ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AIN ARBOR THEORY OF REVERSIBLE iAGNETIC SUSCEPTIBILITY WITH APPLICATION TO FERRITES Technical Report No. 8 Electronic Defense Group Department of Electrical Engineering By: D. M. Grimes Approved by: /i- i]2 4Jf H. W. Welch, Jr. L. W. Orr Project M970 TASK ORDER NO. EDG-4 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 August, 1952

TABLE OF CONTENTS Page ABSTRACTiii 1. INTRODUCTION 1 2. DEIIVATION OF Xrp AID Xt 2 2.1 General Remarks 2 2.2 Derivation of Xrp 5 2.5 Derivation of Xrt 6 3. SPECIFIC APPLICATIONS 8 3.1 Tuned Circuit Frequency Range 8 3.2 Temperature Dependence of the Tuned Circuit Resonant Frequency 12 4. THEORETICAL ASPECTS OF Xo(T) AND J5(T) 13 APPENDIX I - The Derivation of 1/f df/dT 17 BIBLIOGRAPHY 20 DISTRIBUTION LIST 21 ii

ABSTRACT Formulas expressing the reversible susceptibility both parallel to and perpendicular to the biasing field are developed based on Brown's statistical theory. It is concluded that parallel fields give a larger frequency ratio change for a given change in biasing field than do the transverse fields. Expressions giving the temperature coefficient of the frequency ratio in terms of J/Js, Js(T) and Xo(T) are developed. A table is presented showing when each type of field has the lower temperature coefficient. The material under special consideration is the ferromagnetic spinels -- ferrites. iii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN THEORY OF REVERSIBLE MIvAGNETDIC SUSCEPTIBILITY WITH APPLICATION TO FERRITES 1. INTRODUCTION If a small magnetic field LH is applied to a magnetic body the induced magnetization will be nJ. The magnetic flux ZAB = AH + 4-CJ. The incremental susceptibility XL, is defined by: x = LJ* The incremental permeability, /A, is defined by: The rversibl suscptibility, X is defined by= The reversible susceptibility, X,, is defined by: lim X Xr AH -^ 0 If a biasing magnetic field, Ho, is applied to the specimen, Xr will depend upon the magnetic history of the sample, the magnitude of Ho, and the angle a. a is defined as the angle between the vectors Ho and AH. We will designate Xr when a = 0 as Xrp, when a = x/2 as Xrt. In general, Xr can be expressed as: Xr = Xrp cos a + Xrt sin (1) Equations are derived for both Xp and Xrt in isotropic materials and in anisotropic materials magnetized in the [111 direction. Their

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN temperature coefficient is expressed in terms of Xo(T), Js(T) and J/Js. On the basis of these equations the two types of magnetization are compared. The results are applicable to most ferro-spinels, or ferrites since the [111] direction is the direction of easy magnetization in their crystalline structure. It is concluded that for the largest change in permeability ratio parallel fields should be used. This is of significance in the design of magnetic tuning units. 2. DERIVATION OF Xrp AND Xrt 2.1 General Remarks Brownal2,3 derived a theoretical expression for Xrp as a function of the magnetization J for various classes of ferromagnetic crystalline anisotropy. He assumed N domains per unit volume*, each with an equal and fixed volume. He assumed each domain was magnetized to saturation in some direction of "easy magnetization," that is, a direction requiring minimum free energy for magnetization. The partial derivative of the free energy with respect to magnetizing angle in the domain is the generalized anisotropy force. Physically, the anisotropy forces are those forces which act to prevent the magnetization in a given domain from being magnetized in a direction other than the "easy" direction. When Brown's assumption is used, if an external field is applied to the specimen the number of domains favorably oriented with respect to the field increases at the expense of the number of those domains not so favorably oriented. For readers unfamiliar with the theory of magnetic domains, reference No. 9 is recommended. __________________________________ 2 ---------------------------— _____

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Although this model is highly artificial, it can be shown3,4 that the model assumed does not significantly alter the macroscopic results. The microscopic forces are a sum of random and ordered forces both constant for a given domain. When integrated over a macroscopic volume of material,the results depend upon the energy magnitudes and not upon the exact nature of the forces. Thus, as long as one deals with purely reversible phenomena, the mechanism through which the forces act is unimportant. Brownl,2,3 utilizes these assumptions in a statistical approach to obtain the parallel reversible susceptibility and the magnetostriction in a ferromagnetic material. His results may be stated as follows: J = f So S (2) where: S = exp [Lo Hr Jp (9)] SO = f Jp (@) exp [Lo Hr Jp (@)] d l Jp (e) is the component of magnetization parallel to Hr when the domain is magnetized at an angle 9 with respect to the field. dc is an increment of solid angle. Hr is a multivalued function of the applied field and the history of the sample. It is the value of the applied field that would be necessary to produce a magnetization J in the sample if there were no irreversible internal processes, i.e., no hysteresis. It can be defined by the following equation: 3 5

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN J H f 1 dJ r= 7 S 0 Note that Hr, J, and Jp(.) are parallel by these definitions. Lo is a constant for a given material at a given temperature. It is a measure of the internal energy of the material and has the dimension (energy per unit volume)-l. Js is the saturation magnetization. It is the value of magnetization in the material when all domains are aligned. It is also at all times the magnetization in a single domain. For the case of isotropic materials Jp(e) = Js cos 9. Using this, Eq 2 becomes: so S Let 7 = Lo Hr Js and y = cos 9. Then: 1 lsinh s = S e' dr = n -1 dS So = Js d-q Upon solving for J/Js: J = ctnhn - (5) Js This can be expanded, for small values of 77, into: JL 7= X 2 _ +..... (3a) Js 5 45 945 _______~__ 4 __________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN If the magnetic material is assumed to have domains magnetized in the [111] direction, Jp (e) = Z v/3 3 The li's are the direction cosines of the directions of easy magnetization with respect to the crystalline axis. (In this case there are four such directions.) For this condition: Js Js = u tanh du d u. (4) 0 If one puts this in the form of a series solution: j - = 17 1732.... + (4a) Js 3 4 + 9 45 For either the isotropic case or the case of [111] domains J/Js can be expressed as: J = f (77) (5) Js From Eqs 3a and 4a it is observed that the series expansion is identical to the seventh order in 71. Upon differentiating with respect to Hr, both equations give: X LoJs2 (6) where lim X = X Hr-O r o 2.2 Derivation of Xrp From the definition of Xrp, dJp Xrp dr ------------------ 5 -----------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN From Eqs 5 and 6 and the definition of 71, Xp = f' (77). (7) 0 The prime indicates d/d77. Figure 1 shows a plot of Xrp/ Xo (Eq 7) for both the isotropic case (Eq 3) and the case of magnetization in the [111] direction (Eq 4) against the normalized magnetization (Eq 5). From the curve it is seen that the two curves are superimposed over the range 0 < J/Js < 0.6. At this point it becomes necessary to examine the mechanism through which the magnetic field exerts itself in order to obtain a better idea of the validity of the assumptions made. In the isotropic material we are in effect assuming that rotations can and do occur at any value of magnetization. In the case of the anisotropic material we assume no rotation. In a typical B-H loop, when J/J 0.6, rotations begin to occur. As J/Js increases, rotations become increasingly more important. This is the criteria for isotropic materials. Therefore even though the above anisotropic assumptions are made, at high values of applied field the equations for isotropic material must be used. From Eqs 3a and 4a (see Fig. 1), the two curves are identical at low fields. At high fields Eq 3 must be used. Therefore it is assumed that Eq 3 can be used throughout the entire region. (Fig. 1, Curve (1)). In at least the ferrites of iron and nickel, the magnetization direction requiring the least energy expenditure are the [111] directions.5 65 7 Therefore, we will use throughout f(7 ) = ctnh 77- 1/7. 2.3 Derivation of Xrt In a multicrystalline specimen when H, is rotated through a small angle8 J remains parallel to IHr and thus is rotated through the samle small angle. ______-__________________ 6

ZS- ~Z-L.'1V gZ- 09-V OL6-I 1.0 i - 0.8 0.6 xo Q4 02 0 0.2 0.4 0.6 0.8 1.0 J Js FIG. I REVERSIBLE SUSCEPTIBILITY CURVE NO.1 ISOTROPIC MATERIAL CURVE N0.2 MATERIAL MAGNETIZED IN [III] DIRECTION 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Therefore the transverse reversible susceptibility, Xrt, is given by: (see Fig. 2) X - dJ J rt dHr Hr dJ 1, ~ dHr, Hr Fig. 2 The Relationship Between dJt/dHr and J/Hr In order to relate Xrt to Xrp: dHr _ d_ J 1 = 1 dJ d [ j Xrp Upon combining with Eqs 5 and 7, Xrt f 7, ) (8) Xo= 3 / A plot of Xrp/ Xo (Eq 7) and Xrt/ Xo (Eq 8) against J/Js (Eq 5) using the Langevin function for f(7 ) is given in Fig. 3. 3. SPECIFIC APPLICATIONS 3.1 Tuned Circuit Frequency Range These formulas apply to electronic circuits employing variable _______________________________________ 8

N —L 6Z-f — V OL6-1 1.0 0.8 3f) \ \ 0.6 Xr! 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 J Js FIG. 3 THEORETICAL SUSCEPTIBILITY 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN permeability inductive elements where the resonance frequency is varied by the application of a biasing magnetic field. In particular, they apply to radio frequency circuits using ferrite cores with an electronically controlled magnetic biasing field. It is desirable to be able to predict the resultant frequency variation when a given biasing field is applied to a ferrite core whose B-H loop and whose history are known. As a special case it is desirable to know whether the parallel or the transverse fields would give the larger frequency ratio shift. The internal flux in the tnming unit at no time returns to zero. The resonance frequency minimum will be different for the two cases. Thus not only df/d77, but 1/f.df/d77 must be considered to compare the relative value of the two cases. For the case of parallel fields: 1 dfp 2ic 3 Xo(9 1^ <^p - ^ _ ____ ij^ ______ - Jf"(7 ). (9) fp d [1 +4it(f' (7) ) For the case of transverse fields: L. df't = - 2f - ]. (10) ft d77 [+ 4 f[ f 7172 [-f"(X1]and 4'-] are plotted against J/Js in Fig. 4. Lf"( f^7?^d f(_rl ) f(77 )f' (r7 (Curves (1) and (2) respectively.) These equations are applicable when lr>>l in which case the functions of 77 in the denominator are large compared to unity. For large values of r', Lr becomes small, approaching unity. For this circumstance curves (5) and (4), Fig. 4, show plots of [- f"(7 )] and [If(l) 7f' ) respectively. 10

Zfi-eZ-L 7)1X OL-*9-V OL6-N 1.0 i ii! -ll-i i 1.0 - - - -- I- - (I) and (3) PARALLEL FIELDS (2) and (4) TRANSVERSE FIELDS 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0J Js ) - f ) 3) 1L

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Since /0 for a typical ferrite would probably be between 100 and 800, curves (1) and (2) of Fig. 4 should be used until J/Js is about.85. For J/Js > 0.95, curves (3) and (4) should be used. The dashed lines connecting curves (1) and (3), and (2) and (4) represent a possible transition path between the two curves. It is to be noted that the heavy curves cross when J/Js is about 0.9. Since for J/J < 0.9, the curve for the parallel case lies above that for the transverse case, 1 df > 1 dft unmtil J = 0.9 Js. For most magnetic bodies, this represents a biasing field of at least 10 oersteds. Therefore, for practical applications, the maximum frequency change ratio is obtained using parallel fields. 3.2 Temperature Dependence of the Tuned Circuit Resonant Frequency The difference between this section and that of Section 3.1 is that Xo and Js are assumed to be temperature dependent phenomena, although independent of 7. From Appendix I -- For the case of parallel fields: 1 d2 f - 1-(7) d. X + dJs (11) (p dT - dT f"s f(7) ____________________________________ 12 ---------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For the case of transverse fields: 1 dft 1 - (() d X + ( ) dJs (12) -2 t dT = X T Js dT $(17) = 1 — jf(77) The above equations, Eqs 11 and 12, are based on the assumption that 4it Xr>>l. Plots of 5 (7 ) and (17) are shown in Fig. 5. From Fig. 5 it is apparent that: 0 < () ) < 2 o < ( 7)< 1. Using Eqs 11 and 12, together with Fig. 5, it is possible to calculate the temperature coefficient of the frequency change ratange ratio if J(T) and X(T), both readily measured experimentally, are known. In order to design a magnetic tuning unit with the lowest temperature coefficient, it is desirable to know which type of biasing field will give the lowest coefficient. Table I shows the values of the ratio y/x for which the parallel biasing fields give the lowest temperature coefficient using y - dJs /Js dT and x = 1/ XO d Xo/dT A plot of - 2+( r 17) ( ) against J/Js is given in Fig. 6. ^ )+ ((0 4. THEORETICAL ASPECTS OF Xo(T) AMS Js(T) It remains to predict Js(T) and Xo(T) as a function of the chemical composition, the heat treatment, and the impurities. The reversible susceptibilities are impurity dependent magnetic properties.9 Our knowledge of the 13

zg -2 0 - - - -rt to - v OL6- 2.0 1.8 -- 1.6 1.4..... 1.2.8 0 0.2 0.4 0.6 0.8 1.0 J Js FIG. 5 TEMPERATURE COEFFICIENT (1) PARALLEL FIELDS C (X) TRANSVERSE FIELDS 14

Z-Z II IZ-~9-v I Ioz 0.4 ------- - - -- 0.2 ~_____~_____ Js -0.4; —//y - + Cm~)2 -0.6 ____ _______ + -0.8 -1.0 FIG. 6 MINIMUM VALUE OF y/x FOR THE PARALLEL FIELDS TO BE MORE ADVANTAGEOUS THAN TRANSVERSE FIELDS UNDER CERTAIN CONDITIONS. (SEE TABLE I) lT

TABLE I m x-y signs 1 - ) Special Conditions y/x ratio m same > O None <1 Z same < 0 |(1- (7) )x >|(,7 )y any 9 opposite > 0 (1- (())x >| t( )y|; |(1- ('7))x[ > |(77)YI any I opposite > O (1- 9(Q))x >|(( )y; (1-$ (7))x|< [( )y| an incompatible system opposite > 0 (1- (7))xl|< ( )y; | (1- (q))x| > [(q ) none none - H -I O\ nm opposite > 0 I~(l-(R-))x <l (')yl (1-~ ())x; (l(7)yI none opposite < 0 | (1- (7))x > j(+ )yJ _ -_2+__(_7_) + Z () opposite < 0 I (1- ('))x| < j(i)y|j none.< O n I >(

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN physics of impurity sensitive materials is very small. Therefore any quantitative prediction of Xo(T) would be extremely difficult. Js(T) is not an impurity sensitive magnetic property, and thus better theoretical results can be expected. NeellO,ll and Watanabel2'13 have both considered the problem. Neel10 showed the types of magnetization curves possible using different chemical composition. Watanabe12 compares his theoretical results for various percentages of Ni-Zn ferrites with experimental results. The agreement is fair. 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX I The Derivation of 1/f df/dT Upon taking the derivative with respect to temperature of T) and Eqs 6, 7, and 8: dLo = 3 d X _ 6X0 dJs dT Js2 dT Js3 dT =H, L 0 L JI dT Hr JsdT + dT rp 3f' (7 ) X + 3X0 f" ( 7) d( dT 3 ~ ( 7 2 / dT dT r7 Upon simplifying: dXrp = 0 o, rf(7)+ 5-nf"(T7) xaJ ^"(77) (1+) dT dT 3f') + 3 -f(] J3 dT ( d Xrt = dX0 f )+ X j - f'(d771 (14) ardT is- ^7j7 dT r ( The quantity of interest is 1/f df/dT, where f is the resonance frequency of the tuned circuit at any given temperature. 1 df' - XrT ~ 2r p + 3 X ----------------- ig ---------------— 1~~~~r/

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Using the assumption that 47t Xr >> 1, af _ 1 d Xr/dT dT 2 Xr Using Eqs 15 and 14: For the parallel case, ( - t (. f(1)) d X o ~f"i(3 d -2 fp dT = ( (T i _ dT -5(') dXo + (n) dJs ~ Y dT Js dT (15) if 5 (7) = f For the transverse case, _1 dft - f'(17) 1 dXo + I1- fc1 a dJs -2 dT f(dT [ ) ] J s dT -2ft dT.... X ( )o d' + (7l1) Tsd (16) dT Js dT where ( ) = 1-. (-) 19 = - -.f').. e1r/

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN BIBLIOGRAPHY 1. Brown, William Fuller, Jr., "Domain Theory of Ferromagnetics Under Stress: Part I," Phys. Rev. 52, pp. 325-334, 1937. 2. Brown, William Fuller, Jr., "Domain Theory of Ferromagnetics Under Stress: Part II, Magnetostriction of Polycrystalline Material," Phys. Rev. 53, p. 482, 1938. 5. Brown, William Fuller, Jr., "Domain Theory of Ferromagnetics Under Stress: Part III, The Reversible Susceptibility," Phys. Rev. 54, pp. 279-287, 1938. 4. Brown, William Fuller, Jr., "Theory of Reversible Magnetization in Ferromagnetics," Phys. Rev. 55, pp. 568-578, 1939. 5. Bickford, L. R., Jr., "Ferromagnetic Resonance Absorption in Magnitite Single Crystals," Phys. Rev. 78, pp. 449-457, 1950. 6. Healy, Daniel W., Jr., "Ferromagnetic Resonance in Some Ferrites as a Function of Temperature," Cruft Lab. Tech. Rpt. No. 135, 1951. 7. Yager, Galt, Merritt, Wood, "Ferromagnetic Resonance in NiO - Fe203', Phys. Rev. 80, p. 744, 1950. 8. Brown, William Fuller, Jr., Private Communication. 9. Kittel, C., "Physical Theory of Ferromagnetic Domains," Rev. Mod. Phys. 21, pp. 541-583, 1949. 10. Neel, L., "Proprietes Magnetiques des Ferrites; Ferrimagnetisme et Antiferromagnetisme," Ann. d. Phys. (12) 3, pp. 137-198, 1948. 11. Neel, L., "Preuves Experimentales du Ferrimagnetisme et de L'Antiferromagnetisme," Ann. l'Inst. Fourier 1, p. 163, 1950. 12. Watanabe, Hiroshi, "The Temperature Dependency of Magnetization of Ferrites on the Basis of the Theory of Ferromagnetism," J. Phys. Soc. Japan 6, p. 212, 1951. 13. Watanabe, Hiroshi, Private Communication. 20

UNIVERSIT OF MICHIGAN 3 9015 03027 0261 DISTRIBUTION LIST 1 copy 1. Keiser Chief, Countermeasures Branch Evans Signal Laboratory 75 copies Transportation Officer, SCEL Evans Signal Laboratory Building No. 42 Belmar, New Jersey FOR - Signal Property Officer Inspect at Destination File No. 25052-PH-51-91(1443) 1 copy W. G. Dow, Professor Dept. of Electrical Engineering University of Michigan Ann Arbor, Michigan 1 copy H. W. Welch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Michigan 7 copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 1 copy Document Room Willow Run Research Center University of Michigan Ann Arbor, Michigan 1 copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 21