Coercivity and nucleation fields for three-dimensional inhomogenities are calculated and discussed. For soft spherical inclusions in a hard magnetic matrix a (δ /R) 2 dependence of the nucleation field on inclusion radius R and domain wall width δ has been obtained. To calculate propagation fields for spherical inhomogenities, a nonlinear model has been used. The corresponding coercive field can be much higher than the nucleation field, particularly if the magnitude of the anisotropy changes very steeply. Soft inclusions with sharp interfaces are ineffective at producing complete magnetic reversal. For partly nitrided particles coercivity and nucleation field are predicted to be identical, which is confirmed by experiments on high-performance Sm2Fe17N3- δ  magnets

Coey, J.M.D.

Müller, K.-H.

Skomski, Ralph

Wendhausen, P.A.P.

English

DigitalCommons@University of Nebraska

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Ralph Skomski Publications Research Papers in Physics and Astronomy May 1993 Magnetic reversal in Sm2Fe17Ny permanent magnets Ralph Skomski University of Nebraska-Lincoln, rskomski2@unl.edu K.-H. Müller IfW Dresden P.A.P. Wendhausen IfW Dresden J.M.D. Coey Trinity College Dublin, Ireland Follow this and additional works at: https://digitalcommons.unl.edu/physicsskomski  Part of the Physics Commons Skomski, Ralph; Müller, K.-H.; Wendhausen, P.A.P.; and Coey, J.M.D., "Magnetic reversal in Sm2Fe17Ny permanent magnets" (1993). Ralph Skomski Publications. 36. https://digitalcommons.unl.edu/physicsskomski/36 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Ralph Skomski Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Magnetic reversal in SmSFelfMy permanent magnets R. Skomski Phy.sics Dqartmertt. Trinity College Dublin, Dubiirz 2, Ireland K.-H. Miiller and P. A. P. Wendhausen @ W  Dresden, PF 013, O-3027 Dresden Germanic J. M. D. Coey Ph.wics Department, Trinity College Dublin, Dublin 2, Ireland Goercivity and nucleation fields for three-dimensional inhomogenities are calculated and discussed. For soft spherical inclusions in a hard magnetic matrix a (S/R)’ dependence of the nucleation field on inclusion radius R and domain wall width S has been obtained. To calculate propagation fields for spherical inhomogenities, a nonlinear model has been used. The corresponding coercive field can be much higher than the nucleation field, particularly if the magnitude of the anisotropy changes very steeply. Soft inclusions with sharp interfaces are ineffective at producing complete magnetic reversal. For partly nitrided particles coercivity and which is confirmed by experiments on nucleation field are predicted to be identical, high-performance Sm2Fet,N~-s magnets. 1. INTRODUCTION Interstitial introduction of nitrogen into Sm2Fe1, greatly improves the permanent-magnetic properties of the material. Filling the 9~ octahedral interstices causes a lat- tice expansion of about 6% and leads to a Clurie tempera- ture increase of 350 K. Furthermore, the anisotropy due to the crystal field of the nitrogen atoms changes from mod- erate easy plane to strong easy axis.‘jz To obtain magnets with appreciable energy product, the low coereivity of the nitrided powder (jr,#l;!~=O. 1 T) must be improved. A pos- sible way is Zn bonding,” which at present allows the pro- duction of aligned magnets with coercivities up to about 19 T* . . In any case, the observed coercivities are much smaller than expected from the Sm,FeL7N3-,s anisotropy field B, ~22 T.” To explain this behavior, which applies generally to hard magnetic materials, deviations from the ideal crys- tal structure, such as impurities and surface or grain- boundary phases, have to be taken into account.“’ Theo- retical treatment of the coercivity problem is complicated, because it leads to nonlinear micromagnetic equations. There are estimations that predict a dependence propor- tional to S/R of the coercivity on the domain wall width 6 and on the radius R of the inhon~ogenity,7-‘” but the de- tailed coercivity mechanisms remain uneIear.7*8 In Ref. 9, the linearized micromagnetic equations have been used to calculate nucleation fields in one and two dimensions. This approach, which has recently begun to be extended to three dimensions, * t-r3 allows the determination of magnetic properties directly from the microstructure. The question rernsins, however, whether the reversed nucleus can prop- agate and lead to reversal of the whole magnet [cf., e.g., Ref. 7). Here we calculate nucleation fields for three- dimensional inhomogenities in aligned magnets and then use a nonlinear model to determine the corresponding propagation tieids. The theoretical predictions are com- psmd with experimental data on coercivity in homoge- neously and inhomogeneously nitrided Zn bonded Sm2Fe17N3-~ magnets. II. THE LINEARIZED THEORY To describe the magnetic reversal we use the free en- ergy 3 F= c A (VMj)’ T---H&f..-K *f. I 1 I j-1 (1) where Hi(r) denotes the local magnetic field, Miir) with Ma- I&Ii(r) 1 the magnetization, K,(r) the second-order anisotropy constant, and ni(r) the unit vector of the local crystallographic c direction. The exchange parameter A is assumed to be spatially constant (cf., e.g., Ref. 9).14 Hi(r) inc.ludes magnetostatic effects in the form of a macroscopic demagnetization factor. A magnetic configuration &fj( r) may be realized phys- ically if it corresponds to a Iocal minimum of the free energy, Eq. (1). The nucleation field H, is the field at which the free-energy minimum of the initial state disap- pears. At this field the initial configuration &i,(r) becomes unstable, but the state that is reached next. may be sepa- rated from the final configuration by other free-energy bar- riers so that complete reversal does not occur. Eventually, at H&H,, all such barriers are removed, and the magnet is aligned in the -c direction. In the linearized theory, n,(l, ?zv( 1, M,(M,, and I%‘,,+%Y~,, and minimization of the free energy Eq. (1) yields (W Bi%fY=IYt (r)iz,(r)&+~H$#~~ with the operator WHJ = --AV’fk’,(r) +$&$&. 12b) (3) 6047 J. Appi. Phys. 73 (loj, 15 May 1993 0021-6979/931106047-03$06.00 0 1993 American institute of Physics 6047 Downloaded 23 Mar 2007 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspFIG. 1. Spherical soft inclusion in a hard-magnetic matrix. An anisotropy profile K,(r) that can be treated exac.tly is the soft spherical inclusion in a hard-magnetic matrix (Fig. 1) with the anisotropy constants Kr ( r > R ) = KI, and KI ( P< R) =K,<Kh of the hard and soft phases, respec- tively.t3 The nucleat.ion field is given by the lowest eigen- value of B, and using the analogy between B and the Hamiltonian of the Schrodinger equation we obtain 2 J&=2 I+$ , i 1 (4) with S=T(A/‘K,)” being the Bloch wall width” of the soft phase. The quadratic term (S/R )’ indicates a strong decrease of the nucleation field, compared to a S/R depen- dence. The nucleus can actually propagate in the hard- magnetic phase if the lowest-lying eigenstate Y,(r) of B is delocalized. Unfortunately, Eq. (4) corresponds to a local- ized state, and merely describes the switching of the soft inclusion.13 Ill. THE NONLINEAR CASE To obtain a better description of the magnetic reversal, nonlinear effects have to be taken into account. With Ikf~(r)=MO[sin O(r)cosQ?(r)exi+sinO(r)sincP(r)ey~ fCOS @(r)f,-] (5) and rti=eZi we obtain for Hi=H-zzi F= s [LI(VO)~+A sin”@(V@)‘--H&f0 xcos O-KKlcos”O+-]3dr. (6) In the following we restrict ourselves to spherical inhomo- geneities, so minimization with respect to Q, yields @p(r) =const. O(r) can be expanded in spherical harmonics rsj O(r)= c g&9 r;(b,e). (7) m,,z=o If we assume that the nucleus has the same symmetry as the inhomogenity (see Fig. 2), only t.he spherical G term survives, and Eq. (7) becomes F= -I~fif~cos 0 + K,sin”O +K2sin% (8) FIG. 2. Nucleus with spherical symmetu. where the fourth-order anisotropy constant Kz has been included. *’ To minimize the free energy, Eq. (S), we use a trial function O(r) with a linear radial dependence between O(Ro-b)=O and @(R,,+~)=T. Figure 2 shows that S = 26 is the thickness of a domain wall, which actually represents a cross between Bloch and Niel wall. To inves- tigate the propagation of the nucleus, we assume b(Ro and a sufficiently smooth dependence of K, and K, on r, so we finally obtain the propagat.ion criterion H> [4K,Cr) +3&W+ 6 dKl(r) 3 d&(r) 4Mg 4Me ---z-+z(Jp i9) with fiCr)=V\4K,(,)Y3K2(r) * i To illustrate the result Eq. (9) we use the profile r-R0 Ki=Ksi$- hk’i tanh 7 Z (11) where R and 2B can be interpreted as radius and interface thickness, respectively, of a spherical soft inclusion (index s) in a hard matrix with the anisotropy constants Kr -KTf +AK, and K,=Ksz+hKz. In the limit SgB-gR, Ey. (9) yields the coercivity (AK, ++AK,). Equation ( 12) indicates that the coercivity for large inho- mogenities no longer scales as 6/R, which would give ex- tremely small coercivities, but is determined by the “inter- face quality” S/B (cf., e.g., Refs. 7 and 8). The sharper the interface, the larger the coercivity. IV. COMPARlSON WITH EXPERIMENT Sm2Fe17NjV6 provides an excellent basis for the study of magnetic reversal processes, because the local nitrogen 6048 J. Appl. Phys., Vol. 73, No. 10, 15 May 1993 Skomski et al. 6048 Downloaded 23 Mar 2007 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspI... 0.5 1 .o Xl 0’ PARTICLE RADWS (pm) FIG. 3. Experimental (dashed line) and theoretical (solid line) results for the dependence of the coercivity on the radius of the grains at fixed nitrogen&ion conditions (5 h at 450 “C). content, and hence the magnetic properties of the magnet, can be continuously varied by controlling temperature and pressure during the gas-solid reaction. I7 For pure Sm,Fei,, K1 is negative (easy plane), but as nitrogen enters the 9e sites it creates a strong electric field gradient at the Sm 4f shell. The crystal-field coefficients are modified and the anisotropy constants, in particular K,, increase. At typical nitrogenation conditions (500 “C and 1 bar pressure), the anisotropy at the center of spherical particles changes from easy axis to easy plane at the nom- inal composit.ion Sm2Fe1,N,,,.i7 Here we will use this “soft-core phenomenon” to check the validity of Eq. (9), which can be shown to predict that all grains with less than 2.1 nitrogen atoms per formula unit are soft. To avoid interference with surface nucleation effects, which are probably responsible for the low coercivity in untreated and polymer bonded magnets,3 high-coercive Zn-bonded magnets have been used. The experimental re- sults are shown in Fig. 3 (dashed line). After sieving the powder fractions, nitrogenation, Zn bonding, and heat treatment coercivity values up to 1.9 T could be achieved, depending on the particle radius. For 5 h nitrogenation time at 450 “C the calculation predicts a sharp increase of J$ at Rzz7.5 pm (Fig. 3). The smearing out of the exper- imental curve (dashed line) is due to the limited quality of the powder fraction. The plateau value H,- 1.9 T, which cannot be predicted from the core properties, indicates fur- ther, unknown nucleation mechanisms. V. CONCLUSIONS The linearized micromagnetic equations can be used to calculate nucleation fields for three-dimensional inhomo- geneities. The lowest eigenvaluc of the linearized problem yields the nucleation field; but only if the corresponding eigenfunction is delocalized can the nucleation field be in- terpreted as the coercivity field. LocaIized eigenfunctions describe magnetization processes that cannot propagate in the bulk magnet. To obtain more realistic coercivity values, nonlinear equations must be solved. For spherical inhomogeneities a criterion is obtained that predicts, for instance, immediate soft-core propagation in partly nitrided Sm,Fe,,N, mag- nets. The corresponding lowering can be observed experi- mentally and underlines the importance of powder frac- tionation on properties and processing of the 2-17 nitrides, ACKNOWLEDGMENTS This work was supported by the RRlTEiEURAM Programme of the European Commission. It forms part of the Concerted European Action on Magnets. ‘J. M. D. Coey and H. Sun, J. Magn. Magn. Mat. 87, L251 (1990). ‘H. Sun, J. M. D. Coey, Y. Otani, and D. P. F. Hurley, J. Phys.: Con- dens. Matter 2, 6465 ( 1990). ‘Y. Otani, A. Moukarika, H. Sun, J. M. D. Coey, and 1. R. Harris, J. Appl. Phys. 69, 6735 ( 1991). ‘P. A. P. Wendhausen, A. Handstein, P. Nothnagel, D. Eckert, and K.-H. Mtiller, Phys. Status Solidi A 127, Kl21 (1991). ‘K.-H. Miiller, P. A. P. Wendhausen, D. Eckert, and A. Handstein, in Proceedings of the 7th International Symposium on Magnetic Ankotmpy and Coercivity in RE-TM Al1ay.s 1992, edited by L. Folks et ul. (The University of Western Australia, 1992), p. 34. bM. Katter, J. Wecker, C. Kuhrt, L. Schultz, and R. Grossinger, J. Magn. Msg. Mat. 117, 419 ( 1992). ‘D. Givord, Q. Lu, iind M. F. Rossignol, in Science and Technology oj Nanostructured Magnetic Materials, edited by G. C. Hadjipanayis and G. A. Prinz (Plenum, New York, 1991), p. 635. sJ. Fidler, in Magnetic Materials and Their Applications, edited by F. Leccabue and J. L. Sanchez Llamazares (Institute of Physics, Bristol, 1992), p. 55. ‘H. Kronmiiller, Phys. Status Solidi B 144, 385 ( 1987). ‘“H. Kronmiiller, in Science and Technology of IVanawuctured Magnetic Materials, edited by G. C. Hadjipandyis and G. A. Prinz (Plenum, New York, 19913, p. 657. ” R. Skomski, Concerted European Action on Magnets Meeting on Co- ercivity, Schellerhau, April 1992 (unpublished); General Concerted European Action on Magnets Meeting, Dublin, July 1992 (unpub- lished) . ‘zJifan Hu and H. Kronmiiller, Phys. Status Solidi B 172, 673 (1992). 13R. Skomski, Phys. Status Solidi B 174, K77 (1992). ‘“The exchange in SmzFe,, is relatively less atfected by nitrogenation- compared to the enormous change in anisotropy. “D. J. Craik and R. S. Tebble, Rep. Prog. Phys, 24, 116 (1961). leThis equation must not be confused with the quasi-one-dimensional planar approximation, which is sometimes used to describe nonspheri- cal inhomogenities. I75 M D. Coey, R Skomski, and S. Wirth, IEEE Trans. Magn. 28, 2332 ri99;,* 6049 J. kppl. Phys., Vol. 73, No. 10, 15 May 1993 Skomski et aL 6049 Downloaded 23 Mar 2007 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

Magnetic reversal in Sm\u3csub\u3e2\u3c/sub\u3eFe\u3csub\u3e17\u3c/sub\u3eN\u3csub\u3ey\u3c/sub\u3e permanent magnets

UNL | Libraries

University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
Ralph Skomski Publications Research Papers in Physics and Astronomy 
May 1993 
Magnetic reversal in Sm2Fe17Ny permanent magnets 
Ralph Skomski 
University of Nebraska-Lincoln, rskomski2@unl.edu 
K.-H. Müller 
IfW Dresden 
P.A.P. Wendhausen 
IfW Dresden 
J.M.D. Coey 
Trinity College Dublin, Ireland 
Follow this and additional works at: https://digitalcommons.unl.edu/physicsskomski 
 Part of the Physics Commons 
Skomski, Ralph; Müller, K.-H.; Wendhausen, P.A.P.; and Coey, J.M.D., "Magnetic reversal in Sm2Fe17Ny 
permanent magnets" (1993). Ralph Skomski Publications. 36. 
https://digitalcommons.unl.edu/physicsskomski/36 
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Ralph Skomski Publications 
by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 
Magnetic reversal in SmSFelfMy permanent magnets 
R. Skomski 
Phy.sics Dqartmertt. Trinity College Dublin, Dubiirz 2, Ireland 
K.-H. Miiller and P. A. P. Wendhausen 
@ W  Dresden, PF 013, O-3027 Dresden Germanic 
J. M. D. Coey 
Ph.wics Department, Trinity College Dublin, Dublin 2, Ireland 
Goercivity and nucleation fields for three-dimensional inhomogenities are calculated and 
discussed. For soft spherical inclusions in a hard magnetic matrix a (S/R)’ dependence of the 
nucleation field on inclusion radius R and domain wall width S has been obtained. To calculate 
propagation fields for spherical inhomogenities, a nonlinear model has been used. The 
corresponding coercive field can be much higher than the nucleation field, particularly if the 
magnitude of the anisotropy changes very steeply. Soft inclusions with sharp interfaces are 
ineffective at producing complete magnetic reversal. For partly nitrided particles coercivity and 
which is confirmed by experiments on nucleation field are predicted to be identical, 
high-performance Sm2Fet,N~-s magnets. 
1. INTRODUCTION 
Interstitial introduction of nitrogen into Sm2Fe1, 
greatly improves the permanent-magnetic properties of the 
material. Filling the 9~ octahedral interstices causes a lat- 
tice expansion of about 6% and leads to a Clurie tempera- 
ture increase of 350 K. Furthermore, the anisotropy due to 
the crystal field of the nitrogen atoms changes from mod- 
erate easy plane to strong easy axis.‘jz To obtain magnets 
with appreciable energy product, the low coereivity of the 
nitrided powder (jr,#l;!~=O. 1 T) must be improved. A pos- 
sible way is Zn bonding,” which at present allows the pro- 
duction of aligned magnets with coercivities up to about 
19 T* . . 
In any case, the observed coercivities are much smaller 
than expected from the Sm,FeL7N3-,s anisotropy field B, 
~22 T.” To explain this behavior, which applies generally 
to hard magnetic materials, deviations from the ideal crys- 
tal structure, such as impurities and surface or grain- 
boundary phases, have to be taken into account.“’ Theo- 
retical treatment of the coercivity problem is complicated, 
because it leads to nonlinear micromagnetic equations. 
There are estimations that predict a dependence propor- 
tional to S/R of the coercivity on the domain wall width 6 
and on the radius R of the inhon~ogenity,7-‘” but the de- 
tailed coercivity mechanisms remain uneIear.7*8 In Ref. 9, 
the linearized micromagnetic equations have been used to 
calculate nucleation fields in one and two dimensions. This 
approach, which has recently begun to be extended to three 
dimensions, * t-r3 allows the determination of magnetic 
properties directly from the microstructure. The question 
rernsins, however, whether the reversed nucleus can prop- 
agate and lead to reversal of the whole magnet [cf., e.g., 
Ref. 7). 
Here we calculate nucleation fields for three- 
dimensional inhomogenities in aligned magnets and then 
use a nonlinear model to determine the corresponding 
propagation tieids. The theoretical predictions are com- 
psmd with experimental data on coercivity in homoge- 
neously and inhomogeneously nitrided Zn bonded 
Sm2Fe17N3-~ magnets. 
II. THE LINEARIZED THEORY 
To describe the magnetic reversal we use the free en- 
ergy 
3 
F= c 
A (VMj)’ 
T---H&f..-K *f. I 1 I j-1 
(1) 
where Hi(r) denotes the local magnetic field, Miir) with 
Ma- I&Ii(r) 1 the magnetization, K,(r) the second-order 
anisotropy constant, and ni(r) the unit vector of the local 
crystallographic c direction. The exchange parameter A is 
assumed to be spatially constant (cf., e.g., Ref. 9).14 Hi(r) 
inc.ludes magnetostatic effects in the form of a macroscopic 
demagnetization factor. 
A magnetic configuration &fj( r) may be realized phys- 
ically if it corresponds to a Iocal minimum of the free 
energy, Eq. (1). The nucleation field H, is the field at 
which the free-energy minimum of the initial state disap- 
pears. At this field the initial configuration &i,(r) becomes 
unstable, but the state that is reached next. may be sepa- 
rated from the final configuration by other free-energy bar- 
riers so that complete reversal does not occur. Eventually, 
at H&H,, all such barriers are removed, and the magnet is 
aligned in the -c direction. 
In the linearized theory, n,(l, ?zv( 1, M,(M,, and 
I%‘,,+%Y~,, and minimization of the free energy Eq. (1) 
yields 
(W 
Bi%fY=IYt (r)iz,(r)&+~H$#~~ 
with the operator 
WHJ = --AV’fk’,(r) +$&$&. 
12b) 
(3) 
6047 J. Appi. Phys. 73 (loj, 15 May 1993 0021-6979/931106047-03$06.00 0 1993 American institute of Physics 6047 
Downloaded 23 Mar 2007 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
FIG. 1. Spherical soft inclusion in a hard-magnetic matrix. 
An anisotropy profile K,(r) that can be treated exac.tly 
is the soft spherical inclusion in a hard-magnetic matrix 
(Fig. 1) with the anisotropy constants Kr ( r > R ) = KI, and 
KI ( P< R) =K,<Kh of the hard and soft phases, respec- 
tively.t3 The nucleat.ion field is given by the lowest eigen- 
value of B, and using the analogy between B and the 
Hamiltonian of the Schrodinger equation we obtain 
2 J&=2 I+$ , i 1 (4) 
with S=T(A/‘K,)” being the Bloch wall width” of the 
soft phase. The quadratic term (S/R )’ indicates a strong 
decrease of the nucleation field, compared to a S/R depen- 
dence. 
The nucleus can actually propagate in the hard- 
magnetic phase if the lowest-lying eigenstate Y,(r) of B is 
delocalized. Unfortunately, Eq. (4) corresponds to a local- 
ized state, and merely describes the switching of the soft 
inclusion.13 
Ill. THE NONLINEAR CASE 
To obtain a better description of the magnetic reversal, 
nonlinear effects have to be taken into account. With 
Ikf~(r)=MO[sin O(r)cosQ?(r)exi+sinO(r)sincP(r)ey~ 
fCOS @(r)f,-] (5) 
and rti=eZi we obtain for Hi=H-zzi 
F= 
s 
[LI(VO)~+A sin”@(V@)‘--H&f0 
xcos O-KKlcos”O+-]3dr. (6) 
In the following we restrict ourselves to spherical inhomo- 
geneities, so minimization with respect to Q, yields @p(r) 
=const. O(r) can be expanded in spherical harmonics 
rsj 
O(r)= c g&9 r;(b,e). (7) 
m,,z=o 
If we assume that the nucleus has the same symmetry as 
the inhomogenity (see Fig. 2), only t.he spherical G term 
survives, and Eq. (7) becomes 
F= -I~fif~cos 0 + K,sin”O +K2sin% 
(8) 
FIG. 2. Nucleus with spherical symmetu. 
where the fourth-order anisotropy constant Kz has been 
included. *’ 
To minimize the free energy, Eq. (S), we use a trial 
function O(r) with a linear radial dependence between 
O(Ro-b)=O and @(R,,+~)=T. Figure 2 shows that 
S = 26 is the thickness of a domain wall, which actually 
represents a cross between Bloch and Niel wall. To inves- 
tigate the propagation of the nucleus, we assume b(Ro and 
a sufficiently smooth dependence of K, and K, on r, so we 
finally obtain the propagat.ion criterion 
H> [4K,Cr) +3&W+ 6 dKl(r) 3 d&(r) 
4Mg 4Me ---z-+z(Jp 
i9) 
with 
fiCr)=V\4K,(,)Y3K2(r) * i 
To illustrate the result Eq. (9) we use the profile 
r-R0 
Ki=Ksi$- hk’i tanh 7 Z (11) 
where R and 2B can be interpreted as radius and interface 
thickness, respectively, of a spherical soft inclusion (index 
s) in a hard matrix with the anisotropy constants Kr -KTf 
+AK, and K,=Ksz+hKz. In the limit SgB-gR, Ey. (9) 
yields the coercivity 
(AK, ++AK,). 
Equation ( 12) indicates that the coercivity for large inho- 
mogenities no longer scales as 6/R, which would give ex- 
tremely small coercivities, but is determined by the “inter- 
face quality” S/B (cf., e.g., Refs. 7 and 8). The sharper the 
interface, the larger the coercivity. 
IV. COMPARlSON WITH EXPERIMENT 
Sm2Fe17NjV6 provides an excellent basis for the study 
of magnetic reversal processes, because the local nitrogen 
6048 J. Appl. Phys., Vol. 73, No. 10, 15 May 1993 Skomski et al. 6048 
Downloaded 23 Mar 2007 to 129.93.16.206. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
I... 
0.5 1 .o Xl 0’ 
PARTICLE RADWS (pm) 
FIG. 3. Experimental (dashed line) and theoretical (solid line) results 
for the dependence of the coercivity on the radius of the grains at fixed 
nitrogen&ion conditions (5 h at 450 “C). 
content, and hence the magnetic properties of the magnet, 
can be continuously varied by controlling temperature and 
pressure during the gas-solid reaction. I7 
For pure Sm,Fei,, K1 is negative (easy plane), but as 
nitrogen enters the 9e sites it creates a strong electric field 
gradient at the Sm 4f shell. The crystal-field coefficients 
are modified and the anisotropy constants, in particular K,, 
increase. At typical nitrogenation conditions (500 “C and 1 
bar pressure), the anisotropy at the center of spherical 
particles changes from easy axis to easy plane at the nom- 
inal composit.ion Sm2Fe1,N,,,.i7 Here we will use this 
“soft-core phenomenon” to check the validity of Eq. (9), 
which can be shown to predict that all grains with less than 
2.1 nitrogen atoms per formula unit are soft. 
To avoid interference with surface nucleation effects, 
which are probably responsible for the low coercivity in 
untreated and polymer bonded magnets,3 high-coercive 
Zn-bonded magnets have been used. The experimental re- 
sults are shown in Fig. 3 (dashed line). After sieving the 
powder fractions, nitrogenation, Zn bonding, and heat 
treatment coercivity values up to 1.9 T could be achieved, 
depending on the particle radius. For 5 h nitrogenation 
time at 450 “C the calculation predicts a sharp increase of 
J$ at Rzz7.5 pm (Fig. 3). The smearing out of the exper- 
imental curve (dashed line) is due to the limited quality of 
the powder fraction. The plateau value H,- 1.9 T, which 
cannot be predicted from the core properties, indicates fur- 
ther, unknown nucleation mechanisms. 
V. CONCLUSIONS 
The linearized micromagnetic equations can be used to 
calculate nucleation fields for three-dimensional inhomo- 
geneities. The lowest eigenvaluc of the linearized problem 
yields the nucleation field; but only if the corresponding 
eigenfunction is delocalized can the nucleation field be in- 
terpreted as the coercivity field. LocaIized eigenfunctions 
describe magnetization processes that cannot propagate in 
the bulk magnet. 
To obtain more realistic coercivity values, nonlinear 
equations must be solved. For spherical inhomogeneities a 
criterion is obtained that predicts, for instance, immediate 
soft-core propagation in partly nitrided Sm,Fe,,N, mag- 
nets. The corresponding lowering can be observed experi- 
mentally and underlines the importance of powder frac- 
tionation on properties and processing of the 2-17 nitrides, 
ACKNOWLEDGMENTS 
This work was supported by the RRlTEiEURAM 
Programme of the European Commission. It forms part of 
the Concerted European Action on Magnets. 
‘J. M. D. Coey and H. Sun, J. Magn. Magn. Mat. 87, L251 (1990). 
‘H. Sun, J. M. D. Coey, Y. Otani, and D. P. F. Hurley, J. Phys.: Con- 
dens. Matter 2, 6465 ( 1990). 
‘Y. Otani, A. Moukarika, H. Sun, J. M. D. Coey, and 1. R. Harris, J. 
Appl. Phys. 69, 6735 ( 1991). 
‘P. A. P. Wendhausen, A. Handstein, P. Nothnagel, D. Eckert, and 
K.-H. Mtiller, Phys. Status Solidi A 127, Kl21 (1991). 
‘K.-H. Miiller, P. A. P. Wendhausen, D. Eckert, and A. Handstein, in 
Proceedings of the 7th International Symposium on Magnetic Ankotmpy 
and Coercivity in RE-TM Al1ay.s 1992, edited by L. Folks et ul. (The 
University of Western Australia, 1992), p. 34. 
bM. Katter, J. Wecker, C. Kuhrt, L. Schultz, and R. Grossinger, J. 
Magn. Msg. Mat. 117, 419 ( 1992). 
‘D. Givord, Q. Lu, iind M. F. Rossignol, in Science and Technology oj 
Nanostructured Magnetic Materials, edited by G. C. Hadjipanayis and 
G. A. Prinz (Plenum, New York, 1991), p. 635. 
sJ. Fidler, in Magnetic Materials and Their Applications, edited by F. 
Leccabue and J. L. Sanchez Llamazares (Institute of Physics, Bristol, 
1992), p. 55. 
‘H. Kronmiiller, Phys. Status Solidi B 144, 385 ( 1987). 
‘“H. Kronmiiller, in Science and Technology of IVanawuctured Magnetic 
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Magnetic reversal in Sm\u3csub\u3e2\u3c/sub\u3eFe\u3csub\u3e17\u3c/sub\u3eN\u3csub\u3ey\u3c/sub\u3e permanent magnets

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