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Observations of ferromagnetic correlations at high temperatures in paramagnetic iron
243
Journal of Magnetism and Magnetic Materials 30 (1982) 243-248 North-Holland Publishing Company
OBSERVATIONS OF FERROMAGNETIC CORRELATIONS AT HIGH TEMPERATURES IN PARAMAGNETIC IRON P.J. BROWN, H. CAPELLMANN, Imtitut Max eon L.uue Paul Lungevin, Grenoble, France Received
J. DEPORTES, D. GIVORD and K.R.A. ZIEBECK
156X, 38042 Grenoble, France and Luboratoire
Louis Ndel, C.N.R.S.,
166X, 38042
10 May 1982; in revised form 16 July 1982
Polarised neutron scattering with polarisation analysis has been used to obtain a unique measurement of the paramagnetic fluctuations in iron at temperatures befween 1273 and 1573 K. The results clearly demonstrate almost complete ferromagnetic correlation over distances up to 15 A. The average moment per atom taking part in the correlation and giving rise to paramagnetic scattering is about 1.3~~. These findings should lead to a better understanding of paramagnetism in metals.
1. Introduction The ferromagnetism of transition metals (Fe, Co, Ni) is still a very controversial subject and no generally accepted picture has yet developed. No consensus has been reached about the fundamental question of whether amplitude fluctuations or angular fluctuations or both are responsible for the ferromagnetic to paramagnetic phase transition. There is also still considerable disagreement concerning the nature of the paramagnetic phase. The source of the difficulties and the lack of general understanding lie in the fact that magnetism in the transition metals is carried by itinerant electrons. Because of this the translational degrees of freedom are not separable from the magnetic degrees of freedom. This is in constrast to magnetic insulators and rare earth metals. In these materials well localised magnetic moments exist due to localised electrons, which makes the understanding easier. Experimental evidence for a qualitative difference in the paramagnetic phase between itinerant and localized magnetism has been obtained from inelastic neutron scattering [l]: the spin flip scattering function S(qw) was found to be non-diffusive in character (i.e. it did not peak at 0304-8853/82/0000-0000/$02.75
0 1982 North-Holland
o = 0 but rather at finite w) for surprisingly large wavelengths (up to = 20 A) even well above the transition temperature T, in iron and nickel. This has been interpreted as showing the existence of propagating spin flip excitations: “spin waves above Tc”. Further indications of this qualitative difference were found in paramagnetic scattering experiments [2,3] in the itinerant systems CeFe, and MnSi, where considerable short range magnetic order was observed far above T, (up to 20 T, in MnSi), a feature which is absent in the localised ferromagnet Pd 2MnSn [4]. The modem theories [5-91 of itinerant ferromagnetism all postulate that some sort of local magnetisation remains above T,; they have not reached agreement about the nature of the paramagnetic phase. The fluctuating band theory [5,6] is based upon the existence of very strong short range magnetic order well above T,. The short range order is not a critical effect confined to the vicinity of the phase transition, but is due to the itinerancy of the d-electrons. An estimate of a typical wavelength to describe the short range order is 20 A and its thermal variation is expected to be slow. Other theories 17-91 describe the paramagnetic phase as having no appreciable short range order outside the critical region, which would
244
P.J. Brownet al. / Ferromagnetic correlations at high temperatures in paramagnetic iron
agree with the behaviour of a localised Heisenberg model [lo]. A key to understanding of transition metal magnetism may therefore be obtained from a measurement of the spin density - spin density correlation function above T,. This paper describes an experiment in which polarisation analysis has been used to determine this correlation function in paramagnetic iron. A preliminary account of this work has been presented previously [ 161.
2. Paramagnetic scattering of polarised neutrons A rather direct way of studying the space and time correlation of electron spins is by the paramagnetic scattering of neutrons. If polarisation analysis is used a unique separation of the magnetic part from other contributions to the scattering can be obtained. This is possible because it is only the magnetic part of the scattering that is sensitive to the angle between the neutron polarisation direction and the scattering vector (see for instance Marshall and Lovesey (1971) [ 111). Details of the use of the neutron polarisation analysis technique are given by Ziebeck and Brown [ 121 who relate the difference between the spin flip cross-sections for neutrons polarised parallel and perpendicular to the scattering vector to the magnetic response fucntion S(Q, o), thus:
where I’ is the sample volume, V, the unit cell volume, N,,, the number of magnetic atoms per cell, y the neutron magnetic moment and the other symbols have their usual meanings. The measured paramagnetic scattering can then be expressed in terms of an ‘effective moment’ M given by
wheref( Q) is the form factor for the paramagnetic carriers and the integration is over the energy resolution of the instrument. If this energy range is wide enough to include all fluctuations contribut-
ing to the susceptibility then it can be shown that M* = 12 kTX, where x is the magnetic susceptibility.
3. Properties of iron Below 1044 K cw(bcc)iron orders ferromagnetitally with atomic moments of 2.216~, at 4 K [13]. At room temperature spin waves propagate isotropically with a quadratic dispersion and a stiffness constant D = 70 THz A2 [ 11. For large energy transfers z 20 THz the spin wave intensity decreases anomalously consistent with the predictions of band theory and the entry of the spin wave into the Stoner continuum. However, Stoner theory does not describe the transition to the paramagnetic phase correctly. It is based upon the amplitude of the atomic moments vanishing, which leads to an unrealistic value of the Curie temperature T, (an order of magnitude too high). Considerable experimental evidence exists suggesting that some sort of magnetic moment persists beyond T,. The magnetic susceptibility above c has a strong temperature dependence. If fitted to a Curie-Weiss law an “effective paramagnetic moment” may be obtained, 3.13c(,, which suggests that some sort of band splitting remains in the paramagnetic phase. Furthermore, spin flip excitations can propagate at 1.4T, for wavevectors greater than = 0.25 A-’ (“spin waves above T,“), with characteristic energies somewhat reduced from the values at low temperature [I]. Additional evidence for the persistence of exchange split bands is obtained from the very low value of the spontaneous volume magnetostriction (less than 8 X 10e4 [ 141).
4. Experimental The samples used in the experiment were solid solutions of 5 at % silicon in pure iron. This addition of silicon is sufficient to suppress the martensitic transition of a to y iron and to stabilise the bee a-phase up to the melting point at 1750 K. Two samples were used both in the form of cylinders 10 mm dia. X 50 mm long, one a single
P. J. Brown et at. / Ferromagnetic
correlations at high temperatures
crystal with a (110) direction parallel to the cylinder axis and the other a polycrystal. They were put in turn into a furnace which fitted inside the Helmholtz coils used for controlling the polarisation direction on the D5 spectrometer at ILL Grenoble. The sample temperature was measured by a thermocouple which was placed in a small hole drilled in the sample and the temperature uniformity was monitored by observing the change in depolarisation of the neutron beam tansmitted by the sample as it passed through the Curie temperature. On the polarised neutron triple axis spectrometer D5 the neutron beam was monochromated and polarised by reflection from the (111) planes of a Cu,MnAl crystal; a similar cyrstal was used as analyser. Guide fields maintained the neutron polarisation between monochromator and sample and between sample and analyser. Both monochromator and analyser were magnetised in the vertical direction, but Helmholtz coils around the sample position could be energised so as to obtain the neutron polarisation either parallel or perpendicular to the scattering vector, at the sample. The vertical or horizontal field required to rotate the neutron spins and offset the magnetic effect of the furnace was some 300 Oe and had a negligible effect on the sample. An r.f. coil in the incoming beam enabled the incident neutron polarisation to be reversed. At each position of measurement the spin-flip and non spin-flip cross-sections were measured for neutron polarisations successively perpendicular to the scattering plane and parallel to the scattering vector. In some of the inelastic scans it was not possible to obtain the exactly parallel configuration and corresponding corrections have been made to the measured cross-sections.
5. Results With the single crystal specimen at 1273 K and with the spectrometer in the elastic configuration Q-scans along the three principal symmetry directions were carried out from the lowest accessible to beyond the zone angle (Q = 0.23 A-‘) boundary. The results are illustrated in fig. 1: the
in paramagnetic
245
iron
.t
25
.
k
FP 531% 5, 1273K
20
. N”m 2
15
0
D
D,rrcr,on
l
DlrrcllonChm]
.
D,rechonChhh]
+
COOI]
.
N 72 8 2 ;
10
. . e
5
.
‘0
QCA-'I Fig. 1. Paramagnetic scatttering in Fe 5 at 5%Si at 1273 K, along the directions [001], [ hh 0] and 1.
principal features are a strong narrow forward peak which is essentially isotropic, the occurrence of similar strong relatively sharp peaks at the positions of Bragg reflections and the extremely low residual scattering towards the zone boundary. Measurements of the forward peak were also made as a function of temperature and the results are shown in fig. 2. The height of the forward peak drops rapidly with increasing temperature roughly in accordance with the susceptibility. At Q > 0.4 A-’ the rate of fall with temperature is less dramatic and beyond Q = 0.7 A- ’ there is a tendency for the scattering to increase with temperature. To investigate the energy dependence of the paramagnetic scattering, constant T = 1273 K for reduced wave vectors [-0.112, -0.112, 0] and [-0.225, -0.225, 0] from the [llO] position (fig. 3). In the scan at smaller q the observed intensity is strongly peaked at zero energy transfer and is due to the intersec-
246
P.J. Brown et al. / Ferromagnetic
correlations at high temperatures in paramagnetic
tion of the resolution ellipsoid with the observed forwark peak. In the scan at [-0.225, -0.225, 0]fig. 3b, the instrument resolution is shown and it can be seen that the scattering is centred at zero energy transfer. The convoluted energy width is only marginally greater than the spectrometer resolution showing that any inelasticity is small. These results demonstrate that the intensity due to the spin wave scattering [l] is very much less than that due to the quasi-elastic paramagnetic scattering. The scan at [- 0.112, - 0.112, 0] crosses the spin-wave dispersion curve, and in the second scan the resolution included the spin-wave frequency. In both scans the quasi-elastic intensity is sufficient to dominate and obscure a small inelastic component. Measurements made on the polycrystalline sample at 1273 K showed the same features as those made on the single crystal. The relative intensities of the paramagnetic scattering and that of the 110 Bragg peak were used to put all the measurements onto an absolute scale.
.
.
l
iron
1273K
Fig. 2. Paramagnetic scattering in Fe 5 at R; Si at 1273 K, 1373 K, 1473 K and 1573 K along the direction [ hhO].
,
6. Discussion
7
1
t \ \
\
\
The observation of a sharp quasi-elastic forward peak, and similar sharp peaks at the Bragg reflection positions show conclusively that the magnetic carriers giving rise to paramagnetic scattering have ferromagnetic correlations over several interatomic distances even well above T,. In the-following analysis we shall assume that the local variation of the spin density around the iron atom is described within the atomic volume by a 3d-form factor. To demonstrate the range and the nature of the correlations we have plotted Q2A4*(Q) against Q as shown in fig. 4 for two temperatures. These plots were obtained from measurements carried out in the zone centered on [l lo] which enables data to be collected close to the zone center. This figure gives a direct indication of which values of Q contribute significantly to the Fourier spectrum of the spin density-spin density correlation function (the factor Q* is a phase space factor). It is evident from fig. 4 that the correlation function is not of the Omstein-Zemike form (which would yield a result
‘\YP~___~_(_‘~__IIU:nr
A---L---5 FREQUENCY
10
15
[THr]
Fig. 3. Energy dependence of the paramagnetic scattering 5 at 46 Si for reduced wave vectors (b) [ - 0.225, - 0.225,0] (a) (-0.112, -0.112, 0] with respect to the [llO] position. broke curve in fig. (a) is a guide for the eye and in fig. represents the energy resoltuion of the spectrometer.
in Fe and The (b) it
P.J. Brown et al. / Ferromagnetic
Fe 5dx
t
1
o-5
1
1-o QCA-'I
S,
l
1273K
0
1573K
“‘It
correlations at high temperatures in paramagnetic
t
$l I
15
Fig. 4. &144*(Q) as a function of Q in Fe 5 at 16 Si at 1273 K and 1473 K showing which parts of the Fourier spectrum contribute essentially to the spin density-spin density correlation function.
Q2(Q2 + t2)-‘). We observe a peak around Q, = 0.4 A-’ which gives a typical wavelength in the Fourier spectrum X, = 27r/Q, of the order of 16 A, weakly dependent on temperature. The smallness of the scattering for Q > Q, indicates nearly perfect correlation over this distance. This means that the angular misorientation between the local magnetisation directions at different points must be quite small if the distance between the points is of order 16 A or less. As we are unable to make measurements for Q < 0.14 A-’ it is possible that for smaller Q the correlation function finally does follow an Omstein-Zernike form. The limited extent of the data precludes an accurate extrapolation to Q = 0 A-‘, but they appear to be consistent with a value at Q = 0 A- ’ corresponding to the observed susceptibility. The magnitude of the local magnetisation, summed over an atom, which contributes to the observed paramagnetic scattering can be obtained by integration out to the zone boundary of Q2M2(Q) as a function of Q. The value obtained is essentially temperature independent and corresponds to = 1.3~~. This result is not significantly influenced by the uncertainty in the low Q values
iron
241
because of the phase space factor, however the accuracy of the result is limited by the accuracy of the measurements near the zone boundary which are very strongly weighted. The value of the local magnetisation obtained here is significantly smaller than that observed in the ordered phase. This value is an average’ over the characteristic time of the neutron scattering experiment (some lo-l3 s for 0.84 A neutrons) and will not contain components of the magnetisation density which fluctuate faster than this. It is interesting to note that the original measurements of paramagnetic scattering from iron made by Shull [ 151 with unpolarised neutrons and no energy analysis also have a Q dependence which falls off more sharply than that expected for 3d electrons in the localised electron model. The most significant change observed in the paramagnetic scattering with increasing temperature is the decrease of intensity at small Q values. However, from our measurements there appears to be no significant change in the integrated Fe moment, this can be interpreted as due to increasing transverse fluctuations of the magnetisation density within the correlated regions themselves. A tendency for the peak in the Q2M2( Q) plot (fig. 4) to shift towards larger Q values is also observed, corresponding to a slight reduction in the size of the correlated regions.
7. Conclusions It can be concluded that the magnetic properties of iron above T, are dominated by strong ferromagnetic correlations over regions extending up to about five lattice spacings. Within these regions the correlation function does not have the Omstein-Zemike form; the average angular misorientation between the local magnetisation directions at points separated by less than about 15 A must be rather small. These results can provide a basis for a better understanding of the properties of metals in the paramagnetic state. They indicate how the exchange splitting of itinerant electrons in the 3d band may be maintained by short range order in the disordered magnetic phase. The large values of ‘effective’ moments obtained in suscept-
248
P.J. Brown et al. / Ferromagnetic
correlations at high temperatures in paramagnetic
ibility measurements can be attributed to magnetic correlations which persist far into the paramagnetic region. Finally, the existence of correlated regions allows us to understand how short wavelength spin waves can propagate above T,. References (11 J.W. Lynn, Phys. Rev. Bl 1 (1975) 2624. 121J. Diportes, D. Givord and K.R.A. Ziebeck, J. Appl. 52 (1981) 2074. 131 K.R.A. Ziebeck, P.J. Brown, J.G. Booth and J.A.C. J. Phys. F 11 (1981) L127. P.J. Brown and 141 K.R.A. Ziebeck, P.J. Webster, Bland, J. Magn. Mag. Mat. 24 (1981) 258. Z. Phys. B34 (1979) 29; J. Phys. F4 [51 H. Capellmann, 1466.
* On leave from T.H. Aachen, Department of SFB125 Aachen, Jiilich, K&n.
Phys. Bland, J.A.C. (1974)
of Physics, member
[61V.
iron
Korenmann, J.L. Murray and R.E. Prange, Phys. Rev. B16 (1977) 4032, 4048,4058. [71 T. Moriya, J. Magn. Magn. Mat. 14 (1979) 1. PI J. Hubbard, Phys. Rev. B19 (1979) 1504. 191 H. Hasegawa, J. Phys. Sot. Japan 46 (1979) 1504. [lOI B. Sriram Shastry, D.M. Edwards and A.P. Young, J. Phys. Cl4 (1981) L665. 1111 W. Marshall and. S.W. Lovesey, Theory of Thermal Neutron Scattering (Oxford Univ. Press, London, 1971). 1121 K.R.A. Ziebeck and P.J. Brown, J. Phys. FlO (1980) 2015. Proc. Roy. Sot. 321A [I31 J. Crangle and G.M. Goodman, (1971) 477. t141 M. Sbiga, Inst. Phys. Conf. 55 (1981) 241. iI51 M.K. Wilkinson and C.G. Shull, Phys. Rev. 103 (1956) 516. 1161 P.J. Brown, J. Deportes, D. Givord and K.R.A. Ziebeck, J. Appl. Phys. 53 (1982) 1973.
Journal of Magnetism and Magnetic Materials 30 (1982) 243-248 North-Holland Publishing Company
OBSERVATIONS OF FERROMAGNETIC CORRELATIONS AT HIGH TEMPERATURES IN PARAMAGNETIC IRON P.J. BROWN, H. CAPELLMANN, Imtitut Max eon L.uue Paul Lungevin, Grenoble, France Received
J. DEPORTES, D. GIVORD and K.R.A. ZIEBECK
156X, 38042 Grenoble, France and Luboratoire
Louis Ndel, C.N.R.S.,
166X, 38042
10 May 1982; in revised form 16 July 1982
Polarised neutron scattering with polarisation analysis has been used to obtain a unique measurement of the paramagnetic fluctuations in iron at temperatures befween 1273 and 1573 K. The results clearly demonstrate almost complete ferromagnetic correlation over distances up to 15 A. The average moment per atom taking part in the correlation and giving rise to paramagnetic scattering is about 1.3~~. These findings should lead to a better understanding of paramagnetism in metals.
1. Introduction The ferromagnetism of transition metals (Fe, Co, Ni) is still a very controversial subject and no generally accepted picture has yet developed. No consensus has been reached about the fundamental question of whether amplitude fluctuations or angular fluctuations or both are responsible for the ferromagnetic to paramagnetic phase transition. There is also still considerable disagreement concerning the nature of the paramagnetic phase. The source of the difficulties and the lack of general understanding lie in the fact that magnetism in the transition metals is carried by itinerant electrons. Because of this the translational degrees of freedom are not separable from the magnetic degrees of freedom. This is in constrast to magnetic insulators and rare earth metals. In these materials well localised magnetic moments exist due to localised electrons, which makes the understanding easier. Experimental evidence for a qualitative difference in the paramagnetic phase between itinerant and localized magnetism has been obtained from inelastic neutron scattering [l]: the spin flip scattering function S(qw) was found to be non-diffusive in character (i.e. it did not peak at 0304-8853/82/0000-0000/$02.75
0 1982 North-Holland
o = 0 but rather at finite w) for surprisingly large wavelengths (up to = 20 A) even well above the transition temperature T, in iron and nickel. This has been interpreted as showing the existence of propagating spin flip excitations: “spin waves above Tc”. Further indications of this qualitative difference were found in paramagnetic scattering experiments [2,3] in the itinerant systems CeFe, and MnSi, where considerable short range magnetic order was observed far above T, (up to 20 T, in MnSi), a feature which is absent in the localised ferromagnet Pd 2MnSn [4]. The modem theories [5-91 of itinerant ferromagnetism all postulate that some sort of local magnetisation remains above T,; they have not reached agreement about the nature of the paramagnetic phase. The fluctuating band theory [5,6] is based upon the existence of very strong short range magnetic order well above T,. The short range order is not a critical effect confined to the vicinity of the phase transition, but is due to the itinerancy of the d-electrons. An estimate of a typical wavelength to describe the short range order is 20 A and its thermal variation is expected to be slow. Other theories 17-91 describe the paramagnetic phase as having no appreciable short range order outside the critical region, which would
244
P.J. Brownet al. / Ferromagnetic correlations at high temperatures in paramagnetic iron
agree with the behaviour of a localised Heisenberg model [lo]. A key to understanding of transition metal magnetism may therefore be obtained from a measurement of the spin density - spin density correlation function above T,. This paper describes an experiment in which polarisation analysis has been used to determine this correlation function in paramagnetic iron. A preliminary account of this work has been presented previously [ 161.
2. Paramagnetic scattering of polarised neutrons A rather direct way of studying the space and time correlation of electron spins is by the paramagnetic scattering of neutrons. If polarisation analysis is used a unique separation of the magnetic part from other contributions to the scattering can be obtained. This is possible because it is only the magnetic part of the scattering that is sensitive to the angle between the neutron polarisation direction and the scattering vector (see for instance Marshall and Lovesey (1971) [ 111). Details of the use of the neutron polarisation analysis technique are given by Ziebeck and Brown [ 121 who relate the difference between the spin flip cross-sections for neutrons polarised parallel and perpendicular to the scattering vector to the magnetic response fucntion S(Q, o), thus:
where I’ is the sample volume, V, the unit cell volume, N,,, the number of magnetic atoms per cell, y the neutron magnetic moment and the other symbols have their usual meanings. The measured paramagnetic scattering can then be expressed in terms of an ‘effective moment’ M given by
wheref( Q) is the form factor for the paramagnetic carriers and the integration is over the energy resolution of the instrument. If this energy range is wide enough to include all fluctuations contribut-
ing to the susceptibility then it can be shown that M* = 12 kTX, where x is the magnetic susceptibility.
3. Properties of iron Below 1044 K cw(bcc)iron orders ferromagnetitally with atomic moments of 2.216~, at 4 K [13]. At room temperature spin waves propagate isotropically with a quadratic dispersion and a stiffness constant D = 70 THz A2 [ 11. For large energy transfers z 20 THz the spin wave intensity decreases anomalously consistent with the predictions of band theory and the entry of the spin wave into the Stoner continuum. However, Stoner theory does not describe the transition to the paramagnetic phase correctly. It is based upon the amplitude of the atomic moments vanishing, which leads to an unrealistic value of the Curie temperature T, (an order of magnitude too high). Considerable experimental evidence exists suggesting that some sort of magnetic moment persists beyond T,. The magnetic susceptibility above c has a strong temperature dependence. If fitted to a Curie-Weiss law an “effective paramagnetic moment” may be obtained, 3.13c(,, which suggests that some sort of band splitting remains in the paramagnetic phase. Furthermore, spin flip excitations can propagate at 1.4T, for wavevectors greater than = 0.25 A-’ (“spin waves above T,“), with characteristic energies somewhat reduced from the values at low temperature [I]. Additional evidence for the persistence of exchange split bands is obtained from the very low value of the spontaneous volume magnetostriction (less than 8 X 10e4 [ 141).
4. Experimental The samples used in the experiment were solid solutions of 5 at % silicon in pure iron. This addition of silicon is sufficient to suppress the martensitic transition of a to y iron and to stabilise the bee a-phase up to the melting point at 1750 K. Two samples were used both in the form of cylinders 10 mm dia. X 50 mm long, one a single
P. J. Brown et at. / Ferromagnetic
correlations at high temperatures
crystal with a (110) direction parallel to the cylinder axis and the other a polycrystal. They were put in turn into a furnace which fitted inside the Helmholtz coils used for controlling the polarisation direction on the D5 spectrometer at ILL Grenoble. The sample temperature was measured by a thermocouple which was placed in a small hole drilled in the sample and the temperature uniformity was monitored by observing the change in depolarisation of the neutron beam tansmitted by the sample as it passed through the Curie temperature. On the polarised neutron triple axis spectrometer D5 the neutron beam was monochromated and polarised by reflection from the (111) planes of a Cu,MnAl crystal; a similar cyrstal was used as analyser. Guide fields maintained the neutron polarisation between monochromator and sample and between sample and analyser. Both monochromator and analyser were magnetised in the vertical direction, but Helmholtz coils around the sample position could be energised so as to obtain the neutron polarisation either parallel or perpendicular to the scattering vector, at the sample. The vertical or horizontal field required to rotate the neutron spins and offset the magnetic effect of the furnace was some 300 Oe and had a negligible effect on the sample. An r.f. coil in the incoming beam enabled the incident neutron polarisation to be reversed. At each position of measurement the spin-flip and non spin-flip cross-sections were measured for neutron polarisations successively perpendicular to the scattering plane and parallel to the scattering vector. In some of the inelastic scans it was not possible to obtain the exactly parallel configuration and corresponding corrections have been made to the measured cross-sections.
5. Results With the single crystal specimen at 1273 K and with the spectrometer in the elastic configuration Q-scans along the three principal symmetry directions were carried out from the lowest accessible to beyond the zone angle (Q = 0.23 A-‘) boundary. The results are illustrated in fig. 1: the
in paramagnetic
245
iron
.t
25
.
k
FP 531% 5, 1273K
20
. N”m 2
15
0
D
D,rrcr,on
l
DlrrcllonChm]
.
D,rechonChhh]
+
COOI]
.
N 72 8 2 ;
10
. . e
5
.
‘0
QCA-'I Fig. 1. Paramagnetic scatttering in Fe 5 at 5%Si at 1273 K, along the directions [001], [ hh 0] and 1.
principal features are a strong narrow forward peak which is essentially isotropic, the occurrence of similar strong relatively sharp peaks at the positions of Bragg reflections and the extremely low residual scattering towards the zone boundary. Measurements of the forward peak were also made as a function of temperature and the results are shown in fig. 2. The height of the forward peak drops rapidly with increasing temperature roughly in accordance with the susceptibility. At Q > 0.4 A-’ the rate of fall with temperature is less dramatic and beyond Q = 0.7 A- ’ there is a tendency for the scattering to increase with temperature. To investigate the energy dependence of the paramagnetic scattering, constant T = 1273 K for reduced wave vectors [-0.112, -0.112, 0] and [-0.225, -0.225, 0] from the [llO] position (fig. 3). In the scan at smaller q the observed intensity is strongly peaked at zero energy transfer and is due to the intersec-
246
P.J. Brown et al. / Ferromagnetic
correlations at high temperatures in paramagnetic
tion of the resolution ellipsoid with the observed forwark peak. In the scan at [-0.225, -0.225, 0]fig. 3b, the instrument resolution is shown and it can be seen that the scattering is centred at zero energy transfer. The convoluted energy width is only marginally greater than the spectrometer resolution showing that any inelasticity is small. These results demonstrate that the intensity due to the spin wave scattering [l] is very much less than that due to the quasi-elastic paramagnetic scattering. The scan at [- 0.112, - 0.112, 0] crosses the spin-wave dispersion curve, and in the second scan the resolution included the spin-wave frequency. In both scans the quasi-elastic intensity is sufficient to dominate and obscure a small inelastic component. Measurements made on the polycrystalline sample at 1273 K showed the same features as those made on the single crystal. The relative intensities of the paramagnetic scattering and that of the 110 Bragg peak were used to put all the measurements onto an absolute scale.
.
.
l
iron
1273K
Fig. 2. Paramagnetic scattering in Fe 5 at R; Si at 1273 K, 1373 K, 1473 K and 1573 K along the direction [ hhO].
,
6. Discussion
7
1
t \ \
\
\
The observation of a sharp quasi-elastic forward peak, and similar sharp peaks at the Bragg reflection positions show conclusively that the magnetic carriers giving rise to paramagnetic scattering have ferromagnetic correlations over several interatomic distances even well above T,. In the-following analysis we shall assume that the local variation of the spin density around the iron atom is described within the atomic volume by a 3d-form factor. To demonstrate the range and the nature of the correlations we have plotted Q2A4*(Q) against Q as shown in fig. 4 for two temperatures. These plots were obtained from measurements carried out in the zone centered on [l lo] which enables data to be collected close to the zone center. This figure gives a direct indication of which values of Q contribute significantly to the Fourier spectrum of the spin density-spin density correlation function (the factor Q* is a phase space factor). It is evident from fig. 4 that the correlation function is not of the Omstein-Zemike form (which would yield a result
‘\YP~___~_(_‘~__IIU:nr
A---L---5 FREQUENCY
10
15
[THr]
Fig. 3. Energy dependence of the paramagnetic scattering 5 at 46 Si for reduced wave vectors (b) [ - 0.225, - 0.225,0] (a) (-0.112, -0.112, 0] with respect to the [llO] position. broke curve in fig. (a) is a guide for the eye and in fig. represents the energy resoltuion of the spectrometer.
in Fe and The (b) it
P.J. Brown et al. / Ferromagnetic
Fe 5dx
t
1
o-5
1
1-o QCA-'I
S,
l
1273K
0
1573K
“‘It
correlations at high temperatures in paramagnetic
t
$l I
15
Fig. 4. &144*(Q) as a function of Q in Fe 5 at 16 Si at 1273 K and 1473 K showing which parts of the Fourier spectrum contribute essentially to the spin density-spin density correlation function.
Q2(Q2 + t2)-‘). We observe a peak around Q, = 0.4 A-’ which gives a typical wavelength in the Fourier spectrum X, = 27r/Q, of the order of 16 A, weakly dependent on temperature. The smallness of the scattering for Q > Q, indicates nearly perfect correlation over this distance. This means that the angular misorientation between the local magnetisation directions at different points must be quite small if the distance between the points is of order 16 A or less. As we are unable to make measurements for Q < 0.14 A-’ it is possible that for smaller Q the correlation function finally does follow an Omstein-Zernike form. The limited extent of the data precludes an accurate extrapolation to Q = 0 A-‘, but they appear to be consistent with a value at Q = 0 A- ’ corresponding to the observed susceptibility. The magnitude of the local magnetisation, summed over an atom, which contributes to the observed paramagnetic scattering can be obtained by integration out to the zone boundary of Q2M2(Q) as a function of Q. The value obtained is essentially temperature independent and corresponds to = 1.3~~. This result is not significantly influenced by the uncertainty in the low Q values
iron
241
because of the phase space factor, however the accuracy of the result is limited by the accuracy of the measurements near the zone boundary which are very strongly weighted. The value of the local magnetisation obtained here is significantly smaller than that observed in the ordered phase. This value is an average’ over the characteristic time of the neutron scattering experiment (some lo-l3 s for 0.84 A neutrons) and will not contain components of the magnetisation density which fluctuate faster than this. It is interesting to note that the original measurements of paramagnetic scattering from iron made by Shull [ 151 with unpolarised neutrons and no energy analysis also have a Q dependence which falls off more sharply than that expected for 3d electrons in the localised electron model. The most significant change observed in the paramagnetic scattering with increasing temperature is the decrease of intensity at small Q values. However, from our measurements there appears to be no significant change in the integrated Fe moment, this can be interpreted as due to increasing transverse fluctuations of the magnetisation density within the correlated regions themselves. A tendency for the peak in the Q2M2( Q) plot (fig. 4) to shift towards larger Q values is also observed, corresponding to a slight reduction in the size of the correlated regions.
7. Conclusions It can be concluded that the magnetic properties of iron above T, are dominated by strong ferromagnetic correlations over regions extending up to about five lattice spacings. Within these regions the correlation function does not have the Omstein-Zemike form; the average angular misorientation between the local magnetisation directions at points separated by less than about 15 A must be rather small. These results can provide a basis for a better understanding of the properties of metals in the paramagnetic state. They indicate how the exchange splitting of itinerant electrons in the 3d band may be maintained by short range order in the disordered magnetic phase. The large values of ‘effective’ moments obtained in suscept-
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correlations at high temperatures in paramagnetic
ibility measurements can be attributed to magnetic correlations which persist far into the paramagnetic region. Finally, the existence of correlated regions allows us to understand how short wavelength spin waves can propagate above T,. References (11 J.W. Lynn, Phys. Rev. Bl 1 (1975) 2624. 121J. Diportes, D. Givord and K.R.A. Ziebeck, J. Appl. 52 (1981) 2074. 131 K.R.A. Ziebeck, P.J. Brown, J.G. Booth and J.A.C. J. Phys. F 11 (1981) L127. P.J. Brown and 141 K.R.A. Ziebeck, P.J. Webster, Bland, J. Magn. Mag. Mat. 24 (1981) 258. Z. Phys. B34 (1979) 29; J. Phys. F4 [51 H. Capellmann, 1466.
* On leave from T.H. Aachen, Department of SFB125 Aachen, Jiilich, K&n.
Phys. Bland, J.A.C. (1974)
of Physics, member
[61V.
iron
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