits the data is fairly small. No pronounced anomaly can be seen at ¹M"91 K N in agreement with earlier measurements [5]. With the magnetization data re-normalized according to Eq. (5), Arrott plots of the magnetic isotherms have been drawn for the evaluation of s (¹) and s (¹) for ¹(¹, as was shown in Fig. 1 C 1 3 for the data for ¹'¹,. Fig. 3 includes the reciproC cal s data for ¹(¹,. These data have been C 1 multiplied by a factor of 500 for simpler comparison to the data for ¹'¹,. It can be seen that the C critical exponent of the linear susceptibility for ¹(¹, is c@"1 but the absolute susceptibility C values are larger by a factor of 500 compared to the s -data for ¹'¹,. C 1 Fig. 5 gives the experimental s~1(¹) dependence 1 for a larger temperature range below ¹(¹,. C In addition to ¹"¹,, one further anomaly at C ¹*"98 K can be seen. The character of this feature, which can also be identified in the magnetic specific heat measurements, is unclear. At ¹M, N where the magnetic specific heat has an absolute maximum, virtually no anomaly appears in s~1(¹). 1 On account of the smaller variation of the magnetization with field at low temperatures the scatter of the experimental s~1 data increases somewhat with 1 decreasing temperature. For ¹P0 s seems to be 1 finite, in disagreement with the expectation for
Fig. 5. Reciprocal linear susceptibility s~1 versus temperature 1 for ¹)¹, revealing different anomalies at the phase C transitions ¹,"110 K, ¹*"98 K and ¹M"91 K. C N
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a Heisenberg ferromagnet. Since m (¹"0) is only S 0.75 it is logically necessary that s (¹"0) be finite. 1 The temperature dependence of the reciprocal third-order susceptibility s~1 is displayed in Fig. 6. 3 s is definitely finite for ¹P0. At ¹M"91 K, s diN 3 3 verges but since s is finite at this temperature the 1 transition at ¹M is clearly produced by fourth-order N interactions. As will be confirmed with measurements of the perpendicular susceptibility s , the M transverse moment components order antiferromagnetically at ¹M. At ¹*"98 K one further N anomaly appears. Here s exhibits a relative min3 imum. Though it has not yet been shown with adequate experimental accuracy, if we assume s really diverges at ¹* we must attribute this 1 anomaly to bilinear interactions. Fig. 6 shows that s diverges with an exponent of 3 !1 at ¹M"91 K and at ¹,"110 K as well, in C N contrast to the Heisenberg model according to which s should diverge with an exponent of 3 2b!c"!0.66. Fig. 7 shows the low-temperature data for the reduced spontaneous magnetization m plotted verS sus the squared reduced temperature (¹/¹,)2. In C agreement with Ref. [1] these data follow a ¹2 law with high precision until (¹/¹,)2"0.55 or C ¹/¹,"0.74 namely, at ¹"81 K. ¹2 laws have C been observed for the spontaneous magnetization at low temperatures in many systems with fourthorder interactions such as EuS and CrBr [8] and 3
Fig. 6. Reciprocal third-order susceptibility s~1 versus temper3 ature for ¹)¹, showing divergences of s at ¹,"110 K and C 3 C ¹M"91 K. At ¹*"98 K a relative minimum of s can be N 3 noticed.
Fig. 7. Reduced spontaneous magnetization m versus (¹/¹,)2 S C showing the large validity range of the low-temperature ¹2 law. For ¹P0, m is only 0.745. S
also for the low-temperature behaviour of the critical field curves B for EuSe [6] and Eu Sr Te C x 1~x [8]. The reduced longitudinal spontaneous magnetization reaches only 0.75 for ¹P0. Assuming the magnetization component sharing the longitudinal order parameter is perfectly ferromagnetic we may conclude &3/4 of the Gd moment belongs to the parallel order parameter but &2/3 to the perpendicular order parameter. As a consequence, the antiferromagnetic fourth-order interactions which we believe are responsible for the perpendicular configuration of two order parameters are nearly as strong as the ferromagnetic bilinear interactions. The same conclusion may be drawn from the very similar ordering temperatures for the two order parameters, ¹,"110 K and ¹M"91 K. N C The spontaneous magnetization data of the critical temperature range are plotted in Fig. 8 as m2 versus ¹. From this diagram it is clear the S critical exponent of the spontaneous magnetization is b"0.5. The critical temperature range is again very large: 97(¹ (110 K. #3*5. Summarizing the longitudinal magnetization measurements we can say that the critical exponents for the linear and the cubic susceptibility are molecular field-like in GdMg but only for ¹'¹,. C For ¹(¹, the third-order susceptibility diverges C at ¹, in contrast to the molecular field model. This C
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Fig. 8. Squared reduced spontaneous magnetization m2 versus S temperature in the critical temperature range showing that the critical exponent is b"0.5.
indicates the importance of the fourth-order interactions increases discontinuously with the rise of the parallel order parameter at ¹, as a result of the C bilinear exchange interactions.
4. Specific heat measurements The specific heat was measured using a conventional heat pulse method [16]. The contributions of the sample holder and of a minute amount of vacuum grease utilized to mount the sample were measured in a separate run and subtracted. The accuracy is estimated to be better than 2% of c (¹). p Fig. 9 displays the specific heat of GdMg. The most prominent feature of Fig. 9 is the absolute maximum at ¹M"91 K to be identified N with the ordering process of the transverse order parameter. This result for c (¹) conforms perfectly p with earlier data [5]. Based on the existence of this specific heat peak one can be sure the ordering of the transverse moment components is associated with a phase transition different from ¹,. It is then C intuitively clear that virtually no anomalies occur in the longitudinal spontaneous magnetization and the longitudinal linear susceptibility s at ¹M in N 1 spite of the pronounced peak in the magnetic specific heat. Surprisingly, above the 110K phase transition there is no clear trace of a short range order tail, as normally seen in the specific heat of
341
Fig. 9. Specific heat c (¹) of GdMg (d). The continuous line p shows a lattice and electronic (background) specific heat c & fit- % ted to the measured data above 120 K. c is the magnetic .!' contribution (m). Inset: Specific heat c /¹ versus ¹2 for p ¹(10 K. The fit gives H "145 K and c"5.2(4) mJ/mol K2. D
magnetic systems. This conforms to the mean field character of this phase transition and the observation made in the susceptibility measurements that H "114 K and ¹,"110 K are very close to each C 1 other. Latent heat is not found at the phase transitions. Since only s was observed to be dis3 continuous at ¹, any possible latent heat should in C fact be very small. The inset of Fig. 9 shows the specific heat below 10 K in a c (¹)/¹ versus ¹2 representation. The fit p to these data up to 7 K with c (¹)"c¹#b¹3 p gives c"5.2(4) mJ/mol K2 and b"1.28(1)] 10~3 J/mol K4. This b value would correspond to H (0)"145 K, a value much too ‘soft’ for the latD tice of GdMg, indicating strong magnetic contributions. In order to obtain the f-electron related magnetic contribution of GdMg we fitted a Debye model (phonons) plus a linear term (delocalized electrons; anharmonic lattice contributions) to the specific heat data above the phase transitions (¹'120). We found a good representation of the data for H "228 K and a linear term c*¹ with c*" D 14 mJ/mol K2. After subtraction of this background we obtain the magnetic contribution c shown in Figs. 9 and 10. In addition to the .!' three previously mentioned anomalies at ¹,, ¹* C and ¹M a broad hump at around 30 K is observed, N
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Fig. 10. Magnetic specific heat c (¹) of GdMg (left axis) (m) .!' and magnetic entropy S (¹) (continuous line; right axis). The .!' critical temperatures obtained from the susceptibility measurements are indicated.
a feature often found in Gd compounds and in pure Gd metal [17]. The magnetic entropy S (¹) .!' above the phase transition reaches exactly R ln 8, the value expected for the ordering of the 8-fold degenerate J"S"7/2 Gd ground state. The additional sharp shoulder seen at ¹*" 98 K in the specific heat corresponds well with the anomalies at the same temperature seen in s and 1 s in Figs. 5 and 6. This feature appeared less 3 pronounced in the specific heat measurement of Ref. [5]. It is likely that the ¹*"98 K anomaly constitutes some phase change of the parallel order parameter but since the microscopic change at this transition cannot be resolved on the basis of our experiments we will not consider this detail further. At the Curie temperature of ¹, only a rounded C shoulder appears in the magnetic specific heat in Fig. 10. This phase transition can be localized with a much better precision in the magnetization measurements. The results of Fig. 10 make clear there exists no realistic chance to evaluate the critical exponent a at ¹, in GdMg since the anomaly at C ¹, is too much smeared and the proximity of the C ¹*"98 K and ¹M anomalies affects the c (¹) deN p pendence too much at ¹,"110 K. C 5. Measurements of the perpendicular susceptibility Since an antiferromagnetic biquadratic interaction requires a perpendicular moment orientation,
measurements of the perpendicular susceptibility s can be expected to be very revealing. In fact, M these measurements show unambiguously the existence of an antiferromagnetic transverse order parameter. In these experiments the ferromagnetic magnetization component is forced into a direction given by a longitudinal static magnetic field B and the , perpendicular AC susceptibility s is measured M with an alternating magnetic field perpendicular to the static one having an amplitude much smaller than B . In our case the longitudinal magnetic field , was produced by a superconducting coil. The spherical sample was placed inside a small concentric transformer, the axis of which was oriented perpendicular to the static field B . An outer wind, ing section produced the AC magnetic field to periodically excite the transverse magnetization. This field had an amplitude of &1 G. The sample was inside one of a pair of astatic inner secondary coils thus producing an unbalance signal proportional to its susceptibility. After passing the AC signal through a 1 : 100 low noise transformer it was detected with a lock-in amplifier. Fig. 11 shows a set of selected perpendicular susceptibility curves as a function of temperature together with the longitudinal zero-field susceptibility (B "0) measured immediately after cool, down of the superconducting magnet in order to make sure that there was no remanent field in the superconducting coil. In this measurement no definite orientation of the order-parameters is given but since the ferromagnetic one will be preferentially aligned by the small AC field the s(B "0) , curve can be assumed to reflect mostly the behaviour of the ferromagnetic order parameter. This s curve labelled by B "0 shows a distinc, , tive kink at the Curie temperature ¹, with a plaC teau value at lower temperatures given by demagnetization effects, i.e. the formation of magnetic domains. For an ideal Heisenberg ferromagnet which lacks any magnetic anisotropy the demagnetization plateau should continue until ¹P0. This is not the case for GdMg. At about 80 K the susceptibility s begins to decrease , rather abruptly, reaching a value of only 0.6 of the demagnetization plateau value for ¹P0. The cause is the transverse magnetization component
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which increases at the expense of the longitudinal magnetization. In contrast to s , i.e., the curve with B "0, the , , perpendicular susceptibility s shows a rounded M but clear maximum at ¹M but no anomaly at all at N the Curie temperature ¹,. This behaviour proves C that the longitudinal and transverse magnetization components order at different temperatures (¹, and ¹M, respectively). Moreover, the observed N C maximum of s is characteristic of antiferromagM netic ordering of the transverse magnetization component in agreement with the neutron scattering results [2]. It should be noted that the field value of B "0.23 T has been chosen such that the max, imum of s has not been cut off by demagnetization M effects. In other words, a field of this value is needed to orient the ferromagnetic component perfectly parallel to B in order that s really samples the , M perpendicularly oriented antiferromagnetic component. For larger longitudinal magnetic fields the height of the maximum of s decreases and transM forms eventually into a kink. ¹M shifts only slightly N with field as one would expect for the critical field of an antiferromagnet with a Ne´el temperature as large as ¹M"91 K. N Another interesting detail in Fig. 11 should be mentioned: there is evidently an anisotropy be-
Fig. 11. Perpendicular AC susceptibilities s for different values M of the static longitudinal magnetic field B . The curve for B "0 , , giving the longitudinal susceptibility s exhibits a rather sharp , kink at the Curie temperature ¹,. No anomaly is seen at ¹, in C C the s curves (B '0) but a maximum at ¹M"91 K indicates M , N an antiferromagnetic ordering process for the transverse moment components.
343
tween s and s even in the paramagnetic phase. , M The s data of the curve with B "0 should always , , be larger than the s data since the s curve has M , been measured with the smallest field (&1 G). In contrast to this expectation the s curve drops , clearly below the s curves in the paramagnetic M phase. This interesting observation shows there are different interaction energies for both order parameters even in the paramagnetic phase. In the case of the metallic GdMg the third-order susceptibility is nearly constant for ¹'¹,. This means C that the fourth-order interactions change towards ferromagnetic for increasing temperature whereby the importance of s increases relative to s . This 3 1 might explain the observed relative increase of s referred to s . M , This phenomenon will be the subject of a forthcoming publication [12].
6. Conclusions We have shown that the longitudinal and transverse Gd-moment components order at different temperatures in GdMg. At the Curie temperature of ¹,"110 K the longitudinal moment compoC nents order ferromagnetically while at a second critical temperature ¹M"91 K the transverse moN ment components order antiferromagnetically. This has been shown with measurements of the AC susceptibility s perpendicular to a static magnetic M field as well as with magnetic specific heat measurements. Our results conform to earlier specific heat and neutron diffraction studies which revealed an antiferromagnetically ordered magnetization component perpendicular to a ferromagnetic one for ¹)80 K [2,5]. Hence, there are two perpendicular order parameters to be attributed to the two ordering phenomena. This we consider as a plausible consequence of the quite general principle that a new class of magnetic interactions may tend to increase the magnetic order by generating its own order parameter. Because the transverse moment components are the only degrees of freedom left for an ordering process, their order is determined by the fourth-order interactions. In this way the entropy of the system can be further lowered. As a further
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consequence of this the observed phase transitions can clearly be distinguished geometrically. A total antiferromagnetic fourth-order interaction strength of 50 K has been shown by measurements of the third-order susceptibility s to exist at 3 ¹,. This value constitutes an average over biquadC ratic, three-spin and four-spin interaction processes which could not be evaluated separately. Fourth-order interactions not only give rise to ordering of the transverse moment components, they also dominate the spin dynamics of the longitudinal (Heisenberg) order parameter for ¹P0 and ¹P¹,. For instance, the spin order for ¹P0 C differs strongly from what is expected for a conventional Heisenberg ferromagnet. On expense of the transversally ordered moment components the spontaneous longitudinal magnetization is only 0.74 of the theoretical saturation value. This means that nearly equal fractions of the total Gd moment share the longitudinal and transverse spin order [2]. The spontaneous magnetization decreases according to a ¹2 law rather than a ¹3@2 law until a temperature as large as 0.8 ¹ . From the very C large validity range of the ¹2 law one may conclude that this law is exact under the conditions given in contrast to Bloch’s ¹3@2 law which was derived for the Heisenberg ferromagnet and which is only a first-order approximation [18]. The ¹2 law was shown experimentally to hold for both order parameters in a number of materials with fourthorder interactions [8]. It was observed not only for the spontaneous magnetization of ferromagnets such as EuS and CrBr but also for the critical field 3 curves of antiferromagnets such as Eu Sr Te and x 1~x the critical field curve of the metamagnet EuSe [6,8]. Also the critical magnetic behaviour of the longitudinal (Heisenberg) order parameter of GdMg is changed by the presence of the fourth-order interactions which are decisive at ¹,. Since the antiferC romagnetic transverse order parameter is not yet ordered at ¹, the critical behaviour of the longituC dinal order parameter can be measured conveniently. Surprisingly, the critical magnetic behaviour is essentially molecular field-like for both order parameters. At ¹,"110 K the linear susceptibility C s diverges with an exponent of one. At the order1
ing transition of the transverse moment components at ¹M"91 K s diverges also with an expoN 3 nent of one but the linear longitudinal susceptibility s shows virtually no anomaly. The fact that only 1 s diverges at ¹M shows this transition is caused by N 3 fourth-order interactions and the maximum of s at ¹M indicates the transverse moment compoN M nents order antiferromagnetically at this temperature. Since anomalies occur either for s or for s , , M a clear distinction between parallel and perpendicular ordering processes is possible. Even in the paramagnetic phase the order parameters can be distinguished by their different susceptibilities owing to different interaction energies. The magnetic specific heat supports the existence of (at least) two ordering transitions in GdMg. Clearly the specific heat is a scalar quantity and cannot distinguish between ordering phenomena occurring along different spacial directions. The shape of the specific heat anomaly may be characteristic for the transitions of both order parameters. In agreement with earlier investigations [5], c (¹) .!' shows a sharp peak at ¹M but a rounded structure N at ¹,. Whether this is a typical difference for both C ordering processes awaits further theoretical as well as experimental investigations.
Acknowledgements We gratefully acknowledge the technical assistance of Mrs. Ch. Horriar-Esser and Mr. B. Olefs.
References [1] R. Aleonard, P. Morin, J. Pierre, D. Schmitt, Solid State Commun. 17 (1975) 599. [2] P. Morin, J. Pierre, D. Schmitt, D. Givord, Physics Lett. A 65 (1978) 156. [3] K.H.J. Buschow, C.J. Schinkel, Solid State Commun. 18 (1976) 609. [4] D. Kim, P.M. Levy, J. Magn. Magn. Mater. 27 (1982) 257. [5] J. Pierre, A. de Combarieu, R. Lagnier: J. Phys. F 9 (1979) 1271. [6] U. Ko¨bler, R.M. Mueller, L. Smardz, D. Maier, K. Fischer, B. Olefs, W. Zinn, Z. Phys. B: Condens. Matter 100 (1996) 497. [7] E. Mu¨ller-Hartmann, U. Ko¨bler, L. Smardz, J. Magn. Magn. Mater. 173 (1997) 133.
U. Ko~ bler et al. / Journal of Magnetism and Magnetic Materials 188 (1998) 333—345 [8] R.M. Mueller, U. Ko¨bler, K. Fischer, Z. Phys. B, to be published. [9] U. Ko¨bler, Th. Bru¨ckel, A. Hoser, H.-A. Graf, M.-T. Fernandez-Diaz, K. Fischer, Z. Phys. B: Condens. Matter, to be published. [10] U. Ko¨bler, I. Apfelstedt, K. Fischer, W. Zinn, E. Scheer, J. Wosnitza, H. v. Lo¨hneysen, Th. Bru¨ckel, Z. Phys. B 92 (1993) 475. [11] U. Ko¨bler, J. Schweizer, P. Chieux, Th. Lorenz, B. Bu¨chner, W. Schnelle, F. Deloie, W. Zinn, J. Magn. Magn. Mater. 170 (1997) 110.
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[12] U. Ko¨bler, R.M. Mueller, J.P. Brown, Th. Lorenz, B. Bu¨chner, K. Fischer, to be published. [13] A. Kornblit, G. Ahlers, Phys. Rev. B 11 (1975) 2678. [14] B. Stroka, J. Wosnitza, E. Scheer, H. v. Lo¨hneysen, W. Park, K. Fischer, Z. Phys. B: Condens. Matter 89 (1992) 39. [15] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [16] E. Gmelin, Thermochimica Acta 110 (1987) 183. [17] M. Foldeaki, W. Schnelle, E. Gmelin, P. Benard, B. Koszegi, A. Giguere, R. Chahine, T.K. Bose, J. Appl. Phys. 82 (1997) 309. [18] F. Bloch, Z. Phys. 61 (1930) 206.
Critical magnetic behaviour of a ferromagnet with antiferromagnetic fourth-order exchange interactions: GdMg U. Ko¨bler!,*, R.M. Mueller!, W. Schnelle", K. Fischer! ! Institut fu( r Festko( rperforschung, Forschungszentrum Ju( lich, D-52425 Ju( lich, Germany " Max-Planck Institut fu( r Festko( rperforschung, D-70506 Stuttgart, Germany Received 11 September 1997; received in revised form 17 May 1998
Abstract We report on investigations of the critical magnetic behaviour of GdMg. In this intermetallic ferromagnet the sum of all fourth-order interactions (i.e. biquadratic, three-spin and four-spin interactions) is 50 K and antiferromagnetic at the Curie temperature of ¹,"110 K. This has been shown with measurements of the third-order susceptibility s . Although C 3 the Curie temperature can be assumed to be defined mainly by bilinear (Heisenberg) interactions, the critical indices for the linear susceptibility s and the spontaneous magnetization are molecular field like c"c@"1, b"0.5. The 1 third-order susceptibility s is discontinuous at ¹,: for ¹'¹,, s is finite but for ¹(¹, it diverges with a critical 3 C C 3 C exponent of one. The ferromagnetic order—disorder transition is therefore weakly first order. At a second critical temperature of ¹M"91 K the rise of an ordered antiferromagnetic component oriented perpendicular to the ferromagN netic one has been reported two decades ago based on neutron scattering investigations. Here we will show that the third order susceptibility s diverges at ¹M but that the linear susceptibility s stays finite. This is characteristic for a phase 3 N 1 transition driven by the fourth-order interactions. We must therefore assume that the transverse components of the ferromagnetically ordered moments order antiferromagnetically at ¹M. This view is in accordance with measurements of N the susceptibility s perpendicular to an applied static magnetic field. As a result, a second-order parameter perpendicuM lar to the spontaneous magnetization is generated by the fourth-order interactions. As a further consequence of the perpendicular moment configuration, the spontaneous longitudinal magnetization reaches only 0.75 of the theoretical saturation value for ¹P0 and decreases like ¹2 with temperature as has been observed for other ferromagnets with fourth-order interactions, such as EuS and CrBr . ( 1998 Elsevier Science B.V. All rights reserved. 3 PACS: 75.30.Et; 75.40.Cx Keywords: Biquadratic exchange; Critical behaviour; Ferromagnets
1. Introduction
* Corresponding author. Tel: #49 2461 613324; fax: #49 2461 612016.
GdMg is an unconventional metallic ferromagnet [1]. The Gd moments occupy a simple cubic lattice and are not aligned strictly parallel for
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 2 1 6 - 9
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¹P0 [2]. The resulting ferromagnetic moment is only 5.24 l instead of 7 l expected for Gd. FerB B romagnetic saturation is not reached even in fields of 35 T [3]. Since any paramagnet would be magnetically saturated for ¹"4.2 K and B "35 T, 0 strong antiferromagnetic interactions must exist in GdMg in addition to the ferromagnetic ones which are responsible for the high Curie temperature of 110 K. In fact, half-indexed magnetic scattering lines could be observed for ¹(90 K in neutron scattering investigations and a magnetic structure model including an ordered antiferromagnetic component perpendicular to the ferromagnetic one has been proposed [2]. This biaxial magnetic ordering type is also named the flopside spin structure [4]. There is evidently a discrepancy between the high Curie temperature of 110 K and the strong antiferromagnetic interactions at low temperatures which prevent ferromagnetic saturation even in very large magnetic fields. If these antiferromagnetic interactions were of the conventional bilinear (Heisenberg) exchange type (S ) S ) they would add 1 2 to the ferromagnetic interactions and lower the Curie temperature accordingly. Here we will show that the transverse, antiferromagnetically ordered magnetization component is the result of antiferromagnetic fourth-order exchange interactions. There have been two different theoretical approaches to explain the unusual observation of mutually perpendicular ferromagnetic and antiferromagnetic sublattices in a material with pure spin magnetism and therefore with a vanishing magnetic anisotropy [4,5]. Both models require competing interaction mechanisms which destabilize the ferromagnetic order with decreasing temperatures leading eventually to the observed canted spin structure. Here we propose to consider the perpendicular configuration of two order parameters as a consequence of the general principle that an additional class of magnetic interactions will tend to increase the magnetic order further. Though the individual fourth-order interaction processes are sizeable they are prevented from increasing the Heisenberg transition temperature accordingly. Therefore, the magnetic order is increased by inducing an additional order parameter whereby the entropy of the
system can be lowered more efficiently. In the present state of our knowledge this explanation must be viewed as a working hypothesis which awaits a rigorous theoretical justification but which is able to explain the experimental facts from a general point of view without the necessity of making special assumptions on the interaction parameters. Apparently, the fourth-order exchange interactions (i.e. biquadratic, three-spin and four-spin interactions) are such a particular class of magnetic interactions able to generate a new order parameter. Since the transverse moment components are the only degrees of freedom left for an additional ordering process once the longitudinal moment components are ordered it is natural to assume that the fourthorder exchange interactions govern the order of these transverse moment components meaning that the rotational symmetry of the uniaxial magnetic order will be broken. This view is intuitively supported by the fact that an antiferromagnetic biquadratic interaction requires a perpendicular moment orientation but two orthogonal order parameters seem to occur also if other members of the fourth-order interaction class dominate as in the Eu-chalcogenides, in which the ferromagnetic three-spin interactions dominate [6,7]. As a consequence of our hypothesis, we consider the experimental value for the canting angle between the ferromagnetic and antiferromagnetic magnetization component [2] to be essentially 90° in GdMg. In Refs. [6,7] it was shown that in the hightemperature limit the fourth-order exchange interactions give rise to a Curie—Weiss law for the thirdorder susceptibility s which is defined according 3 to 1 1 B " m# m3#2, * s s 1 3
(1)
where B is the applied magnetic field converted to * its value inside the sample and m is the reduced magnetization. The Curie—Weiss temperature H 3 associated with s was shown to be a weighted 3 average over all fourth-order interaction processes [7]. The Curie—Weiss law of s shows that there 3 exists, even in the molecular field approximation, a second ordering transition at H in addition to 3
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the conventional one at H . Both transitions are of 1 the order—disorder type. Since a collinear spin system can order only once it is natural to assume that the ordering process at H affects the disordered 3 transverse moment components. These components order at a finite temperature whenever the fourth-order interaction sum as given by H has 3 a finite value [8]. This transverse order is called ferromagnetic for H '0 and antiferromagnetic for 3 H (0 [8]. 3 The longitudinal and transverse ordering phenomena are mutually independent and occur at clearly different temperatures in GdMg. At ¹,"110 K the longitudinal moment components C order ferromagnetically while at a lower critical temperature of ¹M"91 K the transverse moment N components order antiferromagnetically [2]. This independence of the ordering processes becomes evident from the fact that the spontaneous magnetization of the longitudinal (Heisenberg) order parameter does not exhibit any visible anomaly at the Ne´el temperature of the transverse order parameter. Measurements of the AC susceptibility s M perpendicular to a longitudinal static magnetic field performed in this work further confirm the perpendicular orientation of the antiferromagnetic component. In this communication, we will show by an investigation of the third-order susceptibility s that 3 there are strong antiferromagnetic fourth-order interactions for ¹*¹, in GdMg. These interactions C must be considered as the origin of the antiferromagnetically ordered transverse magnetization component found earlier with neutron scattering studies for ¹)¹M"91 K [2]. The different N observed transition temperatures can clearly be attributed either to second-order (Heisenberg) interactions or to fourth-order interactions. This was demonstrated in the diamagnetically diluted antiferromagnets Eu Sr Te. In this system both x 1~x order parameters are antiferromagnetic for x( 0.85 with definitely different Ne´el temperatures and clearly different critical field values [8—10]. The critical field lines do not touch and no multicritical behaviour occurs. This behaviour allows a clear distinction between second-order phase boundaries and fourth-order phase boundaries. As a further consequence of the perpendicular configuration of
335
two ordered magnetization components, magnetic saturation will not be reached in finite magnetic fields. Both components will not reach the full magnetic moment for ¹P0. The antiferromagnetic total fourth-order interaction strength as identified by means of the thirdorder susceptibility s for ¹*¹, must be conC 3 sidered an average over all individual biquadratic, three-spin and four-spin interaction processes. Comparison with other investigated systems, such as the insulating solid solutions Eu Sr Te and x 1~x Eu Sr S [7], shows that these individual interacx 1~x tion processes are much larger than the Curie— Weiss temperature H which is a measure for the 3 high-temperature average over all fourth-order interactions [7]. In metals, however, no such Curie—Weiss behaviour is observed for s even at 3 temperatures several times larger than the ordering temperature. It appears that in metals s exhibits 3 a strong temperature dependence only near ¹ but C approaches a rather weak temperature dependence in the high temperature limit [11]. Only the observed shift of the experimental s (¹) data relative 3 to the calculated Curie law of s provides some 3 information for the strength of the fourth-order interactions which for metals change inherently with temperature such that s is only weakly tem3 perature dependent and may even cross the calculated Curie line. We must therefore assume there are much larger individual antiferromagnetic and ferromagnetic fourth-order interaction processes responsible for the observed mean value near ¹,. C Though both perpendicular ordering processes are mutually independent and the associated order parameters clearly differ, they show common properties and tend to order at similar temperatures. For instance, it was observed in Refs. [8,12] that both order parameters of EuS are ferromagnetic and both decrease according to a ¹2 law instead with a ¹3@2 law for increasing temperatures for ¹P0. This could be concluded from the fact that the zero-field 153Eu NMR signal, which averages over an isotropic distribution of both order parameters on account of an isotropic orientation of magnetic domains, shows a ¹2 law with high precision [8]. The ¹2 law must therefore be common to both order parameters. This law seems to be
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exact until the critical temperature range near the ordering temperature. The fourth-order interactions also dominate the spin dynamics near ¹, and are found to be the C relevant interactions for defining the asymptotical critical magnetic behaviour of both order parameters, just as they define their behaviour for ¹P0, where they give rise to a ¹2 law instead of a ¹3@2 law. The asymptotical temperature range in which the mean field critical behaviour is clearly observed in GdMg is as large as 10 K. This we attribute to the fact that the resulting total fourthorder interaction is as large as 50 K. Additionally there seems to be no strong competition between the different fourth-order interaction processes. Since the total interaction is antiferromagnetic the ordering of the transverse order parameter is also antiferromagnetic. It is therefore easily possible to orient the longitudinal magnetization component by means of a small magnetic field and to measure its critical behaviour with field-parallel magnetization measurements. In other ferromagnetic compounds such as EuO and EuS the total fourthorder interaction is ferromagnetic [6,12]; therefore, the order of the transverse moment components is ferromagnetic. In such a situation it is virtually impossible to distinguish both order parameters because neither of them will be oriented perfectly parallel to an applied magnetic field. As a consequence, in conventional magnetization measurements a mixture of both order parameters, depending on the strength of the applied magnetic field, will be measured. One may therefore speculate whether or not the same asymptotical critical exponents as observed here for GdMg may apply also for EuO and EuS but only if care is taken to properly discriminate between the two order parameters. In fact, in EuO, the experimental critical exponent for the magnetic specific heat is clearly not consistent with the Heisenberg model prediction [13], compare however Ref. [14]. Of course, the specific heat is a scalar quantity which averages over all space directions. The critical magnetic behaviour of the longitudinal order parameter of GdMg is essentially molecular field like: the linear susceptibility v diverges 1 with the mean field exponent c"c@"1 at ¹,. The C spontaneous magnetization also exhibits the mo-
lecular field exponent of b"0.5. The third-order susceptibility s shows more complicated behav3 iour. It should be recalled that s is the quantity 3 responding most strongly to the fourth-order interactions. Approaching ¹, from higher temperatures C s is finite at ¹, in agreement with the molecular C 3 field approximation but in disagreement with the Heisenberg model according to which s should 3 diverge with an exponent of 2b!c"!0.66. Approaching ¹, from lower temperatures, s diverges C 3 with an exponent of unity. We believe this discontinuous behaviour of s at ¹, shows different C 3 fourth-order interaction types are differently weighted by the rise of the bilinear order parameter. As a consequence of the discontinuous behaviour of s the phase transition at ¹, has to be conC 3 sidered weakly first order. The associated latent heat may be small and in fact no obvious latent heat could be seen in the magnetic specific heat [5]. Evaluation of the AC-susceptibility s measured M perpendicular to a longitudinal static magnetic field reveals clearly the ordering processes of the longitudinal and transverse moment components occur at different temperatures: at ¹,"110 K the C longitudinal components order ferromagnetically which is ascertained from a diverging longitudinal susceptibility s . The perpendicular susceptibility , continues increasing monotonicly at ¹, without C showing any definite anomaly. At ¹M"91 K, on N the other hand, s exhibits a rounded but clear M maximum characteristic for a Ne´el transition but here the longitudinal susceptibility s shows vir, tually no anomaly. The ordering transition at ¹M is N also distinguished by an absolute maximum of the magnetic specific heat [5], whereas at ¹, there is C a rounded shoulder in the magnetic specific heat.
2. Sample preparation The single crystal of GdMg was grown from a nominal composition of GdMg by directional 1.05 freezing. Gd and Mg were sealed in a molybdenum crucible. The crucible, surrounded by a molybdenum block and standing on a cold pin, was RF-heated in vacuum. The upper temperature of the crucible was about 200°C above the melting point of 868°C before slow cooling.
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The crystal did not adhere to the crucible wall and was covered by the excess Mg. It was rather easily cleaved along (1 0 0) planes and showed sharp Laue spots. For the magnetic measurements a nearly perfect sphere of about 1 mm diameter was fabricated.
3. Longitudinal magnetic measurements As a basis for the evaluation of the linear (longitudinal) susceptibility s and the third-order sus1 ceptibility s we make use of the definition 3 according to Eq. (1). For the evaluation of s and s magnetic iso1 3 therms are measured and plotted as m2 versus B /m * (Arrott plot). Fig. 1 shows an Arrott plot for some selected magnetic isotherms with ¹'¹,. In this C representation the slopes of the straight lines give the third-order susceptibility s and the intersec3 tions with the abscissa give the reciprocal linear susceptibility s~1"B /m. 1 * The s -data obtained by extrapolation of the 1 magnetic isotherms in the Arrott plot from the high-field range down to m2P0 (B P0) character* ise the behaviour of the longitudinal magnetization component which is easily turned into the field
Fig. 1. Magnetic isotherms of GdMg for ¹'¹, plotted as C squared reduced magnetization m2 versus B /m (Arrott plot). * The slopes of these lines give the third-order susceptibility s and the extrapolated intersections with the abscissa the recip3 rocal linear susceptibility s~1"B /m of the longitudinal (fer1 * romagnetic) magnetization component.
337
direction in a system in which the transverse magnetization exhibits antiferromagnetic interactions. In small fields with B (0.05 T (not resolved in 0 Fig. 1), orientational processes between longitudinal and transverse magnetization components take place and the susceptibility decreases again with decreasing field (see discussion of Fig. 11). In genuine zero-field measurements no orientation of the longitudinal and transverse magnetization components is ascertained and a mixture of both might be measured. The slopes of the Arrott lines in Fig. 1 change only little with temperature. In particular it is evident that the critical isotherm, i.e. the fictive line passing exactly through the origin, will have a finite slope. s is therefore finite at ¹,. According to the C 3 Heisenberg model s should diverge with an expo3 nent of 2b!c"!0.66 at ¹ . 2b!c results in C zero if the molecular field values of b"0.5 and c"1 are inserted. Moreover, a finite slope of the critical isotherm in the Arrott plot means that the critical exponent d equals 3. A finite value of s at 3 ¹, conforms to the molecular field prediction for C the longitudinal order parameter. Fig. 2 shows the temperature dependence of the reciprocal third-order susceptibility s~1 for tem3 peratures ¹'¹,. In this diagram the calculated C Curie law for s has been added for comparison. 3 The Curie law applies for vanishing fourth-order
Fig. 2. Reciprocal third-order susceptibility v~1 of GdMg ver3 sus temperature for ¹'¹,. The shift of the experimental C s~1 data to the theoretical Curie law s~1"C~1. ¹ is an 3 3 3 indicator for the existence of fourth-order interactions. At ¹,, s is finite coming from ¹'¹,. C 3 C
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interactions. As was shown in Refs. [6,7] for finite fourth-order interactions, s obeys a Curie—Weiss 3 law in the high-temperature limit according to C 3 s " 3 ¹!h 3 with
(2)
10g(S#1)3k B . C " (3) 3 9[(S#1)2#S2]k B The Curie—Weiss temperature H in Eq. (2) is 3 a measure for the high-temperature average of all fourth-order interactions. For metals, however, no such Curie—Weiss behaviour results even at temperatures several times larger than the ordering temperature [11]. Fig. 2 also shows the experimental s data assume nearly 3 temperature independent behaviour at larger temperatures and even cross the calculated Curie law for s at about 210 K instead of approaching a line 3 parallel to C~1 ) ¹ asymptotically for large temper3 atures. The total fourth-order interactions, namely correlations, can therefore change sign as a function of temperature in metallic systems [11]. The only conclusion we can draw from Fig. 2 is that there are appreciable fourth-order interactions in GdMg and these are antiferromagnetic for ¹(210 K but ferromagnetic for ¹'210 K. At the Curie temperature of ¹,"110 K the horizonC tal shift of the experimental s -data to the cal3 culated Curie law for s is 50 K. Therefore, the 3 order—disorder transition occurs in the presence of strong antiferromagnetic fourth-order interactions in GdMg. The critical behaviour of the linear susceptibility s is shown in Fig. 3 where the intersections of 1 the Arrott isotherms with the abscissa B /m" * s~1 from Fig. 1 are plotted versus temperature. As 1 can be seen, the data for ¹'¹, follow with high C precision a (¹!¹,)~1 law until ¹+120 K meanC ing that the critical exponent of s is c"1. Fig. 3 1 includes the Curie slope C~1 calculated using the 1 theoretical value for the Curie constant of the linear susceptibility which following the nomenclature of Eq. (1) is g(S#1)k B. C " 1 3k B
(4)
Fig. 3. Reciprocal linear susceptibility s~1 in the critical tem1 perature range around ¹, showing that c"c@"1. For ¹'¹, C C the theoretical Curie—Weiss line s~1"C~1(¹!¹,) has been 1 1 C added using g"2 and S"7/2 for the calculation of C accord1 ing to Eq. (4).
With g"2 and S"7/2, Eq. (4) gives C " 1 2.015 K/T. The experimentally observed coefficient of the asymptotical (¹!¹,)~1 law is, however, C C!4:.1"2.72 K/T. On the other hand, the ob1 served parameters of the high-temperature Curie—Weiss law are H "114 K and C=" 1 1 2.37 K/T. This value for C= is still much larger than 1 the free ion value and even larger than most reported values for metallic systems. As is well known [15], the effective Bohr magneton number of Gd is always increased in metallic systems. This applies also for alloys in which the gadolinium is strongly magnetically diluted by other diamagnetic metals [3]. It is therefore difficult to relate the conduction band polarization with the strength of the secondorder and fourth-order exchange interactions. Because the existence of two order parameters has been observed also in insulating materials [8,9] the biaxial spin structure of GdMg seems not to be a specifically metallic effect caused by particular properties of the conduction band in GdMg as was proposed in Ref. [5]. In spite of the fact that the critical exponent of the molecular field model of c"1 is observed for ¹P¹, for the linear susceptibility s , H is clearly C 1 1 larger than ¹, and the slope of the asymptotical C (¹!¹,)~1 law is larger than the observed highC temperature Curie constant. It should be noted that H "114 K is unusually close to ¹,"110 K but C 1 that the true molecular field conditions H "¹, C 1
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and C!4:.1."C= are not quite fulfilled. In other 1 1 words, spin fluctuations are unusually small which is confirmed also with the specific heat measurements. In order to evaluate s and s for ¹(¹, in C 1 3 a way consistent with the evaluation for ¹'¹, we C have to replace the reduced magnetization m in Eq. (1) by m!m S. mP (5) 1!m S Eq. (5) means that we subtract from the experimental magnetization value m the spontaneous magnetization m and normalize by 1!m . In this S S way, our new reduced magnetization tends to zero for mPm but approaches unity for mP1 irreS spective of the value of m . The s -data for ¹(¹, C S 1 can then be compared in magnitude with those for ¹'¹,. C Prior to evaluating s and s for ¹(¹, we C 1 3 must know the spontaneous magnetization m (¹). S This quantity was obtained as usual [5] by an extrapolation of the magnetization curves m(B ) to * B P0. Fig. 4 shows the obtained spontaneous * magnetization data as function of temperature. We have included the fitted ¹2 function describing the behaviour of m at low temperatures as well as the S (¹,!¹)0.5 function fitted to the m (¹) data near C S ¹,. The quality of both fits will be shown more C
Fig. 4. Temperature dependence of the reduced spontaneous magnetization m (¹). At low temperatures a ¹2 law has been S fitted to the experimental data. In the critical temperature range m (¹) is described by a (¹,!¹)0.5 law. S C
339
clearly in two consecutive diagrams. Both fitted functions in Fig. 4 reproduce the experimental m (¹) data rather completely. The gap region where S neither of the functions fits the data is fairly small. No pronounced anomaly can be seen at ¹M"91 K N in agreement with earlier measurements [5]. With the magnetization data re-normalized according to Eq. (5), Arrott plots of the magnetic isotherms have been drawn for the evaluation of s (¹) and s (¹) for ¹(¹, as was shown in Fig. 1 C 1 3 for the data for ¹'¹,. Fig. 3 includes the reciproC cal s data for ¹(¹,. These data have been C 1 multiplied by a factor of 500 for simpler comparison to the data for ¹'¹,. It can be seen that the C critical exponent of the linear susceptibility for ¹(¹, is c@"1 but the absolute susceptibility C values are larger by a factor of 500 compared to the s -data for ¹'¹,. C 1 Fig. 5 gives the experimental s~1(¹) dependence 1 for a larger temperature range below ¹(¹,. C In addition to ¹"¹,, one further anomaly at C ¹*"98 K can be seen. The character of this feature, which can also be identified in the magnetic specific heat measurements, is unclear. At ¹M, N where the magnetic specific heat has an absolute maximum, virtually no anomaly appears in s~1(¹). 1 On account of the smaller variation of the magnetization with field at low temperatures the scatter of the experimental s~1 data increases somewhat with 1 decreasing temperature. For ¹P0 s seems to be 1 finite, in disagreement with the expectation for
Fig. 5. Reciprocal linear susceptibility s~1 versus temperature 1 for ¹)¹, revealing different anomalies at the phase C transitions ¹,"110 K, ¹*"98 K and ¹M"91 K. C N
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a Heisenberg ferromagnet. Since m (¹"0) is only S 0.75 it is logically necessary that s (¹"0) be finite. 1 The temperature dependence of the reciprocal third-order susceptibility s~1 is displayed in Fig. 6. 3 s is definitely finite for ¹P0. At ¹M"91 K, s diN 3 3 verges but since s is finite at this temperature the 1 transition at ¹M is clearly produced by fourth-order N interactions. As will be confirmed with measurements of the perpendicular susceptibility s , the M transverse moment components order antiferromagnetically at ¹M. At ¹*"98 K one further N anomaly appears. Here s exhibits a relative min3 imum. Though it has not yet been shown with adequate experimental accuracy, if we assume s really diverges at ¹* we must attribute this 1 anomaly to bilinear interactions. Fig. 6 shows that s diverges with an exponent of 3 !1 at ¹M"91 K and at ¹,"110 K as well, in C N contrast to the Heisenberg model according to which s should diverge with an exponent of 3 2b!c"!0.66. Fig. 7 shows the low-temperature data for the reduced spontaneous magnetization m plotted verS sus the squared reduced temperature (¹/¹,)2. In C agreement with Ref. [1] these data follow a ¹2 law with high precision until (¹/¹,)2"0.55 or C ¹/¹,"0.74 namely, at ¹"81 K. ¹2 laws have C been observed for the spontaneous magnetization at low temperatures in many systems with fourthorder interactions such as EuS and CrBr [8] and 3
Fig. 6. Reciprocal third-order susceptibility s~1 versus temper3 ature for ¹)¹, showing divergences of s at ¹,"110 K and C 3 C ¹M"91 K. At ¹*"98 K a relative minimum of s can be N 3 noticed.
Fig. 7. Reduced spontaneous magnetization m versus (¹/¹,)2 S C showing the large validity range of the low-temperature ¹2 law. For ¹P0, m is only 0.745. S
also for the low-temperature behaviour of the critical field curves B for EuSe [6] and Eu Sr Te C x 1~x [8]. The reduced longitudinal spontaneous magnetization reaches only 0.75 for ¹P0. Assuming the magnetization component sharing the longitudinal order parameter is perfectly ferromagnetic we may conclude &3/4 of the Gd moment belongs to the parallel order parameter but &2/3 to the perpendicular order parameter. As a consequence, the antiferromagnetic fourth-order interactions which we believe are responsible for the perpendicular configuration of two order parameters are nearly as strong as the ferromagnetic bilinear interactions. The same conclusion may be drawn from the very similar ordering temperatures for the two order parameters, ¹,"110 K and ¹M"91 K. N C The spontaneous magnetization data of the critical temperature range are plotted in Fig. 8 as m2 versus ¹. From this diagram it is clear the S critical exponent of the spontaneous magnetization is b"0.5. The critical temperature range is again very large: 97(¹ (110 K. #3*5. Summarizing the longitudinal magnetization measurements we can say that the critical exponents for the linear and the cubic susceptibility are molecular field-like in GdMg but only for ¹'¹,. C For ¹(¹, the third-order susceptibility diverges C at ¹, in contrast to the molecular field model. This C
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Fig. 8. Squared reduced spontaneous magnetization m2 versus S temperature in the critical temperature range showing that the critical exponent is b"0.5.
indicates the importance of the fourth-order interactions increases discontinuously with the rise of the parallel order parameter at ¹, as a result of the C bilinear exchange interactions.
4. Specific heat measurements The specific heat was measured using a conventional heat pulse method [16]. The contributions of the sample holder and of a minute amount of vacuum grease utilized to mount the sample were measured in a separate run and subtracted. The accuracy is estimated to be better than 2% of c (¹). p Fig. 9 displays the specific heat of GdMg. The most prominent feature of Fig. 9 is the absolute maximum at ¹M"91 K to be identified N with the ordering process of the transverse order parameter. This result for c (¹) conforms perfectly p with earlier data [5]. Based on the existence of this specific heat peak one can be sure the ordering of the transverse moment components is associated with a phase transition different from ¹,. It is then C intuitively clear that virtually no anomalies occur in the longitudinal spontaneous magnetization and the longitudinal linear susceptibility s at ¹M in N 1 spite of the pronounced peak in the magnetic specific heat. Surprisingly, above the 110K phase transition there is no clear trace of a short range order tail, as normally seen in the specific heat of
341
Fig. 9. Specific heat c (¹) of GdMg (d). The continuous line p shows a lattice and electronic (background) specific heat c & fit- % ted to the measured data above 120 K. c is the magnetic .!' contribution (m). Inset: Specific heat c /¹ versus ¹2 for p ¹(10 K. The fit gives H "145 K and c"5.2(4) mJ/mol K2. D
magnetic systems. This conforms to the mean field character of this phase transition and the observation made in the susceptibility measurements that H "114 K and ¹,"110 K are very close to each C 1 other. Latent heat is not found at the phase transitions. Since only s was observed to be dis3 continuous at ¹, any possible latent heat should in C fact be very small. The inset of Fig. 9 shows the specific heat below 10 K in a c (¹)/¹ versus ¹2 representation. The fit p to these data up to 7 K with c (¹)"c¹#b¹3 p gives c"5.2(4) mJ/mol K2 and b"1.28(1)] 10~3 J/mol K4. This b value would correspond to H (0)"145 K, a value much too ‘soft’ for the latD tice of GdMg, indicating strong magnetic contributions. In order to obtain the f-electron related magnetic contribution of GdMg we fitted a Debye model (phonons) plus a linear term (delocalized electrons; anharmonic lattice contributions) to the specific heat data above the phase transitions (¹'120). We found a good representation of the data for H "228 K and a linear term c*¹ with c*" D 14 mJ/mol K2. After subtraction of this background we obtain the magnetic contribution c shown in Figs. 9 and 10. In addition to the .!' three previously mentioned anomalies at ¹,, ¹* C and ¹M a broad hump at around 30 K is observed, N
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Fig. 10. Magnetic specific heat c (¹) of GdMg (left axis) (m) .!' and magnetic entropy S (¹) (continuous line; right axis). The .!' critical temperatures obtained from the susceptibility measurements are indicated.
a feature often found in Gd compounds and in pure Gd metal [17]. The magnetic entropy S (¹) .!' above the phase transition reaches exactly R ln 8, the value expected for the ordering of the 8-fold degenerate J"S"7/2 Gd ground state. The additional sharp shoulder seen at ¹*" 98 K in the specific heat corresponds well with the anomalies at the same temperature seen in s and 1 s in Figs. 5 and 6. This feature appeared less 3 pronounced in the specific heat measurement of Ref. [5]. It is likely that the ¹*"98 K anomaly constitutes some phase change of the parallel order parameter but since the microscopic change at this transition cannot be resolved on the basis of our experiments we will not consider this detail further. At the Curie temperature of ¹, only a rounded C shoulder appears in the magnetic specific heat in Fig. 10. This phase transition can be localized with a much better precision in the magnetization measurements. The results of Fig. 10 make clear there exists no realistic chance to evaluate the critical exponent a at ¹, in GdMg since the anomaly at C ¹, is too much smeared and the proximity of the C ¹*"98 K and ¹M anomalies affects the c (¹) deN p pendence too much at ¹,"110 K. C 5. Measurements of the perpendicular susceptibility Since an antiferromagnetic biquadratic interaction requires a perpendicular moment orientation,
measurements of the perpendicular susceptibility s can be expected to be very revealing. In fact, M these measurements show unambiguously the existence of an antiferromagnetic transverse order parameter. In these experiments the ferromagnetic magnetization component is forced into a direction given by a longitudinal static magnetic field B and the , perpendicular AC susceptibility s is measured M with an alternating magnetic field perpendicular to the static one having an amplitude much smaller than B . In our case the longitudinal magnetic field , was produced by a superconducting coil. The spherical sample was placed inside a small concentric transformer, the axis of which was oriented perpendicular to the static field B . An outer wind, ing section produced the AC magnetic field to periodically excite the transverse magnetization. This field had an amplitude of &1 G. The sample was inside one of a pair of astatic inner secondary coils thus producing an unbalance signal proportional to its susceptibility. After passing the AC signal through a 1 : 100 low noise transformer it was detected with a lock-in amplifier. Fig. 11 shows a set of selected perpendicular susceptibility curves as a function of temperature together with the longitudinal zero-field susceptibility (B "0) measured immediately after cool, down of the superconducting magnet in order to make sure that there was no remanent field in the superconducting coil. In this measurement no definite orientation of the order-parameters is given but since the ferromagnetic one will be preferentially aligned by the small AC field the s(B "0) , curve can be assumed to reflect mostly the behaviour of the ferromagnetic order parameter. This s curve labelled by B "0 shows a distinc, , tive kink at the Curie temperature ¹, with a plaC teau value at lower temperatures given by demagnetization effects, i.e. the formation of magnetic domains. For an ideal Heisenberg ferromagnet which lacks any magnetic anisotropy the demagnetization plateau should continue until ¹P0. This is not the case for GdMg. At about 80 K the susceptibility s begins to decrease , rather abruptly, reaching a value of only 0.6 of the demagnetization plateau value for ¹P0. The cause is the transverse magnetization component
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which increases at the expense of the longitudinal magnetization. In contrast to s , i.e., the curve with B "0, the , , perpendicular susceptibility s shows a rounded M but clear maximum at ¹M but no anomaly at all at N the Curie temperature ¹,. This behaviour proves C that the longitudinal and transverse magnetization components order at different temperatures (¹, and ¹M, respectively). Moreover, the observed N C maximum of s is characteristic of antiferromagM netic ordering of the transverse magnetization component in agreement with the neutron scattering results [2]. It should be noted that the field value of B "0.23 T has been chosen such that the max, imum of s has not been cut off by demagnetization M effects. In other words, a field of this value is needed to orient the ferromagnetic component perfectly parallel to B in order that s really samples the , M perpendicularly oriented antiferromagnetic component. For larger longitudinal magnetic fields the height of the maximum of s decreases and transM forms eventually into a kink. ¹M shifts only slightly N with field as one would expect for the critical field of an antiferromagnet with a Ne´el temperature as large as ¹M"91 K. N Another interesting detail in Fig. 11 should be mentioned: there is evidently an anisotropy be-
Fig. 11. Perpendicular AC susceptibilities s for different values M of the static longitudinal magnetic field B . The curve for B "0 , , giving the longitudinal susceptibility s exhibits a rather sharp , kink at the Curie temperature ¹,. No anomaly is seen at ¹, in C C the s curves (B '0) but a maximum at ¹M"91 K indicates M , N an antiferromagnetic ordering process for the transverse moment components.
343
tween s and s even in the paramagnetic phase. , M The s data of the curve with B "0 should always , , be larger than the s data since the s curve has M , been measured with the smallest field (&1 G). In contrast to this expectation the s curve drops , clearly below the s curves in the paramagnetic M phase. This interesting observation shows there are different interaction energies for both order parameters even in the paramagnetic phase. In the case of the metallic GdMg the third-order susceptibility is nearly constant for ¹'¹,. This means C that the fourth-order interactions change towards ferromagnetic for increasing temperature whereby the importance of s increases relative to s . This 3 1 might explain the observed relative increase of s referred to s . M , This phenomenon will be the subject of a forthcoming publication [12].
6. Conclusions We have shown that the longitudinal and transverse Gd-moment components order at different temperatures in GdMg. At the Curie temperature of ¹,"110 K the longitudinal moment compoC nents order ferromagnetically while at a second critical temperature ¹M"91 K the transverse moN ment components order antiferromagnetically. This has been shown with measurements of the AC susceptibility s perpendicular to a static magnetic M field as well as with magnetic specific heat measurements. Our results conform to earlier specific heat and neutron diffraction studies which revealed an antiferromagnetically ordered magnetization component perpendicular to a ferromagnetic one for ¹)80 K [2,5]. Hence, there are two perpendicular order parameters to be attributed to the two ordering phenomena. This we consider as a plausible consequence of the quite general principle that a new class of magnetic interactions may tend to increase the magnetic order by generating its own order parameter. Because the transverse moment components are the only degrees of freedom left for an ordering process, their order is determined by the fourth-order interactions. In this way the entropy of the system can be further lowered. As a further
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consequence of this the observed phase transitions can clearly be distinguished geometrically. A total antiferromagnetic fourth-order interaction strength of 50 K has been shown by measurements of the third-order susceptibility s to exist at 3 ¹,. This value constitutes an average over biquadC ratic, three-spin and four-spin interaction processes which could not be evaluated separately. Fourth-order interactions not only give rise to ordering of the transverse moment components, they also dominate the spin dynamics of the longitudinal (Heisenberg) order parameter for ¹P0 and ¹P¹,. For instance, the spin order for ¹P0 C differs strongly from what is expected for a conventional Heisenberg ferromagnet. On expense of the transversally ordered moment components the spontaneous longitudinal magnetization is only 0.74 of the theoretical saturation value. This means that nearly equal fractions of the total Gd moment share the longitudinal and transverse spin order [2]. The spontaneous magnetization decreases according to a ¹2 law rather than a ¹3@2 law until a temperature as large as 0.8 ¹ . From the very C large validity range of the ¹2 law one may conclude that this law is exact under the conditions given in contrast to Bloch’s ¹3@2 law which was derived for the Heisenberg ferromagnet and which is only a first-order approximation [18]. The ¹2 law was shown experimentally to hold for both order parameters in a number of materials with fourthorder interactions [8]. It was observed not only for the spontaneous magnetization of ferromagnets such as EuS and CrBr but also for the critical field 3 curves of antiferromagnets such as Eu Sr Te and x 1~x the critical field curve of the metamagnet EuSe [6,8]. Also the critical magnetic behaviour of the longitudinal (Heisenberg) order parameter of GdMg is changed by the presence of the fourth-order interactions which are decisive at ¹,. Since the antiferC romagnetic transverse order parameter is not yet ordered at ¹, the critical behaviour of the longituC dinal order parameter can be measured conveniently. Surprisingly, the critical magnetic behaviour is essentially molecular field-like for both order parameters. At ¹,"110 K the linear susceptibility C s diverges with an exponent of one. At the order1
ing transition of the transverse moment components at ¹M"91 K s diverges also with an expoN 3 nent of one but the linear longitudinal susceptibility s shows virtually no anomaly. The fact that only 1 s diverges at ¹M shows this transition is caused by N 3 fourth-order interactions and the maximum of s at ¹M indicates the transverse moment compoN M nents order antiferromagnetically at this temperature. Since anomalies occur either for s or for s , , M a clear distinction between parallel and perpendicular ordering processes is possible. Even in the paramagnetic phase the order parameters can be distinguished by their different susceptibilities owing to different interaction energies. The magnetic specific heat supports the existence of (at least) two ordering transitions in GdMg. Clearly the specific heat is a scalar quantity and cannot distinguish between ordering phenomena occurring along different spacial directions. The shape of the specific heat anomaly may be characteristic for the transitions of both order parameters. In agreement with earlier investigations [5], c (¹) .!' shows a sharp peak at ¹M but a rounded structure N at ¹,. Whether this is a typical difference for both C ordering processes awaits further theoretical as well as experimental investigations.
Acknowledgements We gratefully acknowledge the technical assistance of Mrs. Ch. Horriar-Esser and Mr. B. Olefs.
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