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Interpretation of the susceptibility of paramagnetic nickel
Solid State Communications, Vol. 18, pp. 127—130, 1976.
Pergamon Press.
Printed in Great Britain
INTERPRETATION OF THE SUSCEPTIBILITY OF PARAMAGNETIC NICKEL* P.A. Beck and C.P. Flynn Materials Research Laboratory, University of Illinois, Urbana, IL 61801, U.S.A. (Received 14 July 1975 by A.G. Chynoweth)
The susceptibility of nickel from 1100 to 1 500°Kfits well x Xo + C/(T 0), whereproposed the Curieallowing constantXocorresponds to a nickel A model.is to be interpreted as an moment enof O.7p~. hanced specific heat susceptibility. —
THE STONER-WOHLFARTH band model and the Friedel—Anderson-type local moment description of ferromagnetism differ from each other in regard to the entropy change they predict for the transition from ferromagnetism to paramagnetism. It is well known1 that for b.c.c. iron the predicted difference is large and that the experimental entropy value clearly favors the local moment model. However, for nickel the predicted difference is much smaller and no clear-cut answer results from comparison with the experimental value.1 It is, therefore of considerable interest to examine other relevant evidence.
temperature increases. On the other hand, the deviations observed in the upper half of the temperature range of the data2 suggest the increasing relative importance of a temperature-independant susceptibility component, as the Curie—Weiss component decreases with increasing temperature. In the present work, an attempt was made to represent the temperature variation of the susceptibility in the upper temperature range by x = xo+CI(T—0). (1) The parameter values obtained by least-squares fitting of Xo, C, and 0 to the experimental data2 for various values of the lower limit of the temperature range used, are shown in Table 1. It is seen that the fit becomes excellent (Fig. 1) if the susceptibility data below 1 100°Kare not used; the latter include a measurable magnetic short-range order contribution. Peff was calculated from the Curie constant, and p = gS was calculated from Peff by assuming g = 2.
The excellent susceptibility data for polyciystalline nickel metal obtained by Arajs and Colvin2 between the Curie temperature of 631 and 1512°K show distinct deviations from the Curie—Weiss law. Just above the Curie temperature, the deviations appear to be similar to those resulting in various alloys from the presence of giant moments and the gradual decrease of the latter with increasing temperature.3 Although magnetic clusters in the usual sense are not expected to occur in unalloyed nickel, it is well known that magnetic contributions to the specific heat extend well above the Curie temperature, to at least 1000°K,4and that these magnetic specific heat contributions can be ascribed to the gradual breaking up of short range magnetic order as the
Neutron magnetic scattering results on Ni in the ferromagnetic state5 indicate the presence of a nonlocalized negative polarization amounting to O.lI-LB per Ni atom. It is not clear what role should be assigned to this negative polarization in a localized moment interpretation of the Curie—Weiss term of the susceptibility. At any rate, the Ni atomic moment of O.7p~,calculated from the Curie constant (Table 1), agrees quite well with the Ni moment of °•61PB in
____________
*
This work was supported by grants from the U.S. Energy Research and Development Administration and the National Science Foundation. 127
128
SUSCEPTIBILITY OF PARAMAGNETIC NICKEL
Vol. 18, No. 1
2 The increased root mean square Table 1. Curie— Weiss analysis of x vsranges T for nickel, databelow from1102°Karise Arajs and Colvin deviations (RMSD) for temperature extending from the increasing short-range magnetic order at those temperatures (Fig. 1) Parameters obtained by Temperature range from least-squares fitting 1512°Kto: C 0 Xo RMSD Peff P (emu deg/mole) (°K) (106emu/mole) (lO8emu/g) (PB) (PB) 741°K 919°K 1013°K 1102°K
0.304 0.261 0.248 0.238
658 687 700 712
63.4 110 120 129
20.8 2.75
1.70 1.24
1.56 1.44 1.41 1.38
0.85
0.75 0.73 0.70
in fact, equally consistent with the localized model. Neither magnetooptic nor Fermi surface results at low temperatures are modified in principle by the distinction between a band ferromagnet and an assembly of
I.
aligned Friedel— Anderson-type local moments; merely represent alternative descriptions of the these same
E 0
2
physical system.
0 0
The temperature-independent susceptibility cornponent of Xo = 129 (lO6emu/mole), derived in the present analysis (Table 1), is similar in magnitude to Xs.h. = 98 (l06emu/mole), the Pauli susceptibility
5
/ 800
1000 2~ T (°K)
calculated from the low temperature electronic specific heat coefficient for ferromagnetic nickel, y = 7.1 (mJ deg2mole’).’°The excess ofy 0 Xs.h. = 31(10—6 emu/mole) may be attributed to the fact that many-body enhancement for the suscepti—
FIG. 1. I/(x Xo) vs temperature for Ni. Data points 0 from reference 2. The values of C = 0.238 (emudeg/mole), Xo = 129 (lO_6 emu/mole), and of the Weiss temperature 0 = 712°K,corresponding to the continuous line, we obtained by least-squares analysis of the data in the temperature range of 1102— 1512°K(Table 1). —
ferromagnetic nickel. The simplest interpretation of this agreement, in accordance with Rhodes and Wohlfarth,6 is that localized moments are present in paramagnetic nickel even though, on the basis of older, less accurate susceptibility data, those authors arrived at the alternative conclusion that nickel is a band ferromagnet. Our interpretation is supported by the fact that photoelectron energy distribution studies7’8 above and below the Curie temperature failed to show the difference expected on the basis of the Stoner— Wohlfarth band ferromagnetism, while these results are entirely consistent with the localized model for ferromagnetism in nickel. Magnetooptic results,9 originally presented in support of the band theory, are,
bility is usually larger than that for the electronic specific heat, or, alternatively, to the contribution from a Van Vieck susceptibility term. It agrees well with Xvv estimated2 by Jaccarino for palladium” In either case,eta!. the temperatureand for platinum.’ independent value of Xo = 129 (10-6 emu/mole), corresponding to 2.3 (108pB/Oe) per nickel atom, strongly suggests its interpretation as a band susceptibility. The question arises as to how a band susceptibility component can be reconciled with a localized moment model for a paramagnet. At any one instant, some of the Ni moments will be parallel and the others will be antiparallel with respect to an applied field. For the former the parallel spin d-subband, and for the latter the antiparallel spin d-subband, may be considered completely filled, so that the Pauli susceptibility is zero. A field-induced magnetization does
Vol. 18, No. 1 2.0
SUSCEPTIBILITY OF PARAMAGNETIC NICKEL
related in the usual way to the electronic specific heat, but enhanced by a factor [1 + (U J V)N(EF)]~ Here, U and J have their customary meaning as atomic Coulomb and exchange integrals, and V/2 is the energy release when conduction states screen an added local d orbital. The Xo value obtained in the
4.0
—
129
—
/ ~
1.5 s.h.” /
0 0
~2 1.0
I
II
,)~‘
0
~
0
3.0
/
b x
—
-
explained by thisofmechanism, with andata enhancement present analysis the experimental is thus factor of 1.3.
0.5
0
0.1 0.2 Ni (at. fraction)
The temperature-independent susceptibility per Ni atom obtained above is close to the susceptibility of Ni in paramagnetic Cu-rich Cu—Ni alloys. Graph
2.5 2.0 0.3
in Fig. 2 gives the temperature-independent
FIG. 2. Temperature-independent susceptibility contribution per Ni atom in Cu—Ni alloys (graph X) vs Ni concentration; data 0 from reference 13. Electronic specific heat contribution Ni atom in Cu—Ni alloys (expressed in terms per of susceptibility, graph s.h.) vs Ni concentration; data from reference 15.
0
from reference 14 and £~
become possible if we acknowledge that electrons can be transferred locally between the spin-polarized d orbitals near EF and the essentially unpolarized s-like conduction band. The system finds its lowest energy in a field H when Ni moments aligned parallel to Hundergo a d -÷ s orbital transfer involving minority d spins, while oppositely aligned moments give the reverse transfer. At sufficiently high temperatures,the two spin orientations of the local moments are equally populated. Assuming that charge screening is complete within each atomic cell, so that the orbital transfer introduces no added Ni—Ni interactions, one finds that this mechanism does indeed introduce a temperature-independent susceptibility,
susceptibility per Ni atom, corrected for the diamagnetism of the copper host, for alloys up to 27%Ni. It shows that Ni atoms in dilute Cu—Ni alloys contribute athe susceptibility 2.1 (l08p~/Oe),as compared with value of 2.3 of (108p 8/Oe) derived above for paramagnetic nickel. Ifthe enhancement factors were also nearly the same, one might well suppose that a nonmagnetic Ni solute atom in copper contributes a density of states at the Fermi surface very similar to that which a Ni atom carrying a moment contributes to the Fermi surface of nickel above the Curie ternperature. However, the electronic specific heat contribution per Ni atom in dilute Cu—Ni alloys, also shown in Fig. 2 in terms of the corresponding susceptibility, is much smaller and the enhancement, = 4.7, is much larger than that for nickel. Acknowledgements We wish to thank Prof. S. Arajs for making available to us his susceptibility data in numerical form and F.M. Mueller for reading an early version of the manuscript. The assistance of E.E. Barton in the least-squares fitting of the susceptibility data for nickel is gratefully acknowledged. —
REFERENCES 1.
MOTTN.F.,Adv.Phys. 13, 325 (1964).
2.
AR.AJS S. & COLVIN R.V.,J. Phys. Chem. Solids 24, 1233 (1963).
3.
See for instance: MUKHOPADHYAY A.K., SHULL R.D. & BECK P.A.,J. Less-Common Metals 43,69 (1975). BRAIJN M. & KOHLHAAS R.,Phys. Status Solidi 12,429 (1965); see Fig. 3(c).
4. 5. 6
MOOK H.A. & SHULL C.G.,J. AppI. Phys. 37, 1034 (1966); MOOK H.A.,Phys. Rev. 148,495 (l966)~and SHULL C.G., in Magnetic and Inelastic Scattering of Neutrons by Metals, Met. Soc. Conf (Edited by ROWLAND T.J. & BECK P.A.) Vol.43, p. 15. Gordon & Breach, NY (1968). RHODES P. & WOHLFARTH E.P., Proc. R. Soc. A273, 247 (1963).
130
SUSCEPTIBILITY OF PARAMAGNETIC NICKEL
7. 8.
PIERCE D.T. & SPICER W.E.,Phys. Rev. B6, 1787 (1972). ROWEJ.E.&TRACYJ.C.,Phys. Rev. Lett. 27, 799 (1971).
9.
ERSKINE J.L. & STERN EA.,Phys. Rev. Lett. 30, 1329 (1973).
10.
DIXON M., HOARE F.E., HOLDEN T.M. & MOODY D.E.,Proc. R. Soc. A285, 561 (1965).
11.
SEITCHIK J.A., GOSSARD A.C. & JACCARINO V., Phys. Rev. 136, Al 119 (1964).
12.
CLOGSTON A.M., JACCARINO V. & YAFET Y.,Phys. Rev. 134, 650 (1964).
13.
PUGHE.W. & RYAN F.M.,Phys. Rev. 111, 1038 (1958).
14.
MIZUTANI U., NOGUCHI S. & MASSALSKI T.B., Phys. Rev. B5, 2057 (1972).
15.
KEESOM W.H. & KURRELMEYER B., Physica 7, 1003 (1940).
Vol. 18, No. 1
Pergamon Press.
Printed in Great Britain
INTERPRETATION OF THE SUSCEPTIBILITY OF PARAMAGNETIC NICKEL* P.A. Beck and C.P. Flynn Materials Research Laboratory, University of Illinois, Urbana, IL 61801, U.S.A. (Received 14 July 1975 by A.G. Chynoweth)
The susceptibility of nickel from 1100 to 1 500°Kfits well x Xo + C/(T 0), whereproposed the Curieallowing constantXocorresponds to a nickel A model.is to be interpreted as an moment enof O.7p~. hanced specific heat susceptibility. —
THE STONER-WOHLFARTH band model and the Friedel—Anderson-type local moment description of ferromagnetism differ from each other in regard to the entropy change they predict for the transition from ferromagnetism to paramagnetism. It is well known1 that for b.c.c. iron the predicted difference is large and that the experimental entropy value clearly favors the local moment model. However, for nickel the predicted difference is much smaller and no clear-cut answer results from comparison with the experimental value.1 It is, therefore of considerable interest to examine other relevant evidence.
temperature increases. On the other hand, the deviations observed in the upper half of the temperature range of the data2 suggest the increasing relative importance of a temperature-independant susceptibility component, as the Curie—Weiss component decreases with increasing temperature. In the present work, an attempt was made to represent the temperature variation of the susceptibility in the upper temperature range by x = xo+CI(T—0). (1) The parameter values obtained by least-squares fitting of Xo, C, and 0 to the experimental data2 for various values of the lower limit of the temperature range used, are shown in Table 1. It is seen that the fit becomes excellent (Fig. 1) if the susceptibility data below 1 100°Kare not used; the latter include a measurable magnetic short-range order contribution. Peff was calculated from the Curie constant, and p = gS was calculated from Peff by assuming g = 2.
The excellent susceptibility data for polyciystalline nickel metal obtained by Arajs and Colvin2 between the Curie temperature of 631 and 1512°K show distinct deviations from the Curie—Weiss law. Just above the Curie temperature, the deviations appear to be similar to those resulting in various alloys from the presence of giant moments and the gradual decrease of the latter with increasing temperature.3 Although magnetic clusters in the usual sense are not expected to occur in unalloyed nickel, it is well known that magnetic contributions to the specific heat extend well above the Curie temperature, to at least 1000°K,4and that these magnetic specific heat contributions can be ascribed to the gradual breaking up of short range magnetic order as the
Neutron magnetic scattering results on Ni in the ferromagnetic state5 indicate the presence of a nonlocalized negative polarization amounting to O.lI-LB per Ni atom. It is not clear what role should be assigned to this negative polarization in a localized moment interpretation of the Curie—Weiss term of the susceptibility. At any rate, the Ni atomic moment of O.7p~,calculated from the Curie constant (Table 1), agrees quite well with the Ni moment of °•61PB in
____________
*
This work was supported by grants from the U.S. Energy Research and Development Administration and the National Science Foundation. 127
128
SUSCEPTIBILITY OF PARAMAGNETIC NICKEL
Vol. 18, No. 1
2 The increased root mean square Table 1. Curie— Weiss analysis of x vsranges T for nickel, databelow from1102°Karise Arajs and Colvin deviations (RMSD) for temperature extending from the increasing short-range magnetic order at those temperatures (Fig. 1) Parameters obtained by Temperature range from least-squares fitting 1512°Kto: C 0 Xo RMSD Peff P (emu deg/mole) (°K) (106emu/mole) (lO8emu/g) (PB) (PB) 741°K 919°K 1013°K 1102°K
0.304 0.261 0.248 0.238
658 687 700 712
63.4 110 120 129
20.8 2.75
1.70 1.24
1.56 1.44 1.41 1.38
0.85
0.75 0.73 0.70
in fact, equally consistent with the localized model. Neither magnetooptic nor Fermi surface results at low temperatures are modified in principle by the distinction between a band ferromagnet and an assembly of
I.
aligned Friedel— Anderson-type local moments; merely represent alternative descriptions of the these same
E 0
2
physical system.
0 0
The temperature-independent susceptibility cornponent of Xo = 129 (lO6emu/mole), derived in the present analysis (Table 1), is similar in magnitude to Xs.h. = 98 (l06emu/mole), the Pauli susceptibility
5
/ 800
1000 2~ T (°K)
calculated from the low temperature electronic specific heat coefficient for ferromagnetic nickel, y = 7.1 (mJ deg2mole’).’°The excess ofy 0 Xs.h. = 31(10—6 emu/mole) may be attributed to the fact that many-body enhancement for the suscepti—
FIG. 1. I/(x Xo) vs temperature for Ni. Data points 0 from reference 2. The values of C = 0.238 (emudeg/mole), Xo = 129 (lO_6 emu/mole), and of the Weiss temperature 0 = 712°K,corresponding to the continuous line, we obtained by least-squares analysis of the data in the temperature range of 1102— 1512°K(Table 1). —
ferromagnetic nickel. The simplest interpretation of this agreement, in accordance with Rhodes and Wohlfarth,6 is that localized moments are present in paramagnetic nickel even though, on the basis of older, less accurate susceptibility data, those authors arrived at the alternative conclusion that nickel is a band ferromagnet. Our interpretation is supported by the fact that photoelectron energy distribution studies7’8 above and below the Curie temperature failed to show the difference expected on the basis of the Stoner— Wohlfarth band ferromagnetism, while these results are entirely consistent with the localized model for ferromagnetism in nickel. Magnetooptic results,9 originally presented in support of the band theory, are,
bility is usually larger than that for the electronic specific heat, or, alternatively, to the contribution from a Van Vieck susceptibility term. It agrees well with Xvv estimated2 by Jaccarino for palladium” In either case,eta!. the temperatureand for platinum.’ independent value of Xo = 129 (10-6 emu/mole), corresponding to 2.3 (108pB/Oe) per nickel atom, strongly suggests its interpretation as a band susceptibility. The question arises as to how a band susceptibility component can be reconciled with a localized moment model for a paramagnet. At any one instant, some of the Ni moments will be parallel and the others will be antiparallel with respect to an applied field. For the former the parallel spin d-subband, and for the latter the antiparallel spin d-subband, may be considered completely filled, so that the Pauli susceptibility is zero. A field-induced magnetization does
Vol. 18, No. 1 2.0
SUSCEPTIBILITY OF PARAMAGNETIC NICKEL
related in the usual way to the electronic specific heat, but enhanced by a factor [1 + (U J V)N(EF)]~ Here, U and J have their customary meaning as atomic Coulomb and exchange integrals, and V/2 is the energy release when conduction states screen an added local d orbital. The Xo value obtained in the
4.0
—
129
—
/ ~
1.5 s.h.” /
0 0
~2 1.0
I
II
,)~‘
0
~
0
3.0
/
b x
—
-
explained by thisofmechanism, with andata enhancement present analysis the experimental is thus factor of 1.3.
0.5
0
0.1 0.2 Ni (at. fraction)
The temperature-independent susceptibility per Ni atom obtained above is close to the susceptibility of Ni in paramagnetic Cu-rich Cu—Ni alloys. Graph
2.5 2.0 0.3
in Fig. 2 gives the temperature-independent
FIG. 2. Temperature-independent susceptibility contribution per Ni atom in Cu—Ni alloys (graph X) vs Ni concentration; data 0 from reference 13. Electronic specific heat contribution Ni atom in Cu—Ni alloys (expressed in terms per of susceptibility, graph s.h.) vs Ni concentration; data from reference 15.
0
from reference 14 and £~
become possible if we acknowledge that electrons can be transferred locally between the spin-polarized d orbitals near EF and the essentially unpolarized s-like conduction band. The system finds its lowest energy in a field H when Ni moments aligned parallel to Hundergo a d -÷ s orbital transfer involving minority d spins, while oppositely aligned moments give the reverse transfer. At sufficiently high temperatures,the two spin orientations of the local moments are equally populated. Assuming that charge screening is complete within each atomic cell, so that the orbital transfer introduces no added Ni—Ni interactions, one finds that this mechanism does indeed introduce a temperature-independent susceptibility,
susceptibility per Ni atom, corrected for the diamagnetism of the copper host, for alloys up to 27%Ni. It shows that Ni atoms in dilute Cu—Ni alloys contribute athe susceptibility 2.1 (l08p~/Oe),as compared with value of 2.3 of (108p 8/Oe) derived above for paramagnetic nickel. Ifthe enhancement factors were also nearly the same, one might well suppose that a nonmagnetic Ni solute atom in copper contributes a density of states at the Fermi surface very similar to that which a Ni atom carrying a moment contributes to the Fermi surface of nickel above the Curie ternperature. However, the electronic specific heat contribution per Ni atom in dilute Cu—Ni alloys, also shown in Fig. 2 in terms of the corresponding susceptibility, is much smaller and the enhancement, = 4.7, is much larger than that for nickel. Acknowledgements We wish to thank Prof. S. Arajs for making available to us his susceptibility data in numerical form and F.M. Mueller for reading an early version of the manuscript. The assistance of E.E. Barton in the least-squares fitting of the susceptibility data for nickel is gratefully acknowledged. —
REFERENCES 1.
MOTTN.F.,Adv.Phys. 13, 325 (1964).
2.
AR.AJS S. & COLVIN R.V.,J. Phys. Chem. Solids 24, 1233 (1963).
3.
See for instance: MUKHOPADHYAY A.K., SHULL R.D. & BECK P.A.,J. Less-Common Metals 43,69 (1975). BRAIJN M. & KOHLHAAS R.,Phys. Status Solidi 12,429 (1965); see Fig. 3(c).
4. 5. 6
MOOK H.A. & SHULL C.G.,J. AppI. Phys. 37, 1034 (1966); MOOK H.A.,Phys. Rev. 148,495 (l966)~and SHULL C.G., in Magnetic and Inelastic Scattering of Neutrons by Metals, Met. Soc. Conf (Edited by ROWLAND T.J. & BECK P.A.) Vol.43, p. 15. Gordon & Breach, NY (1968). RHODES P. & WOHLFARTH E.P., Proc. R. Soc. A273, 247 (1963).
130
SUSCEPTIBILITY OF PARAMAGNETIC NICKEL
7. 8.
PIERCE D.T. & SPICER W.E.,Phys. Rev. B6, 1787 (1972). ROWEJ.E.&TRACYJ.C.,Phys. Rev. Lett. 27, 799 (1971).
9.
ERSKINE J.L. & STERN EA.,Phys. Rev. Lett. 30, 1329 (1973).
10.
DIXON M., HOARE F.E., HOLDEN T.M. & MOODY D.E.,Proc. R. Soc. A285, 561 (1965).
11.
SEITCHIK J.A., GOSSARD A.C. & JACCARINO V., Phys. Rev. 136, Al 119 (1964).
12.
CLOGSTON A.M., JACCARINO V. & YAFET Y.,Phys. Rev. 134, 650 (1964).
13.
PUGHE.W. & RYAN F.M.,Phys. Rev. 111, 1038 (1958).
14.
MIZUTANI U., NOGUCHI S. & MASSALSKI T.B., Phys. Rev. B5, 2057 (1972).
15.
KEESOM W.H. & KURRELMEYER B., Physica 7, 1003 (1940).
Vol. 18, No. 1