ANISOTROPY CONSTANTS OF POLYCRYSTALLINE FERROMAGNETIC SUBSTANCES Thesis for the Degree of M. S. MICHIGAN STATE COLLEGE Richard Henry KrOps-zf \ ~31“ 1950 31293 01774 5575 This is to certify that the g . thesis entitled % 1' AnisotrOpy Constanta of Peiyory- 1‘2 stalline Ferromagnetic Materials. 32'" .1 presented by in? 3? Richard H. Kropschot gig has been accepted towards fulfillment " of the requirements for '7! .'I;‘ 5153‘- _H.§_ degree in m .3. ”1‘5 1 ‘3’? 3" ML, Major pro essor Datngrch 6. 1950 . v,’_‘& ‘4' at" :‘ 7 fl" 3’ 31"»4‘péi‘ . '. ,... rr.;) ‘NVL‘ ‘{ J‘ ‘ ) . . g: ‘- . 3,. IL V ‘ .7 ‘2‘. l;"_ ' ”kl . ———-'o I-_ it I _‘ ‘— l '5'". .1..*. 5'- -lwrr'- ' .0: - :- "If" ‘5 If I "’I' V -'* Jo'V' n~—.- .- a :T_ r—-- 1f "-3- Fir—I n- 5-- .‘—~ 1 vs .,.A“ ".c ' l ‘r'. .. 4.? ‘ . . .‘J .A ”H“ _' r2» v . A {r—. . I-._ - "KH ’ . , / h~;_‘ 2 ,Ju- l, o ”0:- ._ ~ .- My ‘ .‘dVyo- 0A:°;&;-'I'L’lh PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. ANISOTRCPY CCESTAITS OF PCLYCRYSEALLIEE FERRCLAGIETIC SJESTAECES BY Richard Henry Kropschot A Thesis Submitted to the School of Graduate Studies of Xichigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of EASTER CF SCIENCE Department of Physics 1950 ACKNOWLEDGMENT I wish to express my appreciation to Dr. Robert D. Spence for his valuable assistance, guidance and helpful eug- geetione during the course of this WJ‘M 535?.{3955 I. II. III. IV. V. TIRED OF CONTENTS INIRODUOTIGN m misomarr consrms m roam MAGNETOMBTER DATA HWRETM‘ION or mm 12 16 20 39 I. XTRODUCTION This thesis deals with ferromagnetic materials. Such materials include the ferromagnetic elements iron, nickle and cobalt and ferromagnetic alloys such as supermalloy, permalloy, alnico, delta- max, silectron, and Heusler alloys. host of the ferromagnetic alloys are composed partially or wholly of the ferromagnetic ele- ments listed above. However, some of the alloys such as Heusler alloys contain entirely paramagnetic elements. In magnetic media the magnetic field intensity H end the magnetic flux density B are related by _B_ = 214-4"; (1) where I is the intensity of magnetization or dipole moment per unit In addition in paramagnetic materials volume. l=%§, (a whore Xlis the magnetic susceptibility. For this case §=(1+’-I1TZ)§' (3) (14) PL " (1+1ITI1I) where'lzis the magnetic permeability. In ferr magnetic materials equations (2), (3) and (H) are not valid in general and the rela- tions between B, H and I must be given graphically. Curves re- lating B and H or I and H are called magnetization curves and serve to define the empirical relationship between the various pairs of veriables. The curves plotted with B as the ordinate and H as the ahcissa are of great interest in practical calculations. Hewever, from the theoretical point of view the curves relating I and H are ust 11y of greater interest. (2) The first quantitative theoretical calculations in magnetism are due to Lengevin who successfully deve10ped an expression for the susceptibility of a mrangentic solution. Langevin considered a large group of non—interacting dipoles in thermal equilibrium at some ter-mperature T. If we consider one dipole of strength“. in an applied field H, then the energ! of the dipole is given by Let the angle between 5.» and I? be 9 , then (5) E 3-}4 H Case. By :pplying the Urxwell Boltzman distribution law, Langevin nation in the obtained an eXpression for the intensity of magneti form 1r [1H 0080 o I l N Q 0036 9 Sinng ’ (7) IT _, NH 0056 £5» 1‘ T Sine (16 where 1‘? is the total nwnber of dipoles per unit volume. This may be written ”. ME 0056 'f a“ k T . (-1 I 8141. ’3? log 9 Sine d6 , (o) 3H o from which one readily obtains I: N'A{ coth Eng; __ ‘1}111‘“}5 NFL (a). (9) "R‘f' In the above I (a) is defined as the Langevin function, that is (3) Ma): (coma-g) a: #3 . (10) If pH< ., 2P3z {map‘N Equating this to (27) .. Z J N” -' 5—53.}? I (36) then from (19) J2 N (29)2 s (s+1)- 2 .125 LS + 1) T 2 A“ V "' c We have shown that by using the Weiss molecular field.theory or its ana10g in quantum mechanics. the exchange energy, we are able to predict a Curie temperature. However, in order to obtain the desired results we must have a condition with the dipoles lined up at all times. We know from experimental results that this is not always true. It is possible, with no applied field, to have the overall magnetization of the sample be zero and with a small applied field the intensity of magnetization is in general less than the saturation magentization. (10) In order to account for the fact that in ferromagnetic materials an extremely small applied field causes a relatively large magnetization, Pierre Weiss assumed the existence of domains. Each domain is at saturation magnetization and in absence of an applied field they are oriented randomly. The Weiss theory does not Justify the existence of the domain type structure but assumes it*‘ in order to account for experimental observations. The domain type structure can, however. be Justified in light of modern theories. The existence of the domain type structure has been verified experimentally by use of powder pattern methods. Fine particles of magnetite in collodial suspension are placed on a prepared sur- face and observed through a.microscope. The pieces of magnetite tend to collect along the domain walls due to the very strong local magnetic fields. The size of the domains seem to be very dependent on the material, the size of the crystal and also on the history of the material. When an external field is applied the domains with the favored orientation can be observed to grow at the expense of the unfavorably orientated domains. The resultant magnetization can be changed in two manners: first, due to favored domains growing at the expense of unfavored ones or. second, by means of rotation of the magnetization vector in the direction of the applied field. The domain structure has its origin in the possibility of lowering the energy of a system by going from a saturated con- figuration such as a single domain structure with high magnetic energy to that of several domains oriented in such a manner that (11) 9 the system has a lower energy. In addition to the two energy terms already mentioned. ex- change and magnetostatic, we must consider two additional energies in order that serious approximations in the domain structure do not have to be made. These two tenns are the anisotropy energy, of which we will say more in a follOWing section, and the magneto- elestic energy. The anisotropy energy tends to direct the mag— netization vector along certain crystalOgraphic axes. The magnetoelastic energy arises from the fact that the crystal dimensions change slightly when a magnetic field is applied. The change in dimensions is called.magnetostrictidn and is related in a direct manner to the anisotropy energy. In the region between two adjacent domains, called a Bloch wall. there exists a transition area where the spin direction changes from parallel to an antiparallel allignment. The change does not occur across just one lattixeplane but rather acroes many. depending on the energy. The exchange energy is inversely preportional to the wall thickness and. herefore. tends to nmke the wall spread throughout the Whole crystal. However, in tEe transition from a parallel to an antiparallel alignment the spin directions are rotated away from the crystalline axis and are, therefore, acted upon by the anisotropy energy. The anisotrOpy energy is directly proportional to the wall thickness. It is a balance between these two energies that determine the thickness of the well. For iron the wall thickness is of the order of 10004 and the total wall energy per unit area is approximately 2 9 l erg/cm . (1?) II. THE ANISOTRCPY CCFSTAYTS All ferromagnetic substances have a crystalline microstruc- ture. As is characteristic of crystalline materials, ferromagnetic materials are anisotropic in the sense that the r physical pro— perties depend on the direction in which they are measured. There- fore, it is to be expected thrt a portion of the internal energy of a ferronegnetic substance will depend on the orientation of the '53 magnetization vector in respect to the crystalline axes. is portion of the internal energy is referred to as the anisotropy eners . The anisotropy energy'of a crystel of cubic symmetry such as iron or nickel may be exPressed in a series expension as E = 10.x,(afm§+afarf +a§af>+K2 (martini) (38) where on. 0",, end as are the direction cosines of the magnetization vector with respect to the x, y and z axes. and K0, K,, and K2 are the anisotrOpy constants. In this expression the odd powers of the direction cosines disappear due to crystalline symmetry and the Clsquered terms ere included in KO due to the fact thet a 2 Z a. + «2+ “3 = 1 It is found exyerimentally that in ferromagnetic materials the constants K1 and K2 are quite large nails the higher oriered terms can be neglected. This means that the ease with which a given sample can be magnetized is dependent on the choice of axes. It is easier to magnetize an iron crystal along the [lOO] direction than it is along the Elli] direction, while the reverse of this is true in a nickel crystal. The origin of the anisotrOPy energy is not immediately obvious. The Heisenburg exchange energy used in explaining (13) ferromagnetism is isotropic since the exchange effect depends only on the relotive orientation of edjecent spins end not on the crystal direction. The magnetic dipole interaction between spins noes gfive a tenn that is dependent on the orientation of the spins with the crystal lattice, but the magnitude of this term is much too small to account for the anisotropy effect. Bloch and 15 Gentile have suggested thrt the spin orbit coupling combined with an electrostatic coupling of tie orbit to the crystel yields the right order of magnitude for the anisotropy constants and at present this is the generally accepted theory. 12 Yen Vleck hes calculated 9 theoretical value for the anisotropy constant Kl, that eqrees with experiment as far as sign, approximate magnitude and temperature dependence. He gives Aw , per atom if due to dinole- 10 K Tc h‘V‘ 4 ‘ dipole interactionS, and of the order of _TFTT:?_ if due to the 31 to be of fl:e order of quadrupole effect. A is the spin—orbit constant, To is the Curie temperature, and hi’is a quantity dependent on the energy levels. The calculated value of K1 is the order of 105 eras/cc. The value for K1 in the expression is also tenpereture denendent, vhich agrees with xperimental evidence. In feet it is found experi- mentally that the constants K1 and K2 may change sign between 9 high and low temperetures. The anisotrOpy constants for various Fe — Co, Fe - Ki, Co - Ii, and Fe — Co—Ni alloys are summarized in e.psper by 5 Bozorth.(1937). I will list briefly some of the results obtained. (11.1) ' Table 1. Values of K1 and K2 for various ferromagnetic elements and alloys COIHPOSj-tion Temp K1 X 103 Kg I 167 Fe Co Ni 00 eras/cc ergs[cc 100 20 #21 150 50 5O 20 ~68 -390 50 50 20 33 -180 50 5o 20 nos Am 100 20 -3t 50 It was the purpose of this research to obtain exterimentel values for the anisotrony constants K1 and K? in eglycrygtalline ferromagnetic substances. The anisotropy energy and the mag- netOstetic energy ere the only portionsof the internal energy that i are dependent on the orientation of the magnetization vector. The other two terms, excha ge end megnotoelestic, are indepenéent of angle. The torque per unit volume on a disk or oblate spheroid seaple due to an applied magnetic field is given by BE T = L I 94> <33) mere ¢ is the tinge between the mgnetization vector end an arbitrary axis in the plane of the sample in which the torque is measured. In a high mecnetic field the meenetiostetic energy becomes a constant along with tbe exchange and magnetoelestic energy. The torque on a sample is then only a function of the anisotropy energy. For tie rotetion of the magnetization vector I in the(1oo) plane the torque on a given sample can be computed by letting a. 2: Cosp .UaB sinBand 05- 0. V'herefl is the angle between I and the x axis. The torque is then given'by T = g!- sin up (140) (15) In a ainilar nannar the torque on a aalplo cut from the (1101 plane in ofthc fora 1' - ain 2((1 - g. 0312’) (1:14» £3 tinge) (41) whore ¢ in tho polar angle. for a aaaplc cut in an arbitrary direction Bitter and Tax-nor“ ; have given the following romlta: 1 fig. 3. Anglca wining orientation of disk I. I, z are cubic ma of the crystal: l in the nornnl to the disk: I. in the direction of tho ”actuation in the plane of the dick; o 1. the angle naaaurod in the plane of the dick. In tor-a of the angina defined in 1‘15. 3 can fl coa'l’ - ainé ain‘P coa'fi , 5 M2 = coa cf ain‘l’ -r “:14 cos? coax. , (1.2) 69 = ain ¢ ain 7C , tho torque on a given sample, for the once of small Xe (assuming :2 no), 1- given by g1.‘1.u2§¢naunh¢ +31coa2 +32ooak¢. (#3) whore 2 2 2 A1=fain 1(1-7coa 7L)-(l+coa1v)col‘+3’ . ( 16) ‘2 a - g sinnx- U? sinhlt- coaa‘x] cos it 1’ . 31 = - $- 311121. cos'x. sin IH’, (N4) 32=-§(1+co.21c)cos¢atnu*P. III. m roman wnmm In ordar to investigate the anisotropy energy torque curves were used. Torque curves are ctrvss relating the angle the direction of rolling aakss with tho magmatic field plotted against tho torqus produced on the sample while alligncd-in that direction. rig. h. The Torqus Mngnotomstor. The torque magnetometer shown in Fig. h was used to unsure tho torqus on disk and oblate spheroid samples. a sample was clamped in the sample holder which is supportsd by a 1/8 inch brass shaft running in glass bearings node from short lengths of capillary tubing. (17) A torsion fiber with a brass indicator at either end was used to suspend the sample holder. the lower indicator reading gives the sample orientation while the difference in the two readings repre- sents the retarding torque on a sample in opposition to the torque produced by an applied magnetic field. The constants of the suspension were evaluated by use of the torsion pendulum. The quantity determined experimentally was the product pan 8 i—E‘J (155) where p is the modulus of rigidity of the wire, a is its radius. and 1 is its length. I is the moment of inertia of the pendulum and P is its period. To evaluate p. an a length of wire 1 - 123.6 cm was used and a cylindrical block of aluminum of mass a = h75.070 grams and radius r 8 3.8137 cm was used as a pendulum. Il'he results of the torsion pendulum measurements are shown below. lumber Trial of Time Osc illat ions Sec . 1 50 MO 2 50 M0 3 25 220 h 25 220 Iron these data it was found that p aq' = 138,600 hue one. (18) The retarding torque produced by the fiber is then given by r = '€;"JET2§ {a (#6) wheree is the angular separation of the indicators and l is the length of the wire used in the instrument. The instrument wire length used during the course of the experiment was 30.1193 on as measured by use of a height gauge. The torque can then be expressed as I (am cm) 8 mo 6 on.) A. uniform magnetic field was applied to the sample by means of flat cylindrical pole pieces of a divided electromagnet. The magi- tude of the field was determined from a calibration curve obtained by taking a large number of readings with a flux meter. The magnetometer ras designed so that a sample about the size of a quarter or smaller, depending on its crystal allignment, would give a siseable deflection. The samples investigated were either transformer laminations or a silicon steel called "silectron'. It was found that samples with a readily observable anisotropy energy could be selected by bending a sample of the material until it broke. If the sample was soft and flexible its anisotropy con- stants were small, however, if it was brittle and showed large orystsllites on the broken edge the sample displayed a anisotropy energy large enough to be measured readin by the torque magistometer. The surface of the metal strips were prepared by polishing then lightly with an abrasive after which an indelible int was applied to the surface so a line could be scribed parallel to the direction of rolling. A compass was used to outline a disk. The disk was cut out as close to the line as possible by use of tin snipe and then (19) ground to the line on an external grinder. Samples of silectron were made in a form which approximated an oblate spheroid by first cutting out a rough sample and soldering to it a brass shaft which was inserted.in a collet on a lathe and.turned.to proper shape. The sample was finished by filing it while it revolved. The procedure for obtaining the torque curves was as follows: The sample was clamped firmly in the center of the sample holder and a strong magnetic field applied. The pointer on the lower indicator was set at predetermined angular positions spaced 5 degrees apart and extending over a range of 180 degrees. At each setting of the lower pointer the upper pointer was rotated to a position at which the torque on the sample was Just balanced by that produced.by twisting the torsion fiber. The difference between the pointer readings gave the angle through which the fiber had been twisted. The torque on the sample was then obtained from Eq. (MB). In plotting the torque data the orientation of the sample in respect to the magnetic field was represented by the angle between the scribed line (along the rolling direction) and the direction of the magnetic field. There are two regions on any one of the curves. from O to 180 degrees. that the readings are found to be unstable. They are located on either side of sore torque where the slepe is positive. The regions extend to a point about half way from sero to the maximum or minimum point. It is believed that the instability is a character- istic of the sample rather than the instrument since the same instabil- ity was observed by Bitter and Tarasovu'using an instrument of quite different design. The errors in the torque measurements arise primarily from the frictional torques in the instrument and errors in reading the (20) indicator scales. The error due to frictional torques appears to amount to less than 0.5 of a degree while the error in reading the indicator scales is estimated at 1.0 degee. These errors amount to about 3.0 percent of the maximum deflection for most samples and since the first order anisotropy constant K1 is directly proportion- al to the maximum deflection, they constitute a most serious limita- tion in the accuracy of the instrument. The second order anisotropy constant K2 is much more sensitive than K1 to small errors in the torque curve and, therefore, the torque magnetometer used in this experiment cannot be used to determine K2. While most of the torque curves were obtained with the sample at room temperature it was thought desirable to measure the torque on a few samples at the same temperature as liquid nitrogen in order to obtain an indication of the temperature dependence of the anisot- ropy constants in polycrystalline materials. This was accomplished by placing the sample in a small cup made of poly-foam which was placed in the sample holder and kept filled with liquid nitrogen while the torque curve was measured. The sample was then allowed to warm up and the torque curve was measured again at room temperature. IV. DATA The torque curves are shown on the following pages followed by their interpre tation . (21) r13. 5. Sup}; 1. Torqm our” of n ”Inn-tailing oblat- Iphoroid of iron uncon, 1a (110! plan. (22) — '-,—— I—C—o-_-‘ Fig. 6. Sample 2. Torque curve of a. polycrystalline oblate spheroid of iron silicon, in (110) plane. (23) Fig. 7. Sample 3. Torque curve of a polycrystalline oblate spheroid of iron silicon, in (110) plane. (2h) v — ——__—- Vi“ Fig. 8. Sample 1!. Torque curve of a polycrystalline disk of iron silicon. in (110) plane. (25) _-- m.‘ Fig.9. Sample 5. Torque curve of disk sample of transformer (26) DEGREES Fig. 10. Sample 6. Torque curve of oblate spheroid sample of transformer iron. (27) _ accuses Fig. 11. Sample 7. 'Borque curve of..d.isk sample of trans- former iron. 0 -L _ ___.A_ (28) . _ . Fig. 12. Sample 8. Torque curve of disk sample of trans- former iron. “-ww.“ . (299 —- —G’:A ~_4'_) Fig. 13. Sample 9. Torque curve of dials sample of transa former iron. (30) Fig. 1’4. Sample 10. Torque curve of disk sample of trans- former iron. (31) Fig. 15. Sample 11. Torque curve of disk sample of trans- former iron. Fig. 16. Sample 12. fomer iron. (32) Torque curve of disk sample of trans- Fig. 17. Sample 13. former iron. (33) if” H T {‘4‘}; “"" J‘ 7 A A." ‘ 0 Torque curve of disk sample of trans- (31$) Fig. 18. Sample 11+. Torque curve of disk sample of trans- former iron. (35) Fig. 19. Sample 15. Torque curve of disk sample of trans- former iron. (36) -——_—'- -_ ‘ ' Fig. 20. Sample'lfit Torque curve oi’ disk sample of trans- former iron. rig. 21'. Sample 17. former iron. (37) Torque curve of disk sample of trans- . (38) Fig. 22. Sample 18. Torque curve of disk sample of trans- former iron. (39) V. Interpretation of Date The torque on a sample out from theillOl plane of a single crystnl is given by liq. (111) to be T 2 sin 2 ¢ (1 - 3/2 einz¢) (K.+ [2/2 sin 24)) . Let us define 3, sin 2 T. (1 - 3/2 sin 2 Q) (W) 52 sin 2¢au - 3/2 sin 2%) where ¢, and (fie-satisfy the conditions 51‘." M ¢, =0 521' : 395 '16. ° ¢a > ¢u (1&8) QT 53750 ‘For' ¢14¢4¢2 , Solving for 4), and (hand inserting the results in the above one obtains ,6. 53 We can then write 112:1 Eli. '12: 246 11+ 11.12.;112: .56109 .21002. (1:9) 9!»? ap- whsre T1 and T2 are the values of the torque at d), and 4’: re- spectively. Solving ,these equations for K1 and [2 one obtains :1: 2.211119% - 1.236 s2 (50) K2 3 13-338 T2 - 11.9923 1‘1 mine. (50) are strictly applicable only for samples out from the (110) plane of a single crystal. However, if one is dealing with (140) a polycrystalline sheet in which most of the crystallites have been so oriented that their (110) planes lie in the plane of the sheet one may reasonably expect that the torque curve will be or the form given by Eq. (hi). I! an experimental torque curve follows this fora one may use Ions. (50) to obtain two constants [1 and la. These constants will not be true anisotropy constants since they do not apply to a single crystal but rather to an aggregate of single crystals. In fact, the departure of [1 and 12 from the values which characterize a single crystal will serve to indicate the percent of the crystallites which are oriented in the assumed manner. If we designate the first order anisotropy cone- stant by K1. and the corresponding quantity for the polycrystalline sample by xlp then the fraction of oriented crystallites is RIP/[1,. The extent to which the torque measured for a given sample agrees with the torque function given in lq. (1&1) can be checked by calculating 11p and [2], and plotting the curve obtained from these values on the experimental curve. Figures 5,6,7 and 8 show curves of a 3% silicon steel called silectron.‘ It is known that the direction of rolling is the [100] axis and that the plane of the sheet coincides with the (110) plane. Subject to the assumption mentioned above it is possible to cal- culate I” by use of lo. (50). Fig. 5 shows the torque curves obtained from a sample of silec- tron out in the fore of an oblate spheroid at temperatures of 295 "' Silectron samples obtained from Allegheny Indlum Steel Corporation, Breckenridge, Pa. (’41) and 77 depees I. The torque on the sample was sero along the roll- ing direction. (i.e. C1001). The torque was also sero at 55 and 90 depees corresponding to the [111] and [110] directions. This corre- sponds to the fact that the energy takes either a maximum or a mini- num value along the aims. (The [111] direction should be at 5&7 de- grees from the [100] direction but the measured values are withm the limits of accuracy of the instrument.) no anisotrow constant for this ample at 295 degrees I is 2.75 x 105 as given by no. (50). Do- sorth6 gives the value of In a 2.87 x 105 for a single crystal of 3% silicon steel therefore one would expect this sample to be 95% alligud. The value of [11, increased about 12% when the temperature of the sample was lowered to 77 deg-see 1. (K11, =3 3.08 x 105) It was ob- served that lowering the temperature shifted the points of sero tor- que. This as: be due to the instruments reaction to the low temper- ature or it may be caused from a change in internal structure or the sample. Figures 6 and 7 show two more oblate spheroid samples cut from the same strip of silectron as sample 1. For sample 2 11? I 2.9lt x 105 at 295°! and = 3.53 x 105 et 77" I. and for sample 3 11 = 2.71; x 105 at 295°: and = 3.27 x 105 at 77°x. The curves also have sero torque at O and 90 degees and. at GWOximately 511.7 degees and can be interpreted in the same manner as sample 1. (152) !he variation in the values of the Pa for the three samples appears to be due either to large localised crystals in the stock the couple was cut from or to a change in crystal structure caused from soldering the brass shaft to the samples. 'L'ho curve of a disk sample of silectron is shown in Fig. 8. In this seaple the points of sore torque are also slightly shifted with a reduction in temperature and the anisotropy constants are lower than those of the oblate spheroid samples. For the disk :11, = 2.61 x 105 at 395° n and . 2.95 x 105 at 77° 1:. These results agree with the results of Bitter and Tarasovll the who found that when oblate spheroid samples were used the curve shifted to a more syn-etrical configuration and the anisotropy constants were larger than those of comparable disk samples. Fig. 9 shows the torque curve of a disk sample cut from a transformer lamination. Examination of the torque curve leads one to believe that this sample lies in the (110) plane since the torque is zero at O, 37 and 92 degees, corresponding to the [110], [111] and [100] directions. The constants are :11, = 5.23 x 10" and KZp =- 1.91: x 10" at 295° x. Using these values for :1 and [2 the theoretical curve was found to be in good agreement with experiment. Gonparing ‘11» with In for a single crystal indicates that the sample is approxinately 12$ alligned with the remaining 88% distributed randomly. The above sample was filed into an approxinate oblate spheroid (M3) and the resultant curve is shown in Fig. 10. The constants are 11p = 6.81 :10" at 295 “x and - 7.98 x 10“ at 77" x. As was observed in the silectron samples 11p increases for an oblate spheroid sample and also increases with decrease in temper- ature. A shifting of the points of more torque with change in tempem- ature was also observed in this sample. l'or the majority of the curves obtained in this experiment the simple type of interpretation previously employed was not found appropriate. However, Akulov and 31'6thth have suggested a method by which one can infer the structure of a sample even when the crystallites have several different states of orientation. They assume that an experimentally measured torque curve may be represent- ed by the expression szlsie ens sin to (51) They next assume that the possible orientation of the crystallitee in the sample fall into five distinct groups which are shown in Table 2. TABLE 2. The Group 1 2 3 ’4 5 Plane containing the Randomly magnetization vector (100) (100) (110) (110) orientated and the rolling crystals direction Rolling direction [100] [no] [100] [no] Relative volume '1 '2 W3 Wu W5 (1th) In this tabla the We indicate the relative volume of the suple occupied by each group and are therefore subject to the restriction Zwi = l (52) The total energy is given by r a i vi :1 (53) Where 11 represents the energy of the 1th group. On differentiat- ing the last expression with respect to 4’ one finds q. r = Z V1 T1 (51+) Is: On inserting in the formula the appropriate formulas for the T1 one obtains an expression for the total torque If : 5%! ("1" We) sin ’4’ 4 :3; (VB—Wu) sin 114+ Q ('3— m) sin 2¢}- (55) Comparing the coefficients in this expression with those of lq.( 51) one obtains 5.[u(w1-w2>+ 3