AFOSR TN 59-272 2L95-13 T ASTIA AD 212 706 T H E UNIV E R S TY OF MI C H I G A N COLLEGE OF ENGINEERING DeparKment of Electrical Engineering Solid State Devices Laboratories Technical Report No. 8 MIAGNETIC PROPERTIES OF POLYCRYSTALLINE MATERIALS D -,Mo Grimes RoDo, Harrington A6L6 Rasmusseni UNRI TProojc.t 2h95 under contract withg AIR FORCE OFFICE OF SCIENTIFIC RES;E.RCH AIR RESEARCH AND DEVELOPMENT COI AND SOLID STATE SCIENCE DIVISION CONTRACT NO, AF 18(603)-8 administered byo THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR February~ 1959

Qualified requestors may obtain copies of this report from the Armed Services Technical Information Agency, Arlington Hall Station, Arlington 12, Virginia. Department of Defense contractors must be established for ASTitA services, or have their Ineed-to-knowt certified by the cognizant military agency of their project or contract,

FOREWORD Messrs. R.Do Harrington and A.L. Rasmussen are with the High Frequency Impedance Standards Section of the National Bureau of Standards, Boulder, Colorado. The spectral data were taken by them,

ABSTRA CT The variation of the magnetic Q with internal magnetization is discussed using both the domain rotation and the domain-wall motion model of magnetization change. The variation of the reversible susceptibilities with magnetic moment is reported on four samples, and the results are compared with results from the frequency spectra in the initial and remanent states. The distribution of magnetic moments in the system as a function of angle between individual and averaged moments is discussed in terms of an infinite series expansion in Legendre polynomials0 The coefficients of the first four terms can be measured0 Experimental data are given for the first three.

INTRODUCTION The magnetic susceptibility is defined to be the ratio of resultant magnetization to magnetic field intensity. In ferrimage nets it is a function of both field strength and frequency. The reversible susceptibility is that susceptibility effective with a differential magnetic field. The differential field can be superimposed upon a finite biasing field. The reversible susceptibility depends upon the angle between biasing and differential fields, upon the magnetization level in the material, and upon the magnetic history of the specimen. The susceptibility-producing mechanism is not completely understood. It is often assumed that either domain moments rotate in unison or that the domain walls move to change the net averaged moment of the sample. The reversible susceptibility and resulting differential magnetostriction for each mechanism were discussed and contrasted in an earlier paper, which will be referred to as I21 In that papers the reversible quantities were discussed in terms of the distribution of magnetic moments. The calculations of predicted behavior of the reversible quantities with magnetic moment were carried out with the assumption that the moments would be distributed in some most probable state determined by a Boltzman type distribution of moments about the unit sphere, This argument has the merit that the direction of the-moments are strongly influenced by the crystallographic orientation, and that the orientation of neighboring grains can be considered to be independent0 This does not means however, that the resulting moments must be randomly

oriented. A factor of considerable interest, therefore, is a calculation of what the distribution of moments is in a magnetic material. The detailed calculation of f(G) is possible, at this time, in only the simplest cases. The basic difficulty is that the solution of this problem must involve the solution of essentially the same number of coupled differential equations as there are atomic moments. If the moments are treated in the aggregate, localized potential minima between neighboring grains and neighboring domains must be included, as must the effect of nonmagnetic inclusions. The dominant role played by this type of surface energy has been stressed by Goodenough.2 The calculation of f(0), then, requires first the development of as realistic as possible a microscopic model of magnetic behavior followed with the distribution of this behavior over the sample, The microscopic models used in I were (A) that the domain moments remained always oriented along the crystallographic directions which minimized the anisotropy energy, and (B) that the static moments remained along "easy" directions, but the susceptibility arose by the rotation of the domain moments away from those directions in the dynamic case, Neither model is completely correct. Both are highly idealized and, for example, contain neither the "curling"t concept discussed by Brown3 and by Frei, Shtrikman and Treves,4 nor the effect of inhomogeneous fields. A more correct formulation awaits the inclusion of more accurate microscopic models. There hasl however, been ample precendent set for considering magnetic susceptibility as due to either domain rotation or to domain wall motion, This lamentable state is continued in this paper,

It was assumed in I that the frequency of the measuring field was much less than the resonance or relaxation frequency of the magneto Under these conditions, the magnetization dependence of the transverse susceptibility on the biasing field is distinctly different for each of the assumed susceptibility sources as long as the anisotropy field remains much larger than the applied field8 The reversible susceptibility, as defined above, is reversible in the thermodynamic sense of zero energy dissipation for only the special case of zero applied frequency. An associated energy loss exists for all nonzero frequencies. This loss is describable as the imaginary part of the complex magnetic susceptibility, ( AX 2K iX")o The magnetic Q is defined to be the ratio of real to imaginary susceptibility. It is convenient in the following section to discuss the loss in terms of Q. Q dependence upon magnetization, field directions, and source of susceptibility is briefly discussed in this paper. Frequency spectra have commonly been used to investigate the susceptibility sources, This extensively used method depends upon the different dependence of the frequency of susceptibility falloff on magnetization mechanism to distinguish between sources,5,6,7y8 Some investigators have also measured the spectra of the material magnetized to remanence and have utilized these results to aid spectral inters pretation99910s11 Epstein12. and Birks13 in particular have utilized spectral information to obtain quantitative determinations of the amount of wallmotional and domain-rotational susceptibility. 1 V P -% %+A I Crn A n n "Y -% V- e It

DISCUSSION Distribution Function f () The macroscopic material is considered composed of randomly oriented crystallites. The crystallites are sufficiently small so that the moment of each crystallite is much less than the aggregate moment of the sample. The crystalline orientation of each grain is taken to be independent of the orientation of its neighbors. For such a model the static moment is always parallel with the static field, The distribution f(Q) sin 0 dQ is defined to be the fraction of theatomic moments in the system oriented at an angle between 0 and 0 + dg with respect to the applied magnetic field. It is convenient to expand f(O) in an infinite series of Legendre functions such that: f(G) k=, AnPn (cos o) (1) n-o where An is a function of the magnetic field present and the magnetic history of the specimen. By definition, 3d&Lcosn Q f(g) <cosn > =(2) fd fEf (() where <cosn 0> represents the average value of <cosn 0) over the sample. Substituting Equation 1 into Equation 2 and integrating over the unit sphere yields: cosn dm()o (3) 2A0 m=o A-1 d( Upon solving for the coefficients Am in Equation 3, using the orthogonality of the Legendre functions,

Am Am (2m + t) <(P(cos 9)> (4) O Thus the coefficient of the m th term in the infinite series expansion for the distribution function can be determined if <cos 0>, <cos2>,,o.' (cosm > are known. So, of course, the first coefficients are: A1/Ao = 3 <cos )> A2/Ao = 5/2 [3 <cos2 > 1 (5) A3/Ao = 7/2 [5 <os3 g> - 3 <cos >] So the distribution function can be experimentally determined if <cosm Q> can be measured. The Am coefficients are functions of both magnetic field and magnetic history. The coefficient A1/A0 can be determined from the M-H loopo The coefficient A2/Ao can be determined from Equation 5 and the magnetostrictionl915 and independently from Equation 9 and 10 of I where it is stated that the- susceptibility due to domain rotation is given by~ (1 - <'cos2e> ) X p 2 X (l <cos2 >) ) x 3 Xl O2 8 (6) 3 4 )(1 + <Cos2 ~) A3/Ao can be determined from a knowledge of A1/Ao and the use of Equations 16 of I, relating the differential magnetostrictions, namelys dp = do( <cos > - <cos3 > ) (7) d L do( <cos > + <cos3 0> ) where d= =O NT_

Magnetic Q. (Domain Rotation) To estimate the magnetization dependence of the magnetic Q when the susceptibility arises from domain rotation, it is convenient to start with a single crystal ferromagneto The domain rotation effects are assumed to obey the differential equationl6 at M at at = _M x H) a M x _(8) where Mo is the magnetic moment, y is the magnetomechanical ratio, Ms is the spontaneous moment, H is the applied magnetic field, and a is a dimensionless parameter proportional to the power loss. The total sample moment is assumed to be the sum of moments from all crystallites. It is thus necessary to first calculate intergranular effects, then to sum up all moments over the polycryptalline sample. Park8 showed that if the magnetic moment of the grains neighboring a given grain averages that of the gross material, then grain size and particle interaction is described by a single constant po p is defined by the equation H = h - pM (9) where H is the effective differential field, h is the applied differential field, and M is the gross magnetization~ p is a function of the localized packing and remains essentially independent of M. p can be regarded as the demagnetizaing factor of the grain partially canceled by the moments of its neighbors~ Upon substituting Equation 9 into Equation 8 and using the additional assumptions that a large static biasing field is oriented

in the Z-direction and sinusoidal time dependence of an additional applied field, the reversible susceptibility matrix is given by(Equation 8 of I.) [1 + <cos2 0> (X + XQ+), 2 <cos o> (C> X+), o X: -2 <cos Q> (X - X+), L1 + <cos2 >j (C_ + i+), O (10) 0, 0, 2 [1- <cos2 o>j (X_ +) where < > represents the average value in the polycrystal and 0 represents the angle between the spontaneous moment and the applied field. The susceptibilities X _+ are defined by the equation Mx + + = M Hx et iHy where Mx and Hx are the components of the differential magnetization and field in the x direction and i = -1. The i operator represents a spatial rotation of 1T/2 radians, The resultant algebra from combining Equations 8,9, and 11 for the Q of the elements on the diagonal of Equation 10, yields: Qr o= O) (..2). (12) axw 2 1 + o2 (1 + a,2) where co is the applied radian frequency and eo = YHto Ht, in turn, is the sum of the static applied and anisotropy fields, Hap and Han, and the effective internal field pM o Thus: Ht= HapH + pM (13) -7

In the low frequency limit:. Qr i * (1L4) ac) Note that Qr is independent of field direction. The low frequency initial susceptibility is obtained by combining Equations 5 and 10 of To The result is: Xo 3@0- (15) where cl = ~Ms. Combining Equations 14 and 15, X Qr - 3_~ (16) 0 3aco (16) The product depends upon a and directly measureable quantities, and does not explicitly contain the biasing magnetic field through coo Magnetic Q. (Wall Motion) The change in magnetic moment due to 180~ wall movement is, in unit volume, given by: AM =L2MsAkxk (17) k where Ak is the area of the k th wall, and xk is the distance through which it is moved by an applied magnetic field H. The wall movement is presumed to be governed by the differential equation: dx d2X 2t~oMsH = fkxk + f3tk + id2k (18) where fk is the restoring constant for the k th wall, P depends upon the structure insensitie properties of the material and the parameter a of Equation 8, while m depends only upon the structure insensitive

properties of the material, po is the permeability of free space. Combining Equations 17 and 18, the wall motional reversible susceptibility under sinusoidal excitation is found to be: (1 - e2 m_) _ j-' /fk In the low freqtuency limit, the real susceptibility and the Q become: o s~Ak-2 Ak k fk QW 1 LAkfk (a) ~ _ ZAk/fk Thus for a given material at a fixed low frequency, the Q varies only with ~, Ak, and fko The reversible susceptibility is given as a function of manetization not only by the detailed mechanistic Equation 20 above, but also by Equations 2 and 18 of I. From these, if Xw = Xw = X for p t 0 virgin material with M = 0 and if Xp decreases montonically with increasing M, then so must Xt~ Indeed the susceptibilities are related by the equation: CX w....... * (22) d lnM -9

If the average stiffness term fk increases with magnetization, then so will Qo Since both Ak and fk vary with field orientation, the parallel and transverse Qts will, in general, differo This feature is distinctly different from domain rotation. Initial and Remanent Susceptibility The initial susceptibility due to domain wall motion has been approximated by Bozorthl7 for sinusoidal internal strains arising from magnetostrictive forces present when annealed material is cooled below the Curie temperature. The result is X'= E (23) where aus is the saturation magnetostriction and E is the Young's Modulus of the material. Likewise the initial susceptibility due to domain rotation can be approximated as X 2_oMS2 (2h) 3K1 where o is the permeability of free space and K1 is the first order anisotropy constant. From Equations 23 and 24 it is apparent that in pure material the relative importance of the two susceptibility mechanisms depends upon the relative magnitudes of the effective anisotropy and magnetostrictive energy densities in the material. For material with susceptibility arising by domain rotation, the relationship between the initial susceptibility and the resonant frequency should satisfy the equation:

f 2s (25) o 2Tr 3X~ where y is a constant, taken to be 2.21 x 105 m/amp-sec, fo is the resonant frequency, Ms is the spontaneous moment of the material, and X is the initial susceptibility. Becker and Doring18 discuss a model of magnetic remanence whereby, as the field is decreased from the saturation value to zero, the moments rotate to occupy the same l"easy" crystallographic directions occupied in virgin material, except that all components initially antiparallel to the saturation direction become parallel. For this model, the remanent magnetization is O,5 Ms and the remanent rotational reversible susceptibilities equal the initial rotational susceptibility. This model is valid for hexagonal material with K1> 0 Fomenko9 used it to interpret his permeability spectrum results, If this is also the position of maximum parallel and minimum transverse field susceptibility, then going around the M-H loop the parallel susceptibility peak occurs for M decreasing in magnitude. The remanent and initial susceptibilities will be equal. The rotational model of gross flux change in cubic crystals would predict moments oriented in the "easy" directions nearest the field directions at remanence. For this case the remanent and rotational susceptibilities taken from the results of I are given in Table 1 for all anisotropy coefficients except K1 equal zero. (These are the expected values from reference 1 for ~.-) ). -11

Table 1 Remanent Results, Mr/Ms Cubic Cubic Hexagonal Hexagonal K 0 K1 > 0 K1 1<0 K1 >O M/Ms.500.866 o831 0785 XrX r 1oO00 o 366.449,500 7( Xr 1o000 1,318 1,276 1.250 An analysis of the relationships between remanent conditions and the reversible susceptibility has been carried out by Frei and Shtrikman19 on the assumption that the reversible susceptibility is due to domain rotation, Their analysis contains the assumption that the reversible quantities are due to rotation and that the expansion of magnetization in terms of applied field must obey the rotational equations through second order in the ratio of applied to anisotropy fields, as can be seen from their Equation 10. Their resulting Equation 29 can be put in the formo (26) s 2M- (Xp X+ 2 Xt)2 for comparison with experimental values, EXPERIMENTAL Samples Measured Four samples were subjected to detailed measurements and are reported here. The compositions of the samples are listed in Table 2. All except the magnesium ferrite sample were fabricated at the University of Michigan~ =12

Table 2 Composition of Ferrites Surveyed Des gnation Composition F-1-2 Ni L67Zn. 33Fe204 F-6-2 Ni.168Zn o533 Co 299Fe20 AA-107-h Fe5Y3012 I-15-1 MgFe204 The samples with designation starting F-l and F-6 were prepared by first ball milling the C.PE oxides weighed to the desired composition in acetone, then decanting, drying and pressing into a toroidal pill. They were then heated rapidly to 11500 C, then heated slowly to 13750 C for one-half hour, then slowly cooled to 12000 and held for two hours. The furnace was then flushed with nitrogen and deenergized. The I-15-1 core was prepared by Dr. D.L. Fresh and is from the same material reported on previously by Rado, et al, as their type F core, The AA-107-4 core was prepared by firing in air at 13500 C for four hours. The mixing and pressing procedure was the same as for the type F-1 and F-6 cores. Procedures and Techniques All measurements were taken at ambient temperatures on toroidal samples, since this geometry avoids the complexities of demagnetizing factors. Coaxial line techniques were utilized for the spectral measurements. The complex susceptibility of the F-6 samples was measured from 1.0 to hO mc using the radio frequency permeameter in conjunction with a Q meter, For the F-1 samples the real susceptibility from 0,1 to ho me and the imaginary susceptibility from 0.1

to 1.5 mc was measured using the permeametero A fixed length coaxial line and a radio frequency bridge was used for the remaining higher loss measurements up to 50 mco From 50 to 5000 mc, variable length coaxial cavity methods were used for complex susceptibility measurements on both types of samples. The susceptibility and Q for the variable field measurements were taken with a Q-meter. For the transverse field measurements, the sample was placed between the pole faces of an electromagnet. The biasing field was along the axis of the toroid, Girdle windings about both outer and inner periphery of the toroid were used to determine the field and magnetization~ An extra winding was placed about the toroid to furnish the biasing magnetic field when the parallel field measurements were taken. F-6 Sampleso Two different samples of F-6 material code designated F-6-1 and F-6-17 were measured for the frequency spectra, The initial spectrum of F-6-1 is shown in Figure 1o Both initial and the remanent, parallel spectra of F-6-17 are shown in Figure 2. All spectra show but a single resonant peak in the real susceptibilltyo The resonant frequency as determined by the peak in the imaginary susceptibility is seen to be about 130 me for F-6-1 with a corresponding low-frequency susceptibility of 470, For F-6-17, the resonant frequency is about 75 me with an initial low=frequency of about 82. The remanent loss for F-6-17 also peaked at 80 mc, with a low frequency susceptibility of 75, An instability of a few percent was noted in both samples between different measuring runso Comparing the initial and remanent curves, it is apparent that the positions of the peak in both real and imaginary susceptibilities -1L4

60 T 71 — T1 IT 50, F 640 30 z I I 00 O 10 000~1 10 00 FREQUENCY, Mc Fi gure 3

F-6-17 100 _ x INITIAL STATE I ~ ~~~~~~ # so.o CS ~6~~-~~~i~~~~~~~1REMANENT STATE x 60h- x x 40H K~~~~~ 20 10. 100 1000 10000 FREQUENCY, Mc Figure 2

I..... i i i i.i -t - - l - 1 - I- i- - - -i"O -8 -.6 -4 -2 O *2 4 6 -.8 1.0 Fi-ure 3,~~ ~ ~ ~ ~ ~~/s. PARLLE AD TANVERE IEL SSCETII~iIE, CREF- ~ ^' —" I.........F.g.reI

occur at essentially the same frequency, and that the ratio of remanent to initial susceptibility is about 0.91. The variation of the low-frequency susceptibility with magnetization is depicted in Figure 3, These data were taken on sample F-6-2, The transverse susceptibility passes through a decided minimum near M = 0. Both the measured Q's are essentially constant over a wide range of magnetization, and indeed were never observed to rise to a value equal to twice the initial value, F-l Samples, The spectra of the sample code designated F-l-5 was tested, Figure 4 shows the complex susceptibility for both initial and parallel remanent states, The initial spectra of F-l1-6 was also measured and found to be similar in all respects to F-l-5. The resonant frequency of F-l-5 was measured as 4,5 me as determined by the peak in the imaginary value, The low-frequency susceptibility was measured as 5610 The ratio of remanent to initial value of susceptibility was 0,66, The peak in imaginary susceptibility at remanence occured at about 6 mc, The susceptibility was measured as a function of the magnetization on specimen F-1-2, see Figure 5, For this core, the transverse susceptibility remained essentially constant over a wide range of magnetization values, The Q's pass through a minimum near M = 0, As in the case of F-6-2, the transverse susceptibility remained always larger than the parallel value, The Q's increased rapidly with increasing field to a value more than 10 times the initial value, I-15-1o The frequency spectrum for this material has been published by Rado, Folen, and Eserson,20 and corresponds to their "tFerrite F"t, They conclude that the lowfrequency susceptibility is predominately -18

600 5000 X' F-1-5 INITIAL STATE 400.......A........ REMANENT STATE <x 300 - X. < I200 100 x:sy o......A...A......x 0.1 1.0 10 100 1000 10000 FREQUENCY, Mc Figure 4

,~~~~~~~~~~~-!....... I~~~~~~~~~~~~~~~~~~~~~~~~~_ - - -oq-...... -0.2....,.-.1....' -,-,,-!, I I...10.0 1.0.2 A ~~~-AA - - 8.0.8 Q+x Q Q, AJ~~~~~~~x _ Q6 fi O- &.6 XO - - / --.-...../ a20 -2 -EQ0 -;8 -.6 -4 0.4.6.8 1.0 M/Ms — PARALLEL ANDTRANSVERSE FIELD SUSCEPTIBILITIES, CORE F-I-2.': g.re 5

41W. I ~ ~ ~.-0 A ~*1 C -I ~ xo 0 rD - 2 0.2.4 6..8 1.0 PARALLELL AND TR~ANSVERSE FIEL SUSCEPTIBILITIES,- CORE 11 Figwre 6 i ~~ ~ ~ ~ ~ ~ ~ - - -e -4 -2 0.2 4,,.6.81. ~~Yi'/' ~,,~ V/iiPAALE AND, TRNVES FIL SUCPIIIIS CORE. 1-5 Figi~~~~~~~~~~~~re 6 ~ ~ ~ ~ ~ ~ ~,

due to the movement of domain walls. The variation of the susceptibilities with magnetization is shown in Figure 6, For this core, both the susceptibilities pass through a maximum in the vicinity of zero moment while the Q's pass through a minimum~ AAD1074, The frequency spectra for this material has been published by two of us21 It was concluded on the basis of spectral information that the susceptibility was predominately due to domain wall motion, The variation of the susceptibilities with magnetization is shown in Figure 7, The transverse susceptibility does not pass through a minimum INTERPRETATION OF RESULTS The values of fo measured experimentally are listed in Table 3, along with the values calculated using Equation 25~ The experimental value of fo was determined by the frequency of the peak value of imaginary susceptibilityo Table 3 Resonant Frequency, mic X (M-O) Sample Type Measured Calculated | pHO Xp(H=O) F-61l 130 179 F-6-17 75 102 o 91 F-1-2 4o5 15 ~66 The expected variation of the parallel susceptibility with magnetization can be computed from the transverse susceptibility dependence. For the case of wall-motional susceptibllity, Equation 22 is the proper -22

....AiC, ii i --, ii i~., II.... r_3.2 3.0 I8L2,i J... A~~~~~~~~~~~~~~ i' k'" -- J,,,!.,,_ Q t o /' I t-2.4~ - 1,0 -J-8 o 00 a6,.~0 I. ~~~LO~~~ -..8-4..20..2.6,8.1.0 PARAL.LEL AND TRANSVERtSE FIELD SUSCEPTIBIlmES, CORE -AA-107-4 Figure 7

equation. For the case of rotational susceptibility, Equation 10 of I is the proper equation. For Equation 10 to be useful, however, the variation of X o with applied field must be known, From Equation 15 and the definition of,o' it follows that in the absence of a detailed knowledge of the effective anisotropy field, which is always the case for polycrystalline material, X2 is known only so long as the biasing field is small compared with the anisotropy field. The magnetic parameters of the four cores are listed in Table h. Table 4 Magnetic Parameters of Four Ferrimagnetic Cores (320 kc) Parallel Transverse Core Xo Q0 Xpo Qpo Xpr pr XtoQto Xtr Qtr F-6-2 - - 65 72.h 39.6 71.7 52.8 60.8 62.0 61.h F-L-2 410 15.2 388 13.2 135 28.1 450 13*5 448 16,7 I 15-1 - 54.2 38.1 3303 53.9 54o2 39.5 43.6 49.8 AA-107-4 36.h 10.9 31.1 7.4 30.0 8,2 36,6 10.4 40~0 9.7'"Values at 500 kco''The subscript t"o" indicates measurements in the virgin state, t"p' indicates parallel and "t" transverse measurements. "tpo"t and "to" represent measurements around the N-H loop at the position M=0 in parallel and transverse fields. "tpr" and "tr" represent similar measurements at the remanent position, Plots of X p computed from Xt for each of the four samples measured are shown in Figures 8 through 11, and compared with the experimentally measured curves, It is apparent that the rotational curve fits F-6-2 well and that the wall-motional curve fits I-15-1 well. The fit is not good for either of the other specimens, but nonetheless the -24

- -EXPERIMENTAL CALCULATED: WALL MOTION 2.0 I-. —'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ____-DOMAIN ROTATION 1L6 N /1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 4 -6 — 4 -.2 0 -.2.4 -6.8 m/m s — )l PA~RALLEL REVERSIBLE SUSCEPTIBILITIES, COR E F- 6 - Figure

t /.CALC LAT.. 4 t.. t-i 1/ DOMAIN ROA6ON I 8-1.0 -. -.6 -4 -.2 ) 2 /4.6.8 1.0 M/Ms*PARALLEL REVERSIBLE SUSCEPTIBILITIES, CORE F-I-2 Figure 9

1~' I t I ~~~~~~~~~~I t I I I t, I If. /,~~~~~ — EXPERIMENTAL CALCULATED-: ---- WALL MOTION -- \!'*~.~ -~ —-DOMAIN ROTATION,I/ 4:-4 ~m// //~~~\ -Jiri _/.6.2,4 8 PARALLEL REVERSIBLE SUSCEPTIBUTIES, CORE AA- 107- 4 Figur~e 10

a\ I i I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I I I - "'-' 1.2,~/ 8.. — EXPERIMENTAL 66 CALCULATED x -— WALL MOTION x x, 0 --— DOMAIN ROTATION.4 ~ t~~~~~~~~~~~~ 12 7 -140 -.8 -%6 -.4 -.2. 0 2.4.6 8 1.0 M/Ms-10 PARALLEL -REVERSIBLE SUSCEPTIBILITIES, CORE I15 -1 Figure IEN

wall-motional curve fits AA-107 closer than the rotational curve, The conclusion that cores F-6-2 and I-15-l have susceptibility arising predominately from rotation and wall motion respectively is substantiated by the spectral interpretation of Figures 1 and 2 and by Rado et al2 The curve calculated for C p for rotation is valid so long as the anisotropy field remains much larger than the applied baising field. F-6-2 contains cobalt, and therefore will have a large anisotropy field. Core F-1-2 is a mixed nickel-zinc ferrite with a corresponding small anisotropy, Thus the lack of agreement with rotational curves is not significant. For the F-1-2 core, there was a slight but real minimum in the Xt in the vicinity of zero moment. This feature is not consistent with wall motion and the accompanying Xp behavior. Therefore it is concluded that domain rotation must be present. No such behavior was found in AA-107o In the spectral data, two peaks in the imaginary susceptibility Twoere observed for AA-107, and only one for F-1-2, There must, therefore, be two important mechanisms for the AA-107, These mechanisms are assumed to be rotation and wall motion. For F-1-2, one can only say that domain rotational effects enter, If wall motional effects are present, the resonant frequency must occur at the same point as the rotational effect, The essential difference between F-1 and F-6 lies in the value of anisotropy, Since rotational effects have been shown to exist in the high anisotropy material, it is expected that they should also exist in the low anisotropy material. Material F-6-2 apparently fits the conditions imposed' in this paper for the static conditions, namely oriented along easy crystallographic directions, and the susceptibility obeys the rotas tional equations~ This material would not satisfy the conditions of

Frel and Shtrikman, Table 5 compares the values of remanent moment calculated from Equation 26 with experimentally measured values. Table 5 Specimen Measured Calculated HC Ms Mr Mr/Ms Mr/Ms x 10 x 10 AA-107-4 o389.259 160 1.06 0.41 F-6-2.660 o041 127 3.58 2.37 i-1-2.610.330 100 3.67 2.24 I-15-1.730.,675 175 1.35 0.98 The agreement is not good, and indeed the best agreement is with the material whose susceptibility arises from wall motion. Upon eliminating Xo from Equation 6 the average value of < cos2 O> is found to be: 2 2 2X -t p <cos2 0> =. (27) 2 Xt + P Solrving for <c os3 03>0 from Equations 7, 2dt + dp Note that (cos2 0> should vary from 1/3 in the demagnetized state to unity when the material is saturated, A similar wall-motional equation yields2Xt Xp 11- 2 -3(dX 2Xt-X~ dn - (29) 2Xt+Xp 3-2 (dln>t -30

If the values of At and M are calculated assuming a Boltzman distribution of moments, then 1t - AMsL(Tn)/r and M = MsL(T)) where A is a constant for the material, L(n) represents the Langevin function of Ar, and q = AHt where Ht is the totalized biasing field, including history, anisotropy, and applied field. Substituting these values into Equation 29 yields a function which goes from 1/3 for M = 0 to unity for saturation. Plots of 2 t X- p as a function of <cos Q> are shown in 2Xt + Xp Figures 12 and 13. In Figure 12, the theoretical curve is the value of < cos2 Q ~ calculated for domain rotation. In Figure 13, the theoretical curve is based upon domain-wall motion. Both curves are based upon Boltzman distribution of moments. The rotational curve attributed to domain rotation can be interpreted in terms of variation from the most probable condition. For F-6-2, as the moment is decreased, moments parallel and antiparallel are initially in excess of the most probable condition. However, as M is further decreased towards zero, the parallelantiparallel components increase to larger than the most probable value. This can be considered the reason for the transverse susceptibility at zero moments being larger than the initial susceptibility, and the corresponding parallel susceptibility at zero moment being smaller than either. Sample F-1-2, if the susceptibility is assumed entirely rotation, maintains at all times a predominate parallel-antiparallel moment configuration. The value of A2 from Equation 5 as a function of magnetic moment can be read directly from Figure 12 for sample F-6-2o More detailed determinations and analyses of the distribution functions on other materials must await simultaneous parallel and transverse susceptibility and differential magnetostriction measurements. -31

F-I2 /~/ *01~ ~~1 F-6 THEORY 27 2L(ii s L( ). 33_ -.8 -.6 -.4 02.2..6.8 M/Ms.~ VARIATION OF- <COSa e> WITH < COS 0 > Figure 12

\ AA-107-4 / - l \ \6 +/ / - 40,.5 \ \4- THEORY 2 L(7) dL(vL7 ) |_'\ /-d7 ii vsL( L) 2 L +dL(w)| -8 -6 -.4 -2 0.2.4.6.8 M/Ms - 2xt xp VERSUS M/ Ms 2t + XFigure 13 figure 13'

CONCLUSIONS The magnetic properties of a ferromagnetic system can be described in terms of a distribution of magnetic moments about the unit sphere. This function cannot be calculated in detail for macroscopic systems. However, assuming idealized models of magnetic behavior such as magnetization by domain-wall motion or magnetization by domain rotation, some reversible properties of ferromagnets can be compared without detailed knowledge of the distribution function. If a detailed distribution function is assumed, then, of course, the magnetic properties can be calculated in detail. The inverse of this calculation can also be made. The procedure consists of firstly expanding the distribution function in an ininite series of Legendre polynomials. Due to the orthogonality properties of these functions, the coefficients of each term in the infinite series can be evaluated if the weighted average value of the proper power of < cos Q> over the sample is known. < cos Q > itself is, of course, proportional to the magnetic moment of the sample. Under certain conditions <cos2 G~ is proportional to the static magnetostriction. It can also be found from a knowledge of the two reversible susceptibilities if domain rotation is the source of the susceptibility~ Further, under the same susceptibility conditions, the differential magnetostriction can be utilized to find <cos3 Q> and one additional coefficient in the expansion. The combination of frequency spectra and magnetization dependence of the susceptibilities on the same types are utilized to examine the source of the susceptibility, In agreement with other investigators, -314

it is found that the major susceptibility source depends upon the type of ferrite, as well as the method of preparation. The nickel zinc cobalt ferrite F-6-2 shows distinctive domain rotational effects. Sample I-15-1 shows wall-motional effect. The authors wish to acknowledge the invaluable assistance of Messrso R.M. Olson, AoHo Voelker, and Mrs. PA. Marchello, -35

BIBLIOGRAPHY 1. Grimes, Do.M, "Reversible Properties of Polycrystalline Ferromagnetse I. Theory of the Expected Variation of the Reversible Properties with Magnetization,t J. Phys Chem. Solids 3., lhl -152 (1957) 2. Goodenough, JB., "A Theory of Domain Creation and Coercive Force in Polycrystalline Ferromagnetics," Phys. Rev. 95, 917-932 (1954). 3. Brown, WF., Jr", t"Criterion for uniform Micromagnetization," Phys. Rev. 105, 1h79-1182 (1957). 4. Frei EoH,, Shtrikman, S., and Treves, D., "Critical Size and Nucleation Field of Ideal Ferromagnetic Particles," Phys. Rev, 106, 4hh6455 (1957). 5. Snoek, JoL, "Dispersion and Absorption in Magnetic Ferrites at Frequencies above One Mc/s, Physica 1, 207-218 (19L8), 6, W1ijn, H.P.J., Gevers, M., and Van der Burgt, CM., "Note on the High Frequency Dispersion in Nickel Zinc Ferrites," RIev. Mod. Phys. 25, 91-92 (1953). 7. Brown, F. and Gravel, C.L., "Direct Observation of Domain Rotation in Ferrites," Phys, Rev. 98, 4L2-4L8 (1955). 8, Park, D,, "Magnetic Rotation Phenomena in a Polycrystalline Ferrite, t Phys. Rev. 97, 60-66 (1955). 9. Fomenko, LA,, "An Investigation of Magnetic Spectra of Solid Solutions of Some NiZn Ferrites in the RadioFrequency Range," Soviet Physics JETP 3, 19=28 (1956)o 10. Harrison, S.E., Kriessman, C.J,, Pollack, S.R.,'tMagnetic Spectra of Manganese Ferrites," Phys. Rev. 110, 8L-84h9 (1958). 11. Rado, GOT.,. Wright, R.W,, Emerson, W.H., ttFerromagnetism at Very High Frequencies. III. Two Mechanisms of Dispersion in a Ferrite," Phys. Rev, 80, 273-280 (1950). 12, Epstein, D J,, "Domain Wall Relaxation in Ferrites," Boston Conference Proceedings, h93-503, T-91, AIEE, February, 1957, -36

13. Birks, J.B, t"Magnetic Spectra," Proc. IEE, Part B Suppl. No 5, 8179-188 (L956). lho Brown, W. Fo, Jr., tDomain Theory of Ferromagnets Under Stress, Part II. Magnetostriction of Polycrystalline Material," Phys. Rev. 53, 482-491 (1938), 15. Lee, EoW, "Magnetostriction Curves of Polycrystalline Ferromagnetics,tt Pro. Phys Soc. Part 2, 72., 249-258 (1958), 16o Gilbert, TL, "The Phenomenological Theory of Ferromagnetism," Armour Research Foundation, 1 May, 1956. 17, Bozorth, R.Mo, Ferromagnetism, D. Van Nostrand, New York (1951), p. 822. 18, Becker, R, and Doring, Ferromagnetismus, Edward Bros., Ann Arbor, 1943o 19. Frei, EoHo,, and Shtrikman, S., "The Remanent State in Ferrites According to the Rotation Model," Boston Conference Proceedings, p. 504-511, T-91, AIEE, February, 1957, 20. Rado, GoTo, Folen, VoJ., and Emerson, W.H,, "Effect of Magnetocrystalline Anisotropy of the Magnetic Spectra of Mg-Fe Ferrites," Proc. IEE Part B, Suppl. No, 5, 198-205, (1956 21. Harrington, R.D, and Rasmussen, Ao,L,, "Initial and Remanent Permeability Spectra of Yttrium Iron Garnet," Proc. IRE 47, 98 (1959)o _ _ _ M~~~f=_7

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