THE TEMPERATURE DEPENDENCE OF THE FERROMAGNETIC RESONANCE LINE WIDTH IN SINGLE CRYSTALS OF IRON AND NICKEL Thesis for the, Degree of Ph. D. MICHIGAN STATE COLLEGE Jerry Arnold Cowen 1954 IIIIIIIIIIIIIIIIIII m... I"l"'l'lllllllll‘lllllllllll" 2 This is to certify that the thesis entitled } 0-th Mam-av. k w'uflk presented bg M new“, has been accepted towards fulfillment of the requirements for M degree in PAM R.D- g‘WbL ‘ Major professor Mew 0169 LIBRARY "idligan Sm. U . . t Vt‘lt‘mm‘ e this checkout from your record‘ fore date due. ate if requested. N BOX to remov lD FINES return on or be ED with earlier due d PLACE IN RETUR TO AVO MAY BE RECALL “-4.9.“- — - | ABSTRACT multff'yQ_fim_ E’I‘llslf-mé'c. ‘-£'..,.‘.-_v.. l'Ll!:I_}_} 1;. ”and- The temperature dependence of the Ferromagnetic Resonance Line Width in single crystals of Iron and Nickel J 0A. Cowen The temperature dependence of the ferromagnetic resonance absorption in single crystals of iron and nickel has been measured from room.temperature to the Eurie point. The g factor and line width were measured and the line width interpreted in terms of a reciprocal parameter l/Tz. l/Tg remains essentially constant from room temperature to near the Curie point of the nickel (350°c) with.a value of approxiamtely 5x109 sec'l. Near the Curie point it appears to rise very rapidly but since the calculated value depends on acc- urate determination of the saturation magnetization.which.is strongly temperature dependent, the effect may not be real. In iron 1/‘1‘2 remained constant at approximately 10x109 sec -1 up to 750°C] The data was taken usingfi several different orientati- ons of the single crystals and also using polycrystalline samples- each sample gave essenttllly the same results. THE TEMPERATURE DEPENDENCE OF THE FERROMAGNETIC RESONANCE LINE WIDTH IN SINGLE CRYSTALS OF IRON AND NICKEL by JERRY ARNOLD COWEN A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and.Astronomy 1954 A .0.) . I. ‘ c ’ , A 1/ p. I 1/ A /\ ‘1 ACKNOWLEDGMENTS The author wishes to thank Professor R. D. Spence for his ever helpful suggestions and his constant interest. Truly, the work would never have been performed without the encouragement and many hours of assistance which he gave. I Be is also indebted to Professor A. J. Smith of the Department of Metallurgical Engineering for his suggestions on the metallurgical phase of the work and for the use of the facilities of his department. To Mr. C. Kingston for his patient help with the machine work involved and to Mr. Joe Mudar whose help and good humor lightened the load immeasurably. manythanks are due. VITA Jerry Arnold Cowen candidate for the degree of Doctor of Philos0phy Final examination: March 22, 1954, 10:00 A. M., Physics Conference Room Dissertation: The Temperature Dependence of the Ferro; magnetic Resonance Line Width in Single Crystals of Iron and Nickel Outline of Studies Major Subject: Physics Minor Subject: Mathematics Biographical Items Born: July 17, 1924, Toledo, Ohio Undergraduate Studies: Denison University 19h2-43 Harvard College 1946-48 Graduate Studies: Michigan State College 1949;54 EXperience: Project Engineer, Alden Products 00., 1948-49; Graduate Assistant, Michigan State College, 1950-52; Research Assistant, Michigan State College, 1953; Assistant Professor, Colorado A & M College, 1953-54 Member of Sigma.Xi, American Association of Physics Teachers, American Physical Society TABLE OF CONTENTS I. INTRODUCTION II. THEORY 8.. b. C. Energy Absorption in a Magnetic Conductor Ferromagnetic Resonance Absorption in an Isotropic Medium Ferromagnetic Resonance Absorption in AnisotrOpic Media III. THEORY OF THE USE OF THE RESONANT CAVITY IV. EXPERIMENTAL METHOD a. b. c. d. e. f. g. Microwave Apparatus Mounting of Samples Heating of Samples The Magnet Preparation of Samples Electrolytic Polishing Domain Structure V. ANALYSIS OF RESULTS VI. RESULTS AND CONCLUSIONS 3. b. 0. LIST OF Nickel Iron Conclusions REFERENCES CITED Page 10 15 20 25 27 3o 31 34 36 39 41 50 59 69 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure l. 2. 3. u. 5. 6. 7. 8. 9. 10. ll. 12. LIST OF FIGURES Precessing Magnetic Dipole in a Large Static and Small RPF Magnetic Field Motion of the Magnetization Vector in Ferromagnetic Resonance Diagram Illustrating AnisotrOpy Torque Equivalent Circuit of Resonant Cavity Block Diagram of Apparatus Cavity Assembly as used in AnisotrOpy Measurements Magnet Calibration Curve Experimental Apparatus for Electrolytic Polishing of Small Iron Samples Electrical Characteristics of the Polishing Cell The Method of Determining Hg and H; from the EXperimental Curve R(H) . and the Theoretical Curve RI and RP Normalized Curve for ElectrOplated Nickel Ho res as a Function of Angle in the (100) Plans of Nickel Ho res as a Function of Angle in the (110) Plane of Nickel Temperature Dependence of 1/T2 for Nickel l/TZ versus Number of Scratches in an 18% Co-Ni Plated Alloy g Values for Nickel Page 11 18 20 26 29 33 37 37 44 51 52 53 55 58 60 Figure Figure Figure Figure Figure Figure Figure 17. [.1 CO o 19. 2c. 21. 22. 23. Ho reg as a function of Angle in the (100) Plane of Iron Ho res as a Function of Angle in the (110) Plans of Iron Diagram Illustrating Effect of Ani— sotrOpy Torque at Low Applied Fields Temperature Dependence of Ho res for Double Peaks Temperature Dependence of 1/T2 for Iron Photograph of Apparatus Photograph of Apparatus 61 62 65 66 72 73 THE TEMPERATURE DEPENDENCE OF THE FERROMAGNETIC RESONANCE LINE WIDTH IN SINGLE CRYSTALS OF IRON AND NICKEL I. INTRODUCTION Ferromagnetic resonance absorption can best be understood by consideration of the motion of a single magnetic dipole in an external magnetic field. Consider an essentially free electron with a spin magnetic moment /k. In the presence of a static magnetic field.H, it will precess with a frequency 75:53:; H known as the Larmor frequency. If a radio frequency magnetic field (of freqé uency f) is applied to the dipole at right angles to the static field (Fig. 1) the dipole will absorb energy from the radio frequency field with an accompanying increase in the angle 9 between the dipole axis and the direction of H. A'ferromagnet can be visualized, in the simplest case, as consisting of a large number of electron spin magnetic moments all strongly coupled together to give a large resultant magnetic moment. In the presence of static.and radio frequency fields of the correct magni- tude and configuration it will absorb energy from the varying field. Quantum mechanically we say that the static field splits previously degenerate energy levels into levels of magnetic quantum numberma and that the radio frequency field can excite transitions between ’ff-“—‘~_ ’ ‘\ K g -— v-- be‘ ‘( H,f FIG.I PRECESSING MAGNETIC DIPOLE IN A LARGE STATIC AND SMALL R-F MAGNETIC FIELD these levels so long as the selection rule d m5:1‘/ is satisfied. The lowest energy level is the most densely pepulated so that the net effect is an absorptive one. The absorption continues because the spins give up energy to the crystalline lattice tending to maintain an excess of electrons in the lower state. This absorption of radio frequency energy is describable by a change in the radio frequency permeability. Whereas the absorption due to a single dipole occurs at essentially one frequency or value of applied field, the absorption in a ferromagnet is appreciable over a broad range of applied fields, giving rise to a resonance type curve. The first eXperimental observations on ferromagnetic resonance absorption were made by Griffiths1 in 1946. He electrOplated thin films of the ferromagnetic elements onto the endplate of a cylindrical microwave resonant cavity. The static magnetic field was applied parallel to this surface and the cavity was excited in such a manner that the radio frequency magnetic field was parallel to the endplate and also perpendicular to the static field. He then measured the Q of the cavity as a function of the applied field and observed a broad resonance type curve. A short time later‘Yager andBozorth2 observed the absorption in sheets of Supermalloy and in 1948 Kip and Arnold3 made measurments on the absorption as a function of the orientation of a 3% silicon iron single crystal A few other papers have been published on resonance in metals?’5 the most complete of which was the work of 6 Bloembergen on the temperature dependence of the reson- ance position and line width in polycrystalline Nickel and supermalloy. 7'10 were investie During this period several workers gating the temperature and anisotrOpy dependence of the resonance line in various ferrites. These materials are magnetic semiconductors of the form XOFeZOB where X is a metallic bivalent ion. They have high resistivity‘ (compared to the metals) and consequently large skin depth and few conduction electrons. The large skin depth means that samples of reasonable size will be uniformly magnet; ized by the microwave field, and their lack of conduction electrons presumably simplifies the theoretical calculations. Because of the simplifications which the ferrites appear to introduce, resonance in metals has been neglected. There have been no measurments on the temperature depend; ence of the resonance line in single crystals and only the few mentioned above on polycrystalline materials. The situation with respect to theory is even less satisfactory. Griffiths attempted to interpret his first results by assuming that the magnetic dipoles precessed in the Lorentz field./7=wfi2+§£77V . The resulting cal; culated Larmor frequency was small by a factor of about, 11 four. Kittel, using phenomenological arguments, suggested a modified theory in which the resonant frequency was I, given by fig (BHJZ1 instead of f‘3%/72 as in the Larmor case. This theory, which will be discussed in detail later on, has been very successful in predicting the position of the resonance line as a function of sample shape, anisotrOpy and temperature. Van Vleck12 has ob- tained essentially the same results as Kittel using a quantum mechanical approach - obtaining eigenfrequencies which are the same as Kittel's resonant frequencies. There have been no satisfactory theoretical eXplan- ations of the temperature dependence of the line width. Bloembergen, using relations first employed by Bloch13 to discuss the problem of nuclear magnetic resonance, was able to represent the line width by means of a spin- spin relaxation time and a spin—lattice relaxation time. Abrahams and Kittelli+ calculated the temperature depend- ence of the spin-lattice relaxation time by considering that the radio frequency field excited spin waves and assuming that the spin system itself came to equilibrium before the spin lattice transition had time to proceed. Unfortunately, it is commonly assumed that the principal contribution to the line width comes from the spin-spin terms and there have been no theoretical calculations on their temperature dependence. Kittel and Abrahams15 have suggested that although at temperatures below one-half of the Curie temperature the spin-spin terms do predominate, at temperatures higher than this the spin-lattice terms take over. If this is so then the assumption that the spin- spin terms make the major contribution may be in error. The object of the present work was to measure the temperature dependence of the ferromagnetic resonance absorption line width in single crystals and.polycrystal- line samples of iron and nickel from room temperature to the respective Curie temperatures and thus obtain infor- mation on the mechanism responsible for line width and the effect of anisotrOpy on the line width. It was heped that this would present a more complete picture of reson- ance in metals and lead to a more satisfactory theory of ferromagnetic resonance. II. THEORY a. Energy Absorption in a Magnetic Conductor Before considering the theory of ferromagnetic reso; nance absorption, it will prove useful to derive the equa— tions for the energy absorbed by a ferromagnetic conductor from an radio frequency field and to show that this can be expressed in terms of an effective permeability'/[7%516 In the Gaussian system of units, Maxwell's equations are given by: ' B) vxlz’. = ‘E 5i D) v-_B_=0 2-4 with the constitutive relations éz/ifl ; P: G .E :Jsa’fi. Assuming that the field quantities vary in a sinusoidal manner with the frequency A), (A) and (I) reduce to CVXH = (47rr+./'WE)§_ CVx£=—jw/afl 22 From this it can be shown: v2 H: klfl 24-3 where K4: (4175!“1/9“ “’2 6,11) 6-2. In general both /L and 6: may be complex,/«=/¢.*J'/u; (“S-«1'51. By assuming the characteristic dimension of the conductor (in this case the thickness) small compared to the wave length in vacuum,the sample can be considered to be in a uniform field Ho, where H0 is the amplitude of the time varying field. The field outside the sample satis- fies Equation 2-3 with K,=c7. Consider an infinite sheet of thickness 2d, then the solution inside is Hg}, = ,4, MK; 7 + 14,2 M K; 7— . The boundary conditions are (HQ;,)£141:: (Ffluzllidy ‘é‘fflr)4un,== Cu HamtJ4wt thus at #:11 I H4”: HM at p=:e{ TH;wt=-I4,ADAAIAII(II Ab Cou4'4?v( at 7.“; Had: -A, Maw-I 41 M ma! Solving the above equations for44,and 4;, we find A,...., A: H» cauthzd 244 8180 Vx§=_l V A gm 3. ‘x/‘wé Ha Cw Kb.?~ ¢ LnuLAxaL PM; = “1:11: H. M /r.' p 2-5 k; L Clair/421 The energy absorbed is given by w= '4 fffrmwenéa" + Wu M‘] W Therefore in this case the power absorbed per unit area from one side of the sample is ' .2 w: E&(t/w/u’ #0 fW/fid) 2;? Yr 2?; For a ferromagnetic metal as used in the resonance eXperi- ments /KId;x>l and therefore the hyperbolic function can ' a be replaced by 1 resulting in w: Re (£2; ”0/ PIT ; for the power flowing into one side of the metal. In a metal the displacement currents are much smaller than the conduction currents and therefore the 6 term in/fij may be drOpped giving . /ys ffie(~/w/“é °___) LU II if‘I/TW = 4 f/Q‘ag’i. FLA/Z.) 377‘ V4W‘r K/fl Let us define an effective permeability/aiét= fitTpf/u such that yFI (, , a , ,,t/ p 67—; 2 JUKI'*/fi. ::'K2‘ ,4. I’J'/‘ 2‘8 therefore 3 Z. , T— , 7A4 4140 /x —.—_—— Y7“ Yfl“av The ferromagnetic resonance experiment measures a quantity prOportional to a) . It is also of interest to ask how'fc and /9_are related to/a' and /M”; solving Equation 2-8 one obtains fl”: l/L‘ILI' fl»; +/a'2v 2.93, #"= W -/e 2-9b b. Ferromagnetic Resonance Absorption in an Isotropic Medium Consider a thin sheet of ferromagnetic material, ly- ing in the xy plane, which is subjected to a large static magnetic field in the z direction and a small radio frequen- cy magnetic field in the x direction (Fig. 2). The mag- netization vector will consist of two parts, a large cone stant M; and a small varying Mxy. The magnetization is related to the total angular momentum by M = X J 2-410 where &= t2, g is the spectroscOpic splitting factor, 9 and m are éhe charge and mass of the electron and c is the velocity of light. The equation of motion is 3.4;]- : W $27—44” 2-11 ,1; . In the case of an isotrOpic ferromagnetic material the torque is Juetflxg‘where fleie the effective field inside the material, that is, the applied field corrected in a suitable manner for the demagnetization effects. The presence of ferromagnetic anisotrOpy will introduce an additional torque which must be added to that already present. In order to simplify the argument the anisoté ropy will be neglected for the present. The components of f/e can be given by an expression of the form 6-8: @‘M@' where f/J'are the components of the applied field, M are the components of the mag- netization, and./M?are the demagnetization factors which depend on the shape of the sample. These factors have rigorous meaning only when the sample is ellipsoidal 10 ‘ ’5- \_ ’ I \ ' Is."— ' 3r \1’ HA? «Y—THHJ SAMPLE FIG. 2' MOTION OF THE MAGNETIZATION VECTOR. IN FERROMAGNETIC RESONANCE 1. and when the applied field lies along one of the principal axes of the figure; for only in this case is the internal field uniform and parallel to the applied field. In this experiment there are demagnetization factors of two types, static and radio frequency. For the static field which penetrates the sample uniformly, Osborne17 has shown that #:= If; ' Aéflfgwhere 4/3 = EC; [I - 77:67; for large m. Here m is the ratio of the diameter of the spheroid to its thickness; thus Ag!!!“ . For the radio m. frequency field which penetrates only very slightly into the sample, ”C: = fir "Aaifl41 H;=-’V7M% where M1 and My, are the radio frequency components of the magnetization and Mr: £3 ) A? = 477‘ {I 1—7:, + ”'34) Here m' is the ratio of the diameter to the effective thiCkness. The thickness is of the order of magnitude of the skin depth 5: raj/f7. and therefore A4 -’—" 0 and 44,5347!“ . If one were to solve the equations of motion as they stand in terms of Hx and H; for a radio frequency sus- deptibility as a function of frequency, the solution would be essentially monochromatic. Since the resonance lines do have width, damping terms must be added to ac- count for this width. We will, following Bloembergené, use the notation and formalism which was applied by Bloch to the case of nuclear resonance. 12 There are two types of interactions which can broaden the resonance lines (still considering only isotrOpic media), those which change the total energy of the spin system and those which can only modify the various components with- out changing the total energy. In the first class are thermal agitation and spin-orbit interactions. These may be represented as a perturbation onIMQ , which in the un- perturbed state and for large fields will be parallel to fig but which may, due to these effects, have components in other directions. If this is so, Mfwill approach M, with a relaxation time 7; given byf] .- - ”it-MD . 7fis known as the spin-lattice relaxation time. 7b The other effect is due to the local magnetic fields of neighboring dipoles or to small inhomogenieties in.fi/. We can represent the whole complex of such effects by an effective irregularity in Mt} of the order of H’ and de- fine I’H’: 7’_ . Here again the perturbation on Mx or M; o? is assumed to decay eXponentially with a relaxation time’T 7o such that M“ “ fl): . 7213 the relaxation time for 73 . all effects disturbing the spin system from within. Separating the vector equation of motion into its component parts, we have ' 8 MX= Nanci],< - 2412 Assuming ”‘42: ’2 ’Mo /and that all time variations are of 13 'wt the form 6" , one obtains J'w M” 3M} [H04 f/V—A/AJMOJ- £47.; 2 _./zv /W' = J’ZLPI [US$1I (A/p /\/z )flVJ7_ A/ [W0.;+ /M9u/7‘ 2-13 ‘kfld ”a = y'fi91[/Tx I'/WN (Aér‘/%n£/ Solving for $21 = K , one obtains 1/)"sz y1M2MO[HO+(/V_/V£,/M,] 24-14 H (you %I[M,(J-w+%)+ {KM—NglMe]+ Wig M, where this is the complex radio frequency susceptibility and is related to the permeability by ‘4TFJL =/4-'I 47T('X,—ij;)='/4,-j,a4 _ Thus fl—IITXMZIHWI’V} N£)M°o_](lu -w1) +1 (10," Ma); 4- 440/7”? 2‘15““ ;& :.I¥WFJ':WO [:Ha I (N, ”V3 )/W° J 3&77' ( ujfw1)1'+'“”77‘2 2-L56 where = 6’{H,+ (NI-map {Ho+//v, gm] I ____ 2‘15 These eXpressions for the real and imaginary parts of the permeability can be combined to give the quantity/u] defined in Equation298. l4 c. Ferromagnetic Resonance Absorption in AnisotrOpic Media It can be shown experimentally that single crystals of magnetic materials are anisotrOpic with respect to an applied magnetic field. In iron, which is a body centered cubic crystal, the [100] directions are most easily mag- netized while the [111] directions are hardest to magnet- ize. If a disk shaped sample of a single crystal is placed in a magnetic field, it will rotate so that the easy dir- ection is parallel to the applied field. This anisotr0py torque is eXpressible in terms of an anisotropy energy and will affect the position and width of the ferromagnetic resonance absorption. The empirical condition which the anisotropy energy must satisfy is that it have the symmetry of the crystal. The energy is thus defined in terms of a set of experi- mentally determined constants, the anisotropy constants, and the direction cosines of the angle between the mag- netization vector and the crystal axes. Thus for a cubic crystal, E = m K1423 s‘aclwc‘e") + *2 “6 ’45“? 2-17 where the «2 are the direction cosines and it and Al— are the first and second order anisotropy constants. The odd power terms are not present because they would not satisfy the requirements of cubic symmetry and the terms in ext}and, «guare missing because of the identity in the direction cosines 0/, ”+ ’91 +43) = / . For large applied fields the magnetization is approximately parallel 15 to the applied field and the direction cosines can be ex- pressed in terms of the angle between the applied field and the axes. Consider a disk cut in the (100) plane, age/445,6 ;a(,~=s.ea 343:0 where e is the angle between the [100] direction and H. Then E:- K, + K, M}9&c29 The torque is given by T: ._aa5 6 =‘3‘2M29 W24=“;€1M¢& A similar analysis for the (110) plane of a cubic crystal gives 7“: —f_. (2mm +3/Lmsé) +12 (M25- —4m 49 -3Mée)2 18 Y 44 ’ In ferromagnetic resonance absorption in iron, nickel, and their alloys the effects of the second order terms are small and therefore have been neglected. The most direct way to introduce the anisotropy energy into the expressions for ferromagnetic resonance is to add the anisotropy torque to the equations of motion. This involves solving the equations of motion for each case. Kittel11 has suggested an artifice whereby the anisotropy torque is considered as exerted by an internal. field HW with the torque being expressed as [ff/X db”? This internal field is represented by demagnetization fact- ors of the form . HXW : - ”Km Mx anus 4M“? In the absence of the radio frequency field there are two 2-19 16 torques which balance one another - the anisotrOpy torque 7": ~15 and the torque exerted by the applied field Msh‘o . We aréfiinterested in torques which change the motion of the magnetization vector,that is, torques exerted when the magnetization vector has been displaced by the radio frequen- cy field. Thus (Figure 3) if M is rotated in the x direction through an angle .09 , there will be a torque in the nega- tive y direction <67}._ _ 3’5. 136 - cast :-— C9D? <36 A7} 77.2. . Ascording to our convention this is due to f/ and there- fore ’4u4 -2;E A6: A7}: -— Mi- #X ”M 27’- bllt M4 1v Mi A g a -— 441.44 therefore g LL 4 9 : M; Nx Jet For the case in which INQ lies in the (100) plane and W * 7:5'M49 . 02"? W49: M! Ma; MWMX 2 then A/ W _ g/(l 54—7415 x _ [14%; 2-20 and similarly ,V W __ 1’6 cowl/9 In the case where fit; lies in the (110) plane a similar analysis (to first order terms) shows W k) 2- _ 32 1‘29 . Ngw“ 33.5 (l—Jm’é -;3_ M2329) 2-23 at) 0 Adding these terms to the equations of motion, we obtain 17 OS-S IS-S 834-8 ES-S BOO} (OOI) PLANE EQUILIBRIUM Ho POSITION FOR M ‘2. {mo} FIG.3 DIAGRAM ILLUSTRATING ANISOTROPY TORQUE. 18 JwM,‘=?[Ho"/+(4 —A/)MO;—jM T: J'wa X;[Ho4(Nx+aQ'-Nx AQMMM -M Al}- M M}; =xI-Hx+{"/X*M ‘ (”WWUM’J Mfi 2‘2“ Now again solving for the radio frequency permeability /& with the same approximations as before. ’0': iWXM/XM[H0+(M} 4‘”)ij(‘01 ‘01) U 2-25 (w —w1)2‘ + (Stu/7;):— 2. 4 M = I?” M0£”°*(”}*”;”‘4)M°] 2‘77: 2-26 (wf— to”); + (aw/7;); where (4):..3/{HMM/1LN WIN/w] “(H +(A/w49- NM)N} 773-27 This allows us to formulate the problem for anisotrOpic media in terms of all of the factors used in the isotrcpic case and the anisotrOpy demagnetization terms. It is of interest to note that this deveIOpment has assumed that the sample is magnetized to saturation for all values of Ho. Under certain conditions this assumption is incorrect. Its effect on the resonance curve will be discussed in Section VI b. 19 III. THEORY OF THE USE OF THE RESONANT CAVITY A'microwave resonant cavity consists of a dielectric filled region completely surrounded by conducting walls except for a small iris or opening. There are electro- magnetic field solutions of particular configuration and frequency which satisfy the boundary conditions and corre- spond to the storage of electromagnetic energy over rather long periods of time. These normal modes, of which there are in general an infinite number. are the resonant fre- quencies of the cavity. We are interested in only the first few of these for which the linear dimensions of the cavity are of the order of the wave length in the guide. In analyzing the action of a cavity, the actual cavity parameters can be replaced by an equivalent circuit made up of lumped elements. Thus the cavity and coupling iris may conveniently be replaced by a parallel resonant circuit which is coupled to the transmission line by a transformer. (Figure 4) ‘ 5 I - <—20 q, .3 “Iv R’ —‘—c ' l v I I FIG.4 EQUIVALENT CIRCUIT OF RESONANT CAVITY 20 We have here assumed that the transmission line is ter- minated on the left by its characteristic impedance 2% and that the line sees an effective resistance If on the right. The transformer has a turns ratio n and is assumed to be ideal, that is, there is no energy lost in the trans- former. If the cavity is tuned to resonance. from the con- #‘U. RI R’zhifi 3-1 and the impedance which the cavity sees is 23’: n12. servation of energy, jfillern*vm' , therefore R The absorbed power is thus I: m~+vR)‘:_xg-_f(,+/v) 3.2 R R R where V is considered to be made up of two components. the voltage of the incident and of the reflected waves on the line, and [7 is the voltage reflection coefficient /”= fig . At resonance the load is purely resistive and n; /—- /7 34-3 [1 2 [6-20 9 thua /-;‘/"v R-I-z-o 532. R Hence at resonance P.1= iL(/+/"1} =~ lo: (/--/72) R” -£5 3; 4 PH "dz 3:1: I” £0 ) '20 where /f, ’i_and ’1,are the incident, absorbed and re- flected powers respectively, and f2: /?-fli . Since both i? and £0 are real numbers, F at reso-'- nance is real and may be positive or negative. $.19 a measure of the coupling between the cavity and the guide. 21 We consider two cases depending on whether {£213 large or small. Case 1 Case 2 Zismall, undercoupled Zglarge, overcoupled [’negative, v,” negative fpositive, Kpositive lam”: guy”, A“: Via—V,” Kym: l/L'i-Va, M=V5~Kv 3-5 The voltage standing wave ratio /> is defined f::Vhaa¢ and therefore ”Haw" f: Vt—Vm = I—F: 339 /’= ”*sz /+/7 2“@346 V;+V4, /+/7 R Vb—K" "f' in We now wish to define the Q of the cavity and relate it to the quantities which we usually measure- fl, IF/ , F”, . The total Q is defined as Qt alw-energy stored in the cavity total energy lost by the cavity per period This total Qt is made up of two parts, the unloaded Qu which represents all of the losses in the cavity proper and the external Qe which represents the energy lost by the cavity to the guide. These are related by __l_.; ___1___+_l.._ Q1; Q3 Qu By considering the energy stored and lost in the equivalent circuits, we can show that qu : WI? and Qe =n22° wt. «IL. (This is only true if the guide is terminated in its characteristic impedance as it is in this case.) Thus 22 Qu is itself made up of two parts, Q0 due to the losses in the capper walls of the cavity excluding the ferromagnetic sample, and Qfer due to the losses in the ferromagnetic sample. These quantities are related in a similar manner by l : l + l Qu Qc Qfer The Qfer is primarily due to the magnetic losses in the sample and only very slightly to the eddy current and dielectric losses. Qrer =‘Zyy-energz stored in the cavity magnetic energy loss per cycle We have shown in section II that the magnetic power loss in a plane sample is prOportional to the square root of /u', the real part of the effective permeability. It has also been shown that /a' is a function of the applied field.f/. Using these results along with the fact that the other losses in the cavity are independent of f/ , we obtain —J-— = ”IV/:7 ,’ / : H’W +3, 4%” a and finally Qe : A W +8 3-6 a We actually measure a power p, proportional to the re- flected power, as a function of the applied magnetic field. 2 or /~,_ 272 where 0(= Vac” ‘- a< We can determine o( by measuring both p and {9 at one point on the curve. For either type coupling,/fl=';L:%fl7 /' f7 23 . I therefore [p].- F-l and hence «K: F " [E—fi) for 3-6a 9+: any point on the curve. Making use of the result ob- tained earlier that ~5§i =;%%. and.also of Equation 3-6 a. we obtain for the two cases undercoupled overcoupled 4/;7 +8e/o =_1__t__/_CJ AV;+3=.L= "/7 "" ”7/ (0 /+/7 7 : /+£——" o ________——-—-'—<:- 3-,"? I— L2 ,I ’< I+~tér’ 3-7 ’/3_ 1“ =;<_+.ZL?_ ,_ 1.21:7 d’fa— difJ-a where A and B involve Qc and Qu and are constants for a given cavity at a given temperature. We have thus shown that a quantity prOportional to the effective permeability can be obtained from measurments made on the cavity. 24 IV. EXPERIMENTAL METHOD a. Microwave Apparatus The apparatus used in the eXperiment is of the con; ventional type as shown in the block diagram (Figure 5). The klystron is a 723AB low power oscillator Operating at about 9000 megacycles per second. It is modulated with a 1000 cycle per second square wave which is supplied, along with the necessary D0 voltages, from a voltage regulated source. The klystron is isolated from the load by a flap attenuator and these two elements are matched to the H-arm of a magic T with an E-H tuner. The E-arm of the T is connected to a matched detector, the output of which is read on a Browning TAA416 twin-tee tuned amplifier peaked to match the frequency of the square wave. The two side arms of the T feed the energy to a matched load and to the resonant cavity through a standing wave detector respectively. Energy from the klystron is coupled into the two side arms but not into the E-arm of the T. That portion which goes into the arm containing the matched load is totally absorbed, the other portion is reflected from the cavity and partially'couplé ed into the E arm. Thus, only power which is reflected from the cavity is measured by the detector and the 25 Zm>O /. >H_>;\\ \ mwowi whoa .maaouoaao S 60306 .605 m 830E 207515002 m n. 0 009 can mObowhwo omxobqs. 5&8 $30.. $538 m><3 025245 L a . l 7 $22. $5.: .232 a 20 55x } rum _ SE . a5... 19 m < 29 eureka}. 26 meter reading is prOportional to this energy. The match— ing devices prevent multiple reflections in any of the arms of the T. The overall matching of the system is to a volt- age standing wave ratio of about 1.2 except in the arm which contains the cavity, in which the actual standing wave ratio depends on the type of cavity used. A mica window was cemented in the guide to prevent leakage of the hydrogen gas which served as a reducing atmosphere. It was found that the window introduced so little reflection that it was not necessary to match out these reflections with a tuning device. The cavity used was three half wave lengths long and was coupled to the guide with a circular iris symmetrically placed. An iris with a diameter of less than 5/16“ re; sulted in an undercoupled cavity. Although data was taken with both undercoupled and overcoupled cavities all of the results reported upon were taken with the undercoupled cavity. The cavity was assembled with silver solder in an atmosphere of hydrogen to prevent excessive oxidation. When oxidation did occur the inside of the guide was pol- ished with a fine grade of carborundum on a wooden swab and washed with a 10% solution of nitric acid. b. Mounting of Samples Three methods of mounting the samples were used. If the sample was large enough to cover the end of the guide - this was true of the plated samples and the rolled poly; 2? crystalline samples i it was soldered directly to the end of the cavity using the hydrogen atmosphere to prevent oxidation. Nickel samples were soldered with pure lead and clamped lightly to keep them fixed in place near the Curie temperature (352°C) where the lead held only due to surface tension. Iron samples were silver soldered. In order to make the anisotrOpy measurements the cry; stale had to be rotated about an perpendicular to the face of the sample. This was accomplished as follows (Figure 6). The bottom of the cavity was closed with a brass block in which a conical hole of 80 taper was bored in such a man- ner that the small end had a diameter of .400", the nomé inal inside width of the X;band guide. The sample, solder— ed with Wood's metal on a matched conical cylinder, was pressed into the end plate and tightly clamped. This made 0a quite reproducible contact and lent itself readily to the anisotropy measurements. In order to measure the angle a protractor was soldered on the conical cylinder and a small pointer was mounted on the end plate. This device was capable of 2° reproducibility, sufficiently accurate for the data which was taken. The temperature dependence data on the single crystals was taken as follows. A brass plate large enough to cover the end of the cavity was bored with a cylindrical recess .005" shallower than the thickness of the sample and with the same diameter as the sample. The sample was then sol- dered into the recess, mechanically polished until the 28 '- i ‘______ x- BAND : _ WAVEGUEDE WATER INLET 4"“ _______, L COOLING ‘h l.-- I P _u 3 JACKET 1 : WATER OUTLET } J:Z——+ i I F L A N G E ———<;: W//”//////////////////‘ 4-—— M I C A W| N DO W I: HYDROGEN . "‘ ' "INLET [—e—o—IRIS RESONANT CAVITY ‘———L _.__._. __SAMPL r1— 5 ,4" r; “v H YDROGE N i ‘ OUTLET 1 ,. +——-CON E Figure 6. Cavity assembly as used in anistropy measurements. 29 __ _____.J sample and plate were flush, electrOpolished and then the plate was soldered to the guide. In addition to these methods of mounting, some com; parison data was taken with samples clamped against win- dows in the side wall of the cavity. This data showed considerable variation due to non-reproducibility of con; tact. Insertion of a thin mica window as suggested by Kip and Arnold only served to decrease the signal strength without adding to the reproducibility. c. Heating the Sample The heating coil consisted of 9 ohms of .112" x .005" Nichrome ribbon wound on the cavity in a non inductive manner and Operated on A. C. from a Variac. It was wound on a base of asbestos paper and held rigid by a thin layer of potters clay. An insulating layer of glass wool was wrapped on the clay and this in turn held in place with a layer of asbestos applied wet and baked dry. The oven was capable of temperatures up to 800°C with a power input of 500 watts. The temperature 'as measured with a call; brated Chromel-Alumel thermocouple inserted in the cavity end plate in such a manner that it read the temperature of the block immediately adjacent to the sample. It was checked with a thermocouple mounted in place of the sample to an accuracy of 2°C from room temperature to 60000. In some cases there were larger thermal lags at higher temp- 30 eratures but these could always be checked by the fact that no ferromagnetic resonance absorption should be ob- served above the known Curie temperature. If there was appreciable absorption above this temperature, the data was discarded. The output of the thermocouple was read on a Leeds and Northrup type K potentiometer. Oxidation of the samples at high temperatures was prevented by passing dried hydrogen gas through the section of the guide con- taining the cavity. The gas entered above the cavity and was passed out at the bottom through a 1/16" cOpper tube, where it was burned. Immediately above the cavity was a section of guide fitted with a water cooling Jacket to insure that the re- mainder of the microwave system was not heated excessively. The mica window to prevent leakage of the hydrogen was mounted on the flange below this cooling coil. d. The Magnet The magnet was constructed of SAE 1020 low carbon ‘steel in the form of a square box 22' x 22' x 9'. The sides of the box, which were 3 1/2" thick, were machined and bolted together with 1/2" bolts. The pole pieces (7“ in diameter), which were machined to slip fit holes bored in the box, were threaded and held in place with retaining rings 1“ thick. Since the ferromagnetic re- 31 sonance line is very wide no special treatment of pole faces to insure a very homogeneous field was undertaken. The cooper used was .072" square formvar insulated wire which was wound on four capper bobbins. The bobbins were made of 3/16 “ thick copper sheet and were designed so that the finished spool was 15“ in diameter and 1 3/4“ thick, each with a resistance of approximately 5 ohms. The coils could be connected in series or parallel as required by the power source which in this case was a bank of 12 marine batteries of 300 ampere hour capacity. In spite of the formvar insulation it was found necessary to insulate the wire from the bobbin with a layer of .010“ asbestos paper. After winding, the coils were dipped in formvar and wrapped in linen. The magnet was controlled with a potentiometer consisting of a water cooled rheo; stat capable of carrying 18 amperes, while the magnet current was read on a Westinghouse Type Px-h meter to an accuracy of about 25 milliamperes in the range 0-5 amperes. In this range the field-current relationship was linear.(Figure 7) The hysterysis of the magnet was approximately 65 gauss and therefore care was taken to read all points in an increasing current. With a pole piece separation of l l/h" the magnet was capable of fields of 12,000 oersteds at a current of 18 amperes. The calibration curve was determined with a Sensitive Research Company model F. M. fluxmeter and was cali- brated at several points with a proton-resonance signal. 32 1,, * , ,, .l. -‘u‘. . . . . k . . . ' O . O . . l . . . 6000* 5000* a) 04000“ 3000‘ - OERSTE H 2000- I000“ G) PROTON RESONANCE CALIBRATION POINT 2 3 4 5 6 7 I - AMPERES FIG.7 MAGNET CALIBRATION CURVE 33 e. Preparation of Samples samples of three types were used: 1. ElectrOplated 2. Rolled polycrystalline 3. Single crystal 1. The nickel samples were plated from a commercial bright nickel (Udylite) which was spectroscopically analyzed to show a small amount of cobalt (probably con- tamination acquired in the laboratory). Samples of a 1% Co-Ni alloy and a 18% Co-Ni alloy were also plated. A typical sample was .001“ thick plated on a brass disk. A dull nickel plate was also tried, but this was not suc- cessful even after electrOpolishing. 2. The rolled samples were of two types, a soft temper commercial type A nickel .004" thick, and a 3% Silicon iron oriented alloy .015" thick. These samples were electrOpolished before use. 3. The single crystal samples came in the form of rods of Armco iron and commercially pure Ni 3/8“ and 1/2“ in diameter respectively. The iron rods were etched and crudely oriented visually by means of light reflection from the crystallographic planes; then oriented more ac— curately by back reflection x-ray methods. The crystals were then cut, on a wet carborundum cut off wheel, into slabs .060“ thick, polished and reetched until they were 3# “k"- Ifliin enough for transmission laue patterns to be taken. After accurate orientation (to less than 1/20 in the best cases) they were mounted on a brass rod with Woods metal and made oblate spheroids with file and emery paper while turning in a lathe. The samples were then re-etched, one surface electrOpolished and were then removed from the rod ready for use. The preparation of the nickel samples was the same as the iron except that it was not possible to find an etching reagent to bring out the crystallographic planes and therefore even the rough orientation was done by x-ray methods. The xéray source was a North American Phillips unit with a Molybdenum target. The cameras which were used were built so that the single crystal rods could be taken off of the camera and put into the saw mounted on a univer- sal head. This was accomplished by use of a holder with two mutually perpendicular axes of rotation on which the rod was mounted in a split collet and held in place with set screws. The base of the holder was mounted on a camera consisting of a film holder and a vertical plate. When the sample was oriented and ready for cutting, the sample holder was taken off of the vertical plate and clamped in the vise. In this way it was not necessary to remove the rod in order to cut it. The orientation of the simple camera was checked by transmission and back reflection Laue patterns of freshly cleaved rock salt. 35 f. Electrolytic Polishing The first work on the electrolytic polishing of capper was done in 1938 by Jacquetla. Elmore19 extended both the theory and technique to iron and cobalt. Al- though it is used extensively in large scale industrial polishing, the technique as applied to small samples is not well known and therefore will be described here in some detail. The process involves the electrolytic re- moval of the anode material in which the rate of removal is determined by the layer of dissolved material surround— ing the anode, rather than the anode potential itself. Figure 8 shows the experimental apparatus. The cell consists of a wide mouth Jar with the bottom out off. A hole is bored in the tap and the sample, mounted on a brass rod, is fixed in the hole with red wax, first in- suring that the brass is protected from the electrolytic action by a thin coat of lacquer. The cathode is a brass ring which fits into the large part of the Jar. The sample is mounted horizontally so that the dissolved iron will not flow away from the surface. The actual currents and voltages observed vary with anode and cathode material and area as well as electrolyte concentration, therefore the numbers are given only for a typical sample of armco iron in the form of a disc .302“ in diameter. 36 ,CATHoa- BRASS RING / f/GLASS‘ IRON SAMPLE_ WAX RU BBER STOPPER BRASS CYLINDER Figure8. Experimental apparatus for electrolytic polishing of small iron samples I ma. IOO - eo - eo _ II 40 — II 20 — ir- i i l l 0 .25 .50 .75 LOO |.25 L50 vous Figure 9. Electrical characteristics of the polishing cell 37 For iron the electrolyte is orthOphosphoric acid of Specific gravity 1.32 in which a dummy iron cathode has been allowed to dissolve for 15 or 20 minutes. This ap- parently insures a certain amount of iron in the bath. When the cell is first connected and the voltage in- creased from 0 to 1 volt the current increases linearly to about 100 milliamperes (Figure 9). If the voltage is increased further the current will continue to increase but much more slowly. If the voltage is now decreased to approximately .6 volt the current increases sharply to 18 milliamperes after which a further decrease in voltage has but slight effect on the current. This plateau is approximately .25 volts wide after which the current drape to zero at zero volts. The best polishing is obtained when the cell is operated on this plateau in the voltage— current curve. Initially there may be slight pitting but this will give way to a high gloss which may have micro- scopic ripples or waves. According to Elmore the current density must be quite uniform at the anode and the cell voltage kept within the narrow.range. The electrolyte must contain some dissolved iron so that the anode is not too readily etched by the acid. The straight portion occurs before polarization at the anode takes place and is essentially the IR drop through the cell. The important factor is the concentra- tion layer of dissolved iron at the anode which serves two purposes: 1) it introduces an opposing emf. and 2) 38 « .Uw . - ., v: r L. i . _ . . 3..— I i _ «It . .4 . . n ,. M. ‘0‘ J 1‘ vflien this back emf. has reached a definite value the layer serves to limit the current through the cell. The layer is produced by slow diffusion of newly dissolved anode material into the main body of the electrolyte. The polishing occurs when the rate of solution of iron is limited by the concentration gradient existing at the anode. Since this gradient is greatest near portions which project, these portions will dissolve more rapidly. No etching takes place because only the concentration layer determines the rate at which the anode dissolves and not the anode material itself. The technique used on nickel samples was that de- scribed by Bloembergens. This method used 10% chromic acid and 90% orthOphosphoric acid used at 90°C and with high current densities (20 amperes per square centimeter). This does not appear to be a true electrOpolishing method but rather an electrolytic etching process. Although the surfaces of the nickel became highly polished it appears that they were still somewhat strained as shown by the results of domain investigations. g. Domain Structure Since all of the factors which determine the line width are not known and since the skin depth for 3 centi- meter microwaves is approximately 10’5 centimeters, every effort was made to obtain surfaces essentially free of strain. In order to do this the samples were cut to orient- ation on a wet wheel .025“ thick, were then polished and 39 .I IIII'I)I|I II) i etched alternately until at least .015" was removed from each face. The samples were then electrolytically polish- ed or etched until very shiny surfaces (surfaces of high specular reflection) were obtained. In order to insure a minimum of strains the surfaces were examined for domains using the colloidal particle technique of McKeehan and Elmorezo. It is to be noted that later writers were not as specific in their directions for the preparation of the colloidal solution as Elmore and that best results are obtained by following his original directions. If the strains are not removed in the etching and polishing process, a domain pattern characteristic of strained sur- faces is observed. If the strains are removed, domain patterns characteristic of the particular crystallographic orientation are observed. In the best cases the armco iron showed regions of strain alternating with unstrained regions. The domain structure on the nickel was very difficult to observe and in most cases the surface showed an anomalous pattern unlike either the strained or un— strained. In these cases the optimum condition: of the surface was determined by taking room temperature curves with successively increasing amounts of electropolishing. 40 V. ANALYSIS OF DATA The actual evaluation of the quantities A and B given in Equation 3-? involves making two completely different types of measurements for each resonance curve and a calculation of Q0. Since the calculated values of Q0 do not compare favorably with experimentally determined values, it is desirable to eliminate the evaluation of A’= and B. Rewriting 3-7 A V/T’ +5 = °(_+£_/2 3-7 2 —-f>*i Let /€Lu be the value of /.’at ferromagnetic resonance, then , = a(+-;64: We *3 7:: 5‘1 where Pres is the power measured at resonance. Inspection of Equation 2-15 shows that at large fieldsI/‘bpproaches 1, hence 2 A 7“ B = g—ég’ 5-2 _. 00 where pan is the power measured at large fields. Combining 347.541, 5-2 we obtain (”a ’i ’1 R: 791/ =(‘4-fw (M) .54-3 fl’/ 4- f” 7’”; ‘ fw’ Since the ratio R inbolves only quantities which can be directly measured, it can easily be determined as a function of Ho. It remains to be shown that a knowledge of R as a function of H0 is sufficient to determine l/Tz. Two 41 methods have been develOped to show this, the first method given is exact, the second is approximate. Rewriting Equations 2-15 /(, :- f’TJzMo [144+ /Aé’/-/4_)Ma](w,l“wz (booz- («091+ {£th 2 3 Writ/470175“ [fil-A‘élflflj “‘77; {wot-fit)” (2%): Let H0; and H, be two values of Ho such that ’- wozl 002: Jed/r fa» H0: ’L/ot1 5-4a wf— when/T fa H.= ”/07, 54b ,1 ¢./ //‘1: Thus from Equation 2-15 it appears that #0:” and M”: 4, a must be solutions of 3. 1 _ :. Ag}. - #0” #27 (H ’6- HOM‘) I f 0 ‘ 5-45 h I " w are '27: (Na/I" NZ-gA/Z)/% 5.5 5: Ju) . XL7; 5-7 and N” includes both anisotrOpy and shape demagnetization factors. Hence 3— [/2- H: + ’7 ;[(H°w+ ’7) + I] ’4. 598a a ’/,_ thus + l - 1 2 Ha +7) ii [pa/4+7) + (ll/071+ all ( a. 5_9 and .1 _' 2 _ 522“) two?“ — (Hog—q) I— W; 540 42 , mu .0 .1 fix, “a + - Thus it is seen that if 647 and ”oz can be determined, J. i/Tz can be obtained. Let R ’and R" be the values of R(H°) corresponding to He; and 1%,; respectively. Combining Equations 24-15, 5'395-‘LI'; 5-6, 5-7: we find * [/1 71 fi+=[0'/o:+/))1+(”,2$A+p)l +(fi/a&+,|)] _F [ m%w+zl)1+ r’ + 40%,,“1- Alf/2T. pa 5-111!L '/ 7 K1 [ W;*’U1+(#”; “up? 4 (fit? M] TF ’- 5;.llb [W +4 {Hem’LAJJI/l“ [6"- where Ig—‘eimo [{Hagmr—(H’Mfw? 5-12a or (g = 4’; M0 MAM?“ ’7Il— (IL/0:21”)? 5-‘-12b and A: {/th ' NE, W” 5-13 To determine l/T2 from the data,a curve of R(H°) against H0 is plotted as shown in Figure 10. Then [€de R-are plotted on the same graph. The intersection of R’andRT with R give the values of Ho; and 14/0,; apprOpriate to the experiment. 1/T2 is then found from Equation 5-10. The preceding method involves some rather tedious calculations and therefore the following approximate method is given. Under most conditions its accuracy is within the limits imposed by the fundamental data. Let us define line width as AH: Hof - ”a; 5-11.» ’2— }_ 43 IIDO 75 R(Ho) .50 .25 FigurelO. The method of determining HOT and H5 from the experimental curve R(H) and the theoretical curve R*and R‘. EXpanding the right hand side of Equations 5—8a and 5—8b by the binomial theorem one finds J” 2 Awe-per MIMI . an + n WWW 5~15 Order of magnitude estimates of the quantities involved show that only the first term need be considered. Hence 5?: (%w+’7)AH 5-16 and —L 21.3 f: «0:1 MW. m/sr/ 7;. .(w 201/ 54-17 Thus we can determine l/Tz by measuring the resonance field and the line width. A Hmay be obtained from a plot of Mac) as follows. Let Ar and A,“ be the values of A, a 4 '- at Ho equal to HJ and 52; respectively. Then from 6. ) Equations 2215, 544a, and 5-4b 7" m - 1‘.“ L! . fl}, " flu??? X fl;/Zw 5‘18 Jr 4. A) / +/ '71-]. X47} 9” fizz/7‘, — (2 «’W 5 9 9’ We shall therefore drOp the superscripts +~and-. In- serting these results in Equation 2-9a, we obtain ”I, g[(’*’/2/’.z/cw);* /’“.&./‘ie)L—7 2? [fife 4. .1 540 Similarly 9; ,af’mc’DH/‘zaljf this, 5-21 I / We insert these approximate expressions for /29 and /u 3.. 1+5 in Equation 5-3 and obtain a ratio 2. ’4 rr [me/oar 150%)”;in -/ = 1 _~ 5.22 a. 1 )1 [4' [0+ 14.2”)4— #3440] The quantity Ra should approximate (”Land A7- at their intersections with R(H°), and therefore the intersections of Ra with R(Ho) give approximate values of 7%: and Hugo We note that R, is a rather slowly varying function of flaw (Table I) and therefore expect that the actual value of TABLE I VALUES 0F 7?, FOR VARIOUS VALUES 0F mm 59sec AZ 45 .756 40 .755 35 .754 30 .753 25 .751 20 .749 15 .747 10 .742 , 5 .738 l/T2 is relatively insensitive to the value of/luuuused. Of course it is necessary to have a value of/a,” with which to obtain Ra' From Equations 24-15 and 54-17 we have ,i) 1 so izzzffle_£12ssu.;:_. ; flM AIL/{1404” +0) 523 1+6 x 'Thus to find l/T2 one proceeds as follows: 1. The data consists of a plot of p as a function of Ho with a single value of /K measured for one point on the curve so that .x’ may be calculated from Equation 3—6a and R(Ho) obtained from Equation 5-3. 2. Assume Race .75. Draw a line on the plot of R(Ho) at .75 and read off the distance (in oersteds) between the intersection points. This is a first approx- imation to a 1‘1. 3. From Equation 5-23 compute an approximate value of /QAAV . From Table I find a second approximation for Ra and then from R(Ho) a second approximation for eff. A. Calculate l/TZ from Equation 5-17. Generally two approximations are sufficient to give all of the accuracy available in this method. In cases in which./¢9:/ is not a good apprdximation at the highest fields available one obtains a better approximation as follows. Let HoL. be the largest field at which measure- ments are made. Then/a'at hQL is given approximately by l __ FL: /+:Z[..ML {1‘ fi,m(%m+4m 5;.24 H #0, (HQ f arm.) 0L This is inserted in place of unity in the numerator and denominator of Equation 5-3. Equation 5-23 is corrected by replacing unity in the numerator and denominator by,k; . It is obvious that the approximate method depends on several assumptions and it will be worth while here to 47 . a. '1 r , I. evaluate these and estimate the error encountered in the evaluation of l/Tz by this method. Let us consider the assumptions: 1. ,a'z/ at highest fields available. Both the exact and approximate treatments depend on this assumption but we have shown in Equation 5-24 how to eliminate it where it is unwarranted and obtain an exact value. 2. A27; 5324/9” There are actually several assumptions inherent in this approximation but since it is possible to calculate this without reference to the assumptions it is not necessary to discuss them explicitly. It suffices to say that in the very worse cases this may introduce an error of 15% in R(Ho). 3. 7chi;;__i<< I Typical calculations on this quantity show that it can never introduce an error of as much as 1%. Thus far we have assumed that .1; is known which is actually not the case. If we examine Equation 2-16 we see that by knowing .TQ;/V, 4? / 4%,afifl(ié , it is possible to calculate JJL. Since, of course, l/TZ is not known, it is necessary to approximate Id by neglecting l/Tz in Equation 2416. If we neglect l/Tz and obtain if; then put this into Equation 5-17 to obtain an approximation to l/TZ , we can approach {Rand l/TZ by successive approx- imations. For small values of l/Tz these corrections are small and even for the largest values observed the error is less than 10%. 48 Thus it appears that this method is susceptible, in the very worse cases, to errors of the order of 20 to 30%. According to Bloembergen6 the method of direct calculation of Qc from known conductivities introduces errors as large as 30% in the total Q, which in turn results in values of u' in error by a factor of three. It is possible to perform a normalization, similar to that which we have performed, which eliminates this error in part. If this is carried out the error in u8 is reduced to a maximum value of 25%. In this discussion we are considering only the error introduced in the evaluation of the data and not the systematic errors incurred in the experiment itself. The temperature dependence of the saturation magnetization was obtained from data in Bozorth's21 book and the temperature dependence of the first order anisotrOpy constants was obtained from Mudarzz. 49 VI. RESULTS AND CONCLUSIONS a. Nickel Figure 11 shows a typical normalized resonance for electrOplated nickel. The dotted line is plotted for #49:] , the solid for,a;g evaluated by Equation 5-2h. it is thus worth while in the interest of good fit to estimate/#4, from the first order approximation to the width and then replot the theoretical curve from this value. It is also obvious from the two curves that the difference in the line width and therefore in the value of l/Tz is very slight, the correction obtained not being worth the effort in performing the multiple approximation in most cases. Figures 12 and 13 show lfimygas a function of the angle between H and the crystallographic axes. These results are as predicted by theory except for the rather large anisotropy shift in H 0,“, in the (100) plane, corresponding to K1: 55 x 103 ergs/cc. whereas the value obtained by Mudar on samples cut from the same rods using the torque magnetometer was about H6 x 103 ergs/cc. The broad resonance line and the consequent indeterminancy in H$,u, may account for this difference. In addition the results further serve to establish the fact that the cry— stals have been properly oriented in the cuttingfr0"': ;. 50 _ , . g .chmhac’ ..«a-H:.:V’*:f 3" 1' Li.» 1 .1331”; Jail: .32: pea—cactus; L8 ciao 23:05.62 wOMHmmwO - o: : 2:9“. 000.? 000m OOON 00MB a . _ _ $25.. 44:32.5..me m>mDQ 440Cmmowrk m2: 030m .omm\oE mmmd u Oman. $36 08» "02.5 0.2.» . .dsme 0.. erE 51 OOI OE ‘ 3on do 203 50: of E 298 we :02”ch o 33.0... «3.59.... 00_ 4 OH mmwmomo i Q 00 00 ON 00 Oil ow « i m 0 _ _ _ _ _ . . _ . . _ _ _ _ _ _ 100m -000 1005 v O O 0) § 80318830- 00: OOE OOn. COS _, ,VL.“ , 0 08 w 52 8 oou .96.... 26:. .0... of c. 296 .o 5:25. o 38.0... .m. 830.... i .. mmmmomo o m 00. , 0w. H ON. 00. m 0m 00 M 0m 0m 0 . I. . . . _ . . _ a I. . _ _ . _ _ _ _ . . _ J 000 . _ . _ _ u _ n u " loom . n _ u _ _ H _ _ _ m . u _ l 000 Q . . . O . u - w. _ 1 00m 8 n. n _ h _ _ . G _ _ n _ u L. 000. S . u _ _ . _ . u _ O . i 00.. _ e. _ W _ _ _ . H u " CON— . n _ n H ” Loom. _ . _ . l Ir l _ . 00¢. 53 r--»‘ W n, will“ Figure 14 is a plot of l/Tz as a function of temp- erature for various nickel samples. The results for the electroplated nickel are in agreement with those obtained by Bloembergenbfor polycrystalline sheets while the single crystal samples show anomalously large widths at room temperature. While the errors in the measurement of l/T2 are very large (see Section VI. c) the difference between the values observed for single crystals and those for plated samples appear to lie within the experimental error. Furthermore the-fact that the three curves agree quite well in the vicinity of the Curie point lends weight to the validity of the curves at lower temperatures. Based on a simple picture it is theoretically expected that the single crystal samples would have narrower widths. This is due to the random orientation of the microcrystals in the polycrystalline material with the result that the effective field acting on each microcrystal is a function of its orientation. The net effect is to broaden the applied field distribution over which resonance takes place. There are at least two factors which could introduce erroneous values of l/T2 in the experiment. The wider lines may be attributable to the possiblity of strain broadening of the line. If all of the strains introduced in preparing the samples were not removed in polishing and etching, the decrease in l/T2 with temperature might be explained as a partial relieving of those strains. If this were the case it might be expected that when the sample 54 NP . ._oxo_c .2 fl ao mocmocmaou SBoEaEfl. E 959.... Do mka m... .9 Saul _ mmxoh