- Email: [email protected]
Minimum-energy configurations of isotropic ferromagnetic shells
Journal of Magnetism and Magnetic Materials 118 (1993)47-51 North-Holland J
Minimum-energy configurations of isotropic ferromagnetic shells J.I. K a t z a a n d D . R . N e l s o n b Department of Physics and McDonnell Centerfor the Space Sciences, Washington University, St. Louis, MO 63130, USA b Department of Physics, Harvard Univ., Cambridge, MA 02138, USA a
Received 18 December 1991; in revised form 23 June 1992
We calculate the exchange and magnetic ('dipolar') energies in various configurations of magnetization of spherical shells of ferromagnetic material with no intrinsic anisotropy. In general, the state of lowest energy is magnetized along lines of constant latitude. We compare this to states in which the magnetization follows lines of constant longitude or is radial, which have higher total energies. A continuum of intermediate states exists, so that in the absence of anisotropy, external fields, or pinning of magnetization, all states may spontaneously decay to latitudinally magnetized states. Although the latitudinal state has no magnetic moment, it may have significant permeability.
1. I n t r o d u c t i o n
Most ferromagnetic materials are crystalline, and have preferential directions of magnetization. Even if the crystal symmetry is cubic there are generally anisotropy energies which are fourth order in the direction cosines of the magnetization vector with respect to the crystal axes. This anisotropy energy, in combination with the magnetic energy (the magnetic interaction between elementary dipoles, called the dipolar energy), determines the distribution of directions of magnetization within a sample of ferromagnetic material [1,2]. The anisotropy energy usually restricts the magnetization to a few preferred directions, except for narrow domain walls whose width and structure are determined by the competition between anisotropy and exchange energy. Amorphous ferromagnets, and crystalline ferromagnets with very small (compared to a domain size) isotropically oriented microcrystals have no bulk anisotropy energy. Their distribution of magnetization is determined by the exchange and
dipolar energies alone. In this note we consider some possible distributions of magnetization within spherical shells (and solid spheres) made of a hypothetical completely isotropic ferromagnet. We assume that the surface introduces no anisotropy, though (as we discuss later) this may be unrealistic, even for amorphous ferromagnets.
2. M a g n e t i c
structures of spherical shells
We describe three possible configurations of magnetization, estimate their energies, and comment on their significance for the effective permeability of a bulk medium composed of shells embedded in a nonmagnetic matrix. In each case the shell has outer radius R and thickness d. The exchange energy is given in the usual classical Heisenberg model as ~I=-JESi'Sj= (ij)
J ~ E (s,- s3 2 (ij)
J
E(S2+S~), 2 (ij)
Correspondence to: Dr. J.I. Katz, Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA.
(1)
where the $i are the individual spins and the sum runs :over all pairs of nearest neighbors i, j,
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.L Katz, D.R. Nelson / Minimum-energy configuration of isotropic ferromagnetic shells
48
counting each distinct pair once. Discarding the final sum (which is constant because all S i are constant), and writing (Si - S ) 2 = S g I Vthl2a 2, where ~ is a unit vector in the direction of magnetization and a is a lattice constant, we have
Jsg f d3x
H = -7" ~ ~
I Vffz12a2,
(2)
where the coordination number and number of spins per unit cell have been absorbed into the definition of J. The magnitude of the magnetization is always M. The magnetic energy may be obtained from the magnetostatic interactions of elementary dipoles. If V x H -- 0 (static configurations without true current) it is [2]
latitude on a globe. Then I Vfftl 2 = Hla
=
CSC20//r 2
4xrx l n ( g R / a ) + 2Ec,
f l n l 2 d3x.
(3)
(1) In the 'hedgehog' configuration ~ = ±~; the magnetization is radial everywhere. Straightforward calculation using eq. (2) yields I Vd~l 2 = 2 / r 2 and
4 rS2oJd Hhh
-
-
a
= 4~x,
(4)
where K-= S2jd/a. This result (as well as the later results eqs. (6) and (8)) does not require d << R, and may even be extended to solid spheres (d = R) if allowance is made for an extra term O(J) resulting from the singularity in ~ at the origin. Because this configuration is spherically symmetric and V • B = 0, B = 0 everywhere. Within the ferromagnetic material H = B - 4"rrM --- -4"a'M, but outside it and in the shell's interior (for r < R - d) H = B = 0. There are equal and opposite magnetic surface charges trM = + M on the inner and outer surfaces of the ferromagnet, and a volume magnetic charge density PM = 2 M / r (resulting from the constancy of M, differentiating this problem from its electrostatic analogue) within the material. The magnetic energy
Uhh = 8w2M2R2d ( d << R ) .
(5)
(2) In the latitudinal configuration ~ = -t-4;; lines of magnetization follow lines of constant
(6)
where the integral over 0 diverges logarithmically at the poles, and has been divided into two terms, the first with a cutoff at a and the remaining portion (at each pole) denoted E c. The value of the core energy E c cannot be calculated from this macroscopic theory, but is O(K). In this configuration V" M = 0, so V- H = 0. Because V × H = 0, H = 0 everywhere. Outside the ferromagnetic material B = H = 0, while within it B -- 4arM. On its surfaces there is poloidal magnetization current density, giving rise to the toroidal field (there is an analogous astronomical configuration involving real current [3]). The magnetic energy U~a = 0.
U = ~1
and
(7)
(3) In the longitudinal configuration ~ = + 0; lines of magnetization follow lines of constant longitude on a globe, running from one pole to the other. As in the latitudinal configuration, IV,hi 2 = C S C 2 0 / / r 2, and
Hlo = 4xrK l n ( 4 R / a ) + 2 E c.
(8)
The magnetization may be described by a magnetic charge density M PM = + - - c o t 0,
(9)
r
which is singular at the poles. Explicit calculation of the fields everywhere is possible using eq. (9). Integrating over rings of constant 0, taking d << R, the magnetic potential outside the sphere (r > R) is ®
Rl
@M( r, O) = +_2wRMdl~ r--7-~et(cos O)
× fP,(cos 0') cos 0' dO';
(10)
an analogous result applies for r < R. Terms in the sum of even I are zero, but all multipoles of odd order contribute. A fair approximation is obtained from the dipole term alone..The dipole moment for d << R is
m = :(-'tr2R2md.
(11)
ZL Katz, D.R. Nelson / Minimum-energy configuration of iaotropicferromagnetic shells
The field energy attributable to a dipole of this magnitude (including the interior energy implied by representing it by a uniformly magnetized sphere) is Ulo = m 2 / 2 R 3 = l'tr4Rm2d2;
(12)
the actual field energy is somewhat larger because of the contributions of the higher multipoles. H is continuous across the material surfaces because of the absence of free current and because there are no magnetic surface charges; hence the contribution to U of the material volume itself is small compared to eq. (12) by a factor O ( d / R ) << 1. The total energy of the longitudinal configuration is Hip + Ulo, where we adopt the dipole approximation eq. (12) for Ulo. For thin shells the magnetic energies are ordered 0 = Uh < Ulo << Uhh, while the exchange energies are ordered Hla = Hip > Hhh. The lowest-energy configuration depends on R, and is
latitudinal, if R > Rt,
(13)
with
R t=
2araM2 In
significant barrier to these decays, and that in the complete absence of anisotropy energy spheres and spherical shells (except the very smallest, for which the magnetic energy is smaller than the exchange energy) decay to the latitudinal configuration on a spin wave time scale.
3. External fields and permeability Application of an external field Bext adds a term Uext = - m "ne~t
- 1+
For So = 1, J = 0.3 eV (correspondin~ to a Curie temperature of about 1000 K), a = 2 A, and M = 103 (cgs) the right-hand sides of the inequalities (13) are = 700 A. Except for extremely small shells (a possibility we henceforth neglect) the latitudinal configuration is energetically preferred. We also consider configurations in which n~ -sin a + 4~ cos a . These form a path from longitudinal configurations (a = ~r/2) to latitudinal configurations (a = 0), with exchange energy independent of a but dipole energy proportional to sin2a, permitting the former to reduce their dipole energy by decaying to the latter. Similarly, hedgehog configurations may reduce their magnetic energy monotonically through a sequence of configurations dz = 4~ cos a + ~ sin a, with energies Hla COS2a + (Hhh + Uhh) sin2a, decaying to latitudinal configurations. It is possible that there is no
(14)
to the energetics, which will favor particles with dipole moments if these moments are free to rotate to the state of lowest energy. The critical field required to equate the latitudinal and longitudinal sphere energies is
Bo = ½ 2Ma/R.
hedgehog, if R < Rt,
49
(15)
For our previous numerical parameters, and spheres of R = 4 × 10 -3 cm, d = 10 -6 cm, we obtain B 0 = 1 G. Eq. (15) shows that the presence of the earth's field is sufficient to convert latitudinal configurations into longitudinal ones for achievable values of the parameters. If biased in this manner, the p e r m e a b i l i t y to additional small fields is anisotropic. B parallel to the biasing field Bbias does not change the magnetization so t h a t / z = 1, while fields orthogonal to Bbias rotate the total external field, and hence the magnetization, with the result that the microscopic permeability of the ferromagnetic shell /x = 1 + 4~M/Bbias.
(16)
In contrast, unbiased longitudinally magnetized particles, if they do not spontaneously decay to the latitudinal state, may have essentially infinite permeability (as is seen in the Bbias ~ 0 limit of the result for biased particles). Even a latitudinal configuration may have significant permeability. Application of an external field Bext = Bext~ tends to align the local magnetization. For values of Rd which are not extremely small the exchange energy may be neglected in
50
J.L Katz, D.R. Nelson / Minimum-energy configuration of isotropic ferromagnetic shells
comparison to the magnetic energy - r a ' B e x t + m 2 / 2 R 3. Minimization yields the resulting magnetic moment m =Bext R3.
(17)
Because only the dipole moment can reduce the total energy, while higher multipoles increase it, only a dipole moment will be induced (in the absence of exchange energy). The ~ component of magnetization will be uniform through the shell, and the angle of rotation of the local magnetization vector will be given by sin a = - B e x t R csc O/4arMd. Because B and H in a uniformly magnetized shell (as opposed to a solid sphere) are nonuniform it is not possible to define a unique local effective permeability. However, the magnetic moment, eq. (17), equals that of an infinitely permeable shell, corresponding to a large effective permeability. The field produced by m is comparable to Be~t in the vicinity of the shell. For a dilute distribution of shells with density n and nR 3 << 1 the volume-averaged permeability is Ix = 1 + 4,rrnR 3.
(18)
If magnetic shells are not very widely separated then their random interactions provide an anisotropic effective demagnetization factor, even if they are individually perfectly spherical. This turns the longitudinally magnetized state into a stable permanent magnet, with substantial ferromagnetic resonance frequency but small effective permeability. In contrast, densely packed latitudinally magnetized shells may acquire a large effective permeability, as their induced magnetization concentrates the magnetic flux into the shells.
4. Discussion
We have neglected the surface energy. Because the surface and its electronic structure are necessarily anisotropic there may be a surface exchange anisotropy energy O ( S 2 j / a 2) ~ 103 erg/cm 2 even in a material without bulk
anisotropy. The ratio of this energy to the characteristic dipolar magnetic energy S 2 j R / 2'rra2d2M 2 ~ 106 for our previous parameters and R = 4 × 1 0 -3 cm, d = 1 0 0 /~. The surface exchange anisotropy energy may exceed the bulk exchange energy by a factor O ( R 2 / a d ) > > 1. There is also a surface magnetic energy resulting from the atomic localization of the magnetization, and the consequent atomic-scale variation of the magnetic field outside the material, but this is O ( M 2 a ) ~ 10 -2 erg/cm 2, much smaller than the characteristic surface exchange energy. In order for our previous results to be valid, the surface anisotropy energy, as well as the bulk anisotropy energy, must be zero to high accuracy. If this is not the case the magnetization will be determined by the surface anisotropy. Depending on the sense of the anisotropy energy, the preferred magnetization will be either normal or parallel to the surface; the former case enforces hedgehog configurations, and the latter latitudinal or longitudinal ones (the dipole energy selecting the latitudinal state if the surface is accurately isotropic in its tangential plane). The presence of a surface anisotropy could reduce the permeability by a large factor. If these conditions on the anisotropy are met, then the lowest energy state of an isolated spherical shell of ferromaguet in no external field depends on its dimensions according to eq. (13). For all but the very smallest shells the lowest energy state is latitudinally magnetized. Other states may decay into latitudinal states through a continuous sequence of intermediate states. Because of the absence of anisotropy, the familiar domain structure is not found, and zero magnetic moment may be obtained by smoothly varying the direction of magnetization across the sphere. The hedgehog and latitudinal states are the simplest examples of this, but more complex magnetic structures are possible, at the price of increased exchange energy. These may have their direction of magnetization given by vector spherical harmonics, and correspond to multi-domain structures with broadened domain wails.
J.l. Katz, D.R. Nelson / Minimum-energy configuration of isotropic ferromagnetic shells
51
Acknowledgements
References
W e t h a n k A . E . B e r k o w i t z a n d H. Suhl for discussions a n d an a n o n y m o u s r e f e r e e for p o i n t ing o u t an i m p o r t a n t e r r o r . This w o r k was supp o r t e d by D A R P A .
[1] C. Kittel and J.K. Gait, Solid State Phys. 3 (1956) 439. [2] W.F. Brown, Jr., Magnetostatic Principles in Ferromagnetism (North-Holland, Amsterdam, 1962). [3] J.I. Katz, Mon. Not. R. Astr. Soc. 239 (1989) 751.
Minimum-energy configurations of isotropic ferromagnetic shells J.I. K a t z a a n d D . R . N e l s o n b Department of Physics and McDonnell Centerfor the Space Sciences, Washington University, St. Louis, MO 63130, USA b Department of Physics, Harvard Univ., Cambridge, MA 02138, USA a
Received 18 December 1991; in revised form 23 June 1992
We calculate the exchange and magnetic ('dipolar') energies in various configurations of magnetization of spherical shells of ferromagnetic material with no intrinsic anisotropy. In general, the state of lowest energy is magnetized along lines of constant latitude. We compare this to states in which the magnetization follows lines of constant longitude or is radial, which have higher total energies. A continuum of intermediate states exists, so that in the absence of anisotropy, external fields, or pinning of magnetization, all states may spontaneously decay to latitudinally magnetized states. Although the latitudinal state has no magnetic moment, it may have significant permeability.
1. I n t r o d u c t i o n
Most ferromagnetic materials are crystalline, and have preferential directions of magnetization. Even if the crystal symmetry is cubic there are generally anisotropy energies which are fourth order in the direction cosines of the magnetization vector with respect to the crystal axes. This anisotropy energy, in combination with the magnetic energy (the magnetic interaction between elementary dipoles, called the dipolar energy), determines the distribution of directions of magnetization within a sample of ferromagnetic material [1,2]. The anisotropy energy usually restricts the magnetization to a few preferred directions, except for narrow domain walls whose width and structure are determined by the competition between anisotropy and exchange energy. Amorphous ferromagnets, and crystalline ferromagnets with very small (compared to a domain size) isotropically oriented microcrystals have no bulk anisotropy energy. Their distribution of magnetization is determined by the exchange and
dipolar energies alone. In this note we consider some possible distributions of magnetization within spherical shells (and solid spheres) made of a hypothetical completely isotropic ferromagnet. We assume that the surface introduces no anisotropy, though (as we discuss later) this may be unrealistic, even for amorphous ferromagnets.
2. M a g n e t i c
structures of spherical shells
We describe three possible configurations of magnetization, estimate their energies, and comment on their significance for the effective permeability of a bulk medium composed of shells embedded in a nonmagnetic matrix. In each case the shell has outer radius R and thickness d. The exchange energy is given in the usual classical Heisenberg model as ~I=-JESi'Sj= (ij)
J ~ E (s,- s3 2 (ij)
J
E(S2+S~), 2 (ij)
Correspondence to: Dr. J.I. Katz, Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA.
(1)
where the $i are the individual spins and the sum runs :over all pairs of nearest neighbors i, j,
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.L Katz, D.R. Nelson / Minimum-energy configuration of isotropic ferromagnetic shells
48
counting each distinct pair once. Discarding the final sum (which is constant because all S i are constant), and writing (Si - S ) 2 = S g I Vthl2a 2, where ~ is a unit vector in the direction of magnetization and a is a lattice constant, we have
Jsg f d3x
H = -7" ~ ~
I Vffz12a2,
(2)
where the coordination number and number of spins per unit cell have been absorbed into the definition of J. The magnitude of the magnetization is always M. The magnetic energy may be obtained from the magnetostatic interactions of elementary dipoles. If V x H -- 0 (static configurations without true current) it is [2]
latitude on a globe. Then I Vfftl 2 = Hla
=
CSC20//r 2
4xrx l n ( g R / a ) + 2Ec,
f l n l 2 d3x.
(3)
(1) In the 'hedgehog' configuration ~ = ±~; the magnetization is radial everywhere. Straightforward calculation using eq. (2) yields I Vd~l 2 = 2 / r 2 and
4 rS2oJd Hhh
-
-
a
= 4~x,
(4)
where K-= S2jd/a. This result (as well as the later results eqs. (6) and (8)) does not require d << R, and may even be extended to solid spheres (d = R) if allowance is made for an extra term O(J) resulting from the singularity in ~ at the origin. Because this configuration is spherically symmetric and V • B = 0, B = 0 everywhere. Within the ferromagnetic material H = B - 4"rrM --- -4"a'M, but outside it and in the shell's interior (for r < R - d) H = B = 0. There are equal and opposite magnetic surface charges trM = + M on the inner and outer surfaces of the ferromagnet, and a volume magnetic charge density PM = 2 M / r (resulting from the constancy of M, differentiating this problem from its electrostatic analogue) within the material. The magnetic energy
Uhh = 8w2M2R2d ( d << R ) .
(5)
(2) In the latitudinal configuration ~ = -t-4;; lines of magnetization follow lines of constant
(6)
where the integral over 0 diverges logarithmically at the poles, and has been divided into two terms, the first with a cutoff at a and the remaining portion (at each pole) denoted E c. The value of the core energy E c cannot be calculated from this macroscopic theory, but is O(K). In this configuration V" M = 0, so V- H = 0. Because V × H = 0, H = 0 everywhere. Outside the ferromagnetic material B = H = 0, while within it B -- 4arM. On its surfaces there is poloidal magnetization current density, giving rise to the toroidal field (there is an analogous astronomical configuration involving real current [3]). The magnetic energy U~a = 0.
U = ~1
and
(7)
(3) In the longitudinal configuration ~ = + 0; lines of magnetization follow lines of constant longitude on a globe, running from one pole to the other. As in the latitudinal configuration, IV,hi 2 = C S C 2 0 / / r 2, and
Hlo = 4xrK l n ( 4 R / a ) + 2 E c.
(8)
The magnetization may be described by a magnetic charge density M PM = + - - c o t 0,
(9)
r
which is singular at the poles. Explicit calculation of the fields everywhere is possible using eq. (9). Integrating over rings of constant 0, taking d << R, the magnetic potential outside the sphere (r > R) is ®
Rl
@M( r, O) = +_2wRMdl~ r--7-~et(cos O)
× fP,(cos 0') cos 0' dO';
(10)
an analogous result applies for r < R. Terms in the sum of even I are zero, but all multipoles of odd order contribute. A fair approximation is obtained from the dipole term alone..The dipole moment for d << R is
m = :(-'tr2R2md.
(11)
ZL Katz, D.R. Nelson / Minimum-energy configuration of iaotropicferromagnetic shells
The field energy attributable to a dipole of this magnitude (including the interior energy implied by representing it by a uniformly magnetized sphere) is Ulo = m 2 / 2 R 3 = l'tr4Rm2d2;
(12)
the actual field energy is somewhat larger because of the contributions of the higher multipoles. H is continuous across the material surfaces because of the absence of free current and because there are no magnetic surface charges; hence the contribution to U of the material volume itself is small compared to eq. (12) by a factor O ( d / R ) << 1. The total energy of the longitudinal configuration is Hip + Ulo, where we adopt the dipole approximation eq. (12) for Ulo. For thin shells the magnetic energies are ordered 0 = Uh < Ulo << Uhh, while the exchange energies are ordered Hla = Hip > Hhh. The lowest-energy configuration depends on R, and is
latitudinal, if R > Rt,
(13)
with
R t=
2araM2 In
significant barrier to these decays, and that in the complete absence of anisotropy energy spheres and spherical shells (except the very smallest, for which the magnetic energy is smaller than the exchange energy) decay to the latitudinal configuration on a spin wave time scale.
3. External fields and permeability Application of an external field Bext adds a term Uext = - m "ne~t
- 1+
For So = 1, J = 0.3 eV (correspondin~ to a Curie temperature of about 1000 K), a = 2 A, and M = 103 (cgs) the right-hand sides of the inequalities (13) are = 700 A. Except for extremely small shells (a possibility we henceforth neglect) the latitudinal configuration is energetically preferred. We also consider configurations in which n~ -sin a + 4~ cos a . These form a path from longitudinal configurations (a = ~r/2) to latitudinal configurations (a = 0), with exchange energy independent of a but dipole energy proportional to sin2a, permitting the former to reduce their dipole energy by decaying to the latter. Similarly, hedgehog configurations may reduce their magnetic energy monotonically through a sequence of configurations dz = 4~ cos a + ~ sin a, with energies Hla COS2a + (Hhh + Uhh) sin2a, decaying to latitudinal configurations. It is possible that there is no
(14)
to the energetics, which will favor particles with dipole moments if these moments are free to rotate to the state of lowest energy. The critical field required to equate the latitudinal and longitudinal sphere energies is
Bo = ½ 2Ma/R.
hedgehog, if R < Rt,
49
(15)
For our previous numerical parameters, and spheres of R = 4 × 10 -3 cm, d = 10 -6 cm, we obtain B 0 = 1 G. Eq. (15) shows that the presence of the earth's field is sufficient to convert latitudinal configurations into longitudinal ones for achievable values of the parameters. If biased in this manner, the p e r m e a b i l i t y to additional small fields is anisotropic. B parallel to the biasing field Bbias does not change the magnetization so t h a t / z = 1, while fields orthogonal to Bbias rotate the total external field, and hence the magnetization, with the result that the microscopic permeability of the ferromagnetic shell /x = 1 + 4~M/Bbias.
(16)
In contrast, unbiased longitudinally magnetized particles, if they do not spontaneously decay to the latitudinal state, may have essentially infinite permeability (as is seen in the Bbias ~ 0 limit of the result for biased particles). Even a latitudinal configuration may have significant permeability. Application of an external field Bext = Bext~ tends to align the local magnetization. For values of Rd which are not extremely small the exchange energy may be neglected in
50
J.L Katz, D.R. Nelson / Minimum-energy configuration of isotropic ferromagnetic shells
comparison to the magnetic energy - r a ' B e x t + m 2 / 2 R 3. Minimization yields the resulting magnetic moment m =Bext R3.
(17)
Because only the dipole moment can reduce the total energy, while higher multipoles increase it, only a dipole moment will be induced (in the absence of exchange energy). The ~ component of magnetization will be uniform through the shell, and the angle of rotation of the local magnetization vector will be given by sin a = - B e x t R csc O/4arMd. Because B and H in a uniformly magnetized shell (as opposed to a solid sphere) are nonuniform it is not possible to define a unique local effective permeability. However, the magnetic moment, eq. (17), equals that of an infinitely permeable shell, corresponding to a large effective permeability. The field produced by m is comparable to Be~t in the vicinity of the shell. For a dilute distribution of shells with density n and nR 3 << 1 the volume-averaged permeability is Ix = 1 + 4,rrnR 3.
(18)
If magnetic shells are not very widely separated then their random interactions provide an anisotropic effective demagnetization factor, even if they are individually perfectly spherical. This turns the longitudinally magnetized state into a stable permanent magnet, with substantial ferromagnetic resonance frequency but small effective permeability. In contrast, densely packed latitudinally magnetized shells may acquire a large effective permeability, as their induced magnetization concentrates the magnetic flux into the shells.
4. Discussion
We have neglected the surface energy. Because the surface and its electronic structure are necessarily anisotropic there may be a surface exchange anisotropy energy O ( S 2 j / a 2) ~ 103 erg/cm 2 even in a material without bulk
anisotropy. The ratio of this energy to the characteristic dipolar magnetic energy S 2 j R / 2'rra2d2M 2 ~ 106 for our previous parameters and R = 4 × 1 0 -3 cm, d = 1 0 0 /~. The surface exchange anisotropy energy may exceed the bulk exchange energy by a factor O ( R 2 / a d ) > > 1. There is also a surface magnetic energy resulting from the atomic localization of the magnetization, and the consequent atomic-scale variation of the magnetic field outside the material, but this is O ( M 2 a ) ~ 10 -2 erg/cm 2, much smaller than the characteristic surface exchange energy. In order for our previous results to be valid, the surface anisotropy energy, as well as the bulk anisotropy energy, must be zero to high accuracy. If this is not the case the magnetization will be determined by the surface anisotropy. Depending on the sense of the anisotropy energy, the preferred magnetization will be either normal or parallel to the surface; the former case enforces hedgehog configurations, and the latter latitudinal or longitudinal ones (the dipole energy selecting the latitudinal state if the surface is accurately isotropic in its tangential plane). The presence of a surface anisotropy could reduce the permeability by a large factor. If these conditions on the anisotropy are met, then the lowest energy state of an isolated spherical shell of ferromaguet in no external field depends on its dimensions according to eq. (13). For all but the very smallest shells the lowest energy state is latitudinally magnetized. Other states may decay into latitudinal states through a continuous sequence of intermediate states. Because of the absence of anisotropy, the familiar domain structure is not found, and zero magnetic moment may be obtained by smoothly varying the direction of magnetization across the sphere. The hedgehog and latitudinal states are the simplest examples of this, but more complex magnetic structures are possible, at the price of increased exchange energy. These may have their direction of magnetization given by vector spherical harmonics, and correspond to multi-domain structures with broadened domain wails.
J.l. Katz, D.R. Nelson / Minimum-energy configuration of isotropic ferromagnetic shells
51
Acknowledgements
References
W e t h a n k A . E . B e r k o w i t z a n d H. Suhl for discussions a n d an a n o n y m o u s r e f e r e e for p o i n t ing o u t an i m p o r t a n t e r r o r . This w o r k was supp o r t e d by D A R P A .
[1] C. Kittel and J.K. Gait, Solid State Phys. 3 (1956) 439. [2] W.F. Brown, Jr., Magnetostatic Principles in Ferromagnetism (North-Holland, Amsterdam, 1962). [3] J.I. Katz, Mon. Not. R. Astr. Soc. 239 (1989) 751.