Magnetic field in a cavity

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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 (2-3), pp.88-92.

�10.1051/jphysrad:01959002002-308800�. �jpa-00236073�

MAGNETIC FIELD IN A CAVITY

By ^{W. H.} MEIKLEJOHN,

General Electric Research Laboratory, Schenectady, ^New York, ^{U. S. A.}

Résumé. 2014 La valeur du champ classique ^de Lorentz dans

cavité sphérique (403C0 Ms/3)

n’est pas atteinte dans

matériau ferromagnétique ^à

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

matériau ayant

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

résultats montrent que les cavités

les inclusions non-magnétiques perturbent considérablement l’approche ^à la saturation dans les matériaux ferromagnétiques.

Abstract.

The classical Lorentz field in

spherical cavity ^of 403C0 Ms/3 ^{does not}

ⁱⁿ

ferromagnetic material which has reached technical saturation. This effect is due to the formation of closure domains in the material

the surface of the cavity. Measurements of

material of low saturation magnetization shows that the field in the cavity approaches ^403C0 Ms/3 only ^{at fields}

that

five times the maximum demagnetizing field in the material. These results show that cavities

non-magnetic inclusions will greatly effect the approach ^to saturation in ferromagnetic materials,

pointed ^out by ^Néel.

PHYSIQUE 2(), FÉVHIER t949,

The 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 satu-}

ration.

Experimental procedure.

^The cavity ^{in the} ferromagnetic ^{material was made} by machining ^a

half spherical ^{hole in an end of two} rods which

were from 1-1/2 ^to 5 inches in diameter by ^12"

long ^{as shown in} Fig. 1. ^{The two rods were butted} together ^{to form a} spherical cavity. ^A hole 0.100"

in diameter was drilled into this cavity ^{for inser-}

tion of a gauss ^meter probe. ^The gauss ^meter

probe, ^as shown in the cross-sectional 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 http://dx.doi.org/10.1051/jphysrad:01959002002-308800

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. The permanent magnet ^{is a} rod less than 1/8" long which is magne- tized ^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. The field Ha}

FIG. 2.

A schematic diagram

of the rods assembled in the electromagnet.

is the field measured ^at the position ^{of the} cavity

with the ferromagnetic rod removed. The 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 1 000 oersteds.

The 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. ^{The 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

=1700.

The 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,

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.

The expected behavior is deri- ved from two calculations that approximate ^the

low field and the high ^field regions. ^{The low} field

approximation ^{is a} calculation of the field in a

cavity ^{of a} paramagnetic ^{material of} semi-infinite

extent with no external demagnetizing ^surface.

FIG. 3.

The measured magnetic ^field in

_spherical

cavity ⁱⁿ

^{iron rod}

compare with high field and low field calculations.

The calculation involves the solution of Poisson’s

equation ^{with the} given boundary conditions.

The 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

fields of about 2 000 oersteds where the _permea-

bility ^{of iron} decreases below 10. This expected

behavior is shown in Fig. ^3.

The high ^field approximation ^{is the} result of a

calculation of the field in a cavity ^{of a} uniformly polarized ferromagnetic material of semi-infinite extent with. no external demagnetizing ^surfaces.

The result of this calculation is

This expected ^{behavior is shown for the} high

field regions ⁱⁿ Fig. ^3.

Comparison ^of experiment.

^The 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 beha- vior bécasse the material is not paramagnetic ; ^{it is}

neither homogeneous ^nor isotropic. Being ^{a ferro-} magnet, ^the material has ^a spontaneous ^magne-

tization due to the Weiss molecular field. There

are magnetic domains in the material, ^{and there-}

f 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. The}

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. The strain and crystalline ^aniso- tropy ^{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. The fields in the material for the polarized

case is given by

The maximum demagnetizing ^{field will be for}

r = a and 0

whence

Hd

^8r Mg/3 :::::! 14,000 ^oersteds (iron). (5) Since this field is much greater ^{than the}

4 000 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 ^{4 500 oersteds we chose nickel as a better}

material to use to check the calculations. Calcu- lations using Eq. (5) ^{show that} experimental ^data

on nickel should agree with Eq. (4) ^above

4 200 oersteds.

The experimental ^{results for nickel are shown}

in Fig. ⁴ with the calculated results shown for

comparaison. ^{The field in the} cavity ^{is less than}

that predicted by Eq. (4) ^{at a} field where ^one

FIG. 4.

The measured magnetic ^{field in}

spherical cavity in nickel rod

compared ^with high field and low field calculations.

expects agreement, i.e., ⁴ 200 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

FIG. 5.

The measured magnetic ^{field in}

spherical cavity ⁱⁿ

^{monel rod}

compared ^with high ^{field and}

low field calculations.

has a total demagnetizing ^field (Eq. 5) ^of approxi- mately 1 000 oersteds. The results for monel are

shown in Fig. ^{5 where} again ^a comparison ^{is made}

91

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.

The field inside ^a 1" diameter sphere ^{of iron}

-

containing ^a 3/8" ^diameter cavity ^was also inves-

tigated. ^{The field in the} cavity ^at saturation is

given by

where N1

demagnetizing factor of the external

surface,

N2

demagnetizing factor of the internal surface.

The 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 demagne- tizing ^{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 1000 ^{oersteds these}

materials have reached a least ⁹⁹ % ^{of their satu-}

ration magnetization.

The effective demagnetizing ^{factors for} spherical

cavities is shown in Fig. ^{7 as a} function of the

FIG. 6.

The measured magnetic ^{field in}

spherical cavity ⁱⁿ

^iron spherical ^shell

compared ^{with the} high field and low field calculations.

where the low field and high field calculations are

given ^for comparison. ^The 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 = 7 000 oersteds, i.e., Be

2 550 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

FIG. 7.

The effective demagnetization factors of

sphe-

rical cavity

^{function of the} applied magnetic ^field.

applied field, which ^is essentially ^{the field in the}

material far from the cavity.

These experimental results show that cavities or

non-magnetic 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].

The author would like to acknowledge. ^the very

helpful discussions with J. S. Kouvel and the

experimental assistance of R. E. Skoda,

1?REFERENCES

[1] ^NÉEL (L.),

^{The Law} Convergence ^for a/H ^and

New Theory ^of Magnetic Retentivity ", J. Physique Rad., 1948, 9, 184-192.

[2] ^NÉEL (L.),

Relationship Between the Anisotropy

Constant and the Law ^of Approach ^to Saturation of

Ferromagnetic Materials ", ^J. Physique Rad., 1948, 9, ^n° 6,193-199.

DISCUSSION

’

Mr. Kurti.

Referring ^to nomenclature,

I should like to suggest ^{that the} expression

"

Lorentz-field " should not be used in the case of

spherical cavities. Lorentz’s " sphere

^{was a}

mathematical artifice ^{and did not involve} thé

"

scooping-out

of any material.

Mr. Meiklejohn.

1 agree. However, ^{1 believe}

that these measurements indicate that the dema-

gnetizing ^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 boun- ding ^{the one} crystal being considered will not allow M to be in the direction of H.

Mr. Bates.

Has Mr. Meiklejohn any special

kinds of closure domains in mind ?

Mr. Meiklejohn.

^{As 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

factor. At high fields there ^will probably ^be many small closure domains.

°

Mr. Wohlfarth.

^{Would it be} possible ^{to inves-} tigate ^the suspected closure domains by using garnets ?

Mr. Meiklejohn.

^{1 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 tech-

nique ^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).

^{The results were not corrected} for the finite diameter of the rod. The magnitude ^of

the correction depends ^{on the ratio of the diame-}

ter of the cavity ^to the diameter of the rod. Expe- rimentally, ^{1 found} that with 1 inch diameter cavi- ties in rods of 3 and 5 inches diameter the data

were the same above 2 000 oersteds. The sign

of the correction is such that the curves in Fig. ⁷

will be lower when the correction is made, parti- cularly ^{at the lower fields.}

Mr. Kondorskij.

^{Would 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.

^{In the case} of , ^a polycrystal-

line material as used in these experiments, ^{1 believe}

the vector distribution of the _easy axes of _magne- tization 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. ^The

dimension of the cavity ^{in this case} is less than a

closure domain will be.

Mr. Meiklejohn.

1 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

material, ^{due to the} individual anisotropies ^{of the}

particles.