^ecknical ^iot&

DEMAGNETIZING FACTORS FOR OBLATE

SPHEROIDS USED IN FERRIMAGNETIC

RESONANCE MEASUREMENTS

L B. Schmidt, W. E. Case, and R. D. Harrington

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^ote 22f

Issued September 4, 1964

DEMAGNETIZING FACTORS FOR OBLATE SPHEROIDS DSED

IN FERRIMAGNETIC RESONANCE MEASUREMENTS

L. B. Schmidt, W. E. Case, and R. D. Harrington

Radio Standards Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price: 20 cents

TABLE OF CONTENTS

Page

Abstract 1 Discussion 1 References 8

Appendix I 9

Table 1 10

in

Demagnetizing Factors for Oblate Spheroids Used in

Ferrimagnetic Resonance Measurements

L. B. Schmidt, W. E. Case, and R. D. Harrington

Demagnetizing factors for oblate spheroids

magnetized along the short axis are given for

aspect ratios from 25. to 35. in increments

of 0. 1, from 35. to 55. in increments of 0. 2,

from 55. to 80. in increments of 0. 5, from

80. to 129. in increments of 1.0. The values

of all demagnetizing factors given in the tables

have been rounded off to 6 decimal places and

are accurate to ±5 units in the seventh

place. The tables are presented in a form

convenient for use in ferrimagnetic resonance

measurements on disk shaped samples. A

brief discussion of the effect of accuracy of

demagnetizing factors on measurements of

this type is included.

DISCUSSION

It is well known that the applied field H required for ferrimagnetic

z resonance in finite specimens at a frequency, u, may be obtained from

Kittel's [1948] equation,

- = f[H - (N - N ) M ] [H - (N - N ) M ] Y , (1)

•y Vzz xz z z y z /

where y is the gyromagnetic ratio, M is the magnetic moment per unit

z volume, and N , N , and N are the demagnetizing factors along the corx y z responding axes. The equation is often used to calculate the gyromagnetic ratio, v, from resonance measurements made on spherical specimens. For this situation N = N = N and equation (1) reduces to

x y z
£ = H

However, in some situations, it may be more advantageous to use

other geometries. For example, low field losses encountered in ferrimagnetic resonance measurements at lower microwave frequencies

may be avoided by using thin disk shaped samples with the dc field perpendicular to the face of the disk. The dc and rf field orientations for

such measurements are shown in figure &

1. For this situation,* N = N

x y

and Kittel's equation becomes

- = H - (N - N ) M . (2)

y z z x z

In a typical measurement, H is determined experimentally and

(N - N ) is calculated from relations developed by Stone r [ 1945] which

z x

express the demagnetizing coefficient as a function of aspect ratio for

spheroidal ellipsoids. Stoner expresses the demagnetizing factors as D , D , where D = N /4tt and D = N /4tr. Equation (2) then becomes

x z xx z z

- = H - (D - D ) 4ttM . (3)

y z z x z

The value of the magnetization, M , may be determined independently

using a magnetometer.

We have recently carried out a study involving measurements of H

on several materials at 1100 MHz, using disk shaped samples [Case,

Harrington, and Schmidt, 1964] . This study indicated that the measurement of disks of several sizes is often advantageous in obtaining data of
RF

MAGNETIC FIELD

FIGURE 1 Field and Sample Orientations for Ferrimagnetic

Resonance Measurements on a Disk.

(Demagnetizing factors in the table are applicable for this geometry. )
this type. Each sample of each material was measured at a number of

different aspect ratios (diameter/thickness) by grinding the same diameter sample to successively smaller thickness for each measurement.

By using these results a plot of — as a function of aspect ratio is readily

obtained. Figure 2 shows results of this type on one material used in

the study. The variation from a smooth curve in this example was less

than one percent. It is apparent that D - D must be known to considerably better than one percent if this quantity is not to contribute to the

CO observed variation in — . In fact, it can be shown that D - D must be

y

z x

known to a few hundredths of a percent for typical materials if the conco . tribution of any error in D - D to this one percent variation in — is to z x y

be less than 0. 1%. (See Appendix I. ) These demagnetizing factors may

be calculated to sufficient accuracy using Stoner's equations. When

measuring many different samples as mentioned above, it is apparent

that tables of demagnetizing factors, given in finer aspect ratio increments than is presently available, would be very convenient.

Stoner has given tables of D , but only in increments of 5 and 10 in

z the region of aspect ratios of 25 to 129. Osborn [1945] prepared graphs

of N/4ir which equals D , but again the resolution was not adequate in

z this region. We have therefore prepared Table I which gives demagnetizing factors that meet the minimum requirements for increments

of aspect ratios needed to prepare graphs within the precision represented by figure 2 and the calculations of — mentioned above. The tables

Y

are shown in six decimal place s-, however, this accuracy is not required

for these calculations.

Table I was condensed from tables prepared on a computer, calculated to eight significant figures and rounded off to six significant

figures which provided a convenient comparison to Stoner's tables.
<y>

COCD

i-hOh o a oo

CO

•f-i Q o ^ r— o o <+H

ho < PS CO

oc

ro o w <1) rt Oh h- p

CO < o O CO o1—

1

to UJ h

0. COu

if)

<

CD >

o ;>* ^J- "3*u

<=> Q

ro

i—

i

rto

•i-i o Oh CVJ H

*—• o ©^ oo r— £©

^" «^- ro ro *0 ro

(saBisaao)-^-
Kittel's equation is written in terms of N , N , and N when N + N

x y z x y

+ N = 4tt. It is more convenient, as noted by Stoner, to express the

z demagnetizing factors in terms of D , D , and D when

N N N

x 4ir y 4tt z 4tt

In the tables prepared by the computer, D was first calculated

z using Stoner 's equation 4. 5 for an oblate spheroid as follows

1 Ti m -1 1 D = =- 1 7~T~ cos m z

1 -m2

L (1 -m2

) 2 J

where m = l/aspect ratio. Then D = D = (1 - D )/Z was calculated to

x y z give the values of D - D which are also given for convenience in using

Z X

equation (3). The computer program was written to generate its own

input in increments of aspect ratio as required. Using this method, the

increments could be in integers, in tenths, or in hundredths. In cases

where measurements are made on disks with very small changes in

aspect ratios, tables in increments of 0. 01 are most efficient. The

table was readily prepared on the computer in increments of 0. 01 from

1. 01 to 130. 00 to satisfy this requirement and possible future requirements.

Where applicable, the values of this table were compared with the

corresponding values given in Table II in Stoner 's paper. In two instances,

the two disagree by one part in the sixth place. Stoner noted that his calculations were carried to the seventh place and rounded to the sixth place.

The computer calculated our tables to the eighth significant figure and

rounded to the sixth significant figure. At a third place, the aspect ratio

of 60, checking by hand calculations confirmed our value, so that Stoner's

value appears to be in error.
The accuracy of Table I was spot checked by hand calculations.

In addition, the accuracy of the entire table was verified by checking

by differences [Miller, 1950] . The values of all demagnetizing factors

given in the tables have been rounded off to 6 decimal places and are

thus accurate to ±5 units in the seventh place.

The tables were studied for the possibility of using linear interpolation for values between the aspect ratios given. The accuracy of

this interpolation was verified by comparing the midpoint between increments in the tables calculated by linear interpolation with the corresponding point in the tables prepared by the computer in increments

of 0. 01 in aspect ratio. It was found that the values of D may be interpolated linearly, accurate to one unit in the sixth place, for all

aspect ratios. The values of D - D may be interpolated linearly,

accurate to one unit in the sixth place, except within the range of aspect

ratios between 80. and 129. 0. Within this range, values of D - D

may be interpolated linearly accurate to two units in the sixth place.

It was previously noted that this accuracy is more than adequate in the

use of equation (3).
REFERENCES

Case, W. E. , R. D. Harrington, and L. B. Schmidt (April -June

1964), Ferrimagnetic resonance in polycrystalline ferrite and

garnet disks at L-band frequencies, J. Res. NBS 68C , No. 2,

85-89.

Kittel, C. (1948), On the theory of ferromagnetic resonance absorption,

Phys. Rev. 73, 155-161.

Miller, J. C. P. (1950), Checking by differences -I, Mathematical tables

and other aids to computation, 4, 3-11.

Osborn, J. A. (1945), Demagnetizing factors of the general ellipsoid,

Phys. Rev. 67, 351-357.

Stoner, E. C. (1945), The demagnetizing factors for ellipsoids, Phil.

Mag. (7), 36, 803-821.
APPENDIX I For a fixed value of H and M , the percent error in — given bv z z v

(J A -

Y J.11L1 (JU.U.

(J lcu uy dii error a \.u - l z Mini

X

J - z D IS O

X

Y

equation (2)

4).

Y

A(D - z D - z

D ) X

D

X

r (D - z D ) X

4-ttM _ z

Y

- (4)

The percent error in— introduced by an error in D - D is deY z x

pendent on a factor that is inversely proportional to tJ. Any percent

error in (D - D ) will thus have a greater influence on the percent z x ^ error in— as the frequency decreases. For example, if data were desired at 1100 MHz on a material such as a commercially available substituted garnet we have 4ttM = 575 gauss, — = 390 oersteds, and D - s z 6

y z D = . 976735 at an aspect ratio of 100. If we desire the percentage

A -

Y rrl error in D - D to cause an error — ^ 0. 1%, then substituting in

z x W

A(D - D ) v

z x

equation (4) we have —— — = 0. 07%. In many cases, u would be

z x

greater which would increase the allowable percent error of D - D to

hold its contribution of error in— to 0. 1%; however, at the same time,

M could also be larger to produce the opposite effect. Thus D - D

z z .X. must be known to a few hundredths of a percent in order to cover all

situations.
TABLE 1

Demagnetization Factors of Oblate Spheroids Magnetized Along the Short

. . ,,- * ^ . / Diameter

.

Axis Versus Aspect Ratio (

,

. . ) Thickness

and

Tables are given for D and D - D where z is along the short axis

D = D

x y

and

D + D + D = 1. x y z

10
Aspect

Ratio

D

z

. 940224

D -D

z X

. 910336

Aspect

Ratio

30.0

D

z .949778

D -D

Z X

Aspect

Ratio

35.

D

z .956700

D -D

z X

25. .924667 .935049

25. 1 . 940450 .910676 30. 1 .949938 .924907 35. 2 .956937 .935405

25. 2 .940675 .911013 30. 2 .950097 . 925146 35.4 .957172 .935758

25. 3 .940898 .911348 30. 3 . 950255 .925383 35. 6 . 957404 .936106

25.4 .941120 .911680 30.4 .950412 . 925618 35.8 .957634 .936451

25.5 .941340 .912010 30. 5 .950568 .925853 36.0 .957861 .936792

25.6 .941558 .912337 30.6 .950723 . 926085 36. 2 .958086 . 937129

25. 7 .941775 .912662 30. 7 .950877 .926316 36.4 .958308 .937463

25.8 .941990 .912984 30.8 .951031 . 926546 36.6 .958528 . 937793

25.9 .942203 .913305 30.9 .951183 . 926774 36.8 .958746 .938119

26.0 .942415 .913622 31.0 .951334 .927001 37. .958962 .938443

26. 1 .942625 .913938 31. 1 .951484 .927226 37. 2 .959175 .938763

26. 2 .942834 .914251 31. 2 .951634 .927450 37.4 .959386 .939079

26. 3 .943041 .914562 31.3 .951782 .927673 37.6 .959595 .939393

26.4 .943247 .914870 31.4 .951930 .927894 37.8 .959802 .939703

26.5 .943451 .915177 31.5 .952076 .928114 38.0 .960007 .940010

26.6 .943654 .915481 31.6 .952222 .928333 38. 2 .960209 .940314

26. 7 .943855 .915783 31. 7 .952367 .928550 38.4 .960410 .940615

26.8 . 944055 .916083 31. 8 .952511 .928766 38.6 .960608 .940912

26.9 . 944254 .916381 31.9 .952654 .928981 38.8 .960805 .941207

27. .944451 .916676 32.0 .952796 .929194 39. .960999 .941499

27. 1 . 944647 .916970 32. 1 .952938 .929407 39.2 .961192 .941788

27. 2 . 944841 .917262 32. 2 .953078 .929617 39.4 .961383 .942074

27. 3 .945034 .917551 32. 3 .953218 . 929827 39.6 .961572 .942358

27.4 .945226 .917838 32.4 .953357 .930035 39.8 .961759 .942639

27. 5 .945416 .918124 32.5 .953495 . 930242 40.0 .961944 .942916

27.6 .945605 .918407 32. 6 .953632 .930448 40. 2 .962128 .943192

27. 7 .945793 .918689 32. 7 .953769 .930653 40.4 .962310 . 943464

27.8 .945979 .918969 32. 8 .953904 .930856 40.6 .962490 .943734

27.9 .946164 .919246 32.9 .954039 .931059 40.8 .962668 . 944002

28.0 . 946348 .919522 33.0 .954173 .931260 41.0 .962844 .944267

28. 1 .946531 .919796 33.1 .954307 .931460 41.2 .963019 .944529

28.2 .946712 .920068 33.2 .954439 .931659 41.4 .963193 .944789

28.3 .946892 .920338 33.3 .954571 .931856 41.6 .963364 .945046

28.4 .947071 .920607 33.4 .954702 .932053 41.8 .963534 .945302

28.5 . 947249 .920873 33.5 .954832 .932248 42.0 .963703 .945554

28.6 . 947425 .921138 33.6 .954962 .932442 42.2 .963870 .945805

28.7 .947601 .921401 33.7 .955090 .932635 42.4 .964035 .946053

28.8 .947775 .921662 33.8 .955218 .932827 42.6 .964199 .946299

28.9 . 947948 .921922 33.9 .955346 .933018 42.8 .964362 .946543

29.0 .948120 .922180 34.0 .955472 .933208 43.0 .964523 .946784

29.1 .948291 .922436 34.1 .955598 .933397 43.2 .964682 .947023

29.2 . 948460 .922690 34.2 .955723 .933585 43.4 . 964840 .947260

29.3 .948629 .922943 34.3 .955848 .933771 43.6 .964997 .947495

29.4 .948796 .923194 34.4 .955971 .933957 43.8 .965152 .947728

29.5 .948962 .923444 34.5 .956094 .934142 44.0 .965306 .947959

29.6 .949128 .923692 34.6 .956217 .934325 44.2 .965459 .948188

29.7 .949292 .923938 34.7 .956338 .934508 44.4 .965610 .948415

29.8 . 949455 .924183 34.8 .956459 .934689 44.6 .965760 .948640

29.9 .949617 .924426 34.9 .956580 .934870 44.8 .965908 .948863

11
Aspect D D -D

Aspect D D -D

Aspect D D -D

Ratio z .966056

z X Ratio

55.0

z .972087

z X

.958131

Ratio

80.0

z .980673

z X

45. .949084 .971009

45. 2 .966202 .•949303 55. 5 .972333 . 958500 81. .980908 . 971362

45.4 .966347 .949520 56.0 , 972575 .958862 82.0 .981137 .971706

45. 6 .966490 .949735 56. 5 .972812 . 959218 83. .981361 .972041

45.8 .966633 .949949 57. .973045 . 959568 84.0 .981580 .972369

46.0 .966774 .950161 57.5 .973275 .959912 85.0 .981793 .972690

46.2 .966914 .950371 58.0 ,973500 .960250 86. .982002 .973003

46.4 .967053 .950579 58.5 .973722 .960582 87. .982206 .973308

46.6 .967190 .950785 59.0 .973940 .960909 88. .982405 .973607

46.8 .967327 .950990 59.5 .974154 .961231 89.0 .982600 .973900

47. .967462 .951193 60.0 974365 .961547 90.0 .982790 .974186

47. 2 .967596 .951394 60.5 .974572 .961859 91.0 .982977 .974465

47.4 .967729 .951594 61.0 .974777 .962165 92.0 .983159 .974739

47. 6 .967861 .951792 61. 5 974977 .962466 93. .983338 .975007

47.8 .967992 .951989 62.0 975175 . 962763 94.0 .983513 .975269

48. .968122 .952183 62. 5 . 975370 .963055 95.0 .983684 .975526

48. 2 .968251 .952377 63.0 975561 .963342 96.0 .983852 .975778

48.4 .968379 .952568 63.5 975750 .963625 97.0 .984016 .976024

48. 6 .968506 .952759 64.0 .975936 .963904 98.0 .984177 .976266

48.8 .968632 .952947 64.5 976119 .964178 99.0 .984335 .976503

49.0 .968756 .953135 65.0 976299 .964448 100.0 .984490 .976735

49.2 .968880 .953320 65. 5 .976476 .964714 101.0 .984641 .976962

49.4 .969003 .953504 66.0 976651 .964977 102. .984790 .977185

49.6 .969125 .953687 66.5 .976823 .965235 103.0 .984936 . 977404

49.8 .969246 .953869 67.0 976993 .965490 104.0 .985079 .977619

50.0 .969366 . 954048 67.5 .977160 .965741 105.0 .985219 .977829

50. 2 .969485 .954227 68.0 .977325 .965988 106.0 .985357 .978036

50.4 .969603 .954404 68.5 .977488 .966232 107.0 .985492 .978239

50.6 .969720 .954580 69.0 ,977648 .966472 108.0 .985625 .978438

50.8 .969836 .954754 69.5 977806 .966709 109.0 .985756 .978633

51. .969952 .954927 70. 977961 .966942 110.0 .985884 .978825

51.2 .970066 .955099 70. 5 978115 .967172 111. .986009 .979014

51.4 .970180 .955270 71.0 978266 .967400 112.0 .986133 .979199

51.6 .970293 .955439 71.5 .978416 .967624 113.0 .986254 .979381

51.8 .970405 .955607 72.0 978563 . 967844 114.0 .986373 .979560

52.0 .970516 .955773 72.5 978708 .968062 115. .986491 .979736

52. 2 .970626 .955939 73.0 978852 .968277 116.0 .986606 .979909

52.4 .970735 .956x03 73. 5 978993 .968489 117.0 .986719 .980079

52.6 . 970844 .956266 74.0 .979133 .968699 118.0 .986830 .980246

52.8 .970952 .956428 74.5 979270 .968905 119.0 .986940 .980410

53.0 .971059 .956588 75.0 ,979406 .969109 120.0 .98 7048 .980571

53.2 .971165 .956748 75.5 .979540 .969310 121. .987153 .980730

53.4 .971271 .956906 76.0 ,979673 .969509 122.0 .987258 .980887

53.6 .971375 .957063 76. 5 .979803 .969705 123.0 .987360 .981040

53.8 .971479 .957219 77.0 .979932 . 969898 124.0 .987461 .981192

54.0 .97158.2 .957374 77. 5 .980060 .970089 125. .987560 .981341

54. 2 .971685 .957527 78.0 .980185 .970278 126. .987658 .981487

54.4 .971787 .957680 78.5 .980310 .970464 127.0 .987754 .981632

54.6 .971888 .957831 79.0 . 980432 .970648 128. .987849 .981774

54.8 .971988 .957982 79.5 980553 .970830 129.0 .987942 .981914

12