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Spin waves in Pd3Fe
Solid State Communications,
Vol. 11, pp. 271-274, 1972.
Pergamon Press.
Printed in Great Britain
SPIN WAVES IN Pd3 Fe W.G. Stifling and R.A. Cowley Department of Physics, University of Edinburgh, The Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
(Received 21 April 1972 by M.F. Collins)
The spin wave dispersion relation in ordered Pd 3 Fe has been measured at 296°K by neutron inelastic scattering techniques. The results are compared with various models for the spin wave spectrum.
THE NATURE of the exchange interactions between Fe ions in a Pd host has been the object of considerable study2 In particular, the spin wave dispersion relations in the small wavevector region have been determined by neutron diffraction techniques in both dilute F e / P d alloys z and in ordered 3 and disordered 4 Pd3Fe. In this paper we report measurements by neutron inelastic scattering techniques of the spin wave spectrum in Pd s Fe. Because of its low Curie temperature a fairly complete dispersion curve for the spin waves has been obtained, particularly in the [0, 0, 1] direction, which enables us to compare our results with different models for the exchange interactions.
[o.o.~]
/ tO
•
|
6
0 4
4
2
0 0.1
Pd~ Fe is a ferromagnet with a Curie temperature of 540°K. It crystallises in the ordered Cu3Au structure below 1000°K. The specimen was aligned with a [0, 0, 1] axis vertical and mounted at room temperature on the triple-axis crystal spectrometer at the Pluto reactor of A.E.R.E., Harwell. The spin waves were studied by operating the crystal spectrometer mainly in its constant energy mode, with the incident neutron energy held fixed at either 82.9, 99.0 or 127.7 meV. One of the main difficulties in this type of experiment arises from the necessity of distinguishing between the scattering from the spin waves and from the phonons. In this particular case we made a prior study s of the phonon
0.3
~
0.4
O~
0
0.1
0.2
0.3
0.4
0.5
FIG. 1. Spin wave dispersion curve for [0, 0, ~] direction of P d s F e . (a) Compared with simple, single spin, Heisenberg Model (full line). (b) Compared with localised spin models with spins on both Fe and Pd ions. The data of reference 3 is shown by the full line while the heavy dotted lines are the dispersion curves obtained using the model fitted to this data. The model fitted to our data is shown by the fine dotted lines.
spectrum in Pd3Fe and so are confident that the results reported here arise from spin wave scattering. The intensity of the scattering from the 271
272
SPIN WAVES IN Pd3 Fe
spin waves was maximised in comparison to that arising from the phonons by making most of the measurements for momentum transfers, Q, around the reciprocal lattice point (1, 0, 0), although less extensive measurements were also made around the (2, 0, 0), (3, 0, 0) and (1, 1, 0) lattice points. In Fig. 1 are shown the results obtained for the spin waves propagating along the [0, 0, 1] direction. The typical uncertainties in most of the measurements, obtained with the constant energy transfer technique, are shown by the horizontal error bars. For comparison, a 'Constant _(2' measurement is also indicated. Measurements obtained around the different reciprocal lattice points are not shown separately, but were in agreement with one another. Corrections have been calculated to allow for the finite resolution of the crystal spectrometer. The resolution calculations were performed using the formalism derived by Cooper and Nathans s and the uncertainty in these corrections accounts for a large part of the error associated with the measurements at the smallest wavevectors.
IO /J 8
f
z
2
0
0
0.1
0.2
0.3
WAVE VECTOR
0.4
0.5
C2n/a)
FIG. 2. Spin wave dispersion curve for [~, ~, 0] direction of Pd 3 Fe compared with (1) simple, single spin, Heisenberg model (full line). (2) 1ocalised spin model fitted to the present data (dashed line).
Vot. 11, No. 1
Similar measurements have also been made of the spin waves propagating in the [1, l, 0] direction, as shown in Fig. 2. A large part of the uncertainty in these measurements arises because the experiments performed around the (1, 0, 0) and (1, 1, 0) lattice points gave systematically slightly different results. We have as yet been unable to find the origin of these discrepancies The dispersion curve shown in Fig.2 is very similar in form to that shown in Fig. 1 and in fact is within error almost isotropic or dependent only on Iql. The spin wave dispersion curve of Pd3Fe has also been studied by Antonini e t al. 3 using the polarised neutron diffr action technique. 7 They obtained results for wavevectors ~(= a q / 2 7 r ) < 0.18 which are shown by the solid line in Fig. l(b). Our results are in substantial agreement with theirs at the smallest wavevectors where ours are most uncertain, but are significantly different at the largest wavevectors obtained in their experiment, where our results are more reliable. Antonini et al.3 obtained from their results the parameters of a model proposed by Leoni and Natoli. 8 In this model, localised spins, SFe and Spa , on the Fe and Pd ions are assumed to interact with Heisenberg exchange interactions between Fe and Pd nearest neighbours, J~, and between Fe and Fe next nearest neighbours, Jz. There are then 4 branches of the spin wave dispersion relation for each wavevector. The spins SFe and Spa were taken from the polarised neutron diffraction measurements of Shirane e t al. 9 as 1.43 and 0.17 respectively, and Antonini et al. deduced from their measurements that J~ = 1.58 -+ 0.17meV and g2 = 6.85 + 0.09meV. The results for the spin wave spectrum are then shown in Fig. l(b); the lowest and highest frequency branches are non-degenerate while the non-dispersive branch is doubly degenerate. Clearly the model is not consistent with our measurements. Using the same model, we have performed a least-squares fit to our results and obtain the values J~ = 3.48 +0.06meV and Ja = 5.98 +_0.12meV. As is seen in Fig. l(b), these values give a good representation of the [0, 0, 1] direction frequencies, but are poorer for the [1, 1, 0] direction, Fig. 2. The
Vol. 11, No. 1
SPIN WAVES IN Pd3 Fe
Table i . Frequencies (THz) for symmetry points obtained using a model with localised spins on each ion. q(-~)
Acoustic
Optic I, II
Optic III
(0, O, O) (0, O, ½)
0 8.65
9.63 9.63
13.10 20.94
(~, ~,~ O)
9.63
9.63
36.49
(½, ~, ~)
9.63
9.63
53.02
frequencies predicted by this model for the principal symmetry points of the Brillouin zone are listed in Table 1. The value of D, where cg(q) = Dq2 for small q, is 213 meV ~2, whereas Ant'onini et al. 3 obtain 226 +--6 meV ~k2 . Also shown in Fig. l(a) is a comparison of our results with the form of dispersion relation expected for a simple Heisenberg ferromagnet with one spin in each formula unit; ho)(_q) = 2J(3 - cos(q:a) - cos(q,ja) - cos(qza)), with J = 11.6meV chosen to fit the ~ = 0.25 point of the dispersion curve. Although this curve gives a value of D = 167meVA whmh ls conslderabl~ smaller than that obtained by Antonini et al., the curve is qualitatively very similar to our results. Figure 2 shows the dispersion curve calculated with our simple Heisenberg model for the [1, 1, 0] direction. There are considerably larger discrepancies than in the [1, 0, 0] direction. •
2
•
•
273
groups were not obtained above 8.5 THz but this might be due to the increasing technical difficulties of the experiments. In particular no evidence was obtained for other branches of the spin wave dispersion relation even though some of the scans searched the frequency range of the flat branch shown in Fig. l(b). Further work is, however, in progress to confirm this result. We conclude therefore that a localised spin model is unsatisfactory. In particular, the observed fiequencies are considerably lower than those predicted by this model for the higher frequencies in the [1, 1, 0] direction. We suggest that the single branch and its behaviour is more like that expected from an itinerant electron model for Pd~Fe. ~1 Similar measurements have recently been reported by Antonini and Minkiewicz 12 for the isomorphous material Pt3Mn. They observed the acoustic spin wave excitations propagating in the [1, 0, 0], [1, 1, 0] and [1, 1, 1] directions and showed that their results also could not be fitted with Leoni and Natoli's model. They also reported 2 points on an optical spin wave branch. We have been unable to obtain similar results in Pd 3Fe.
.
The intensity of the scattered neutron groups was approximately consistent with that expected from a simple Heisenberg model. They varied as If(O._)12(n(q) + 1), where f(Q) is the form factor 1° and n(q) the occupation number. Well defined
We intend to continue our measurements on Pd~ Fe to further test our suggestion about the nature of the dispersion curves, and hope that our suggestions will stimulate studies of the energy band curves in this material.
Acknowledgements - We have benefitted from helpful discussions with Dr. H. Montgomery and from a particularly helpful set of notes on the spin wave theory of Pd3Fe by Dr. M.W. Stringfellow. The work was supported by the Science Research Council to whom we are very grateful.
REFERENCES 1.
SCHRIEFFER J.R., J. appl. Phys. 39, 642 (1968).
2.
STRINGFELLOW N.W., J. Phys. C. 1, 1699 (1968).
3.
ANTONINI B., MEDINA R. and MENZINGER F., Solid State Commun. 9, 257 (1971).
4.
MENZINGER F., SACCHETTI F. and TEICHNER R., Solid State Commun. 9, 1579 (1971).
274
SPIN WAVES IN Pd3Fe
5.
STIRLING W.G., COWLEY R.A. and STRINGFELLOW M.W., J. Phys. F: (lletal Physics), to be published.
6.
COOPEP, M.J. and NATHAN~ R., Acta Crystallogr. 23, 357 (1967).
7.
ALPERIN H.A., STEINSVOLL 0., NATHANS R. and SHIRANE G., Phys. Rev. ]54, 508 (1967).
8.
LEONI F. and NATOLI C., Nuovo Cirn. 55B, 21 (1968).
9.
Vol. 11, No. 1
SHIRANE G., NATHANS R., PICKART S.J. and ALPERIN H.A., Proc. Int. Conf. on Ilagnetisrn,
Nottingham, p. 222, (1964). 10.
WATSON R.E. and FREEMAN A.J., Acta CrystaIlogr. 14, 27 (1961).
11.
MATTIS D.C., The Theory of lfagnetisrn, Harper & Row (N.Y.) (1965).
12.
ANTONINI B. and MINKIEWICZ V.J., Solid State Commun. 10, 203 (1972).
La courbe de dispersion des ondes de spin dans le Pd 3 Fe ordonn4 a 6t6 d6termin6e fi 296°K, par la diffusion in41astique de neutrons. Les r6sultats sont compsr6es avec les modules divers du spectre des ondes de spin.
Vol. 11, pp. 271-274, 1972.
Pergamon Press.
Printed in Great Britain
SPIN WAVES IN Pd3 Fe W.G. Stifling and R.A. Cowley Department of Physics, University of Edinburgh, The Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
(Received 21 April 1972 by M.F. Collins)
The spin wave dispersion relation in ordered Pd 3 Fe has been measured at 296°K by neutron inelastic scattering techniques. The results are compared with various models for the spin wave spectrum.
THE NATURE of the exchange interactions between Fe ions in a Pd host has been the object of considerable study2 In particular, the spin wave dispersion relations in the small wavevector region have been determined by neutron diffraction techniques in both dilute F e / P d alloys z and in ordered 3 and disordered 4 Pd3Fe. In this paper we report measurements by neutron inelastic scattering techniques of the spin wave spectrum in Pd s Fe. Because of its low Curie temperature a fairly complete dispersion curve for the spin waves has been obtained, particularly in the [0, 0, 1] direction, which enables us to compare our results with different models for the exchange interactions.
[o.o.~]
/ tO
•
|
6
0 4
4
2
0 0.1
Pd~ Fe is a ferromagnet with a Curie temperature of 540°K. It crystallises in the ordered Cu3Au structure below 1000°K. The specimen was aligned with a [0, 0, 1] axis vertical and mounted at room temperature on the triple-axis crystal spectrometer at the Pluto reactor of A.E.R.E., Harwell. The spin waves were studied by operating the crystal spectrometer mainly in its constant energy mode, with the incident neutron energy held fixed at either 82.9, 99.0 or 127.7 meV. One of the main difficulties in this type of experiment arises from the necessity of distinguishing between the scattering from the spin waves and from the phonons. In this particular case we made a prior study s of the phonon
0.3
~
0.4
O~
0
0.1
0.2
0.3
0.4
0.5
FIG. 1. Spin wave dispersion curve for [0, 0, ~] direction of P d s F e . (a) Compared with simple, single spin, Heisenberg Model (full line). (b) Compared with localised spin models with spins on both Fe and Pd ions. The data of reference 3 is shown by the full line while the heavy dotted lines are the dispersion curves obtained using the model fitted to this data. The model fitted to our data is shown by the fine dotted lines.
spectrum in Pd3Fe and so are confident that the results reported here arise from spin wave scattering. The intensity of the scattering from the 271
272
SPIN WAVES IN Pd3 Fe
spin waves was maximised in comparison to that arising from the phonons by making most of the measurements for momentum transfers, Q, around the reciprocal lattice point (1, 0, 0), although less extensive measurements were also made around the (2, 0, 0), (3, 0, 0) and (1, 1, 0) lattice points. In Fig. 1 are shown the results obtained for the spin waves propagating along the [0, 0, 1] direction. The typical uncertainties in most of the measurements, obtained with the constant energy transfer technique, are shown by the horizontal error bars. For comparison, a 'Constant _(2' measurement is also indicated. Measurements obtained around the different reciprocal lattice points are not shown separately, but were in agreement with one another. Corrections have been calculated to allow for the finite resolution of the crystal spectrometer. The resolution calculations were performed using the formalism derived by Cooper and Nathans s and the uncertainty in these corrections accounts for a large part of the error associated with the measurements at the smallest wavevectors.
IO /J 8
f
z
2
0
0
0.1
0.2
0.3
WAVE VECTOR
0.4
0.5
C2n/a)
FIG. 2. Spin wave dispersion curve for [~, ~, 0] direction of Pd 3 Fe compared with (1) simple, single spin, Heisenberg model (full line). (2) 1ocalised spin model fitted to the present data (dashed line).
Vot. 11, No. 1
Similar measurements have also been made of the spin waves propagating in the [1, l, 0] direction, as shown in Fig. 2. A large part of the uncertainty in these measurements arises because the experiments performed around the (1, 0, 0) and (1, 1, 0) lattice points gave systematically slightly different results. We have as yet been unable to find the origin of these discrepancies The dispersion curve shown in Fig.2 is very similar in form to that shown in Fig. 1 and in fact is within error almost isotropic or dependent only on Iql. The spin wave dispersion curve of Pd3Fe has also been studied by Antonini e t al. 3 using the polarised neutron diffr action technique. 7 They obtained results for wavevectors ~(= a q / 2 7 r ) < 0.18 which are shown by the solid line in Fig. l(b). Our results are in substantial agreement with theirs at the smallest wavevectors where ours are most uncertain, but are significantly different at the largest wavevectors obtained in their experiment, where our results are more reliable. Antonini et al.3 obtained from their results the parameters of a model proposed by Leoni and Natoli. 8 In this model, localised spins, SFe and Spa , on the Fe and Pd ions are assumed to interact with Heisenberg exchange interactions between Fe and Pd nearest neighbours, J~, and between Fe and Fe next nearest neighbours, Jz. There are then 4 branches of the spin wave dispersion relation for each wavevector. The spins SFe and Spa were taken from the polarised neutron diffraction measurements of Shirane e t al. 9 as 1.43 and 0.17 respectively, and Antonini et al. deduced from their measurements that J~ = 1.58 -+ 0.17meV and g2 = 6.85 + 0.09meV. The results for the spin wave spectrum are then shown in Fig. l(b); the lowest and highest frequency branches are non-degenerate while the non-dispersive branch is doubly degenerate. Clearly the model is not consistent with our measurements. Using the same model, we have performed a least-squares fit to our results and obtain the values J~ = 3.48 +0.06meV and Ja = 5.98 +_0.12meV. As is seen in Fig. l(b), these values give a good representation of the [0, 0, 1] direction frequencies, but are poorer for the [1, 1, 0] direction, Fig. 2. The
Vol. 11, No. 1
SPIN WAVES IN Pd3 Fe
Table i . Frequencies (THz) for symmetry points obtained using a model with localised spins on each ion. q(-~)
Acoustic
Optic I, II
Optic III
(0, O, O) (0, O, ½)
0 8.65
9.63 9.63
13.10 20.94
(~, ~,~ O)
9.63
9.63
36.49
(½, ~, ~)
9.63
9.63
53.02
frequencies predicted by this model for the principal symmetry points of the Brillouin zone are listed in Table 1. The value of D, where cg(q) = Dq2 for small q, is 213 meV ~2, whereas Ant'onini et al. 3 obtain 226 +--6 meV ~k2 . Also shown in Fig. l(a) is a comparison of our results with the form of dispersion relation expected for a simple Heisenberg ferromagnet with one spin in each formula unit; ho)(_q) = 2J(3 - cos(q:a) - cos(q,ja) - cos(qza)), with J = 11.6meV chosen to fit the ~ = 0.25 point of the dispersion curve. Although this curve gives a value of D = 167meVA whmh ls conslderabl~ smaller than that obtained by Antonini et al., the curve is qualitatively very similar to our results. Figure 2 shows the dispersion curve calculated with our simple Heisenberg model for the [1, 1, 0] direction. There are considerably larger discrepancies than in the [1, 0, 0] direction. •
2
•
•
273
groups were not obtained above 8.5 THz but this might be due to the increasing technical difficulties of the experiments. In particular no evidence was obtained for other branches of the spin wave dispersion relation even though some of the scans searched the frequency range of the flat branch shown in Fig. l(b). Further work is, however, in progress to confirm this result. We conclude therefore that a localised spin model is unsatisfactory. In particular, the observed fiequencies are considerably lower than those predicted by this model for the higher frequencies in the [1, 1, 0] direction. We suggest that the single branch and its behaviour is more like that expected from an itinerant electron model for Pd~Fe. ~1 Similar measurements have recently been reported by Antonini and Minkiewicz 12 for the isomorphous material Pt3Mn. They observed the acoustic spin wave excitations propagating in the [1, 0, 0], [1, 1, 0] and [1, 1, 1] directions and showed that their results also could not be fitted with Leoni and Natoli's model. They also reported 2 points on an optical spin wave branch. We have been unable to obtain similar results in Pd 3Fe.
.
The intensity of the scattered neutron groups was approximately consistent with that expected from a simple Heisenberg model. They varied as If(O._)12(n(q) + 1), where f(Q) is the form factor 1° and n(q) the occupation number. Well defined
We intend to continue our measurements on Pd~ Fe to further test our suggestion about the nature of the dispersion curves, and hope that our suggestions will stimulate studies of the energy band curves in this material.
Acknowledgements - We have benefitted from helpful discussions with Dr. H. Montgomery and from a particularly helpful set of notes on the spin wave theory of Pd3Fe by Dr. M.W. Stringfellow. The work was supported by the Science Research Council to whom we are very grateful.
REFERENCES 1.
SCHRIEFFER J.R., J. appl. Phys. 39, 642 (1968).
2.
STRINGFELLOW N.W., J. Phys. C. 1, 1699 (1968).
3.
ANTONINI B., MEDINA R. and MENZINGER F., Solid State Commun. 9, 257 (1971).
4.
MENZINGER F., SACCHETTI F. and TEICHNER R., Solid State Commun. 9, 1579 (1971).
274
SPIN WAVES IN Pd3Fe
5.
STIRLING W.G., COWLEY R.A. and STRINGFELLOW M.W., J. Phys. F: (lletal Physics), to be published.
6.
COOPEP, M.J. and NATHAN~ R., Acta Crystallogr. 23, 357 (1967).
7.
ALPERIN H.A., STEINSVOLL 0., NATHANS R. and SHIRANE G., Phys. Rev. ]54, 508 (1967).
8.
LEONI F. and NATOLI C., Nuovo Cirn. 55B, 21 (1968).
9.
Vol. 11, No. 1
SHIRANE G., NATHANS R., PICKART S.J. and ALPERIN H.A., Proc. Int. Conf. on Ilagnetisrn,
Nottingham, p. 222, (1964). 10.
WATSON R.E. and FREEMAN A.J., Acta CrystaIlogr. 14, 27 (1961).
11.
MATTIS D.C., The Theory of lfagnetisrn, Harper & Row (N.Y.) (1965).
12.
ANTONINI B. and MINKIEWICZ V.J., Solid State Commun. 10, 203 (1972).
La courbe de dispersion des ondes de spin dans le Pd 3 Fe ordonn4 a 6t6 d6termin6e fi 296°K, par la diffusion in41astique de neutrons. Les r6sultats sont compsr6es avec les modules divers du spectre des ondes de spin.