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Induced 3D magnetism in the ternary intermetallic compound Pd2GdIn
Solid State Communications, Vol. 84, No. 5, pp. 577-580, 1992. Printed in Great Britain.
0038-1098/92 $5.00 + .00 Pergamon Press Ltd
INDUCED 3D MAGNETISM IN THE TERNARY INTERMETALLIC COMPOUND Pd2Gdln K.-U. Neumann,* J. Crangle, t R.T. Giles,* D. Visser,* N.K. Zayer* and K.R.A. Ziebeck* *Department of Physics, Loughborough University of Technology, Loughborough, LEI 1 3TU, U.K. *Department of Physics, University of Sheffield, Sheffield, $3 7RH, U.K.
(Received 29 April 1992 by G. Gfintherodt) Magnetisation measurements on Pd2Gdln chemically disordered into the B2 structure reveal a magnetically ordered state below ,,~ 7 K. The saturation magnetisation obtained in fields up 5T using a squid magnetometer yields an extrapolated moment at 0 K significantly larger than that expected for a Gd + ion. Above ,,, 15 K the Curie-Weiss susceptibility produces an effective moment closer to that expected for Gd 3+ . On the basis of these measurements it is suggested that the local moment on the Gd atoms polarises the Pd atoms through an indirect exchange mechanism. Below -,~ 7K the. magnetic isotherms reveal a field-induced transition ferromagnetism, suggesting that the magnetic order in zero applied field may be complex. The results provide new information relevant to the interpretation of recent neutron diffraction data and to the occurrence of superconductivity as reported in related Heusler alloys containing rare earth elements. 1. INTRODUCTION THE COEXISTENCE of long-range magnetic order and superconductivity has been established in a series of ternary compounds containing rare earth ions [1,2]. It is thought that this coexistence arises from the relative weakness of the exchange spin flip (pair breaking) interaction between the closed 4f shells of the magnetic atoms and the conduction electrons. This behaviour and the underlying mechanism are different from those occurring in heavy Fermion superconductors in which the magnetism and superconductivity are believed to be carried by the same felectrons. The materials which have been the most extensively studied are the ternary borides of the type RRh4B4 and the Chevrel phases of the type RMotSes, where R represents a rare earth element. These materials have relatively complicated crystallographic structures and are sometimes not even single phase. Mostly the magnetic order is antiferromagnetic, in which the propagation vector is much shorter than the coherence length of the Cooper pairs. A new series of ternary intermetallics has been found based on the formula Pd2RSn which have Heusler L21 structure [3,4]. On the basis of a.c. susceptibility, specific heat and electrical resistivity measurements down to 1.3K, only the compounds containing Tm, Yb or Lu were found to be
superconducting; the transition temperatures which are in the range of 1.5-3 K, are significantly higher than those observed in the Chevrel compounds. Those compounds containing Gd through to Er were found to order magnetically below 5K. A magnetic transition was also identified in the superconducting phase of the Yb compound. From the variation of lattice parameter it was concluded that the rare earth ions are in a trivalent state. The separation of the rare earth ions is typically 0.47 nm, which is much larger than the distances occurring in the Chevrel phases. Consequently, the estimated exchange coupling is approximately five times smaller and involves an indirect mechanism generally similar to RKKY. The importance of the Pd sublattice in establishing the superconducting and magnetic properties in these compounds was emphasised by Jorda [5]. Heusler alloys form ideal systems for studying the effects of electron concentration [6,7] and atomic order on the physical properties such as magnetism. In the series Pd2Mnln the antiferromagnetic structure changes as a function of the chemical disorder between the Mn and In sites. Changes in the magnetic structure also occur if In is replaced by Sn, which increases electron to atom ratio. It was therefore decided to determine the effects of electron concentration and atomic order on the magnetic and
577
578
I N D U C E D 3D M A G N E T I S M IN P d , G d I n
superconducting properties reported in the series Pd2RSn and hopefully gain a deeper insight into the underlying mechanisms. The first o f these studies is concerned only with Pd2Gdln which, since Gd 3+ is expected to be in an S state, may be the most straightforward o f the series to interpret. The consequences o f these results on the interpretation of recent neutron diffraction data and the reported observation of superconductivity in related Heusler alloys is discussed. 2. E X P E R I M E N T A L A 10g ingot o f Pd2Gdln was prepared by repeatedly melting the appropriate amounts of spectrographically pure (5N) constituent elements in an argon arc furnace. From the ingot, which had been furnace-cooled, specimens suitable for susceptibility and resistivity measurements were cut from different parts and the remainder was crushed to a particle size less than 50#m. X-ray diffraction measurements confirmed that the compound was disordered, having the B2 structure in which the In and G d atoms are statistically distributed between the two sites. AC resistivity measurements using a standard four probe technique were employed to reveal a transition close to 7 K above which the resistivity increased linearly to room temperature. The magnetic measurements were made on the Sheffield University SQUID magnetometer, which was manufactured by Quantum Design. Available fields were up to + 5.5 T. Temperatures were from below 2 K up to 300K. Field could be set and controlled to within + 1 0 - r T and temperature could be set and controlled to within +10 -2 K. 3. R E S U L T S The susceptibility measurements indicate C u r i e Weiss behaviour down to approximately 15 K and suggest a transition to a magnetically ordered state in the vicinity of 7 K (Fig. 1). However, the magnetic isotherms obtained below 6 K and in magnetic fields up to 5 T show that the ordered state is not simply associated with ferromagnetic order of the trivalent G d atoms. Saturation magnetisation extrapolated to 0 K indicates a saturation ferromagnetic Bohr magneton number per formula unit of 8 #B. This value is significantly larger than the 7/z s (gJ) value expected for a trivalent Gd ion. However, the paramagnetic effective Bohr magneton number (Pctr = g { J ( J + 1)} 1/2) obtained from the observed Curie-Weiss susceptibility, 8.22 #B per formula unit is much closer to that observed for a Gd 3+ ion, i.e., 7.94#B. The increase in the Bohr magneton number
Vol. 84. No. 5
~m15-
:5
#
_.E 0
1 O0
200
300
Temperature K
Fig. 1. The inverse susceptibility as a function o f temperature for disordered Pd2Gdln. The data shown in the inset were taken in an applied field of IT. would suggest that states other than the 4 f levels on the rare earth atoms are involved. The isotherms which are shown in Fig. 2 were obtained by increasing the applied field in steps of 0.5 T. Below 6 K there is a crossover among the isotherms for fields less than 1.5T, consistent with a field induced transition to a higher magnetisation state at low temperatures. The transition is more apparent in the Arrott plots [8, 9] (shown in Fig. 3) which reveal a substantial degree o f polarisability at low temperatures. Only at temperatures greater than about 15 K do the isotherms approach the linear form expected for homogenous magnets (Fig. 3) i.e., where C u r i e Weiss behaviour is observed. The ,,~ 1 #n observed above the value expected for
8o|
2
r 0
Magnetic Field T
Fig. 2. Magnetic isotherms of disordered Pd2Gdln. The temperatures are marked on the curves.
Vol. 84, No. 5 q~ | x~ ~. ~_ 6 % 4 ..
I N D U C E D 3D M A G N E T I S M IN Pd2GdIn 2 ~ ,-o
K -" " ~4 s6
579
associated with the f electrons, but if the conduction band is polarised to produce magnetic order on the Pd lattice, then the nature o f this order will significantly influence the occurrence o f superconductivity, and account for why it has only been reported in those compounds containing Tm, Yb and Lu.
6 10 is
~N 2
0
20
2
4
6
8
Field/magnetization x 10.2 T2J "1 kg
10
Fig. 3. Arrott plots for disordered Pd2Odln showing isotherms below 20 K.
4. D I S C U S S I O N Since the Pd and the G d atoms are separated by 0.29 nm, the exchange interaction between them must be predominantly of an indirect nature. The features observed can be adequately described by a Hamiltonian of the form
H = - ~ ' ~ J i j S i . S j - ~-'~jtmMt. M,n - ~"~ CitSi. M,, ij
the saturated moment (g J) of a Gd 3+ ion is probably associated with the palladium atoms. It is well known [10] that small amounts of magnetic impurities in Pd can polarise the matrix, producing giant magnetic moments. The large susceptibility of Pd arises from the high density of electron states at the Fermi level. However, in the case of Pd2Gdln, local moments on the Gd atoms are concentrated. The X-ray measurements reveal that the compound is disordered into the B2 structure. In the B2 structure the In and Gd atoms are completely disordered and randomly occupy the body-centred sites within the simple cubic Pd sublattice. The Pd and Gd atoms are therefore separated by ,-, 0.29nm. Consequently, if it is the effective field from the Gd atoms which is polarlsing the Pd, then it must be via an indirect mechanism such as the R K K Y . F r o m the magnetic isotherms (Fig. 1) it would appear that below 6 K the applied field stabilises a ferromagnetic state for fields greater than approximately 1.5T. In the absence of an applied field, or in low fields, the magnetic order may be complex depending upon long-range interactions which are known to be important in stabilising the magnetic structure of Heusler alloys [11]. Neutron diffraction measurements are required to establish the details of the magnetic order. However, in the case of Pd2ErSn a doubling of the Er lattice consistent with antiferromagnetic type 2 order has been reported to co-exist with a modulated magnetic structure [12]. The origin of the modulated magnetic structure was not established, but in the light of the present measurements it would seem to be associated with the Pd lattice. Certainly magnetisation measurements on Pd2DySn [13] reveal a field dependent magnetisation at low temperatures, which suggest that the phenomena which we attribute to the Pd lattice is a general feature of these materials. Until now [5], it has been assumed that the magnetism is uniquely
Irn
il
where S is the spin on the G d atom, M is the moment o f the Pd atom. The summation ij is taken overall G d sites and Im runs over all sites occupied by Pd. d# is the R K K Y exchange constant between Gd moments, Ju, is the coupling between the Pd moments and Ca is the coupling between the Pd and Gd moments. In a mean field approximation, this Hamiltonian results in an effective field which acts on the Pd sites and is given by
Hdf = ~-~j,,,(Mnl + ~ C,t(S,). n
l
Since only the G d atoms possess a spontaneous magnetic moment, the effective field gives rise to an induced moment on the Pd. In this model the effective field essentially goes to zero at the Curl, temperature, where the Gd moments disorder. Consequently, above Tc the induced moment on the Pd atoms disappears in the paramagnetic phase and the paramagnetic state is characterlsed only by the Gd moments and their interactions. In the magnetically ordered state, the ( S ) is non-zero and because o f the effective field the ( M ) is also non-zero. Thus if an external field is applied so that the moments are aligned ferromagneticaUy, an average magnetic moment {+er formula unit in excess of that expected for a G d + a t o m occurs. This is in agreement with observation. Owing to the polarisation of the Pd atoms, an additional set of competing interactions are introduced in the ordered star,. Consequently, in the absence of an applied field the ferromagnetic state may not necessarily b¢ the ground state o f the system. A more detailed account o f this model calculation will be given elsewhere.
Acknowledgements - We are grateful to B. Chavda for his assistance in preparing the specimens.
580
INDUCED 3D MAGNETISM IN Pd,Gdln REFERENCES
1. 2. 3.
4. 5. 6.
13. Fischer, A. Treyvand, R. Chevrel & M. Sergent, Solid State Commun. 17, 721 (1975). B.T. Matthias, E. Corenzwit, J.M. Vandenberg & H. Barz, Proc. Natl. Acad. Sci. USA 74, 1334 (1977). M. Ishikawa, J.L. Jorda & A. Junod, in Proc. Int. Conf. on Superconductivity in d-and f-based Metals. (Edited by W. Buckel & W. Weber), p. 141. K F Z Karlsruhe (1982). S.K. Malik, A.M. Umarji & G.K. Shenoy, Phys. Rev. B34, 3144 (1986). J.L. Jorda, Thesis, University of Geneva (1983). P.J. Webster & R.S. Tebble, J. Appl. Phys. 39, 471 (1968).
7. 8. 9.
10. 11. 12. 13.
Vol. 84. No. 5
P.J. Webster & M.R.J. Ramadan, J. Magn. Magn. Mat. 13, 301 (1979). A. Arrott, Phys. Rev. 108, 1394 (1957). P.E. Brommer & J.J.M. Franse, Article in Ferromagnetic Materials, 5, (Edited by K.H.J. Buschow & E.P. Wohlfarth), p. 323. Elsevier (1990). J. Crangle & W.R. Scott, J. Appl. Phys. 38, 921 (1965). P.J. Webster & K.R.A. Ziebeck, J. Magn. Magn. Mat. 50, 7 (1985). H.B. Stanley, J.W. Lynn, R.N. Shelton & P. Kalvins, J. Appl. Phys. 61, 3371 (1987). J. Crangle (work in progress).
0038-1098/92 $5.00 + .00 Pergamon Press Ltd
INDUCED 3D MAGNETISM IN THE TERNARY INTERMETALLIC COMPOUND Pd2Gdln K.-U. Neumann,* J. Crangle, t R.T. Giles,* D. Visser,* N.K. Zayer* and K.R.A. Ziebeck* *Department of Physics, Loughborough University of Technology, Loughborough, LEI 1 3TU, U.K. *Department of Physics, University of Sheffield, Sheffield, $3 7RH, U.K.
(Received 29 April 1992 by G. Gfintherodt) Magnetisation measurements on Pd2Gdln chemically disordered into the B2 structure reveal a magnetically ordered state below ,,~ 7 K. The saturation magnetisation obtained in fields up 5T using a squid magnetometer yields an extrapolated moment at 0 K significantly larger than that expected for a Gd + ion. Above ,,, 15 K the Curie-Weiss susceptibility produces an effective moment closer to that expected for Gd 3+ . On the basis of these measurements it is suggested that the local moment on the Gd atoms polarises the Pd atoms through an indirect exchange mechanism. Below -,~ 7K the. magnetic isotherms reveal a field-induced transition ferromagnetism, suggesting that the magnetic order in zero applied field may be complex. The results provide new information relevant to the interpretation of recent neutron diffraction data and to the occurrence of superconductivity as reported in related Heusler alloys containing rare earth elements. 1. INTRODUCTION THE COEXISTENCE of long-range magnetic order and superconductivity has been established in a series of ternary compounds containing rare earth ions [1,2]. It is thought that this coexistence arises from the relative weakness of the exchange spin flip (pair breaking) interaction between the closed 4f shells of the magnetic atoms and the conduction electrons. This behaviour and the underlying mechanism are different from those occurring in heavy Fermion superconductors in which the magnetism and superconductivity are believed to be carried by the same felectrons. The materials which have been the most extensively studied are the ternary borides of the type RRh4B4 and the Chevrel phases of the type RMotSes, where R represents a rare earth element. These materials have relatively complicated crystallographic structures and are sometimes not even single phase. Mostly the magnetic order is antiferromagnetic, in which the propagation vector is much shorter than the coherence length of the Cooper pairs. A new series of ternary intermetallics has been found based on the formula Pd2RSn which have Heusler L21 structure [3,4]. On the basis of a.c. susceptibility, specific heat and electrical resistivity measurements down to 1.3K, only the compounds containing Tm, Yb or Lu were found to be
superconducting; the transition temperatures which are in the range of 1.5-3 K, are significantly higher than those observed in the Chevrel compounds. Those compounds containing Gd through to Er were found to order magnetically below 5K. A magnetic transition was also identified in the superconducting phase of the Yb compound. From the variation of lattice parameter it was concluded that the rare earth ions are in a trivalent state. The separation of the rare earth ions is typically 0.47 nm, which is much larger than the distances occurring in the Chevrel phases. Consequently, the estimated exchange coupling is approximately five times smaller and involves an indirect mechanism generally similar to RKKY. The importance of the Pd sublattice in establishing the superconducting and magnetic properties in these compounds was emphasised by Jorda [5]. Heusler alloys form ideal systems for studying the effects of electron concentration [6,7] and atomic order on the physical properties such as magnetism. In the series Pd2Mnln the antiferromagnetic structure changes as a function of the chemical disorder between the Mn and In sites. Changes in the magnetic structure also occur if In is replaced by Sn, which increases electron to atom ratio. It was therefore decided to determine the effects of electron concentration and atomic order on the magnetic and
577
578
I N D U C E D 3D M A G N E T I S M IN P d , G d I n
superconducting properties reported in the series Pd2RSn and hopefully gain a deeper insight into the underlying mechanisms. The first o f these studies is concerned only with Pd2Gdln which, since Gd 3+ is expected to be in an S state, may be the most straightforward o f the series to interpret. The consequences o f these results on the interpretation of recent neutron diffraction data and the reported observation of superconductivity in related Heusler alloys is discussed. 2. E X P E R I M E N T A L A 10g ingot o f Pd2Gdln was prepared by repeatedly melting the appropriate amounts of spectrographically pure (5N) constituent elements in an argon arc furnace. From the ingot, which had been furnace-cooled, specimens suitable for susceptibility and resistivity measurements were cut from different parts and the remainder was crushed to a particle size less than 50#m. X-ray diffraction measurements confirmed that the compound was disordered, having the B2 structure in which the In and G d atoms are statistically distributed between the two sites. AC resistivity measurements using a standard four probe technique were employed to reveal a transition close to 7 K above which the resistivity increased linearly to room temperature. The magnetic measurements were made on the Sheffield University SQUID magnetometer, which was manufactured by Quantum Design. Available fields were up to + 5.5 T. Temperatures were from below 2 K up to 300K. Field could be set and controlled to within + 1 0 - r T and temperature could be set and controlled to within +10 -2 K. 3. R E S U L T S The susceptibility measurements indicate C u r i e Weiss behaviour down to approximately 15 K and suggest a transition to a magnetically ordered state in the vicinity of 7 K (Fig. 1). However, the magnetic isotherms obtained below 6 K and in magnetic fields up to 5 T show that the ordered state is not simply associated with ferromagnetic order of the trivalent G d atoms. Saturation magnetisation extrapolated to 0 K indicates a saturation ferromagnetic Bohr magneton number per formula unit of 8 #B. This value is significantly larger than the 7/z s (gJ) value expected for a trivalent Gd ion. However, the paramagnetic effective Bohr magneton number (Pctr = g { J ( J + 1)} 1/2) obtained from the observed Curie-Weiss susceptibility, 8.22 #B per formula unit is much closer to that observed for a Gd 3+ ion, i.e., 7.94#B. The increase in the Bohr magneton number
Vol. 84. No. 5
~m15-
:5
#
_.E 0
1 O0
200
300
Temperature K
Fig. 1. The inverse susceptibility as a function o f temperature for disordered Pd2Gdln. The data shown in the inset were taken in an applied field of IT. would suggest that states other than the 4 f levels on the rare earth atoms are involved. The isotherms which are shown in Fig. 2 were obtained by increasing the applied field in steps of 0.5 T. Below 6 K there is a crossover among the isotherms for fields less than 1.5T, consistent with a field induced transition to a higher magnetisation state at low temperatures. The transition is more apparent in the Arrott plots [8, 9] (shown in Fig. 3) which reveal a substantial degree o f polarisability at low temperatures. Only at temperatures greater than about 15 K do the isotherms approach the linear form expected for homogenous magnets (Fig. 3) i.e., where C u r i e Weiss behaviour is observed. The ,,~ 1 #n observed above the value expected for
8o|
2
r 0
Magnetic Field T
Fig. 2. Magnetic isotherms of disordered Pd2Gdln. The temperatures are marked on the curves.
Vol. 84, No. 5 q~ | x~ ~. ~_ 6 % 4 ..
I N D U C E D 3D M A G N E T I S M IN Pd2GdIn 2 ~ ,-o
K -" " ~4 s6
579
associated with the f electrons, but if the conduction band is polarised to produce magnetic order on the Pd lattice, then the nature o f this order will significantly influence the occurrence o f superconductivity, and account for why it has only been reported in those compounds containing Tm, Yb and Lu.
6 10 is
~N 2
0
20
2
4
6
8
Field/magnetization x 10.2 T2J "1 kg
10
Fig. 3. Arrott plots for disordered Pd2Odln showing isotherms below 20 K.
4. D I S C U S S I O N Since the Pd and the G d atoms are separated by 0.29 nm, the exchange interaction between them must be predominantly of an indirect nature. The features observed can be adequately described by a Hamiltonian of the form
H = - ~ ' ~ J i j S i . S j - ~-'~jtmMt. M,n - ~"~ CitSi. M,, ij
the saturated moment (g J) of a Gd 3+ ion is probably associated with the palladium atoms. It is well known [10] that small amounts of magnetic impurities in Pd can polarise the matrix, producing giant magnetic moments. The large susceptibility of Pd arises from the high density of electron states at the Fermi level. However, in the case of Pd2Gdln, local moments on the Gd atoms are concentrated. The X-ray measurements reveal that the compound is disordered into the B2 structure. In the B2 structure the In and Gd atoms are completely disordered and randomly occupy the body-centred sites within the simple cubic Pd sublattice. The Pd and Gd atoms are therefore separated by ,-, 0.29nm. Consequently, if it is the effective field from the Gd atoms which is polarlsing the Pd, then it must be via an indirect mechanism such as the R K K Y . F r o m the magnetic isotherms (Fig. 1) it would appear that below 6 K the applied field stabilises a ferromagnetic state for fields greater than approximately 1.5T. In the absence of an applied field, or in low fields, the magnetic order may be complex depending upon long-range interactions which are known to be important in stabilising the magnetic structure of Heusler alloys [11]. Neutron diffraction measurements are required to establish the details of the magnetic order. However, in the case of Pd2ErSn a doubling of the Er lattice consistent with antiferromagnetic type 2 order has been reported to co-exist with a modulated magnetic structure [12]. The origin of the modulated magnetic structure was not established, but in the light of the present measurements it would seem to be associated with the Pd lattice. Certainly magnetisation measurements on Pd2DySn [13] reveal a field dependent magnetisation at low temperatures, which suggest that the phenomena which we attribute to the Pd lattice is a general feature of these materials. Until now [5], it has been assumed that the magnetism is uniquely
Irn
il
where S is the spin on the G d atom, M is the moment o f the Pd atom. The summation ij is taken overall G d sites and Im runs over all sites occupied by Pd. d# is the R K K Y exchange constant between Gd moments, Ju, is the coupling between the Pd moments and Ca is the coupling between the Pd and Gd moments. In a mean field approximation, this Hamiltonian results in an effective field which acts on the Pd sites and is given by
Hdf = ~-~j,,,(Mnl + ~ C,t(S,). n
l
Since only the G d atoms possess a spontaneous magnetic moment, the effective field gives rise to an induced moment on the Pd. In this model the effective field essentially goes to zero at the Curl, temperature, where the Gd moments disorder. Consequently, above Tc the induced moment on the Pd atoms disappears in the paramagnetic phase and the paramagnetic state is characterlsed only by the Gd moments and their interactions. In the magnetically ordered state, the ( S ) is non-zero and because o f the effective field the ( M ) is also non-zero. Thus if an external field is applied so that the moments are aligned ferromagneticaUy, an average magnetic moment {+er formula unit in excess of that expected for a G d + a t o m occurs. This is in agreement with observation. Owing to the polarisation of the Pd atoms, an additional set of competing interactions are introduced in the ordered star,. Consequently, in the absence of an applied field the ferromagnetic state may not necessarily b¢ the ground state o f the system. A more detailed account o f this model calculation will be given elsewhere.
Acknowledgements - We are grateful to B. Chavda for his assistance in preparing the specimens.
580
INDUCED 3D MAGNETISM IN Pd,Gdln REFERENCES
1. 2. 3.
4. 5. 6.
13. Fischer, A. Treyvand, R. Chevrel & M. Sergent, Solid State Commun. 17, 721 (1975). B.T. Matthias, E. Corenzwit, J.M. Vandenberg & H. Barz, Proc. Natl. Acad. Sci. USA 74, 1334 (1977). M. Ishikawa, J.L. Jorda & A. Junod, in Proc. Int. Conf. on Superconductivity in d-and f-based Metals. (Edited by W. Buckel & W. Weber), p. 141. K F Z Karlsruhe (1982). S.K. Malik, A.M. Umarji & G.K. Shenoy, Phys. Rev. B34, 3144 (1986). J.L. Jorda, Thesis, University of Geneva (1983). P.J. Webster & R.S. Tebble, J. Appl. Phys. 39, 471 (1968).
7. 8. 9.
10. 11. 12. 13.
Vol. 84. No. 5
P.J. Webster & M.R.J. Ramadan, J. Magn. Magn. Mat. 13, 301 (1979). A. Arrott, Phys. Rev. 108, 1394 (1957). P.E. Brommer & J.J.M. Franse, Article in Ferromagnetic Materials, 5, (Edited by K.H.J. Buschow & E.P. Wohlfarth), p. 323. Elsevier (1990). J. Crangle & W.R. Scott, J. Appl. Phys. 38, 921 (1965). P.J. Webster & K.R.A. Ziebeck, J. Magn. Magn. Mat. 50, 7 (1985). H.B. Stanley, J.W. Lynn, R.N. Shelton & P. Kalvins, J. Appl. Phys. 61, 3371 (1987). J. Crangle (work in progress).