ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR REVERSIBLE SUSCEPTIBILITY IN FERRIMAGNETIC MATERIALS Technical Report No. 24 Electronic Defense Group Department of Electrical Engineering By: Da M.' imes Approved jby: /" D. W. artin H. W. Welch Project 1970 TASK ORDER NO. EDG-6 CONTRACT NO. DA-36-039 sc-15358 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 29-194B-0 May, 1954

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv 1. INTRODUCTION AND DEFINITIONS 1 2. THEORETICAL DEVELOPMENT 2 2.1 The Initial Susceptibility 2 2.2 The Parallel Reversible Susceptibility 6 2.3 The Transverse Reversible Susceptibility 7 3. EXPERIMENTAL METHOD 12 3*1 Specimen Shape 12 3.2 General Procedures 13 3.3 Parallel Case 14 33.31 Susceptibility 14 3.3.2 Magnetization 19 3.24 Transverse Case 19 3.4.1 Susceptibility 19 3.4*2 Magnetization 22 4. EXPERIMENTAL RESULTS 28 4.1 General Properties 28 4.2 Susceptibility versus Magnetization 29 4.3 Errors 39 5. EXTENSION OF THEORY 39 5.1 General Discussion 39 5.2 The Irreversible Magnetization 41 5.3 Modifications of the Reversible Model 45 ~53.1 Contribution of Metastable Volume 45 5.3.2 High M/M8, decreasing IMI 45 5.3.3 High M/M, increasing IMI 47 6* COMPARISON WITH EXPERIMENT 49 6.1 The Parallel Case 49 6.2 The Transverse Case 50 7. ACKNOWLEDGMENTS 51 APPENDIX 52 BIBLIOGRAPHY 59 DISTRIBUTION LIST 60 ii

LIST OF ILLUSTRATIONS Page Fig. la, lb, and lc Possible Mechanisms of Magnetization Changes 3 Fig. 2 Specimen Energy vs Wall Spatial Coordinate U Fig. 3 Relationship for Determining the Transverse Reversible Susceptibility 7 Fig. 4 Theoretical Parallel Reversible Susceptibility vs Magnetization 9 Fig. 5 Theoretical Transverse Reversible Susceptibility vs Magnetization 10 Fig. 6 Effect of Toroidal Geometry on M-H loop 18 Fig. 7 Distortion Due to Toroidal Geometry 19 Fig. 8 Compromise Curve for Parallel Averages 20 Fig. 9 Test Core Windings 25 Fig. 10 Graphical Construction Xrt/Xo vs M/Ms 27 Fig. 11 Sample Transverse Magnetization vs Magnet Current 30 Fig. 12 Reversible Susceptibility vs Magnetization, 1-8, 250 31 Fig. 13 Reversible Susceptibility vs Magnetization, G-5, 25~ 32 Fig. 14 Reversible Susceptibility vs Magnetization, E-3, 250 33 Fig. 15 Reversible Susceptibility vs Magnetization, E-3, 45~ 34 Fig. 16 Reversible Susceptibility vs Magnetization, E-3, 75~ 35 Fig. 17 Reversible Susceptibility vs Magnetization, E-3, 100~ 36 Fig. 18 Comparison of Susceptibility Loops 37 Fig. 19 Susceptibility vs Magnetization for Different Peak Magnetizations 38 Fig. 20 Spatial Coordinate of a Domain Wall 42 Table I f(n) for Isotropic and [111] Oriented Material 8 Table II MIMs and Xr/Xo vs? for f(r) = L(nl) and f(q) - E(n) 11 Table III Magnetic Parameters of E-3, G-5, and I-8 29 Table IV M/Ms and Xr/Xo vs' for f(n) = tanh n 46 Table V M/ns and X r/Xo vs r for f(r) -( 1 ) iii

ABSTRACT Expressions for the reversible susceptibility both normal to and parallel with an applied biasing field are given. The theoretical basis for these expressions is considered. Experimental data for three different ferrite specimens are given. Qualitative deviations from the reversible equations are considered assuming domain wall motion to be retarded by large potential holes. The averaging effect of toroidal geometry is considered in some detail. iv

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN REVERSIBLE SUSCEPTIBILITY IN FERRTITAGNETIC M.ATERIAL 1. INTRODUCTION AND DEFINITIONS The parallel reversible susceptibility in ferromagnetic material has been the subject of several authors since it was discussed by R. Gansl in 1911. A recent paper which includes a review of the experimental status is that of Tebble and Corner2. The reversible susceptibility Xp is customarily defined ast X = lim aM ( arp AH -- 0 AH- where AH has a sense opposite to that of the change in H which brought the specimen to the point (M,H). The differential susceptibility, X d differs from the reversible susceptibility in that AH is taken in the same sense as the change in H which brought the material to the point (M,H). X d is, in general, larger than Xrp. It is assumed that if a vanishingly small ac field is applied such that AH is alternately parallel and antiparallel with the aforedescribed direction of H the resulting susceptibility will be the parallel reversible susceptibility. The transverse reversible susceptibility, Xrt, is hereby defined as: Xrt H imO ( (2) -. 1'

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where AH is perpendicular to the direction in which H was changed to bring the specimen to the point (M,H) and thus also perpendicular to the direction AH used to measure the parallel reversible susceptibility. The initial susceptibility X is defined ass Xo = M_ o Xr (3) 2. THEORETICAL DEVELOPMENT 2.1 The Initial Susceptibility Since the magnetic moment per unit volume is given by S.H (40* M = ZE H g (H all atoms evidently changes in M occur by means of changes in the net spin S aligned in the direction of the field. Three possible mechanisms of change are illustrated in Figs. la, lb, and lc. In Fig. la, before the application of an external field a domain wall is located at the position x. When the field is applied the wall moves to a position x + Ax. All atoms between x and x + Ax must undergo a 180~ reversal in direction of their magnetic moment. In Fig. lb the solid arrow may rotate to the position shown by the dashed arrow. For the same AM as in la, a much larger number of atoms must rotate through a small angle Q. Fig. lc shows still another possibility. *. is the Bohr magneton, g is the Lande g factor, S is the net ionic spin ih units of i, H is the applied magnetic field, 2

',I / I / x x + Ax FIG la FIG lb 1^/ FIG Ic POSSIBLE MECHANISMS OF MAGNETIZATION CHANGES 3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The question of whether the low frequency initial permeability in ferrites arises from wall movement (Fig. la) or rotation (Figs, lb, Ic) was rather convincingly answered by Rado3, et al. They measured the spectrum of permeability versus frequency for a ferrite, then ground it to single-domain-size particles and remeasured. They concluded that the low frequency initial permeability is primarily due to wall movement as shown in Fig. la. The variation with the position of a domain wall in the energy of a ferromagnetic body appears similar to Fig. 2. We now wish to determine the relationship between the shape of the potential curve and the initial susceptibility. z> w W CL | <Ln SPACIAL CO-ORDINATE NORMAL TO WALL FIG 2. SPECIMEN ENERGY VS WALL SPACIAL CO-ORDINATE In the absence of an external field, the wall will come to rest at a minimum energy position such as xl. If the energy function is assumed to be a continuous function of position it can be expanded into a Taylor seriest V(x) = V(xi) + 1p (x-xl)2 (5) where Pr is the radius of curvature of V(x) at the point x = x1. When a reversible field is applied, the potential V*(x) is given by: V*(x) = V(x) - y Mx dH (6) for a wall of unit area. y is a constant depending upon the type of wall considered.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Combine Eqs. 5 and 6 and differentiate to obtain: d _ 1_ (x-x) - Ms dH (7) Pr The minimum energy condition is then: x - x1 = Pr YMs dH (8) The magnetization and susceptibility associated with this initial reversible field for a wall of unit area is then: dMr = M (x-xl) Pr 2 M2 dH Xo Pr 2 MS (9) The problem of evaluating Xo thus becomes one of determining Pr and the wall area located at each potential minimum. Becker and Doring4assumed a system of cubical domains, 90~ walls, and a sinusoidal variation in internal stresses and obtained: Xo =. (10) 3TXs o-i where Xs is the saturation magnetostriction and oi is some average value of internal stresses. This is found to agree quite well with experiment both in parameter dependence and in magnitude. However, this does not substantiate an assumption that X O must have its origin at least predominately in wall movement for Kersten obtained a result: ---------— 5 — i

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 9 Xsa Ci assuming only rotational processes. 2.2 The Parallel Reversible Susceptibility Gans first suggested the parametric equations M X =f(s ); X 3f(N) (12) S X, where f(n) = L(n), the Langevin function: L(n) = (ctnh n - n ) (13) d The prime indicates d. It was found that this equation fit quit3 well the data for iron and nickel. This was first put on a semitheoretical basis by Brown6 in 1938. To do this Brown assumed ferromagnetic material to consist of N domains per unit volume all of fixed and equal volume. These domains were considered to be subjected to random but fixed forces in such a manner that the ordinary techniques of statistical mechanics could be applied. His development of Eq. 12 is given in the appendix. He later7 attempted to derive these equations with a model allowing variation in domain size. However a large number of assumptions were necessary. Some are so stringent as to question whether the work is really an extension of his original papers. 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2.3 The Transverse Reversible Susceptibility If the parallel reversible susceptibility is given by Eq. 12 where f(n) is any arbitrary continuous function of ), Xrt/ X can be developed as follows: dM dHrt Hrt M FIG 3. RELATIONSHIP FOR DETERMINING TRANSVERSE REVERSIBLE SUSCEPTIBILITY In a multicrystalline specimen when Hr is rotated through a small angle,.+dM) remains parallel to Hr + dHrt. Thus, see Fig. 3, (Hr is defined in the appendix). X dM, M (1i) Xrt- d ir Hr Now 1 dHr d M X r dM dM X rt where the first equality is the definition of Xrp, and the second follows from Eq. 14. Substituting for Xrp, M, from Eq. 12, and using dM = Ms f'( T) dn, 1, ( 3Xo 0 dn X rt --------------------— ~~~~~, 7 --------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN or, upon integrating X rt = 3 f()(15) Xo T since Xrt = Xrp = Xo when M = 0. Brown obtains Eq. 12, from which Eq. 15 follows, for spherical and cubic symmetries. The form F(I) varies with the exact anisotropy. Table I shows the result for two such assumptions (see appendix for derivation). TABLE I f(-r) FOR ISOTROPIC AND [11l] ORIENTED MATERIAL Anisotropy f(T) isotropic L(T) 1 ctnh - [111] E(nr) = 3 fV / tanh p[d'2 o The functions L(r) and E(n), for small values of r, can be expanded as follows' L(TI) 3 X - 2 2 TI 7 + e....... (16) E() = - + - 7 17 +........ These expressions are identical up to r7. Fig. 4 shows two plots of the parametric Eq. 12 where in the upper curve f(Ti) - L(T), and in the lower f(r)) = E(T). Fig. 5 shows Eq. 15, once again in the upper curve f(Tr) = L(n), in..... -- 8.............8 -

f( f77f()=L(X)7) =___ _ E (_\ ^_____|__.8 |l.6 Xrp 3f'(1,) Xo.4i ___ _ _ _ (0 f('r) = E(T)) __________________^.2 1.0 -.8 -.6 -.4 -.2 0.2.4.6.8 1.0 M = f(() Ms FIG 4 THEORETICAL PARALLEL REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION

fWrl s L(i?) ss L (77)~~~~~~~t f(7.6 Xrt 3f('TN.4 -1.0 -.8-.6 -.2 0 *4 M=f Ms FIG 5 THEORETICAL TRANSVERSE REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE II IM/Ms and Xr/X vs r for f(L) = L(n) and f(r) = E(P) T _n L(n) 3L'() 3 () E() 3E'(r) 3 E(X 0 0 1.000 1.000 0 1.000 1.000.450.148.960.987.148.960.987.650.211.920.974.211.920.974.820.262.878.959.262.878.959.960.302.840.944.302.840.944 1.09.338.801.930.338.801.930 1.22.371.761.912.371.761.912 1.35.403.719.896.403.719.896 1.47.431.680.880.431.680.880 1.60.460.640.863.460.640.863 1.73.487.600.845.487.600.845 1.86.512.561.826.511.537.824 2.01.539.519.804.538.516.803 2.15.562.481.784.560.480.781 2.31.586.441.761.585.435.760 2.49.612.400.737.610.390.735 2.68.636.360.712.634.354.710 2.90.661.329.684.658.309.681 3.15.686.280.653.682.267.650 3.44.711.240.620.706.225.616 3.82.739.200.580.731.177.574 4.32.769.160.534.757.135.526 4.98.799.120.481.782.096.471 5.35.813.105.456.793.078.445 6.10.836.081.411.809.054.398 7.10.859.059.363.824.036.348 8.10.877.046.325.833.024.309 9.20.891.035.291.841.015.274 11.00.909.025.248.848.009.231 13.00.923.018.213.853.06.197 15.00.933.013.187.857.003.171 20.00.950.007.143.861.001.129 30.00.967.003.097.86.000.086 75.00.987.001.040.866.000.035 1.000 0 0.867 0 0..11 —----------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN the lower f(') = E(P). It is evident that both equations give identical results for the two functions at low values of magnetization. Since E(r) is obtained for the function f('r) in Eqs. 12 and 15 under the assumption that the magnetization vectors can lie only in [11] directions, we expect the true behaviour to approximate this case only at low fields. At higher fields domains will begin to leave [1l1 directions so as to more nearly align with the field, and we expect the behaviour to approach that of the isotropic case. Therefore, since there is little difference between the two cases at low fields where the anisotropy is important, we expect that f(nr) = L(Tj) should be a good approximation throughout. Numerical values are given in Table II. 3, EXPERIMENTAL METHOD 3.1 Specimen Shape In their commonest applications ferrites are generally used in the form of toroidal cores. The advantage of this shape is that closed flux paths around the ring eliminate surface poles and macroscopic demagnetizing fields. However the magnetic properties of such a core still contain certain shapedependent factors superimposed upon the properties of the core material itself, for which it is not always possible to correct in an unambiguous way. For this reason most studies in which the properties of materials are to be measured have used other special shapes adapted to the particular experiment. For instance the parallel susceptibility is often measured on very long, thin, cylindrical specimens. The shape best suited to one particular measurement may be highly impractical for another, as is the case for the "parallel" and "transverse" reversible susceptibilities. Yet it is desirable that both these measurements be _________________________ 12 _____...

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN done on the same specimen if a comparison is to be significant, because it is very difficult to obtain reproducibility with ferrites between even supposedly identical specimens. Specimens of different shapes, even if made from the same batch and treated identically as far as possible, might have quite different properties. The toroidal shape eliminates the demagnetizing effects for fields directed around the ring, while at the same time the lateral extent of the specimen is confined so that uniform perpendicular fields may be more readily applied. For this reason we elected to perform all of the present measurements on toroidal cores. The special difficulties which the shape introduces are taken up as they arise in the following discussion. 3,2 General Procedures The specimens are cycled manually around their major hysteresis loops by a battery-powered do field, hereinafter called the "bias" field. Magnetization as a function of this field is deduced from flux changes measured with a General Electric fluxmeter. The whole B-H loop is read point by point by increasing the bias field quickly from zero to the desired H, reading the flux changes, then continuing up to saturation and on around the major loop back to zero. This entire cycle is repeated for each point, so that cumulative errors in the fluxmeter readings are avoided. An array of fixed resistors and copper knife switches is used to control the bias field in order to provide quick positive changes in the driving current and maximum reproducibility. While the bias field is held steady at each point, a 5 Kc/sec alternating field of amplitude of order 10-3 oersted is applied to measure the reversible susceptibility. A signal generator output passes in series through a high impedance, a non-inductive decade resistance box, and the core winding, A VTVMI 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN is provided to measure voltage across either the decade box or the winding. The first serves to measure the current, proportional to the applied field, while the second measures the vector sum of the small IFR drop in the winding and the induced voltage, proportional to the ac flux in the core. It is assumed that the induced flux is in phase with the current since other studies in this laboratory and elsewhere show that losses in ferrites are negligible at 5 Kc/sec. The measured permeability is deduced from the ratio of these two voltages, although in practice the decade box is always adjusted to make the ratio unity to remove the dependence of the result on voltmeter calibration. The high impedance in series serves to make the current independent of the VTIV connections and of the value of the decade resistor. The signal generator is operated at low output level to assure a pure waveform. Further decrease of the signal voltage is found not to effect the value of the result, assuring that the signal is small enough to read the true limiting incremental permeability for zero applied field. In these experiments the incremental field for measuring susceptibility is in all cases applied along the toroid by a toroidal winding. For the parallel case the bias field is applied in the same direction by means of another toroidal winding, while for the transverse case it is applied parallel to the toroid axis by means of an electromagnet. When measuring the parallel susceptibility it is necessary to maintain an effective infinite impedance in the primary circuit to the measuring signal. This was accomplished by placing an iron core choke in series with the primary windings. 3.3 Parallel Case 3.3.1 Susceptibility. Discussion. The essential difficulties with this ase arise from the fact that the magnetic field produced at a point within a 14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN toroidal winding varies inversely with radius of the point from the toroid axis. So the magnetization and susceptibility vary with radius at given applied fields, and the observed values of these quantities are averages over the cross sectional area. Given the experimental values of the averages, it is not in general possible to solve explicitly for the unaveraged quantities to be compared with theory. Instead we must calculate geometrical averages of the theoretical curves, which can then be directly compared with the experimental data. However, we need to know how the magnetization varies with position in the sample to compute these averages. The only quantity whose geometrical dependence is known is the applied field H. We must make an additional assumption about the relation of M to H before the calculation can be done. Thus we cannot make an ideal check of Eqs. 12 and 13 independent of any additional assumptions with data obtained from toroidal cores. However, it is found that average X vs. M curves computed under several rather different simple assumptions for the dependence of M on H are similar in general features. In the following discussion, a compromise curve is arrived at which is believed to have qualitative validity. Averaging Equations. In measuring magnetization, we measure a total flux 0 = f BdA = AB, where B = A JBdA is the arithmetic mean of B. So if one defines H =- HdA and =1 - (B- H), then r2 i 1 N;LfMdA=(-) fMdr (17) M A J(Ar2 rl)' I r1 is the simple arithmetic mean of M over the cross section. 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The averaging of susceptibility involves a weight factor. Experimentally we observe a ratio of two voltages. One is proportional to the average incremental ac induction amplitude AB, the other is proportional to the average incremental ac field amplitude AH. The observed average incremental permeability is thus not the average of the ratio AB/ AH, but is the ratio of the averages AB and AH. So: --- /~dr - = fA=dA= 1. f~~ —-— T -(18) AH AHdA f A HdA f! dr r since A H is proportional to l/r. Then trivially, the susceptibility -X - A dr r l _-dr r 711 / l1dr r r Ir So 1 X = 1 -- dy (19) n y y r1 where a = r2 Theoretical Average Curves, Two extremely idealized assumptions for the dependence of M on H, which we believe bracket all our actual cases, are: (A) Let the M-H loop be a parallelogram -. ( - 1) for - 1< < + 1 s c here Ms is the saturation magnetization, Hc the coercive force, and j3 is the ratio f the remanent magnetization to M......_______________________ 16 ___________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (B) Let the M-H loop be a pair of laterally displaced Langevin functions M = L H + 1 where K is a constant. Each assumption contains one parameter, which must be assigned some definite value in the calculations, and which is to be chosen to fit the ideal loop to some feature of actual ones. The ideal loops were fitted to the criterion that the remanent magnetization is MR 0.2 Ms. P 5 S in (A) K = 0,615 in (B) Another parameter occurring in the calculations is the ratio a = rl/r2. We take a = 1/2 for definiteness, which is the nominal ratio for all of our cores. Fig. 6 shows the averaged magnetization i/Ms vs. H/Hc for these two assumptions (solid lines). The approach to saturation of these two curves clearly represents opposite extremes to the general features of observed curves. For (A) the ideal relation itself (not averaged) is shown, where the averaged curve differs from it, by the dashed line. For (B) this difference is everywhere too small for the separate curves to be distinguished on this scale. The given curve is the averaged one, and the curve A represents the difference M/Ms - M/MI on an expanded scale. For each of these two assumptions we have calculated the integral X/X o q. 19 for several values of H/Hc, using Eqs. 12 and 13. The results are plotted gainst Ti/Ms, taken from Fig. 6, together with the ideal relation itself, in 17...

-n-.0 ^'' -- O -- /-Ai O _ _ _ - - / Aav Ms.6 Ba, ov~~~~~~~F _ _ _ \ \\^A___ ___ _ _Hc oo -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 4 16 -.4 AB x40 FIG 6 EFFECT OF TOROIDAL GEOMETRY ON M-H LOOP Ai, Bi: IDEALIZED BEHAVIOR FOR INFINITE SPECIMEN AOv Bv: AVERAGED BEHAVIOR FOR TOROIDAL GEOMETRY Li = Bovay - B

.8.2 -1.0 -.8 -. -.4.2..4 6.8 0 M Ms FIG 7 DISTORTION DUE TO TOROIDAL GEOMETRY IDEAL CURVE AVERAGE OVER TOROID for -- for M =L615 H.615

N..6 Xrp /' X~ -1.0 -.8 -.6 -.4 -.2 0.2.4.6.8 1.0 M FIG 8 COMPROMISE CURVE FOR PARALLEL AVERAGES

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Fig. 7. It is seen that the averaged curves each differ from the ideal curve in essentially similar respects. The labor of preparing such a theoretical curve from an M vs. H curve especially fitted to each experimental case is felt to be excessive, since the integrals yield only to numerical methods. Since the results of these two extreme assumptions are so similar, it is felt that a "compromise" curve based on them has enough qualitative validity to suffice. Fig. 8 shows such a curve, and this is the "theoretical" curve presented for comparison with experimental data in all the graphs of Section 4.1. 3.3.2 Magnetization. The magnetization was measured in accordance with the description of Section 3.2. A special difficulty arises only for the measurement of the saturation magnetization. The measurement of the saturation magnetization is inherently difficult. It must be known for to compare the susceptibility curves with the theory it is necessary to normalize the magnetizations. Our method of measuring Ms was to trace out the B-H loop to as high an applied field as could easily be applied around the toroid. This was about 225 oe. The data were corrected for the finite wire size of the B winding, then Ms was plotted against H. A Langevin function was fitted to the two highest points, and from this a value for Ms was calculated. This value is quoted in Section 4.1. 3.4 Transverse Case 3..1 Susceptibility. There is no a priori distortion of the susceptiility measurements in this case, even though they are made in the same direction nd with the same toroidal windings as before. For now the bias field, which is applied parallel to the toroid axis by an electromagnet, is nominally uniform. he resulting magnetization is assumed to be essentially homogeneous so that the esulting susceptibility has a constant value throughout the sample. Therefore, 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN even though the incremental field AH still varies with radius the resulting incremental induction AB is everywhere proportional to it, and the ratio of their averages is the same as the ratio of their values at any point. Note that Eq. 19 reduces properly in agreement with this: f dr X dr X 1 1 = x (20) i ldr Accordingly, the theoretical curve appearing on all experimental plots for the transverse case is just that given by Eqs. 13 and 15. 3.4.2 Magnetization. Demagnetizing Factor. The essential difficulty of the transverse case is the measurement of the magnetization when a non-zero demagnetizing factor is present. The true internal field H is the sum of the applied field Ha and the demagnetizing field NM, i.e., H = Ha - NM. As always, 4wM = B-H = (p-l)H. Thus: H Ha [H N B (21) 1 N 1 a NL a 4NTH N and (V-1) Ha t4M = (~-1)H p ) Ha (22) 1 + (~-1) nd for the apparent magnetization Ma, defined by 4TMa' (B - Ha) one obtains? Ma-' (1-i) M (23).......___________________________ 22 _

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN When the "demagnetizing factor" T4 approaches unity, as for a thin disk magnetized normal to its area, and if L is fairly large, then: (a) From Eq. 21, Ha A L H = - B. To produce a given field H in the sample will require an applied field many times larger, in fact of the same order of magnitude as B. Further, the true field may be essentially unknown experimentally because it is given as a small difference of large terms, (b) Although Eq. 22 shows that the sample can still be magnetized by a sufficiently large applied field, the experimental apparent magnetization will approach zero according to Eq. 23. For magnetization parallel to the axis our toroids appear as disks with a length/diameter ratio of 1/4 or less, for which the demagnetizing factor is approximately 0.8. However, if the field on the core is applied by an electromagnet whose pole faces are close to the core surface the effective demagnetizing factor becomes smaller and should ideally become zero if the pole faces make perfect magnetic contact with the core surface. We have measured the apparent saturation magnetization of a core as a function of the electromagnet gap and found that it increased sharply as the gap approached zero. The curve became so steep that its limiting value at the ideal zero was rather indeterminate. Therefore this method is not suitable for the determination of saturation magnetization. However, one sees from Eq. 23 that the ratio of the apparent magnetization Ma to the true magnetization M is given by (1 - N) which is assumed to be constant independent of field strength for a fixed geometry. Thus for a given Ma Msa oint - where the subscript s denotes the saturation value. So, if the core M M, s is subjected to a given applied field one can determine the value of s- correctly 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN even though he has no idea what the true magnetization and true net field might be. If then one measured X~ simultaneously, the plot of these pairs of value xo contains no dependence on the demagnetizing factor. This simplification is, of course, subject to the assumption that N is actually constant so one should choose the geometry to keep N as small and the applied field as homogeneous as possible. Measurement of Magnetization. The magnet used in these measurements was built with accurately parallel plane pole faces about O0% larger than. the test cores and a continuously adjustable gap. The core windings are made flat and uniform, and the magnet gap is closed down so as to clamp the core in place. The windings on a test core are shown in Fig. 9. The cores are provided with a girdle winding embracing the core cross section only, designated the "B" winding. A second flat winding on a teflon plug fits into the hole in its center and is called the H winding. The flux measured by the H winding, divided by its area and number of turns, is the measure of the applied field Ha. The B winding, although of No. 34 wire, actually embraces an area several percent larger than the core cross section. This area carries a flux density equal to Ha, and since Ha is of the same order as B this extra flux must be subtracted from the total measured by the B winding. The driving current of the magnet is cycled about a given set of values, as described in section 3.2, while the flux changes in the B and H windings are separately measured by the flwnmeter. Then 4nM, = (B - Ha). The magnet is itself an iron system having some hysteresis and remanence o both the applied field Ha and the flux density B in the core will have certain emanent values -I and -p respectively as the driving current passes through zero |hile increasing. The fluxmeter gives the relative values (Ha+Y)) and (B+p) for 24

H WINDING /L WINDING V "* " ^-z2- J — 1^ ^ 1'1 — "/ FIG 9 TEST CORE WINDINGS 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN each point of the loop, so the apparent M then also contains an additive constant V. (B + ) - (Ho + ro) = Ad (Mo + v); hUv = -o A typical core reaches saturation at several times smaller a field than does the magnet, so that the cycle of current values used is only a minor loop for the magnet itself. There is no assurance that this minor loop is symmetrical about the origin so the constants n and 3 are essentially not known without some absolute field measure. Any sort of probe to make such a measure would interfere with optimum geometrical arrangements. One can in principle evaluate v by plotting a loop of Ma vs. magnet current extending to saturation in both directions and choosing the zero of Ma halfway between the two saturation values. However, it is more convenient in these experiments to evaluate it in the following way. A plot of incremental permeability vs. magnetization gives in general a different curve, as M increases from zero to saturation along the major loop, from the curve obtained as M decreases from saturation to zero. These two curves necessarily meet and cross at M = 0 for the whole graph must by symmetrical in the two opposite directions of M. Thus the place where the "ascending" and "descending" experimental curves of ha cross each other identifies the zero of M. The experimental procedure goes as follows. The magnet current is taken repeatedly around a fixed cycle of values, starting always at zero and using always the same peak values but stopping in between at successive intermediate values. Flux readings relative to the starting point are read from both the B and H windings for each of these intermediates with the fluxmeter, and Xr is read at each point from the toroidal winding as in Section 3.2. The corrected difference of the B and H readings is hT(Ma + v ), obtained as a function of magnet current I. l ________________________ _- 26

W I IX I 1 1S 111 W1_ 2 \ \ <~ 0.M0 27 w~~~~~~r - -— GRAPICALCON -RUCTON - S --- ----. - - -- ~~~X Ms Q > _______________ __ ___________ _ __________ 2 7 _ _ 0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN he data for Xr is plotted vs. I to determine the current I1 at which the two ranches cross. The two values of (Ma + v ) for this current (one with ascending current, other with descending current) must then be equal but opposite in sign, hich gives an equation for v. One then determines from the data the value Io of current at which Ma is zero on the ascending curve. The value of Xr on the scending curve at the current Io is Xo. Msa is determined from the experimental Ma X r alue of (Ka + V)s. The data are renormalized to give and —, each vs. I. Mal Xo Finally these values are plotted against one other to give the result. Fig. 10 ummarizes this procedure graphically. Note that points are obtained for only one ign of the applied field. We have measured points on the other side in a core r two as a check, and have found that the method of determining v is valid. 4. EXPERIMENTAL RESULTS.1 General Properties. Data were taken on three different ferrite specimens that were purchased from the General Ceramics and Steatite Corporation, Keasby, New Jersey, and represent their types E, G, and I. Room temperature data only are given on cores G-5 nd 1-8, data at about room temperature and at 25~ intervals to 100~ C on core -3. A summary of the measured values is given in Table III. H is the coercive force in oersteds, HI is that applied field at which he parallel susceptibility attains its maximum value. Hc is always of the same ign as, but is always less than Hc. Values of Hc, Hc and remanance Mr/is are givn for measurements around the toroid. It would have been desirable to know H' for he transverse susceptibility but unfortunately our data does not yield this information. 28

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE III MAGNETIC PAPAiETERS OF E-3, G-5, and I-8 Parallel Transverse ~lr a Sample Temperature H HX M MsI 5 0 max s 0 max 00..~C oe oe cgs cgs I-8 25.115.081.371 216 89.8 94.4 177 84.4 86.4 G-5 25.260.22.383 270 30.5 31.0 214 29.1 30.2 E-3 25.595.38.431 294 37.0 39.3 296 37.1 37.5 50.590.48.377 261 40.6 41.9 263 39.4 40.0 75.524.30.342 232 40.7 43.2 236 41.4 42.0 100.467.30.325 207 41.7 44.2 206 42.4 43.0 4.2 Susceptibility versus Magnetization. A sanple curve showing 4n(Ma + v) versus current applied to the electromagnet is shown in Fig. 11 for the transverse field measurements. The value of Msa is obvious from the figure. The experimental curves showing the variation of the reversible susceptibility with magnetization for both parallel and transverse fields are shown in Figs. 12 thru 18. Fig. 18 is a replot of Fig. 12 and the Xrp vs. H curve from which Fig. 12 was plotted. The solid lines represent the experimental data, the dashed lines the theoretical curves. It is to be noted from Figs. 12 thru 17 that the maximum value of both susceptibilities occurs for decreasing IMI before M reaches zero. Except for the region of the maximum the experimental curve is lower than the theoretical curve. Fig. 18 is plotted to emphasize that the parallel susceptibility always has a larger value for decreasing than for increasing IMI. On the contrary there is a crossover in the transverse susceptibility loop, the curve for decreasinglMI being higher at low fields and lower at high fields. 29

(/) 16 0 0 -8 320) /~ ~30 16 FIG I1 0:> 30

....L- - -\-.8 -.6 M Xr.4 _ l/,7 -.3, -1.0 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 0.1.2.3.4.5.6.7.8.9 1.0 M Ms FIG 12 REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION f EPo TRANSVERSE EXPERIMENTAL v. —— V PARALLEL GC-I-8; 25~C TRAN THEORETICAL. — PARALLEL

____________ __ 11!__[.._..... __ - 0 0 _ —— /^f^ — 9 - ^ ^ —.6 --- yi"/r9v'-. vKo., / / / i, /,X M FIG 13 REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION EXPERIMENTAL 0 o TRANSVERSE GC-G-5; 250C THEORETICAL. TRANSVERS 1.-. _PARALLEL ~ ~ ~ ~ ~___ \l __ Ms~~~ ~~~~~~~~~FI' 13 REESIL SUCPIBLT VMANTIATO'/~~~~~~~~XEIMNA^ — TRASVRS hX~~tKII~~~lfcIM ~ 1 \L^ - ^P R L E GC-G-5,25~C~ ~ ~ ~ ~~"'%'~}

,0FI 4I _/ _ = T'V __Z7 - - _ _ _ _ _ _ _S-S,_Xr -. 4,!/. /. — ^.3 _ _ -.2 V -1.0 -.9 -.8 -.7 -.6 -.5 -.4 -3 -.2 -.1 0.1.2.3.4.5.6.7.8.9 1.0 FIG 14 Ms REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION EXPERIMENTAL ~ —-~ TRANSVERSE /v —v PARALLEL GC-E-3, 250C THEORETICAL - TRANSVERSE THEORETICAL PARALLEL

V -^ y _ ^ ^ "S. ^ ^'_.8 ___ -| -.9 -.8V -... —. 6 _...3 4 7.6.. 9 -x-.5/ -.3/ _ _,E -9.0 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -2 -.9 0.1 2.3.4.5.6.7.8.9 9.0 EXPERIMENTAL — - V^ —- PARALLE GC-E- x, 450 THEORETICAL -- TRANSE THEORETICAL {_.-_ PARALLEL

—.9-.8-.7-.6-.5-.4-.3 -.8 -.7 _ _.3..6. #7. 7 \vM/ _ _ _ /_/, L/ V.,0\-,,xX FIG 16 Ms REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION THEORETICAL -- RAE I /.4 X EXPERIMENTLo ( PARALLEL

0001 --- - - - - 0 -- --- Lo -~X _ _ _ _ _ _~~~ 77 ^ 9 ___ _0_ _ _ _ 7_____ /_, -.80 __ __ ___LV _^____ __^ ____ __7 _ _ _\_ 0~~~~.2 I~~~ -.4 0 1.0 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 0.1.2.3.4.5.6.7.9 1.0 M FIG 17 Ms REVERSIBLE SUSCEPTIBILITY VS MAGNETIZATION EXPERIMENTAL o — o TRALLEL v-~-vPARi L -GC-E-3; 1000C THEORETICAL- ------ TRANSVERSE t —----- PARALLEL

Xrt o M Xrp O< M Xrp O' H FIG 18 COMPARISON OF SUSCEPTIBILITY LOOPS CORE E-3 37

SNOI1VZ113N9VIW >V3d IN383-I1C SOd SW/W SAOX/djX SWl Aw" 61 91S 8' 9'' 7' 0V- " - 9 — 8'9i A tli'~~~~~ I'A 0 ox - 9. - ___L L I I I I I Vi N I I, I I I I _ V / / ____Xi, ___^__ Vx ~~ ~V' 0og Jo9't7- = XDUH. __ __ __ _ ___ __ _____ ____ ____ ____ ___I____I_____I I

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Fig. 19 shows the variation of the maximum value of X / X vs. M/Ms with the peak value of M reached during the cycling process. Similar curves are available for both 1-8 and G-5. The peak value of susceptibility increases and shifts toward zero M as the peak M is diminished. 4.3 Errors The accuracy of our susceptibility measurements was limited by the noise reading on our instrument for measuring the voltage drop across the toroid, (Hewlett Packard 400C VTVM) and our ability to read the meter. The normalized data are considered accurate within + 5%, the absolute values accurate within + 8% With the exception of the parallel field measurement of Ms, our magnetization data are limited in accuracy by fluxmeter drift and errors in meter reading. These are considered to be well within + 5,. It is difficult to estimate the error in the value gotten for Ms from fields applied around the toroid. This value must never be less than the value read from the transverse measurements. Table III shows the parallel measurement to be larger for G-5 and I-8 and the same for E-3 as the apparent value along the toroidal axis. The curves for Xr/Xo vs. M/MS should approach zero at M = Ms. That they do so would indicate our measurement would probably be within 10%. 5. EXTENSION OF THEORY 5.1 General Discussion For a perfect single crystal with zero demagnetizing factor the application of a small field parallel with an easy direction of magnetization should result in an infinite susceptibility. That the susceptibility is noninfinite in polycrystalline material can be considered to be the result of two different types of forces acting to retard wall movements. These are reversible and irreversible forces. 39

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In polycrystalline material the reversible forces can be considered as arising from intragranular potential minima and from intergranular demagnetizing factors. In the discussion of section 2.1 we described the forces as arising from some potential energy function that was a continuous function of spatial coordinate. We now wish to discuss hysteresis and susceptibility using statistical arguments to describe the trapping of domain walls in potential holes, i.e. in metastable positions. Each hole is to be characterized by a single number, loosely called its "depth" f. A wall encountering such a hole will be trapped and assumed to be held rigidly if the total net force of the field plus reversible forces on the wall is less than f, but will break free if this force exceeds f. Actually these potential minima must surely, for finite wall areas, have a finite radius of curvature. However, for purposes of this discussion we consider them to be infinitely sharp i.e. the walls held rigidly when under the influence of a potential hole. A wall thus rigidly held in an infinitely narrow hole would contribute nothing to the susceptibility. The behaviour of a ferromagnet must lie somewhere between this model and the model assuming only reversible behaviour. The magnitude and direction of the reversible forces arising frora localized demagnetizing factors on a particular grain will depend upon the size, shape and neighbors of that grain. This will be rather independent of the magnetization of the material and must surely be distributed in some random fashion. Although the exact nature of this force distribution is not known its resulting variations in energy magnitude with direction must be nearly the same as for other reversible cases so the resulting distribution of magnetic moments throughout the material must approximate that given by an extremum of Eq. Al. This equation is very stable with respect to different force distributions as can be seen by 4o

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN observing that in the neighborhood of the extremum value given by a en W/a N = 0 higher derivations of fn W with respect to N form a polynomial in negative powers of NY. An illustration of the stability of this equation would be to apply the equations for isotropic material to [111] material as discussed in section A,3. The equality to high values of qr can be directly attributed to the slowly varying character of Eq. Al. We therefore consider the distribution of magnetic moments in a ferromagnetic material when only reversible forces are applied to be described by Eq. A6. The resulting magnetic disorder could be considered as analogous to a magnetic entropy, and should be considered when dealing with magnetic energy problems. 5.2 The Irreversible Magnetization In the appendix a quantity n was defined as the number of domains of fixed and equal volume whose moments are oriented in the y direction. Since we are now speaking of domain wall movement, this definition is no longer applicable. In this section nj will be the fraction of the total number of atoms whose perA manent dipole moments are oriented in the direction of the unit vector y., If no potential holes or "snags" were present, then under an applied field H according to the reversible equations: iks (H —~j) nj = eAM(2-4i) (24) esAM(H Ti) i In terms of wall positions, this must also be equal to (See Fig. 20) h

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN a.. X Xo([Po )' /xo(p0 l) x[~a'J (pJ - piJ)] O x2 (p2i) FIG 20 SPATIAL CO-ORDINATE OF A DOMAIN WALL n - 2 + E ai x0 (25) inj Yi2 i o ijr y i aj is the area of a particular wall separating domains oriented in the directions A A Yi and yj respectively, x0 is the spatial coordinate of the wall relative to a position at which the total net magnetization is zero and Eq. Al is an extremum and Z is a sum over all walls of type ij in unit volume. If there were no snags, x would be a function only of the pressure which the applied field exerts on the walls. This pressure is given by p i = 3 [f-(i - sj)] (26) A given wall would find an equilibrium position for which the reversible forces just balanced this pressure. When snags are present walls may become caught in them, and x is no longer a simple function of pressure. ie can express the average x for a large....... 42....-2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN number of walls in terms of the distribution of these snags by arguments of the nature of a "mean free path" discussion. Let the number of snags per unit volume whose depth, in the sense of section 5.1, lies between f and f + df be given by $(f). The total number of snags per unit volume is then 0 z = (f)df Suppose that initially there is a field Ho present, and that all the walls occupy positions xo determined only by the reversible forces and the field. That is, none of the walls are held in metastable positions. We will focus attention only on those walls of a particular class ij with area a"ia lying in a particular range between a and a + da. When the field is changed to H2, each wall will tend to move to a definite new equilibrium position x2. As it moves toward x2, there is at each intermediate position x a net force f(x) of the field plus reversible forces, this force diminishing to zero when x2 is reached. According to the assumed model, if at x a wall encounters a snag bigger than f(x) it will be held there. If there are No walls in the group considered, which we imagine to start moving together over the interval xo to x2, the number Na(x) still free to move at a position x in the interval will be diminished by the number so far caught. Thus we can write dN'(x) = -N (x) a?) dx f ((f) df f(x) --------------------- --------------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Upon integrating: N (x) = No exp - aij f dx (f) df Xo f(x') The fraction of the original group of walls which become snagged in the interval between x and x + dx is given by: 4 (x) dx -= IdN( = [exp (aiOfdx f(f) df a) df a (f) d dx I0 \ f(x') f(x) The average position xv at which the group of walls considered will become snagged when the field is changed to H2 is then. x Xo This average value of x is the quantity to be substituted into equation (25) when snags are present. Eq. 24 is used to predict the magnetic properties of our experimental system. If we define a vector H' for which there is an equation analogous to 24 but which corresponds to a system described by x of Eq. 27 substituted into Eq. 25, it is apparent that H' must have properties different from that of H. The product Hij determines the fraction of the atoms with their moments oriented in the.j direction. Using Eq. 27 a different fraction of the atoms will have their moments oriented in the yj direction because of the dependence of A (x) on direction. Therefore if H' is to be considered as a vector it must be considered as a special type of vector whose magnitude will be given by IHI g(j) where g(j) is some function of the angle the field makes with the atoms in question. ---------- -- u -----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It would be possible to describe this same phenomenon in terms of a different distribution function rather than a variable H. Either of these methods would be quite involved. A third alternative is to describe this redistribution in terms of a different anisotropy type. This is done in the next two sections. 5.3 Modifications of the Reversible Model 5.3.1 Contribution of Metastable Volume. It was seen in section 5.1 that the reversible susceptibility of a ferromagnetic material is decreased by the presence of intragrain potential holes which are occupied by domain walls. In general these holes would have a finite radius of curvature. The greater the depth of a minimum the smaller is this radius, and therefore the less a wall occupying it will contribute to the susceptibility. This would always act to decrease the susceptibility from the value given by the reversible equations. This decrease from maximum should increase as the peak M reached during a cycle increases causing walls to cross potential barriers capable of retaining them in deep metastable states when the applied field is decreased. In general it would not be the same holes that would decrease the two different susceptibilities. 5.3.2 High M/Ms, decreasing IMI. When IMI Ms = 1 all the material is necessarily aligned with the applied field. As IMI is decreased from Ms the metastable volume will be predominately oriented in the direction of the macroscopic magnetization, the remainder we take to be oriented in accordance with the reversible equations. Thus at a specified value of r the number of atoms with their magnetic moments parallel with the biasing field will be increased by the action of the potential holes. Therefore for a specified value of M there will be more atoms aligned parallel than would otherwise be the case. This necessarily means the component antiparallel with the field is correspondingly increased. Since the number of atoms is fixed this must also mean that there ____________ u? -----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN will be fewer atoms with their moments aligned normal to the field than would be the case with only reversible forces. To find qualitatively how this affects the susceptibilities consider the extreme case of all magnetic moments oriented either parallel or antiparallel with the field. Eq. A7 then becomes: eAMHr -AMsHr - _ e.s..r.-M =Hr tanh n M e r +e s where, as usual,y=AMsIHr This leads to: XrP 2 X tanh j __ = sech2 rt th (28) Xo Xo Values of Xp/Xo and Xrt/Xo vs. Iv/M1s are given in Table IV. The result is that both curves are higher than the corresponding curve for isotropic material, but are of the same order and of the same general form. TABLE IV M/Ms and X r/X vs n for f(r) = tanh' )n tanh n = M/Ms sech2 rT = Xp/ Xo anh = Xrt/ X 0 0 1 1.3.291.914.971.6.537.711.895.8.664.559.830 1.0.762.420.762 1.h.885.216.632 2.0.964.071.482 5.0.9999.000.200 co 1.000 0 0 - The integration constant in X r/Xo must be zero for symmetry to exist about Tr = 0. 46

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 5.3.3 High M/Ms, increasing IMI. In a randomally oriented state none of the volume is in a metastable condition. After the application of an external field a finite fraction of the material will be maintained in metastable positions. Since domain walls would move the farthest when located between atoms with moments parallel and antiparallel with respect to the field a larger fraction of the material oriented antiparallel will be held in metastable states than in any other direction. This fraction decreases to zero for parallel alignment. As the magnetization increases the amount of material available to be held in these metastable states is smallest for antiparallel and increases to maximum at parallel alignment. As a result one would, for high values of M/Ms, expect the total number of atoms whose magnetic moments are held in metastable conditions to go through a maximum at some angle between 0 and I radians. Thus for a specified T] the normal component of magnetization will be increased. It then follows that for a specified M the total magnetic moment both parallel and antiparallel with the field must be decreased but the difference remains the same, so the fraction of the atoms antiparallel must decrease. The extreme condition of this model would be for this fraction to be zero, i.e. no atoms possess magnetic moments which make angles of greater than n/2 with the applied field. For this case the magnetization would be given by: (See section A.2) 1 M o/~ d y eTnl ~~whe ~~re: where Y = cos Q = AM H s7 47

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This leads to: M J eti 1i Ms e-Ll For small fields, M = 1 + n +.... Ms? 12 The resulting susceptibilities are given by: r = 12 1 - ~~~~Xo = [^'1~ 2 (e~1)2](2) (29)1' Xrt 12 [ e rl 1 XXO Le-hl j j.X is the value of the parallel reversible susceptibility when r = 0. Using this model Xrt X when r = 0. Values of ), M/Ms, X p Xo and Xrt/ Xo are given in Table V. Obviously Xrt/Xo is much greater using this model than using the isotropic model. * To evaluate the integration constant note that Xh must become infinite at some value of I/Ms. It will be positive above tht value and negative below. The only point where such a discontinuity can exist is for ~r = O. Thus the integration constant must be zero. 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE V M/,s and Xr/ Xo vs for f(, ) = - l -e'r rJ 1NM/MS Xr/Xo Xt/ Xo 0.500 1.000 o 1.582.943 6.98 2.656.828 3.94 3.720.671 2.88 5.806.397 1.93 7.857.234 1.46 10.900.012 1.08 15.933 0.744 20.95 0.576 50.980 0.240 X- 1.000 0 0 6. COMPARISON WITH EXPERIMENT 6.1 The Parallel Case. From Figs. 12 thru 18, as described in section 4.2, one sees that for decreasing IMI the experimental curves start, at high IMI, below the theoretical curve. The experimental curve becomes larger than the theoretical curve, passes through a maximum before /IMI = 0 is reached, then remains less than the theoretical curve for increasing IMI. Fig. 19 illustrates the variation of the curves for small IMI /Ms as the peak value of M reached during the cycle is altered. In Fig. 19 the ordinates have been normalized to the highest value of susceptibility measured. These differences can be correlated with the modifications of section 5.3.1. Consider in the region of small decreasing ) IMI a fraction D of the material to be held in metastable positions that contribute nothing to the susceptibility. The maximum value of X will occur when the total M of the remaining material (l-D) is zero. There will still be a net magnetization in the direction of the last previous peak value contributed by D, and as seen before D is expected to increase with the magnitude of IMI at that peak. ---------------------- 9 -------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Thus the value of M/Ms for which X is a maximum will increase with increasing peak M during the cycling. The value of Xmax should also decrease proportional to (l-D). This can be seen experimentally in Fig. 19. Thus Figs. 12 thru 17 should actually be normalized to the value of susceptibility occuring when the material is carefully demagnetized and H=M=O. This value can never be exceeded. The corrections of section 5.3.2, Table IV, are small for the parallel case, so will have little effect. The hysteresis loop of Xr/ Xo, see Fig. 18, is traced in the opposite direction when plotted against M than when plotted against H. This arises formally because Hct as defined in section 4 is less than Hc. All measurements we have made has given Hc' of the same sign as Hc. Qualitatively, the arguments of section 5.3 for constant r instead of constant M would give the proper hysteresis loop for Xrp vs. H. This variation of Xrp with M and with history is in qualitative agreement with the results of Tebble and Corner2. 6.2 The Transverse Case. There are two opposing effects which determine the hysteresis loop for this case. Firstly, for the same reasons as the parallel susceptibility, the transverse susceptibility will go through a maximum before reaching M = O for decreasing IMI. Secondly, upon comparing the susceptibility curves for the isotropic distribution of magnetic moments and for the distribution of the model used in section 5.3.3, Table V, it is seen that the latter yields considerably higher values of Xrt. This would produce a larger Xrt for increasing fields than for decreasing fields. The first effect would predominate at low fields, the second at igh fields. The result is an intersection of the susceptibility curves as can --

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN be seen on Figs. 12 thru 17 and is plotted to be especially evident in Fig. 18. The value of M/M at which the intersection occurs should then decrease with des creasing peak M reached during the cycle. Unfortunately we were unable to obtain experimental data for minor loops using a transverse field, 7. ACKNOWLEDGEMENTS The authors with to thank Professor H. W, Welch, Jr. for encouraging this work; Messrs. John Newton and Ralph Olson for assistance in recording data; Professor E.. Katz, Dr. L W. Orr and Mr. M. H. Winsnes for aid during the early stages of the work. __ —--------------- --- 1 —-----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX Derivation of the relationship between Xp and M. (after Brown) rp A.1 General Formulation Consider a ferromagnet to be composed of N domains per unit volume, each of fixed and equal volume. These domains are acted upon by internal forces of a reversible nature probably arising from microscopic demagnetizing factors or from walls held in potential holes with a finite radius of curvature. If these were negligible and if the entropy were negligible the system would occupy the minimum energy condition determined by the interaction of the remaining surface and volume energies. Certainly the entropy of the system is negligible. However, a pseudoentropy whose disordering arises not from thermal agitation but localized randomally oriented stresses is not negligible, as is obvious from Eqs. 10 and 11. Brown states that the detailed mechanism of the microscopic forces are unimportant, but their energy magnitudes are important. Therefore one does not need to know the exact distribution of these localized stresses for similar results should be obtained from any of a large number of possible distributions. One such possible distribution could be gotten by considering the number of macroscopically indistinguishable ways N domains of fixed and equal volume could be microscopically arranged. This is given by: W - N rJ (N) where y is a possible direction for orientation of the domain moment. Using Stirling's approximation: ------------------ 2 --------------------— 52

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN nn W a N $n N - N Ny en N~ (Al) The potential of the material can be written as v = Z N (Vv + VS) (A2) where V7 involves an energy proportional to the number of domains involved, VS depends upon the surface energy. The system is also subject to the constraint on the total number of domains: N = E NY (A3) If the system is considered to occupy a most probable position then en W should be stable with respect to variations in N. The similarity between this distribution and that due to the conventional entropy is obvious. V7 is just the customary equation for the magnetostatic energy of a dipole in an applied field: V = -Hr Ms cos Q (A) 9 is the angle between Hr and the direction of orientation of the domain. The term Vs involves the surface energy. This ill depend upon the particular geometry involved and must be considered constant. Taking variations of Eqs. Al, A2, A3 with respect to N and introducing Lagrange multipliers, one obtains: [-n NV + A(V + V) + B] = 0 (A5) 53

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - or N = exp [ B a(V + V)] N If n. = -! thenm = eAV, exp (AHrMscos ) (6) eAV: exp (AHrMscos 8) Y G The resulting magnetization must then be given by: M N cos 8 exp(AHrMscos 8) -- 8_ _-.. ----— (A7) Ms E exp (AHrMscos 8) This formulation considers only magnetization by reversible processes. Therefore Hr is not the actual applied field, but is that value of field which would be necessary to bring the material to a magnetization M if no irreversible processes were involved, i.e. Hr o Xr _ (A8) o Xr A.2 Isotropic Material For the case of isotropic material Eq. A7 becomes: M r d n s d8 cIs a AHrMscos 9 F Ej; of 0 in d eAHIrMCos Ms f o/sin d8 e s

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN If cos Q = y and AHrMs =, 1 M fdy yeTY fdre.y] -I fdy elT 1 (el - e6-) 2 sinh n Therefore: M I cosh ) - ~ sinh r1 -- =..... - = ( ctnh? - ~ ) = L() (A9) Ms 1 sinh T The parallel reversible susceptibility is given by: Xrp d Msd L(r) al AM2 L'() (A10) rp = 8 r (Ala) d where the prime indicates -d. The initial susceptibility is defined by Eq. 3. Using Eq. 16 for small values of Mr one sees that: AM2 Xo = -2 (All) 3 Combining Eqs. A10 and All, —' p 3 L'Q(r) (A12) Xo

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A.3 [1i ] Anisotropy Here each domain must be located in such a manner that its direction of magnetization is in the [ll1 direction of the crystal. If ai is the di1 + rection cosine between Ms and the axis Oxi, then ai = Pi where i= 1 for each value of i. If ji is the direction cosine between the applied field and the crystalline axis Oxi, then: cos = _ Z i i (A13) V3 i 1 After combining Eqs. A7 and A13, for a single crystal: M _1 Z Z PJj exp [ -' Z Pii ] (Alh) /3p j J3 i Ms, exp [EPi To evaluate these sums, first consider the denominator of Eq. A14. The outside sum over p is over the eight possible combinations of (P1, P2, P3) = (1,1,1); (1,1,-1); *~*~~*" (-1,-1,-1). Upon simplifying this is seen to be: exp [ L Z pii] = 8 n cosh [- - k 1 (A15) P V3i k 3 From Eq. A15 one obtains: __________________ % — - -- - -- - - - - - --

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN V'j 8~' j p exp( Pii 3 ji P V = Z Z Pej exp ( - Piei) (A16) P j i3 i =8 Z sinh ( 4 j) IT' cosh ( k) k where the prime denotes k f j. The first expression on the right side is exactly the numerator of Eq. A14. By substituting the second expression for the numerator, and Eq. A15 for the denominator in Eq. All, one obtains' M = 1 jt MX EZ tanh 4 (A17) Ms A J 3 for a single crystal. To obtain the value of M/Ms for polycrystalline material it is necessary to integrate over all possible values of Zj M, ES zf j tanh ej (A18) Ms V3 J J 4 3J where dc is the differential solid angle. Each of the three tj must be equivalent so: ---------------------- 7 ------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN M- "-^ m d5 Z tanh f z = m f fdQ sin @ cos 9 tanh(- cos ) o O Let u =- cos, so: f3 M 3Xa u tanh u d u = E(n) (A19) Ms 2 o % rp d= Ms v -A 2 E'(n) (A20) From Eq. 17, as for the isotropic case, AM2 X = - (A21) 3 Therefore, for [111 anisotropy, x- = 3E'() (A22) Xo ----------------------- $~~58 —----------

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN BIBLIOGRAPHY 1. Gans, R., "Equation of the Curve of Reversible Susceptibility", Physik. Z. 12, pp. 1053-1054, 1911. 2. Tebble, R. S. and Corner, W. D., "Investigation on the Reversible Susceptibility of Ferromagnetics", Proc. Phys. Soc. 63, pp. 1005-1016, 1950. 3. Rado, G.E.; Wright, R.W. and Emerson, W.H., "Ferromagnetism at Very High Frequencies. III. Two Mechanisms of Dispersion in a Ferrite", Phys. Rev. 80, pp. 273-280, 1950. Rado, G.E.; Wright, R.W.; Emerson, W.H. and Terris, A., "IV. Temperature Dependence of the Magnetic Spectrum of a Ferrite", Phys. Rev. 88, pp. 909-915, 1952. 4. Becker, R. and Doring, W., "Ferromagnetisms", Edward Bros. Ann Arbor, pp. 154, 1943. 5. Kersten, M., "The Effect of Elastic Tension on the Magnitude of the Initial Permeability", Z. Tech. Physik 12, pp. 665-669, 1931. 6. Brown, W.F., Jr., "Domain Theory of Ferromagnetics under Stress", Phys. Rev. 52, pp. 325-334, 1937. "II. Magnetostriction of Polycrystalline Material", Phys. Rev. 53, pp. 482-491, 1938. "III. The Reversible Susceptibility", Phys. Rev. 54, pp. 279-287, 1938. 7. Brown, W.F., Jr., "Theory of Reversible Magnetization in Ferromagnetics", Phys. Rev. 55, pp. 568-578, 1939. 8. Brown, W.F., Jr., Private correspondence 59

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