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High field magnetization of Pr1–xGdxNi single crystals
Physica B 406 (2011) 2281–2283
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Physica B journal homepage: www.elsevier.com/locate/physb
High field magnetization of Pr1–xGdxNi single crystals Katsuhiko Nishimura a,n, Wayne D. Hutchison b, Yuta Tajiri a, Yosikazu Isikawa a, Keisuke Sato c, Koichi Kindo c a
Graduate School of Science and Engineering, University of Toyama, Toyama 930-8555, Japan School of Physical, Environmental and Mathematical Sciences, The University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia c Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan b
a r t i c l e i n f o
abstract
Article history: Received 29 January 2011 Accepted 21 March 2011 Available online 29 March 2011
Magnetization measurements were carried out on Pr1 xGdxNi single crystals for x ¼ 0.1, 0.2, 0.3 and 0.4 using pulsed magnetic fields up to 55 T and at 4.2 K. For the data observed along the c-axis there are clear indications of transitions, of the combined system of Pr and Gd moments, from a ferrimagnetic to a ferromagnetic state. The observed saturation magnetizations at 55 T and intermediate magnetizations at 1 T were well modeled assuming ferro- and ferri-magnetic structures, respectively. & 2011 Elsevier B.V. All rights reserved.
Keywords: PrNi High-field magnetization Spin rearrangement Antiferromagnetic coupling
1. Introduction Studies of the magnetism in intermetallic compounds containing both rare earth (RE) and transition (T) metal elements, in which the magnetic structures can be described in terms of exchange interactions and crystal electric field effects (CEFs), are numerous. It is well known that the sign of coupling of the RE moments with the T moments (such as Fe, Co) changes between light and heavy RE elements [1–4]. The conduction electron polarization due to a RE ion is negative with respect to the spin moment at the ion site and therefore the coupling of the spins of a RE ion and the conduction electrons is antiferromagnetic (antiparallel). Consequently, it follows that there is an antiferromagnetic coupling of the total magnetic moments of heavy RE ions with transition metal ions, whereas the coupling is ferromagnetic for the light RE with transition metal ions. These observations reflect Hund’s rule, which has J¼ LþS and J¼L–S for the heavy and light rare earths, respectively, and are consistent almost universally with data for RE-T compounds in literature. For example, the cobalt moment is coupled antiparallel to the heavy RE moment (parallel to the light RE moment) in RECo2 systems [4–8]. As a direct consequence of the above, in the case of intermetallic compounds, which mix light and heavy REs, the arrangement of the magnetic moments of the heterogeneous RE ions is also antiferromagnetic (leading to a ferrimagnetic structure). This antiferromagnetism, originating from the opposite sense of the
n
Corresponding author. Tel./fax: þ 81 76 445 6844. E-mail address: [email protected] (K. Nishimura).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.03.054
spin-orbit interactions of heavy and light rare earths, is illustrated by a number of experimental studies [1–4]. Explicitly, a minimum ordered state magnetization (compensation point) is observed part way between the end point compounds of mixed light and heavy RE compounds [1,2,7–9]. Our recent reports on the magnetic properties of Pr1 xGdxNi [10], Nd1 xGdxNi [11], Pr1 xTbxNi and Nd1 xTbxNi [12], using single crystals, all documented this type of behavior, with a minimum in magnetization at certain intermediate concentrations x. Amongst these mixed compounds, those containing Pr also showed evidence of a field dependence of the magnetic structure, which has not been reported in the literature previously. The magnetization data along the c-axis (easy-axis of PrNi), indicates a critical field Bc for a transition from a ferrimagnetic to ferromagnetic state having an x dependence, which reflects the low temperature saturation magnetization (MS) rather than the ordering temperature. In this paper we pursue this behavior of the magnetization of Pr1 xGdxNi single crystals to high externally applied field.
2. Experimental The single crystal samples of Pr1 xGdxNi (x¼0.1, 0.2, 0.3 and 0.4) were grown using the Bridgeman method. Polycrystalline starting material was prepared by argon arc melting, then packed in an alumina crucible and vacuum-encapsulated in a quartz tube before it was passed through a Bridgeman furnace over a week. The crystallographic orientations of the single crystals were determined by the Laue x-ray back reflection method, with the observed Laue patterns consistent with the samples being single
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K. Nishimura et al. / Physica B 406 (2011) 2281–2283
phase and being of the CrB-type (Cmcm) orthorhombic crystal structure [13]. Small samples, cubic in shape, were cut from the crystal boles for use in the various measurements. The low-field DC magnetization data (M) were measured using a SQUID magnetometer (MPMS) with external fields (B) up to 7 T. The high-field experiments with Pr1 xGdxNi were carried out in the International MegaGauss Science Laboratory at the Institute of Solid State Physics, University of Tokyo. Magnetization was measured at 4.2 K and in applied magnetic fields up to 55 T using a long-pulse magnet.
3. Results and discussions The low-field magnetization data at a temperature of 2 K and along the c-axis of Pr1 xGdxNi samples, for x ¼0.1–0.4 are shown in Fig. 1(a). A detailed description of the low-field magnetic properties has been given elsewhere [10]. The onset of a more rapid increase in M towards the right hand side of the plot is apparent for x ¼0.2 and 0.3. In the high-field magnetization data of Fig. 1(b), which was collected at 4.2 K and up to 55 T along the c-axis, the full extent of changes in M associated with a rearrangement of magnetic moments may be seen for all four samples. A critical field (Bc) value is estimated from the point maximum slope (dM/dBmax) for each data set. The estimates of Bc are 10.5, 6.9, 8.5, and 10.5 T for x¼0.1, 0.2, 0.3 and 0.4, respectively. These transformations are quite broad in field terms and we define Bc1 and Bc2 as the transition start (lower) and finish (upper) critical field values, respectively. Examples of how these values are estimated from the intersection point of the two extrapolated lines from the data in the lower and higher field regions can be seen in Fig. 1(b), while all the Bc, Bc1 and Bc2 values are plotted in Fig. 4. The observed values of saturation magnetization (MS) at 55 T are 3.08, 3.46, 3.87, and 4.22 mB/f.u. for x¼0.1, 0.2, 0.3 and 0.4, respectively, which correlate well with the theoretical average moment expected at each x. From Fig. 1(b) it can also be seen that the magnetizations saturate at about 20 T, which is rather small compared with those reported for Gd-Co system ( 4100 T) [14]. Since the c-axis is the easy magnetization axis of PrNi, the small saturation fields in this case suggest that the Pr-Gd exchange interactions are relatively weak. Unlike that at the other directions, the observed variation of the c-axis magnetization data is due to the interplay between Zeeman and Pr-Gd exchange energies alone. Measurements along the other principle crystalline directions do not have the same features. As an example the M curves for Pr0.8Gd0.2Ni along the a, b and c-axis are shown in Fig. 1(c). For the a-axis, intermediate in magnetic anisotropy for PrNi, M develops more or less monatonically to saturate at the same levels as for the c-axis, but at a slightly
higher field. While along the hard b-axis although M develops at a similar rate with applied field until around 10 T; after that the rate slows markedly and indeed full saturation is not obtained by 55 T. The observed c-axis MS values are plotted with solid-triangle symbols in Fig. 2. The magnetization at B¼1 T ( in Fig. 1(a)), which can be regarded as the M values for the antiferromagnetic arrangements of Pr and Gd moments (MAF), are also plotted with solid-circle symbols for x ¼0.1–0.4. The variations of MS and MAF can be explained neatly by assuming ferromagnetic and antiferromagnetic coupling between the Pr and Gd magnetic moments, respectively. The open-triangle and open-circle symbols plotted in Fig. 2 are the calculated values taking 2.35 mB/(PrNi) [15] and 7.0 mB/(GdNi) [16] as saturation magnetizations for the ferromagnetic and antiferromagnetic arrangements, respectively. Both these estimations model the experimentally observed values of MS and MAF along the c-axis against x very well, and thus support a simple picture that the Pr and Gd moments are aligned parallel above 20 T, but anti-parallel in low applied field (around 1 T). Adopting this model for the structures at the extremes, estimates of Bc for the various x can be made. We assume Bc is the point at which the average Zeeman energy generated by the field on the antiparallel sublattice exceeds that due to the heterogeneous (Pr-Gd) exchange. Explicitly we apply a molecular field approach, defining average fields at the Pr and Gd sites as He(Gd)¼ amGd–bmPr and He(Pr)¼ gmPr–bmGd, where a, b, g are positive constants reflecting the interaction strength Gd-Gd, Pr-Gd and Pr-Pr respectively. Here mGd ¼xNMGd and mPr ¼ (1–x)NMPr are the sublattice magnetizations where N is the number of RE spins per unit volume and MGd and MPr are the
Fig. 2. Magnetization of Pr1 xGdxNi at 1 T and 2 K (solid circles) from Fig. 1(a), and saturation magnetization at 55 T and 4.2 K (solid triangle) from Fig. 1(b) are plotted against Gd fraction x. Open circles and open triangles are the results of calculations assuming antiferromagnetic and ferromagnetic arrangements of Pr and Gd moments, respectively (see text).
Fig. 1. (a) Magnetization of Pr1 xGdxNi, with x¼ 0.0–0.4, observed at 2 K with external field along the c-axis. (Lines are a guide for the eye.) (b) High field magnetization of Pr1 xGdxNi, with x¼ 0.0–0.4 along the c-axis at 4.2 K. (c) High field magnetization of Pr0.8Gd0.2Ni along the a, b and c-axis at 4.2 K.
K. Nishimura et al. / Physica B 406 (2011) 2281–2283
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Fig. 3. Results of the observed magnetic transition temperatures TM and the calculated ones with the exchange parameter b (see text).
Fig. 4. The observed lower critical field Bc1, upper critical field Bc2, the field of maxmum of dM/dB (Bc), and the theoretical values of Bc (Btheo.) (see text).
RE moments (expressed in SI units in this instance). The average exchange energy can be written as Eex ðxÞ ¼ m0 axm2Gd þ bmGd mPr þ gð1xÞm2Pr h i 2 2 ¼ m0 N2 ax3 MGd þ bxð1xÞMGd MPr þ gð1xÞ3 MPr
experimental data reasonably well, lying neatly midway between Bc1 and Bc2. The transition is of course broad due to the random distribution of Pr and Gd ions, whereas the theory assumes the average molecular field. In this paper we have presented high field magnetization measurements on the light-heavy rare earth mixed compounds Pr1 xGdxNi. Within these data we have identified a more abrupt spin rearrangement (from ferrimagnetic to ferromagnetic) occurring only when the field is along the c-axis and depending on the value of x. The saturation magnetizations of all the samples were well reproduced by simply summing the weighted MS values of end point RENi compounds. The variations of the Bc with x in Pr1 xGdxNi correlate well with a simple model whereby the average Zeeman term becomes equal to the heterogeneous exchange at Bc.
where N is the number of RE spins per unit volume and MGd and MPr are the RE moments. It follows that the increase in exchange energy for inversion of one sublattice (to align with the other) is
DEex ðxÞ ¼ 2m0 N2 bxð1xÞMGd MPr In this expression the crucial unknown is the heterogeneous exchange strength b, which we can estimate by modeling TM(x). Explicitly, we define separate Curie constants CGd and CPr for the Gd and Pr sublattices respectively, then for an applied field Ha in the paramagnetic phase, we have mGd T ¼ CGd ðHa þ amGd bmPr Þ mPr T ¼ CPr ðHa þ gmPr bmGd Þ
Acknowledgement
For the case Ha ¼0, these equations can be solved to give an expression for TM(x) in terms of a, b, g, CGd and CPr. All these parameters except b can be evaluated from susceptibility data for the end point compounds PrNi and GdNi (i.e. x ¼0 and 1). We set aCGd ¼TM(x ¼1)¼72.8 K, a ¼18.8, gCPr ¼TM(x ¼0) ¼20.9 K and g ¼ 26.8. It then remains to determine the value of the heterogeneous exchange b. We have done this by choosing the b value, which allows the TM(x) expression to best fit our experimental TM data. This process, illustrated in Fig. 3, leads to b ¼ 19 which means the heterogeneous exchange is similar in magnitude to the homogeneous values (a,g) above. The final step is then to equate the change (increase) in exchange energy with the reduction in Zeeman energy due to inversion of the minority spin sublattice (Gd for x o 0.25 and Pr for x4 0.25) to the applied field direction. We have
This work was performed using facilities of the Institute for Solid State Physics, the University of Tokyo.
DEZeeman ¼ 2xNMGd B, x o0:25 DEZeeman ¼ 2ð1xÞNMPr B, x 40:25 and setting DEZeeman ¼ DEex at Btheo. gives Btheo: ¼ m0 Nbð1xÞMPr , Btheo: ¼ m0 NbxMGd ,
x o0:25 x 40:25
These theoretical expressions are compared with the experimental values of Bc (defined as a field at the maximum of dM/dB), Bc1 and Bc2 in Fig. 4. We note that the theory matches the
References [1] K.N.R. Taylor, Adv. Phys. 20 (1971) 598 and references therein. [2] W.E. Wallace, E. Segal, Rare Earth Intermetallics, Academic Press, New York and London, 1973 111. [3] I.A. Campbell, J. Phys. F2 (1972) L47. [4] J.J. Franse, R.J. Radwanski, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, vol. 7, Elsevier, 1993 Chapter 5, and references therein. [5] N.H. Duc, P.E. Brommer, J.J.M. Franse, Physica B191 (1993) 239. [6] N.H. Duc, T.D. Hien, P.E. Brommer, J.J.M. Franse, J. Magn. Magn. Mater. 104–107 (1992) 1252. [7] Z.W. Ouyang, G.H. Rao, H.F. Yang, W.F. Liu, J.K. Liang, Appl. Phys. Lett. 81 (2002) 97. [8] J.J. Liua, P.Z. Si, W.S. Zhang, X.C. Liu, J. Alloys Compd. 474 (2009) 9. [9] W.M. Swift, W.E. Wallace, J. Phys. Chem. Solids 29 (1986) 2053. [10] Y. Tajiri, K. Nishimura, L. Li, W.D. Hutchison, Solid State Commun. 148 (2008) 365. [11] K. Nishimura, K. Mori, Y. Narita, W.D. Hutchison, J. Magn. Magn. Mater. 310 (2007) 1730. [12] Y. Tajiri, K. Nishimura, Lingwei Li, T. Kawabata, K. Sato, K. Kindo, W.D. Hutchison, J. Alloys Compd 487 (2009) 5. [13] A.E. Dwight, R.A. Conner Jr., J.W. Downey, Acta Crystallogr 18 (1965) 835. [14] R.J. Radwan´ski, Physica B142 (1986) 57. [15] G. Fillion, D. Gignoux, F. Givord, R. Lemaire, J. Magn. Magn. Mater. 44 (1984) 173. [16] K. Uhlı´rˇova´, J. Prokleˇska, J. Poltierova´ Vejpravova´, V. Sechovsky´, K. Maezawa, J. Magn. Magn. Mater. 310 (2007) 1753.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
High field magnetization of Pr1–xGdxNi single crystals Katsuhiko Nishimura a,n, Wayne D. Hutchison b, Yuta Tajiri a, Yosikazu Isikawa a, Keisuke Sato c, Koichi Kindo c a
Graduate School of Science and Engineering, University of Toyama, Toyama 930-8555, Japan School of Physical, Environmental and Mathematical Sciences, The University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia c Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan b
a r t i c l e i n f o
abstract
Article history: Received 29 January 2011 Accepted 21 March 2011 Available online 29 March 2011
Magnetization measurements were carried out on Pr1 xGdxNi single crystals for x ¼ 0.1, 0.2, 0.3 and 0.4 using pulsed magnetic fields up to 55 T and at 4.2 K. For the data observed along the c-axis there are clear indications of transitions, of the combined system of Pr and Gd moments, from a ferrimagnetic to a ferromagnetic state. The observed saturation magnetizations at 55 T and intermediate magnetizations at 1 T were well modeled assuming ferro- and ferri-magnetic structures, respectively. & 2011 Elsevier B.V. All rights reserved.
Keywords: PrNi High-field magnetization Spin rearrangement Antiferromagnetic coupling
1. Introduction Studies of the magnetism in intermetallic compounds containing both rare earth (RE) and transition (T) metal elements, in which the magnetic structures can be described in terms of exchange interactions and crystal electric field effects (CEFs), are numerous. It is well known that the sign of coupling of the RE moments with the T moments (such as Fe, Co) changes between light and heavy RE elements [1–4]. The conduction electron polarization due to a RE ion is negative with respect to the spin moment at the ion site and therefore the coupling of the spins of a RE ion and the conduction electrons is antiferromagnetic (antiparallel). Consequently, it follows that there is an antiferromagnetic coupling of the total magnetic moments of heavy RE ions with transition metal ions, whereas the coupling is ferromagnetic for the light RE with transition metal ions. These observations reflect Hund’s rule, which has J¼ LþS and J¼L–S for the heavy and light rare earths, respectively, and are consistent almost universally with data for RE-T compounds in literature. For example, the cobalt moment is coupled antiparallel to the heavy RE moment (parallel to the light RE moment) in RECo2 systems [4–8]. As a direct consequence of the above, in the case of intermetallic compounds, which mix light and heavy REs, the arrangement of the magnetic moments of the heterogeneous RE ions is also antiferromagnetic (leading to a ferrimagnetic structure). This antiferromagnetism, originating from the opposite sense of the
n
Corresponding author. Tel./fax: þ 81 76 445 6844. E-mail address: [email protected] (K. Nishimura).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.03.054
spin-orbit interactions of heavy and light rare earths, is illustrated by a number of experimental studies [1–4]. Explicitly, a minimum ordered state magnetization (compensation point) is observed part way between the end point compounds of mixed light and heavy RE compounds [1,2,7–9]. Our recent reports on the magnetic properties of Pr1 xGdxNi [10], Nd1 xGdxNi [11], Pr1 xTbxNi and Nd1 xTbxNi [12], using single crystals, all documented this type of behavior, with a minimum in magnetization at certain intermediate concentrations x. Amongst these mixed compounds, those containing Pr also showed evidence of a field dependence of the magnetic structure, which has not been reported in the literature previously. The magnetization data along the c-axis (easy-axis of PrNi), indicates a critical field Bc for a transition from a ferrimagnetic to ferromagnetic state having an x dependence, which reflects the low temperature saturation magnetization (MS) rather than the ordering temperature. In this paper we pursue this behavior of the magnetization of Pr1 xGdxNi single crystals to high externally applied field.
2. Experimental The single crystal samples of Pr1 xGdxNi (x¼0.1, 0.2, 0.3 and 0.4) were grown using the Bridgeman method. Polycrystalline starting material was prepared by argon arc melting, then packed in an alumina crucible and vacuum-encapsulated in a quartz tube before it was passed through a Bridgeman furnace over a week. The crystallographic orientations of the single crystals were determined by the Laue x-ray back reflection method, with the observed Laue patterns consistent with the samples being single
2282
K. Nishimura et al. / Physica B 406 (2011) 2281–2283
phase and being of the CrB-type (Cmcm) orthorhombic crystal structure [13]. Small samples, cubic in shape, were cut from the crystal boles for use in the various measurements. The low-field DC magnetization data (M) were measured using a SQUID magnetometer (MPMS) with external fields (B) up to 7 T. The high-field experiments with Pr1 xGdxNi were carried out in the International MegaGauss Science Laboratory at the Institute of Solid State Physics, University of Tokyo. Magnetization was measured at 4.2 K and in applied magnetic fields up to 55 T using a long-pulse magnet.
3. Results and discussions The low-field magnetization data at a temperature of 2 K and along the c-axis of Pr1 xGdxNi samples, for x ¼0.1–0.4 are shown in Fig. 1(a). A detailed description of the low-field magnetic properties has been given elsewhere [10]. The onset of a more rapid increase in M towards the right hand side of the plot is apparent for x ¼0.2 and 0.3. In the high-field magnetization data of Fig. 1(b), which was collected at 4.2 K and up to 55 T along the c-axis, the full extent of changes in M associated with a rearrangement of magnetic moments may be seen for all four samples. A critical field (Bc) value is estimated from the point maximum slope (dM/dBmax) for each data set. The estimates of Bc are 10.5, 6.9, 8.5, and 10.5 T for x¼0.1, 0.2, 0.3 and 0.4, respectively. These transformations are quite broad in field terms and we define Bc1 and Bc2 as the transition start (lower) and finish (upper) critical field values, respectively. Examples of how these values are estimated from the intersection point of the two extrapolated lines from the data in the lower and higher field regions can be seen in Fig. 1(b), while all the Bc, Bc1 and Bc2 values are plotted in Fig. 4. The observed values of saturation magnetization (MS) at 55 T are 3.08, 3.46, 3.87, and 4.22 mB/f.u. for x¼0.1, 0.2, 0.3 and 0.4, respectively, which correlate well with the theoretical average moment expected at each x. From Fig. 1(b) it can also be seen that the magnetizations saturate at about 20 T, which is rather small compared with those reported for Gd-Co system ( 4100 T) [14]. Since the c-axis is the easy magnetization axis of PrNi, the small saturation fields in this case suggest that the Pr-Gd exchange interactions are relatively weak. Unlike that at the other directions, the observed variation of the c-axis magnetization data is due to the interplay between Zeeman and Pr-Gd exchange energies alone. Measurements along the other principle crystalline directions do not have the same features. As an example the M curves for Pr0.8Gd0.2Ni along the a, b and c-axis are shown in Fig. 1(c). For the a-axis, intermediate in magnetic anisotropy for PrNi, M develops more or less monatonically to saturate at the same levels as for the c-axis, but at a slightly
higher field. While along the hard b-axis although M develops at a similar rate with applied field until around 10 T; after that the rate slows markedly and indeed full saturation is not obtained by 55 T. The observed c-axis MS values are plotted with solid-triangle symbols in Fig. 2. The magnetization at B¼1 T ( in Fig. 1(a)), which can be regarded as the M values for the antiferromagnetic arrangements of Pr and Gd moments (MAF), are also plotted with solid-circle symbols for x ¼0.1–0.4. The variations of MS and MAF can be explained neatly by assuming ferromagnetic and antiferromagnetic coupling between the Pr and Gd magnetic moments, respectively. The open-triangle and open-circle symbols plotted in Fig. 2 are the calculated values taking 2.35 mB/(PrNi) [15] and 7.0 mB/(GdNi) [16] as saturation magnetizations for the ferromagnetic and antiferromagnetic arrangements, respectively. Both these estimations model the experimentally observed values of MS and MAF along the c-axis against x very well, and thus support a simple picture that the Pr and Gd moments are aligned parallel above 20 T, but anti-parallel in low applied field (around 1 T). Adopting this model for the structures at the extremes, estimates of Bc for the various x can be made. We assume Bc is the point at which the average Zeeman energy generated by the field on the antiparallel sublattice exceeds that due to the heterogeneous (Pr-Gd) exchange. Explicitly we apply a molecular field approach, defining average fields at the Pr and Gd sites as He(Gd)¼ amGd–bmPr and He(Pr)¼ gmPr–bmGd, where a, b, g are positive constants reflecting the interaction strength Gd-Gd, Pr-Gd and Pr-Pr respectively. Here mGd ¼xNMGd and mPr ¼ (1–x)NMPr are the sublattice magnetizations where N is the number of RE spins per unit volume and MGd and MPr are the
Fig. 2. Magnetization of Pr1 xGdxNi at 1 T and 2 K (solid circles) from Fig. 1(a), and saturation magnetization at 55 T and 4.2 K (solid triangle) from Fig. 1(b) are plotted against Gd fraction x. Open circles and open triangles are the results of calculations assuming antiferromagnetic and ferromagnetic arrangements of Pr and Gd moments, respectively (see text).
Fig. 1. (a) Magnetization of Pr1 xGdxNi, with x¼ 0.0–0.4, observed at 2 K with external field along the c-axis. (Lines are a guide for the eye.) (b) High field magnetization of Pr1 xGdxNi, with x¼ 0.0–0.4 along the c-axis at 4.2 K. (c) High field magnetization of Pr0.8Gd0.2Ni along the a, b and c-axis at 4.2 K.
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Fig. 3. Results of the observed magnetic transition temperatures TM and the calculated ones with the exchange parameter b (see text).
Fig. 4. The observed lower critical field Bc1, upper critical field Bc2, the field of maxmum of dM/dB (Bc), and the theoretical values of Bc (Btheo.) (see text).
RE moments (expressed in SI units in this instance). The average exchange energy can be written as Eex ðxÞ ¼ m0 axm2Gd þ bmGd mPr þ gð1xÞm2Pr h i 2 2 ¼ m0 N2 ax3 MGd þ bxð1xÞMGd MPr þ gð1xÞ3 MPr
experimental data reasonably well, lying neatly midway between Bc1 and Bc2. The transition is of course broad due to the random distribution of Pr and Gd ions, whereas the theory assumes the average molecular field. In this paper we have presented high field magnetization measurements on the light-heavy rare earth mixed compounds Pr1 xGdxNi. Within these data we have identified a more abrupt spin rearrangement (from ferrimagnetic to ferromagnetic) occurring only when the field is along the c-axis and depending on the value of x. The saturation magnetizations of all the samples were well reproduced by simply summing the weighted MS values of end point RENi compounds. The variations of the Bc with x in Pr1 xGdxNi correlate well with a simple model whereby the average Zeeman term becomes equal to the heterogeneous exchange at Bc.
where N is the number of RE spins per unit volume and MGd and MPr are the RE moments. It follows that the increase in exchange energy for inversion of one sublattice (to align with the other) is
DEex ðxÞ ¼ 2m0 N2 bxð1xÞMGd MPr In this expression the crucial unknown is the heterogeneous exchange strength b, which we can estimate by modeling TM(x). Explicitly, we define separate Curie constants CGd and CPr for the Gd and Pr sublattices respectively, then for an applied field Ha in the paramagnetic phase, we have mGd T ¼ CGd ðHa þ amGd bmPr Þ mPr T ¼ CPr ðHa þ gmPr bmGd Þ
Acknowledgement
For the case Ha ¼0, these equations can be solved to give an expression for TM(x) in terms of a, b, g, CGd and CPr. All these parameters except b can be evaluated from susceptibility data for the end point compounds PrNi and GdNi (i.e. x ¼0 and 1). We set aCGd ¼TM(x ¼1)¼72.8 K, a ¼18.8, gCPr ¼TM(x ¼0) ¼20.9 K and g ¼ 26.8. It then remains to determine the value of the heterogeneous exchange b. We have done this by choosing the b value, which allows the TM(x) expression to best fit our experimental TM data. This process, illustrated in Fig. 3, leads to b ¼ 19 which means the heterogeneous exchange is similar in magnitude to the homogeneous values (a,g) above. The final step is then to equate the change (increase) in exchange energy with the reduction in Zeeman energy due to inversion of the minority spin sublattice (Gd for x o 0.25 and Pr for x4 0.25) to the applied field direction. We have
This work was performed using facilities of the Institute for Solid State Physics, the University of Tokyo.
DEZeeman ¼ 2xNMGd B, x o0:25 DEZeeman ¼ 2ð1xÞNMPr B, x 40:25 and setting DEZeeman ¼ DEex at Btheo. gives Btheo: ¼ m0 Nbð1xÞMPr , Btheo: ¼ m0 NbxMGd ,
x o0:25 x 40:25
These theoretical expressions are compared with the experimental values of Bc (defined as a field at the maximum of dM/dB), Bc1 and Bc2 in Fig. 4. We note that the theory matches the
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