THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COTIEGE OF ENGINEERING REVERSIBLE PROPERTIES OF FERROMAGNETS D. M.- Grimes October, 1956 EP-i86

ACKNOWLEDGEMENTS The writer wishes to express his appreciation for the help of Mr. Paul Nace, Mr. Ralph Olson and Joan Kuiper. This work was carried out under the auspices of the Engineering Research Institute. The initial stage of the work was supported by the U. S. Army Signal Corps, the balance by the U. SO Air Force Office of Scientific Research and Development Command. ii

REVERSIBLE PROPERTIES OF FERROMAGNETS The magnetic moment of a piece of ferromagnetic material is equal to the weighted-averaged value of the cosine of the angle between the direction of magnetization of the sample and that of each atomic magnetic moment. If it be assumed that the atomic moments, even though grouped into domains, remain oriented according to a Boltzmann distribution,'23 quantitative relationships between the magnetization and the reversible properties of ferromagnets can be given. Let r be a dimensionless parameter proportional to the applied magnetic field H plus a history dependent constant. Under these circumstances, M 1 M cos G = ctnh - - L () (1) s The reversible susceptibility parallel to the biasing field is then given by: w M cos 3 G w dL(j) ( rpas H o d(2) The parametric relationship involving the r was given as early as 1911 by Gans. The derivation of the Boltzmann distribution was put on a more quantitative basis by Brown,l15 in the late 30's. Although reasonable agreement with experiment was noted in many cases, as was pointed out by Tebble and Corner6 in 1950, the susceptibility is not a single valued function of the magnetization particularly when going from one hysteresis loop to another. In 1954 Grimes and Martin7 extended the work of Brown5 to obtain the expression for low frequencies that: M cos 0 w s L(1s ) 3'X3X p (3) rt (H + D) where D is a history dependent term with the dimensions of H. Early this year 23 it was shown that equations 2 and 3 would be approximated for the case where the reversible susceptibility has its origin in domain-wall motion. It was also shown that for the case where the susceptibility is due to the rotation of all the moments of a domain that the susceptibilities were given by: r3 2X. r(1- G) =o3 r L(X) (4) nrp -2 1o

and for low frequencies or infinite material~ r 3 r 3Kr os/\) 3 r (1 L ) (5) -~ r t= 4 (1 +Cos 9) = ( (5)o The exact form of the parametric equations depends upon the material remaining always oriented in "easy" crystallographic directions, thus they can be considered to have quantitative significance only for M < ~ 0.5 Ms s In a forthcoming paper3 it is shown that the variation of the susceptibility matrix with magnetization when it results from domain rotation is given by: (1 + cos 9), K cos iG 0 K L(), 0 4 2 = - K os, 2 X ( + os) K 2 (1 X ) O (6) 0, o', 3 (o os o 3 X i In the usual form for which the matrix is written for microwave application the susceptibility term? - - %22 =3. The off-diagonal terms are proportional to M, a fact previously pointed out by Rado. In the same paper, the differential magnetostriction "d" for each of the four cases where equations 2, 3, 4, and 5 are applicable is described. The results are based upon the equation: 2 Xs (Cos - (7) where Xks = X10 = 11l1 is the saturation magnetostriction and 9 is the angle between the applied magnetic field and the magnetization. The results are that "dit for the case of domain-wall motion and parallel fields is given by2d =H 3 da~ dL( (8)

for domain-wall motion and transverse fields bydW 2o 3Xs (cos _- ) -(r ( ) t 2 (H + D) 3 for domain rotation and parallel fields bye 9 Xs/ r3 dr 3 ) dr 9 s 0 (cos G ) 3 (o ( (10) p 2 M5 and for domain rotation and transverse fields by: r 9 s 3 Xr dr d= (o os 9 + cos ) = [L()( + ) - (11) t 4 Ms.2 where Ms Independent of the exact form of the distribution, so long as the effective magnetic field can be considered as the sum of the applied field and a history dependent field, the wall-motion susceptiblities are related by: d ew w w 1d _ + 1 0rt (13) rp'rt n dr Thus if Xw is a monotonic decreasing function of M, then Xw must also rp rt be monotonic decreasing if the two susceptibilities are equal when M 0. Dependent only upon the applied magnetic field being much smaller than the anisotropy field, the following relationship holds: 3 tr 2Xr + r (14) o 0rt rp So if rP is a monotonic decreasing function of M, Xrt must be monotonic increasing function. These differences point out a new technique for the separation of the magnetization mechanisms. Namely, the measurement and comparison of the parallel and transverse reversible susceptibilities -3

Fisgure 1 shows the expected variat;ion of the wall-motional suseeptibilities according to equations 1, 2, and. 3; Figure 2 shows the expected variation of the domain-rotational susceptibilities according to equations 1, 4, and 5; Figure 3 shows the expected wallmotional differential magnetostrietions according to equations 8 and 9; Figure 4 shows the expected domain-rotational differential magnetostrictions according to equations 1, 1.0, and 11. Note that the largest value of d is expected from drt and remember no quantitative correlation is expected for M > ~0~5 Ms, Experimental data have been taken on several ferrite samples to compare with the theoretical c-urves. Figire 5 shows suseeptibility data from specimen F-6-2, which was made in our laboratory by mixing to the composition Nio682o o992Zno5326Fe204, firing at 1375~0 for 1i/2 ho-urn 12000~C for 2 hours and then slowly cooling —-all in a N2 atmosphere. The shaded area of Figure 5 represents the area between the susceptibility curves calculated from equations 2 and 3 and from assuming all moments to be either parallel o3r antiparallel with the applied magnetic field, From the data of Figure 5 it is concluded that the susceptibility of F-6-2 is very predominantly due to domain rotation. To cheek this point the relaxation frequency was calculated from the equation: f a 2Ms (15) and found to be on the order of 150 me/seco Experimentally, the peak in the loss portion of the susceptibility was found to be P40 mc/sec,9 and the frequency at which the suseeptibility was one-half of the znitial va.lue was 160 mc/sec, verifying the rotational susceptibility mechanisme Figure 6 shows a similar curve for co-'re i-15.lo.It is from a batch of material prepared by Dr. Do Fresh at the UoSo Bureau of Mines, College Park, Maryland. The frequency dependence of this samrple has been measrued by Rado10 et.al,, and will be analysed by him at the forthcoming London conference,~ The composition is assumed to be the same as that of their "Ferrite Ftt" Mg 97Fe 03Fe204o The measurement of the reversible susceptibility of this particular material was initiated by request of Dr-o Rado. The conclusion from Figure 6 is that one-half or more of the'nitial susceptibility is due to domain wall motion, Figure 7 shows still another similar set of data on core F-10-2. F-10-2 was fired with F6-29 but is believed to be of the composition 13 Ni3800Fe C1282Co0330Zn 4588Fe2~L. From the data of Figure 7 it Is inferred that over one half of the suaseeptibility is due to domain-wall motion. Figure 8 shows a comparison of the experimental differential magnetostriction data of Bozorth and Williams11- wth the results of equations 1, 8, and 12,o do was taken from their values for Bs and Xs while o was -'4"

taken from Bozortho12 There have been no arbitrary scale correctionso Equation 8 is deemed superior to their corresponding equation for (a) as they pointed out, they used a domain-rotation model where the susceptibility was due to domain wall motion, and (b) they substituted cos 93 for os3 G. In conclusion, I wish to thank Dr. Rado for making their manuscript available to me before the conference. -5

REFERENCES 1. Brown, W. F., Jr., Phys. Rev., 52, 325 (1937). 2. Grimes, D. Mo, Thesis, University of Michigan, 1956; Bull, Am, Physo Soc. 1, 25 (1956). 3. Grimes, D. M., Submitted to Physics and Chemistry of Solids, 4. Gans, R., Phys. Z o, 12, 1053 (1911). 5. Brown, W. F., Jr., Physo Rev., 54, 279 (1938). 6. Tebble, R. S., and Corner, W, D., Proc. Phys. Soc. 63B, 1005 (1950). 7. Grimes, D. M., and Martin, D. W., Phys. Rev,, 96, 889 (1954) o 8. Rado, G. T., Phys. Rev,, 89, 529 (1953) 9. The frequency measurements were made by the High Frequency Impedance Standards Section, National Bureau of Standards, Boulder, Colorado. 10. Rado, G. T., Folenr, V. J,, and Emerson, W. H. To be presented at the October 29 to November 2 London Conference and published by the Institution of Electrical Ergineers. 11o Bozorth, R. M., and Williams, H. J., Rev, Mod, Physo, 17, 72 (1945). 12. Bozorth, R. M., Ferromagnetism, D. Van Nostrand, 1951, p,541. 13. Jefferson, C. F,, Tech. Rep. No. 66, Electronics Refuse Group, Engineering Research Institute, University of Michigan, June, 1956o -6

Figure 1o Theoretical reversible susceptibility due to domain-wall motion~ Curves 1, 2, and 3 are for transverse fields curves 4, 5, and 6 are for parallel. fields, Curves 3 and 6 are for six possible directons of orientation of the magnetic moment in the s igle cr'ystal. Curves 2 and 5 are for eighbt possible directions, and curves 1 and 4 are for an infinite ntrlmber of suc.h dir.-ections.

i.6, r.2 1.0.8.6.4.2.2.4.6.8 10 1.0.8.6.4.2 0.2.4.6.8 1.0 M/Ms

Figure 2. Theoretical reversible susceptibility due to domain rotation. Curves 1, 2, and 3 are for transverse fields, curves 4, 5, and 6 are for parallel fields. Curves 3 and 4 are for six possible directions of static orientation of the magnetic moment in the single crystal, Curves 2 and 5 are for eight possible directions and curves 1 and 6 are for an infinite number of such directions.

- Q _ _ _ _ _ 0 0 c~i. ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I _t I I_ 10 (1 C I I I II -0,

~.O.18.... ct ca c — 0 CI CD cnN'd t — I-,C+/ ~ —-,, ~ dtCDC 0' ct -.1 —-- dW _,: o CD HJ ~ L o d — C+ o') - f,,'.03 - - H D HH F- P 0 Pi (D t-om ct cd ~ 0.I. 2..3.4.5.6 7.0 o ~~~~~~M/M (D (1) i-~ (D ct C.4 (Dc~

a o (D QD ~o.6.5 1.5 0 0.5~~~~/ 3 P-6 d I p) IP H C D bt~ft.' P i (LEF(RIGHT SCALE) d kip" d.3 d I-f 0 (), (D c_ Hd c~ ~~ 4 12O I m' 0/ d O) tj r ( L EFT SCALE) _ _ dId _ _ _ 9d FPt ctO. ~,~d.: 5 CD ct(D HO) 0~0 ~~~~~~~~C~~~~~~~d~ Pi 0 2.3.4 5.6.7 8.9 M/Ms 0~~~~~~~~~~~~~~ ct CD )-j. I-I. ct

Figure 5. Figure 5a. represents the variation of the reversible susceptibility with magnetization. Curves 2 and 5 represent the experimental data taken on Core F-6-2 for transverse and parallel fields respectively at both 10 and 320 kc/sec. Curves 1 and 4 represent the theoretical curves for transverse fields for domain rotation and domain-wall motion respectively; curves 4 and 7 represent the variation for parallel field.s, domain rotation and domain-wall motion respectively. Curves 3 and 6 represent transverse and parallel fields for all moments either parallel or antiparallel with the applied field when the susceptibility is due to domain-wall motion. The arrow indicates the direction of the change in magnetization. Figure 5b. shows the corresponding theoretical curves, curve 2 represents the symmetrical part of the experimental transverse field data, curve 9 represents the antisymmetrical part; curve 4 represents the symmetrical part of the experimental parallel field data; curve 8 represents the antisymmetrical part.

1.5 1.4 1.4 1.3 a 1 ~-".. l l.~ I- I. I 0.2 1.0.8.6.4.2 0.2.4.6.8 1.0 0.1.2.3 4.5.6 7.8.91.0 M M Ms Ms

, F-9-,E o5 o aarrE; Xtq paqslvqsnTT' W~t~ s9 T-5T-I uacOoads ao$'s2p aurs aq; sxuasaaxdaa axr2Tu sT;'9 amruT, II;6/.)~-s, W,>, ~ ~, 0 N) I I I0 __ __ _1! _ I i/__ o, "1 1 1 1 o!b

0 Oj 1 1 I I II'I 1r 1,.0-8 -6 -4 -.2 0.2.4.6.8 I.0.2.4.6.8 1 * a f.M, _ M_ p MS M god78 -.6 -.2 0.2.6.8 1.0 - 4 ctu'Msf Ms 0D Ct+

-o(00 l -J 1 X(0 w a- q._1 X OD - 0( N - 0 0 Figure 8. A comparison of the theoretical curve for the variation of the parallel differential magnetostriction due to domain-wall motion with that reported by Bozorth and Willieams for Permalloy 45. There have been no arbitrary scale corrections.