AFOSR-TN-57-209 ASTIA Document No. AD 126 506 ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR REVERSIBLE PROPERTIES OF POLYCRYSTALLINE FERROMAGNETS II. EXPERIMENTAL VARIATION OF THE REVERSIBLE SUSCEPTIBILITY AND Q WITH MAGNETIZATION Technical. Repbort No'. 2 Solid State Deviqces Laborattory Department of Elrcdtri-all Eng!ineering By: D. M. Grimes Approved by: ) H. W. Welch, Jr. Project 2495 AIR FORCE OFFICE OF SCIENTIFIC RESEARCH AIR RESEARCH AND DEVELOPMENT COMMAND CONTRACT NO. AF-18(603)-8 May, 1957

INTRODUCTION The reversible susceptibility and the differential magnetostriction of ferromagnetic and ferrimagnetic material has been considered from a statistical standpoint in a previous paper,l which will be designated I wherever referred to in this present paper. The variation of both quantities with magnetization was considered for differential fields applied both parallel with and normal to the effective biasing field and for models assuming that the reversible susceptibility had its origin in domain-wall motion and in domain rotation respectively when the static moments were oriented along directions of minimum anisotropy energy. The magnetic Q is considered to be the ratio of the real to imaginary part of the reversible susceptibility when that susceptibility is measured at a finite frequency. The present paper considers the variation with magnetization of Q for biasing and differential fields parallel and normal respectively. Detailed measurements are shown for four different ferrite materials.

THE MAGNETIC Qo REVERSIBLE SUSCEPTIBILITY BY DOMAIN ROTATION. The variation of the magnetic Q with magnetization and effective biasing field can be seen from Eq, 9 of Io Substituting the values of X and X, the susceptibilities of the positively and negatively rotating field, from Eq. 5 of I into Eq. 9 of I, the resultant magnetic Q can be written as C2D l 1 Q: %) (, ++ c + = gl +, (e) for both parallel and transverse fields, and for values of applied angular frequency small compared to:d, Q=( (:1 + e2). (2) The symbols are as defined in I, where X is the radial frequency, I = a Han with H the effective anisotropy field and y the magnetomechanical ratio, and E is a constant proportional to the loss term of the differential equation describing the motion of the magnetic moments. From Eq0 7 of I, vl of Eq. 1 can be replaced by l)1 + Gpo = 7Ms, where p is described in I. From the form of Eq. 2 it is apparent that any variation of Q with magnetization is dependent upon the p and e variation. The applied field Hap acts as an addition to the effective anisotropy field. The parameters e and p must be, in general, allowed to vary with both H and MN e is dependent upon the intrinsic nature of the material and, as such, is relatively insensitive to the magnetization level. p is nearly constant. In the low frequency limit, the initial susceptibility X (X + X is given by: (see Eqs. 5 and 10 of I) 1 -

x 2@2 o = 3,(3) so the product X Q can be written as: 2c XQ = o (l + e2) (4) o 3C This expression depends upon H and M only through the e variation, which must surely be slowly varying for at least low magnetic fields. Combining Eq. 4 with Eoo 9 of I:.Xrt (l + E2)(1 + <cose>); QX = (. + )(1 <cos2e>). (5) Eqg 5 should be valid to larger values of M than Eq. 9 of I and would be limited by the static moments remaining aligned along crystallographic easy directions. According to Amar2 this is valid to higher field strengths in polycrystalline material than would be expected from the single crystal parameters, For domain rotation the Q will thus be the same monotonic increasing function for parallel and transverse fields. REVERSIBLE SUSCEPTIBEEILITY BY WALL MOTION. Eoq. 21 of I is the differential equation of motion utilized to described the motion of a domain wall. The change in magnetic moment due to the movement of 1800 walls is given by: = -Z2M' A-xI (6),where Ao is the area of the itth wall and x. is the distance through which it is moved funer the influence of the applied magnetic field M and V is the volume of materialI From E.L 6 and Eqo 21 of I it follows that

-3and that the magnetic Q is given by: A. Z,xa (1a 2 ) The low frequency limits are X = 2 ZMs Ai/ai, and Q _ A(9)a The constant P is related to e through a series of magnetization and field independent parameterso3 The remaining variable portions of Eq. 9 are the terms involving Ai and aio The expected variation of Z Ai/ai can be seen fron Eqs. 2 and 18 of I. ai would presumably increase with moment for the parallel fields, so the Q would also increase. For the case of transverse magnetic fields the case is not so clear cut although from Eq. 18 of I, Ai/ai must decrease with increasing field. The variation of the ai would presumably depend on the origin of the restoring force and for large applied field ai would increase with Hap. However, there is no obvious reason to expect the ai to increase and indeed they might decrease with Hap for small values of Hap It is thus to be expected that the magnetic Q for wall.motional susceptibility will increase with M4 for parallel fields but increase more slowly or even decrease with M for low M until finally increasing with M for high M in transverse fields. HYSTERESIS EFFECTS

-4Remaining hysteresis effects in the susceptibility versus magnetization plots, when the susceptibility is due to domain-wall motion, has been discussed4 in terms of the metastable volume of material with moment remaining oriented in the direction of the previous saturation magnetization. This metastable volume is considered to contribute a net magnetization in the direction of the previous saturation, but to contribute nothing to the parallel field susceptibility, Thus, the susceptibility goes through a peak value always less than the susceptibility of the virgin material at a value of magnetization finite and in the direction of the previous saturation field. If the gross magnetization is presumed to change by rotation of the domain moments, then after the material has been saturated as the field is decreased the moments again rotate in easy directions. If, as the saturating field is decreased, the domain moments rotate until the component perpendicular to the field is the same as in the virgin material, but all components originally parallel and anti-parallel with the field remain parallel, then the value of remanent magnetization will be 0.5Ms and the susceptibility due to domain rotation, and thus proportional to [1 - <cos2e>], will, at this point, just equal that of virgin material. This is the model used by Fomenko5 to interpret his results. If this is also the position of the maximum number of atomic moments normal to the field direction, then it will be, in turn, the position of maximum parallel and minimum transverse field susceptibility. For parallel fields, then, the maximum would occur for finite decreasing M, but would equal in value the susceptibility of virgin material. If the hysteresis action is such as to increase the parallel-antiparallel component, then Xrt would always exceed X p Reviewing, in both the rotational and rt rp wall-motional models the peak parallel susceptibility would occur for a decreasing magnetization, but the ratio of peak to initial should be less for rotation thatn

for domain wall motion. These results are based upon the assumption that in static fields all moments are in crystallographically "easy" directions. Thus the changes in finite fields are considered to be either wall motional or to be rotational between easy axes. That there is often a difference between the type of processes depending upon the size of signal has been demonstrated06 SAMPLESo To test the hypothesis of I regarding the variation of the susceptibility and of the previous section regarding the magnetic Q, it was decided to measure a series of ferrites with widely varying anisotropic and magnetostrictive constants. Now the room temperature saturation magnetization is positive for magnetite and negative for cobale and nickel ferrite.7 The first order anisotropy constant of cobalt ferrite is positive, that of nickel and iron ferrite is negal tive.7 Small zinc ferrite additions increase the susceptibility of each of the above ferrites. These constants effect the susceptibility in the manner to be described The initial susceptibility due to wall motion has been approximated by Becker8 for sinusoidal internlal strains arising from magnetostrictive forces which arose when the material was cooled through its Curie temperature0 The result is 8vi~ 9X2E I where X is the saturation magnetostriction and E is the Young's modulus of the material0 Likewise the initial susceptibility due to domain rotation can be approximated by9 Xo= (il) 0 3Ka where ~o is the permeability of free space and K1 is the first order anisotropy

constant o In an attempt to vary the ratio of wall notional to domain rotational susceptibility the ratio of naickel and cobalt was varied in mixed iron-cobaltnickel-zinc ferrites. To test the equations of I describing the effect of a biasing magnetization on the domain rotational transverse susceptibility, it is necessary to have the anisotropy sufficiently high to assure that the moments are aligned along the easy crystallographic directions and still small enough to allow a measurable susceptibility due to the rotation against the effective anisotropy fieldso To accomplish the manufacture of such ferrites, a series of fifteen compositions (shown in Table I) with different amounts of iron, nickel and cobalt were manufactured under the following conditions: The material was weighed out, ball milled in a steel mill with a thin acetone slurry for six hours, pressed into a toroidal shape of one inch OoDo, 5/8 inch IoDo and about 1/4 inch thick, placed on an alundum tray which had been rubbed with Fe 03, and heated rapidly to 1150 C. They were then heated at about 600C/hr to 1375 C, held for 1/2 hour, then the temperature reduced to 12000 and held for two hourso They were then slowly cooled to 11000, flushed with nitrogen and furnace cooled in a nitrogen atmosphere0 The resultant ferrites were black and of about 0.95 X-ray density0 They were toroidally wound, and then placed' between the pole faces of a battery powered electromagnet. The pole faces were, tightened snugly to an insulating sheath over the windings and the toroidal windings taken to a Boonton Type 260-A Q-Metero For a preliminary survey, the capacitances and Q were measured at 500 kc/sec as a funcs tion of the magnet currentso From these fifteen specimens, F-1-2, F-6-2 and F-10-2 were chosen for detailed measurements0 In addition, a sample of magnesium ferrite designated E!15l1 was measured in detail. Magnesium ferrite was chosen as a test ease since it was in this material that Rado obtained striking evidence of the existeance of both wall notional and domain rotational susceptibility resonances0lO

-7TABLE I COMPOSITION OF -THE FERRITES SURVEYED Moles of oxide added to mix. Designation Ni0 C90 ZnO Fe 0 F-1.2337 0.2263.5000 F-2.2314.0023 t t F-3.2290.0047 t F':4.2150.0187 F-5.1963.0374 " F-6.0843.1496 " " F-7.2065 0.2294.5641 F-8.2044.0021 It It F-9.2024.0041 " F-10 o 1900 o0165 " F-ll.1735.0330 F-13 ~ 0743.1322 It F-14.1613 0.1791.6596 F-15.1484 0129 " " F-16.1355.0258 tt The amounts are listed for CoO for comparison, an equivalent amount of Co20 was added. All oxides were commercially available C. P. grades. EXPERE TA.L TECHNIQUE' The experimental technique utilized to determine the susceptibility and the magnetic Q is the sazme as that described earlier, except the high frequency measurements were taken with the toroidal leads connected to a Q-meter instead of a VTVMo The frequencies used were 320 kc/sec and 500 kc/ seco The lower frequency was used for all except F-1-2, to avoid insofar as possible transverse field magnetostrictive resonances. For a check on the susceptibility variation, certain data were repeated at 10 kc/sec using the previous technique. RESUiLTSo The results of the measurements are presented in Figs. 1 through 4. Plots of the normalized susceptibility and the Qd are shown as the ordinates with the magnetizatino as abscissa. The direction of change of3 matgnetizatio is

-8indicated by the arrow. Figs. 5 through 8 show the symmetric and antisymnetric parts of the susceptibility and the susceptibility-Q product as a function of magnetization~ Table II shows the measured values of Q and susceptibility at zero magnetization, the saturation magnetization, the remanent magnetization, and the coercive force@ TABLE II PARAMETERS OF THE MEASURED FERRITES Values for M = 0 320 kc/sec Parallel Transverse Specimen H m Ms m Mr m Xo X0 c m sm r m o 0 0 O x 10 5 x 04 F-6-2 125 2.79 1.84 72.4 46.5 60.8 52.8 4906+ F-l-2 20.3 3 10 1.89 13.2 388* 13.5 450* F-10-2 193 2~38 1.21 71.5. 153 68.8 163 -15-1 175 1o 23 0.98 34.6 51.2 38,8 54.2 42.4+ 45~ 4+ * f 500 kc/sec + f 10 kc/sec F-6-2o Data from specimen F-6-2 are shown in Figs. 1 and 5. The transverse susceptibility fits the expected curve for domain rotation to greater than O.6 Ms This fact and the nearly constant Q leads to the conclusion that if only wall motional and domain rotational processes exist then the susceptibility in low fields must be due almost exclusively to domain rotation and H > Hap for at an ap least M <.5 M o The slight decrease in Q with magnetization for transverse fields is believed due to magnetostrictive losses still present at 320 kc/sec, The para

-9llel field susceptibility peaks at about a decreasing 0.3 Mso The lack of symmetry in the parallel case is difficult to interpret in view of Eq. 10 of I and the symmetry of the transverse field data. Fig. 9 compares the experimental parallel field susceptibility with that calculated from the experimental data for transverse fields using Eqo 10 of I, Some possible sources of the lack of symmetry are l(1) that the domain structure may be different for magnetizations around the toroid and parallel to its axis, so a real difference in the hysteresis may exist in the two cases, Thus some wall motional susceptibility may exist in parallel fields, not present. in transverse fields. (2) It is also possible that the moments are not located in easy crystallographic directions, so the restoring force is asymmetric, (3) The fact that the applied alternating signal was of finite magnitude might effect the two cases differently. Simultaneous differential measurements of parallel and transverse susceptibilities on a sphere would seem to be in order. The relaxation frequency due to domain rotation can be written as Y 21 fS 3 ~(12) 21f 3 X Any sizeable contribution to the susceptibility from domain wall motion should invalidate an experimental check of Eqo 12. The frequency calculated for F-6-2 is found to be about 150 inc/sec. The spectrum of a specimen of this material was measured and found to have but one resonance which occurred at about 125 inc/sec.11 F-lw-2. Data for specimen F-l-2 are shown in Figso 2 and 6o Since this specimen is a nickel-zinc ferrite, it was expected that the anisotropy energy would be smaller than for F-6-2,7 so rotational susceptibility was expected. From Figs. 3 and 10 it is apparent that the susceptibility does not follow the predicted curve for domain rotation based upon the static moments being oriented along easy crystallographic directions. However, the curve XQ does follow the expected curve to about M = 0.5 Ms. and, further, the magnetic Q increases monotonically

10 from a broad minimum value at about M = 0.2 Ms (decreasing), as does the Q for parallel fields. This behavior is consistent with that expected of domain rotation (for a small value of H an)O It is therefore concluded that the susceptibility is probably due to domain rotation, but the effective anisotropy fields are not sufficiently large to maintain the static moments along easy crystallographic directions~ The anisotropy field can be estimated as that field for which the Q is twice its initial value. Thus the averaging equations for the susceptibility as derived in I can only be expected to be valid to relatively small values of M/Ms. The parallel field data shown in Fig. 4 is consistent with the above description, as is the large ratio of peak Q to that near M 0.= O This material is found to have but a single relaxation at about 35 mc/sec.11 F-10-2. Data for specimen F-10-2 are shown in Figs. 3 and 7. The striking features of this material are that both susceptibilities go through a single maximunm, and that both maxima are displaced to the left of center by about 0.3 Ms* The Q increases with field more rapidly with parallel than transverse fields, and the shape of the susceptibility curves as well as the susceptibility-Q curves are indicative that the mechanism of change in gross and reversible moments is the same. The Q variation indicates that the reversible susceptibility is due to wall motion. The material exhibits a striking drift in the susceptibility and Q after the biasing field has been changed, for magnetizations up to about 0.7 MsO For lower magnetizations it was necessary to delay all readings by a minimum of two minutes after altering the field. This drift is indicative of changes in the gross moment by wall motion, a fact not inconsistent with the results of Epstein.2 The initial susceptibility frequency spectra shows a pronounced peak at about 15 mec/sec and a resonant frequency of about 50 mc/sec,ll but no second peak is observed. If the susceptibility is due to wall motion, then the rotational contribution must be completely negligible, This implies that the anisotropy constant

K1 must be very large and the magnetostrictive constant very small. It is concluded that the wall motional concept is the only concept consistent with all the observed facts. This is not inconsistent with the data of reference 7o Fig. 10 compares the parallel susceptibility calculated from the transverse field data using Eqo 20 of I in the formx /X w ^ /= rp d(InX</X0 ) d A tl/nM) with experimental values. I-15-1o The magnesium ferrite I1-5-1 is of the same material reported on by Rado, Folen and Emerson0o3 The transverse field magnetic Q increases with applied field,, but less rapidly than does the parallel field Q. The susceptibilities both have but a siangle maxima, but Xrt Q vs M shows the double peak characteristic of rotation0 The symmetrized values of the transverse field variables lie between that expected by domain rotation and by wall motion0 Thus it is concluded that both mechanisms are of importance in this sample, in agreement with conclusions based upon the frequency spectrum013 DISCUSSIONO An analysis of the relationships between the remanant condition and the reversible susceptibility has been carried out by Frei and Shtriknan4 on the assumption that the reversible susceptibility is due to domain rotation0 One itemn they did not point out is that their analysis assumes not only that the reversible quantities be due to domain rotation but also the expansion of magnetiFrei and Shtri1huan independently derived Eq0 10 of I at the remanant point0

- 12 - zation in terms of applied field must obey the rotational equations through at least second order in the ratio of applied to anisotropy fields. This can be seen in their Eq. 10 from which the remainder of their work is derived. The validity of the averaging equations of I for wall motions rest upon the domains remaining always oriented in easy crystallographic directions. Conversely, if the anisotropy were zero then the equations for wall motion could be immediately carried over for domain rotation. Sample F-6-2 apparently fits the conditions of I but not the conditions of Frei and Shtrikman. Sample F-l-2 did not fit the conditions of I, yet had susceptibility predominately due to domain rotation. It must therefore be expected that this sample would more nearly fit the conditions assumed by Frei and Shtrikman. This is verified in Table III. The theory would not be applicable for 1-15-1 and presumably not for F-10-2. The measured ratio of remanent to saturation magnetization is compared in Table III with that calculated by Frei and Shtrikman's Eq. 29. The striking disagreement for F-6-2 and I-15-1 is also obvious. TABLE III COMPARISON OF MEASURED AND CALCUIATED REMANANCE Specimen Measured Calculated F-6-2.66.ll F-l-2.61.51 F-10-2.51.76 I-15-1.80 1.53 Calculated using Frei and Shtrikman's Eq. 29 CONCLUSIONS The expected variation of the magnetic Q with magnetization has been dis

- 13 - cussed for the cases where the susceptibilities have their origin in domain rotation and in domain-wall motion, and the nonzero static magnletization exists with moments along easy crystallographic directions. Detailed experimental results on four samples are reported and analyzed in tems of the expected Q variation as well as the expected susceptibility variation described in I. It is concluded that two of the samples have susceptibilities arising very predominately from domain rotation, one from wall motion and one from both. It is assumed that no other mechanisms are operative. It is concluded that Frei and Shtrikman's analysis of the remanent position is valid under only the more restricting conditions described herein,

BIBLIOGRAPHY lo Grimes, Do M,, "Reversible Properties of Polycrystalline Ferromagnets: I. Theory of the Expected Variation of the Reversible Properties with Magnetization," J, Phys, Chem, Solids, In Press (1957). 2. Amar, Henri, Bullo Am, Physo Soc. 2, 8 (1957). 3. Galt, J. K., Bell Sys. Tech, JO 33, 1023 (1954), Eq. 13. 4. Grimes, D. M. and Martin, Do. W., Phys. Revo 96, 889 (1954). 5. Fomenko, L. A., Soviet Physics, Jept 3, 19 (1956). 6. Went, Jo J. and Wijn, H. P. J., Phys. Revo 82, 269 (1951). 7. American Institute of Physics Handbook, McGraw-Hill, New York, 1957, po 5-221,2. 80 Becker, R. Physik, Z. 33, 905 (1932). 90 Bozorth, R. M., Ferromagnetism, D. Van Nostrand, New York, 1951, p. 821, 10. Rado, G. T., Rev. Mod. Phys. 25, 81 (1953) 11. Harrington, R. D. and Rasmussen, A. L,, J. Research N. B. S., To be submitted0 12o Epstein, D. J,, Boston Conference on Magnetism, T-91, AIEE, New York, 1957, p. 493. 13. Rado, G. T,, Folen, V. J. and Emerson, W. H,, Proco Inst. Elect. Engrs. 104B, (1957), To be published. 14. Frei, E. H, and Shtrikman, S., Boston Conference on Magnetism, T-91, AIEE, New York, 1957o - 14 -

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Q X 1.2 X.8 1K F-6-2 K _ _ _ _ _ _ _ _ _ _ _ _ _ _ I I I| 1.0.8.6.4.2 0.2.4.6.8 1.0 M/Ms Fig. 1o The Variation of the Normalized Susceptibilities and Q with Magnetization for F-6-2. The arrow indicates the direction of change of magnetization.

2 -rXrt 8.O t'.;c3~~~~~~/Ir 4.0 4 I Q~~~~~~~Xp Qt/ 1.0.8.6.4.2 0.2 4.6.8 1.0 M/ M Fig. 2. The Variation of the Normalized Susceptibility and Q with Magnetization for F-l-2.

tI I I I I I 11 Q X F-IO-2 2.4' 1.2 Xrt Q t 1.0.8.6.4.2 0.2.4.6.8 1.0 M/Ms N Fig. 3. The Variation of the Normalized Susceptibility and Q with Magnetization for F-10-2.

[ I. I I I I 1- 1 1 Q-X 2.4 1.2 2.0 1_ x 1.0.8.6 4.2 0.2 4.6.8 1.0 M/Ms Fig. 4. The Variation of the Normalized Susceptibility and Q with Magnetization for I-15-1.

~yXrt Qt 1.2 Xrt / *8 -Xrp.4 F-6-2.2.4.6.8 1.0 M/MS Fig. 5. The Symmetrized X and XQ Curves for F-6-2. The top curves represent the symmetric part and the bottom curves the antisymmetric parts.

~XrpQp Qt 1.2 2 4 6 8 1.0 Fig. 6. The Symmetrized andXQ Curves for F-1-2. The top curves represent the symmetric part and the bottom curves the antisymmetric parts.

1.2 8.2 O -.2.4.6 8 1.0 Fig. 7. The Symmetrized X and 7Q Curves for F-10-2. The top curves represent the symmetric part and the bottom curves the antisymmetric parts.

1.2 rt Qt 1.0.8 0, 2.4.6.. [1.0 -M/s MS Fig~ 8. The Symmetrized X and XQ Curves for I-15-1. The top curves represent the symmetric part and the bottom curves the antisymmetric parts.

I! i I I i I. Xrp Experimental Theoretical.6 F-6-2 -.8 -6 4 -.2 0.2.4.6.8 M /Ms Fig. 9. The solid points represent experimental data, the hollow points parallel susceptibility calculated from the transverse data and Eq. 10 of I.

rp I I 1. F-10-2 1.2 1.0 Theoretical.6 Experimental I II I I I I.6.8 -.8 -.6 -.4 -.2 0.2.4.6.8 M /Ms- - Fig. 10. The solid points represent experimental data, the hollow points the parallel susceptibility calculated from the transverse data and Eq. 13. The dashed curves show the expected approximate limits of validity.