Magnetic field in a cavity

Transcription

1 Magnetic field in a cavity W.H. Meiklejohn To cite this version: W.H. Meiklejohn. Magnetic field in a cavity. J. Phys. Radium, 1959, 20 (23), pp < /jphysrad: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1959 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 TOME LE JOURNAL DE PHYSIQUE ET LE RADIUM 2(), FÉVHIER t949, 88 MAGNETIC FIELD IN A CAVITY By W. H. MEIKLEJOHN, General Electric Research Laboratory, Schenectady, New York, U. S. A. Résumé La valeur du champ classique de Lorentz dans une cavité sphérique (403C0 Ms/3) n est pas atteinte dans un matériau ferromagnétique à son niveau de saturation technique. Ceci est dû à la formation de domaines de fermeture dans la substance à la surface de la cavité. Les mesures effectuées sur un matériau ayant une saturation magnétique faible montrent que le champ dans la cavité n approche de la valeur 403C0 Ms/3 que pour des champs égaux à cinq fois le champ démagnétisant maximum de la substance. Comme l a montré Néel ces résultats montrent que les cavités ou les inclusions nonmagnétiques perturbent considérablement l approche à la saturation dans les matériaux ferromagnétiques Abstract. classical Lorentz field in a spherical cavity of 403C0 Ms/3 does not occur in a ferromagnetic material which has reached technical saturation. This effect is due to the formation of closure domains in the material near the surface of the cavity. Measurements of a material of low saturation magnetization shows that the field in the cavity approaches 403C0 Ms/3 only at fields that are five times the maximum demagnetizing field in the material. se results show that cavities or nonmagnetic inclusions will greatly effect the approach to saturation in ferromagnetic materials, as pointed out by Néel. magnetic field in an isotropie, homogeneous, uniformly magnetized material is given by the two relations For the case of a spherical cavity in an infinitely long rod these relations give the field in the cavity (Be) as because 4rc M 0 in the cavity and the only NM is that due to the,divergence of M at the surface of the cavity. One finds that for a practical ferromagnetic material this relationship does not hold except as a limiting case for very high values of Ha. Closure domains in the material near the surface of the cavity cause large deviations from this relationship for magnetic fields that produce technical saturation. Experimental procedure. cavity in the ferromagnetic material was made by machining half spherical a hole in an end of two rods which were from 11/2 to 5 inches in diameter by 12" long as shown in Fig. 1. two rods were butted together to form a spherical cavity. A hole 0.100" in diameter was drilled into this cavity for insertion of a gauss meter probe. gauss meter probe, as shown in the crosssectional view of Fig. 1, contains a small permanent magnet. A FIG. 1. A view of the cavity machined in the metal rods and the gauss meter used to mesnre the magnetic field. Article published online by EDP Sciences and available at

3 A 89 torque is exerted on the permanent magnet by the magnetic field when the assembeyisrotated. This torque is bucked against the torque of a spring and the scale is calibrated in gauss. permanent magnet is a rod less than 1/8" long which is magnetized across its diameter and is contained in tubing of.090" outside diameter. A schematic diagram of the rods assembled in the electromagnet is shown in Fig. 2. field Ha M magnetic moment per unit volume 1 700, b radius of the ferromagnetic rod 3fi inch, a radius of the cavity 1/4 inch. Calculations. expected behavior is derived from two calculations that approximate the low field and the high field regions. low field approximation is a calculation of the field in a cavity of a paramagnetic material of semiinfinite extent with no external demagnetizing surface. FIG. 2. schematic diagram of the rods assembled in the electromagnet. is the field measured at the position of the cavity with the ferromagnetic rod removed. field at the cavity when the ferromagnetic rod is in the electromagnet is slightly less than Ha due to the elimination of the polarization over the area of contact between the rod and the electromagnet pole face. This field was calculated to be 60 oersteds and therefore quite negligible in this experiment where the différences are in the order of oersteds. influence of the polarization of the butted surfaces of the rods (there is some separation due to surface roughness) and the hole for inserting the probe are easily shown to be negligible. field at the center of the cavity due to the polarization on the butted surfaces is calculated to be where a radius of thé spherical cavity 1/4 inch, b radius of the rod 3/4 inch, L separation of the butted surfaces 10 3 cm,.all magnetic moment per unit volume field at the center of the oavity due to the polarization on the surface of the hole used to insert the probe of the gauss meter is easily calculated to be where d diameter of the drilled hole 1/8 irich, FIG. 3. cavity in an iron rod as compare with high field and low field calculations. calculation involves the solution of Poisson s equation with the given boundary conditions. result is : where y B /H. For low magnetic fields the ferromagnetic material will act somewhat like.a paramagnetic material and the permeability will be high such that 2u» 1. Hence we have His behavior will be expected in iron up to

4 90 fields of about oersteds where the permea of iron decreases below 10. This bility expected behavior is shown in Fig. 3. high field approximation is the result of a calculation of the field in a cavity of a uniformly polarized ferromagnetic material of semiinfinite extent with. no external demagnetizing surfaces. result of this calculation is in Fig. 4 with the calculated results shown for comparaison. field in the cavity is less than that predicted by Eq. (4) at a field where one This expected behavior is shown for the high field regions in Fig. 3. Comparison of experiment. experimental measurements of the field in a cavity in iron is shown in Fig. 3. In the low field region one obtains large deviations from the calculated behavior bécasse the material is not paramagnetic ; it is neither homogeneous nor isotropic. Being a ferromagnet, the material has a spontaneous magnetization due to the Weiss molecular field. re are magnetic domains in the material, and theref ore B is not in the direction of H which was an assumption in the derivation of Eq. (3). Because of these domains, the vector distribution of the polarization on the surface of the cavity will be quite different from the paramagnetic case. detailed calculation will depend upon the domain configuration at the surface of the cavity. In the high field region one should expect much better agreement between the calculated result given by Eq. (4) and the experimental result (Fig. 3). For magnetic fields greater than all anisotropy fields present in the material one expects the magnetization to be in the direction of the applied field. strain and crystalline anisotropy fields amount to about 500 oersteds in iron while the demagnetizing field in the material due to the surface polarization of the cavity is quite large. fields in the material for the polarized case is given by FIG. 4. cavity in nickel rod as compared with high field and low field calculations. expects agreement, i.e., oersteds. It appears that the closure domains near the surface of the cavity are not eliminated at fields equal to the total demagnetizing fields in the material. In order to determine the fields at which the effects of the closure domains are eliminated measurements were made on monel. This material maximum demagnetizing field will be for r a and 0 7r whence Hd 8r Mg/3 :::::! 14,000 oersteds (iron). (5) Since this field is much greater than the oersteds used in this experiment, one does not require that there be experimental agreement with the high field calculations. Since the maximum field that we could attain was only oersteds we chose nickel as a better material to use to check the calculations. Calculations using Eq. (5) show that experimental data on nickel should agree with Eq. (4) above oersteds. experimental results for nickel are shown cavity in a monel rod as compared with high field and low field calculations. FIG. 5. has a total demagnetizing field (Eq. 5) of approximately oersteds. results for monel are is made shown in Fig. 5 where again a comparison

5 with calculations. It appears that a field of about five times the maximum anisotropy fields will eliminate the closure domains in the neighborhood of cavities. However, the actual ratio of applied field to anisotropy field will depend on the material and its metallurgical treatment. field inside a 1" diameter sphere of iron was also inves at saturation is containing a 3/8" diameter cavity tigated. field in the cavity given by where N1 demagnetizing factor of the external surface, N2 demagnetizing factor of the internal surface. experimental results are shown in Fig. 6 expect much better bqhavior at this surface because the polarization tends to produce a uniform and not a diverging magnetic field as occurs in the material near the surface of the cavity. One can think in terms of an effective demagnetizing factor of a spherical cavity which is derived from this experiment by the relationship where 1Ve effective demagnetizing factor, B, magnetic field in the cavity, Ha magnetic moment per unit volume of the material. In calculating thèse results, we have used M Ms since for fields of oersteds these of their satu materials have reached a least 99 % ration magnetization. effective demagnetizing factors for spherical cavities is shown in Fig. 7 as a function of the cavity in an iron spherical shell as compared with the high field and low field calculations. FIG. 6. where the low field and high field calculations are given for comparison. deviation from the high field approximation could be due to both surfaces or only the interior surface. If we take N2 1.5 as determined from the data in Fig. 3 and substitute the values from data of Fig. 6 at H oersteds, i.e., Be and take Il mes we get This is very close to the expected value of 4?r/3 and indicates that there are no detrimental closure domaine near the external surface. une should effective demagnetization factors of a sphe FIG. 7. rical cavity as a function of the applied magnetic field. applied field, which is essentially the field in the material far from the cavity. se experimental results show that cavities or nonmagnetic inclussions will greatly effect the approach to saturation in ferromagnetic materials as has been pointed out by Néel [1]. In addition, these results show that the classical Lorentz field of 41t MsJ3 does not occur in ferromagnetic materials at technical saturation which has a direct bearing on calculation of the magnetization of polycrystalline materials [2]. author would like to acknowledge. the very helpful discussions with J. S. Kouvel and the experimental assistance of R. E. Skoda,

6 Has Referring Would 1 As 1 Would In ?REFERENCES [1] NÉEL (L.), " Law Convergence for a/h and a New ory of Magnetic Retentivity ", J. Physique Rad., 1948, 9, [2] NÉEL (L.), " Relationship Between the Anisotropy Constant and the Law of Approach to Saturation of Ferromagnetic Materials ", J. Physique Rad., 1948, 9, n 6, DISCUSSION. Mr. Kurti. to nomenclature, I should like to suggest that the expression " Lorentzfield " should not be used in the case of spherical cavities. Lorentz s " sphere " was a mathematical artifice and did not involve thé " scoopingout " of any material. Mr. Meiklejohn. However, 1 believe agree. that these measurements indicate that the demagnetizing field experienced by one crystal in a polycrystalline material will be smaller than that calculated by Poisson s equation, for finite fields. This will be true because the vector distribution of the easy axis of magnetization of the crystals bounding the one crystal being considered will not allow M to be in the direction of H. Mr. Meiklejohn any special Mr. Bates. kinds of closure domains in mind? Mr. Meiklejohn. a first approximation 1 have considered the domain to be conical. At low fields the diameter of the base of the cone may be equal to the diameter of the cavity, making the effective shape of the cavity that of an ellipsoïd. This would account for the low demagnetizing At high fields there will probably be many factor. small closure domains. Mr. Wohlfarth. it be possible to investigate the suspected closure domains by using garnets? Mr. Meiklejohn. think it would be possible. Since the garnet must be thin, the cavity would necessarily be small in order to eliminate the effect of the surface polarization due to the divergence of Lez on the surface of the cavity. Prof. Rathenau suggested that the Bitter technique be used and let the water evaporate. A few quick experiments were tried without success, but a more careful experiment may show the domains. Mr. Meiklejohn (in answer to Mr. Foner and Mr. Kaczér). results were not corrected for the finite diameter of the rod. magnitude of the correction depends on the ratio of the diameter of the cavity to the diameter of the rod. Experimentally, 1 found that with 1 inch diameter cavities in rods of 3 and 5 inches diameter the data were the same above oersteds. sign of the correction is such that the curves in Fig. 7 will be lower when the correction is made, particularly at the lower fields. Mr. Kondorskij. it be approximately correct, in the low field region, to estimate the effective demagnetizing factor as the ratio of the demagnetizing factor of the sphere to the magnetic permeability of the rod (in the low field region)? Mr. Meiklejohn. the case of, a polycrystalline material as used in these experiments, 1 believe the vector distribution of the easy axes of magnetization of the crystals at the cavity surface is more important than the permeability ; and therefore I would not expect such a relationship to exist. Mr. Zi jlstra. What will the field in the cavity be in the case of a fine particle magnet? If a particular particle is removed from a fine particle magnet, it is important to know what the field of the other particles will be in the cavity. dimension of the cavity in this case is less than a closure domain will be. Mr. Meiklejohn. don t know what the field will be, but 1 expect it will be less than 4n M/3, and may be worse than in the case of the solid of the material, due to the individual anisotropies particles.