R 263 Philips Res. Rep. 10, 113·130, 1955 FERROMAGNETIC RESONANCE ABSORPTION IN, BaF~12019'A HIGHLY ANISOTROPIC CRYSTAL *) by J. SMIT and H. G. BELJERS 538.221 Summary The ferromagnetic resonance absorption has been measured at 24 000 Mcfs for a single crystal of hexagonal BaFe 12 0 19 • The resonance conditions are severely influenced by the crystalline anisotropy and, for fields too small for saturation, also by the Weiss-domain structure. The theory predicts, for a varying magnetic field perpendicular to the hexagonal axis, at most three absorption peaks, which have been observed at elevated temperatures. The gyroscopic splitting Iactorg and the auisotropy field are evaluated. The g-factor has the spin-only value. The crystalline anisotropy is suggested to be caused by dipole- dipole interaction. Résumé L'absorption de résonance ferromagnétique d'un monocristal hexa- gonal du BaFe 12 019 a ëtë mesurée à 24 000 MHz. Les conditions de rësonance sont influencées très gravement par l'auisotropie cristalline et si le champ magnétique est trop petit pour la saturation, elles le sont aussi par les domaines de Weiss. La thëorie prévoit au plus trois maxima d'absorption pour un champ magnétique variable perpen- diculaire à l'axe hexagonal, maxima qui ont été observés aux tem- pératures ëlevëes. Le facteur de sëparation gyroscopique g et le champ d'anisotropie sont évalués. Ce facteur g a la valeur du spin seul. Il est probable que l'anisotropie est causëe par l'interaction magnëti- que entre les dipoles. Zusammenfassung Die ferromagnetische Resonanzabsorption wurde bei 24 000 MHz an einem Einkristall des hexagonalen BaFe 12 019 gemessen.Die Resonanz- bedingungen werden wesentlich beeinfluBt von der Kristallanisotropie und - sofern die Felder zur Erzielung einer Sättigung zu niedrig sind- auch von den Weissschen Bezirken. Die Theorie liefert höchstens drei Absorptionsmaxima für ein variabeles Magnetfeld senkrecht zur hexagonalen Achse, welche bei hohen Temperaturen beobachtet wurden. Der gyroskopische Spaltungsfaktor g und das Anisotropie- feld sind ausgewertet worden. Der g-Faktor hat den nur-Spin Wert. Die Kristallauisotropie wird wabrscheinlich von der magnetischen Dipolwechselwirkung verursacht. I. Introduction A ferromagnetic substance, when exposed to a microwave field whose magnetic vector is perpendicular to a static magnetic field H, shows resonance phenomena if the frequencyis equal to the Larmor frequency, which in an isotropic crystal equals , e COr = 2nfr = yH = -gH, (1) 2mc *) A summary of this paper was given by the first author at the Conference on Ferrimagnetism, at the Naval Ordnance Laboratory, Maryland, U.S.A., 11-12Oct. 1954.
SummaryThe ferromagnetic resonance absorption has been measured at 24 000Mcfs for a single crystal of hexagonal BaFe12019• The resonanceconditions are severely influenced by the crystalline anisotropy and,for fields too small for saturation, also by the Weiss-domain structure.The theory predicts, for a varying magnetic field perpendicular tothe hexagonal axis, at most three absorption peaks, which have beenobserved at elevated temperatures. The gyroscopic splitting Iactorgand the auisotropy field are evaluated. The g-factor has the spin-onlyvalue. The crystalline anisotropy is suggested to be caused by dipole-dipole interaction.RésuméL'absorption de résonance ferromagnétique d'un monocristal hexa-gonal du BaFe12019 a ëtë mesurée à 24 000 MHz. Les conditions derësonance sont influencées très gravement par l'auisotropie cristallineet si le champ magnétique est trop petit pour la saturation, elles lesont aussi par les domaines de Weiss. La thëorie prévoit au plus troismaxima d'absorption pour un champ magnétique variable perpen-diculaire à l'axe hexagonal, maxima qui ont été observés aux tem-pératures ëlevëes. Le facteur de sëparation gyroscopique g et le champd'anisotropie sont évalués. Ce facteur g a la valeur du spin seul.Ilest probable que l'anisotropie est causëe par l'interaction magnëti-que entre les dipoles.ZusammenfassungDie ferromagnetische Resonanzabsorption wurde bei 24 000 MHz aneinem Einkristall des hexagonalen BaFe12019 gemessen.Die Resonanz-bedingungen werden wesentlich beeinfluBt von der Kristallanisotropieund - sofern die Felder zur Erzielung einer Sättigung zu niedrig sind-auch von den Weissschen Bezirken. Die Theorie liefert höchstensdrei Absorptionsmaxima für ein variabeles Magnetfeld senkrechtzur hexagonalen Achse, welche bei hohen Temperaturen beobachtetwurden. Der gyroskopische Spaltungsfaktor g und das Anisotropie-feld sind ausgewertet worden. Der g-Faktor hat den nur-Spin Wert.Die Kristallauisotropie wird wabrscheinlich von der magnetischenDipolwechselwirkung verursacht.
I. Introduction
A ferromagnetic substance, when exposed to a microwave field whosemagnetic vector is perpendicular to a static magnetic field H, showsresonance phenomena if the frequencyis equal to the Larmor frequency,which in an isotropic crystal equals
, eCOr = 2nfr = yH= -gH, (1)
2mc*) A summary of this paper was given by the first author at the Conference on
Ferrimagnetism, at the Naval Ordnance Laboratory, Maryland, U.S.A., 11-12Oct. 1954.
in which {}is the angle between the magnetization vector and the hexag-onal axis. This axis is the easy direction of magnetization, so that K> 0;at room temperature K~3.106 erg/cm3• The stiffness in the easy directionis equal to that of a magnetic field of magnitude HA = 2 K/ M, in whichM is the saturation magnetization; this field has a value of about 17 000oersteds, which corresponds to a wavelength of 6 mm. We shall prove,however, that the resonance frequency can become vanishingly small bythe application of a magnetic field in a difficult direction of magnetizationwhich is here perpendicular to the hexagonal axis of the crystal.
Both in the easy and the difficult orientations of the magnetization theenergy has an extreme value. This remains so for the difficult direction if weapply a magnetic field in this direction. For small fields it is a maximum, but
f for strong :fields,when the magnetization vector is,turned over into this direc-tion, apparently it is a minimum. Consequently for a field strength at 1vhich
! .the magnetization vector has just arrived in this orientation the stiffnessI 'in the direction from which it arrives is zero. The stiffness for movementsin the perpendicular direction can remain finite, but this does not preventthe resonance frequency from becoming zero for this particular value ofthe field strength. At higher :fields(J)r increases with H. Thus, for a frequencywhich is not high enough to excite ferromagnetie resonance with the mag-netization vector in the easy direction, one should expect to find resonanceat two values of a field in the difficult direction. From these values one candeduce both th~ g-factor and the anisotropy field H~. A more detailedtheory will be given in section 2.
Such resonance experiments were carried out on a BaFe12019 singlecrystal but the situation is there more complicated owing to the fact thata :fieldperpendicular to the easy direction leaves the Weiss-domain pattemundisturbed. This is always the case if there are several easy orientations
114 J. S~UT and H. G. JJELJERS
where y is the gyromagnetic ratio and g the spectroscopie splitting factor:the latter is equal to 2 for electron spins. This means that the resonancefrequency is proportional to the stiffness with which the magnetic momentis bound to the equilibrium orientation. If the crystal is magneticallyanisotropic, so that the energy depends upon the orientation of the mag-netization with respect to the crystallographic axes, an extra stiffness willbe added, which may be either positive or negative. In the cases so far
, studied this extra anisotropy stiffness was small compared to that due tothe external magnetic field at microwave frequencies. This is not the casefor the hexagonal crystal BaFe12019, which has a very high crystallineanisotropy 1). The anisotropy energy can in this case be described with theformula
Ec = K sin2 {} , (2)
__ ._--
FERROMAGNETIC RESONANCE ABSORPTION IN BaF ... O"
which make equal angles with the applied field in the difficult direction.This applies to cubic crystals as well, where only the 1800 walls disappear.It will be shown in sectio_n 3 that the domain structure changes the reso-nance conditions completely for a hexagonal crystal.
The experimental technique will be treated in section 4 and the resultsof the measurements will be given in sectio~ 5.
2. Theory of ferromagnetic resonance conditions
The magnetic free enthalpy G(T,H) of 1 cm" of a magnetized crystal is,for constant Tand H, a function of the orientation of the magnetizationvector. This energy includes that of the static magnetic field, crystallineanisotropy, demagnetization energy, etc. Let the equilibrium direction ofthe magnetization vector be the '-direction, and the small angles of de-viation in two perpendicular directions ç and 'f). The equations of motionarethen
-M~ = y àG/àç, ~Mç = Y àG/à'f). ~
For small deviations from the equilibrium position we may use for G thefirst terms of a Taylor series: .
G = Go + t (G~~ç2 + 2G~rl'f) + G'Y/rJ'f)2),
so that (3) becomes-,M~ = r (G~~ç + G~'Y/'f)), ~
Mç = Y (GrJ~ç + GrJrJ'f))·~
·This pair of equations has solutions which vary harmonically in time ifthe angular frequency t» satisfies
For an arbitrary orientation of the equilibrium position with respect to acoordinate system with polar angles (if, cp) one obtains
where the second derivatives have to be taken in the equilibrium positionfor which Gf} = G,p= O.The expression does not always hold for sinif = O.We see that the resonance frequency is proportional to the geometricalmean stiffness in two perpendicular directions which coincide with the
, principal .axes of the energy.We shall now apply this equation (7) to the case of a uniaxial crystal of
ellipsoidal form with the z-axis as the preferred direction. The demagneti-zation coefficients are Nx, Ny. and Nz. A static magnetic field H is applied
"
115
(3)
(4)
(5)
(6)
(7)
116 J. SMIT and H. G. BELJERS
in the direction -&= cp = n/2, and the sample is assumed to be homogene-ously magnetized. The free enthalpy is then the sum of the crystallineanisotropy energy, the magnetostatic and the demagnetization energies,~~& .
The first curve is a circle and the second one a hyperbola (see fig. 1, dashed
- curve).
----_................ ,
",-,
"'\Z. without __'>,
ss domains\
H \ /Y \ /
\ /\ I
X·' \I
Fig. 1. Dependence of the resonance frequency on the strength of the applied magneticfield H perpendicular to the hexagonal axis for various orientations of the microwavefield It and of the Bloch walls with respect to the static field. The values of the angle abetween the Bloch walls and H are indicated (0, 7&/2).For II>HA + NMthe specimenis saturated in the y-direction. The (J)r/'Y~ and H-scales are the same, The graph appliesapproximately to BaFe1201Dàt room temporature. ,
FERROMAGNETIC RESONANCE ABSORPTION IN BaFe"O" 117
3. Influence of Weiss-domain structure
If there is no static magnetic field present, the ellipsoid will be dividedinto Weiss domains, with the Bloch walls parallel to the preferred axis.This Weiss-domain structure will persist on the application of a per-pendicular field, and will presumably not be changed very much. The fieldonly rotates all spins in the domains. The resultant magnetization inthe z-direction remains zero, so that there are no demagnetizing fields inthis direction. Moreover the component of the magnetization perpendicularto the walls is continuous (no magnetic poles on the walls). It has beenshown 2) that this does not necessarily remain the case for the dynamiccomponent of the magnetization vector. In order to investigate this forthe present case we write down the expression for the free enthalpy whichis now a function of the orientations (-&1'<PI) and (-&2'<P2) of the twomagnetization vectors of the two kinds of Weiss domains. This energyper cm" reads
It is assumed that the Weiss domains are thin slabs, as is suggested bythe powder pattern of fig. 2, where the surface is in a basal plane. It is takenon a much thicker crystal.
Fig. 2. Powder pattern of a BaFe12019 single crystal. The surface is a basal plane.Magnification 350 x.
118 J: SMIT and H. G. BELJERS
The last term of (11) represents the demagnetization on the walls ofthese very thin Weiss domains (with demagnetization coefficient 4n) and
. is equal to
. (Mln - M2n)22n 2 '
·wher~ Mln and M2n are the components of MI and M2 normal to the wall.The angle between the Bloch wall and the static field is a. The equilibriumposition is given by
The value of sin'!9-corresponds to that of (9) with N,= O. In this case thebinding to the z-axis is always increased by the demagnetization.
Proceeding in the same way as in section 2 one now gets a four by fourmatrix for the equations of motion. The problem has some resemblanceto that of antiferromagnetic resonance 3). In our case the two magnetizationvectors are coupled by demagnetizing fields, instead of by Weiss molecularfields.
For convenience we put LI'!9-l= Xl' Llfjll= X2' LI-&2= X3' Llfjl2 = X4 andwM sin-&jy = z, with '!9-= '!9-1= n - -&2'In the experiments the a.c. mag-netic field was always in the basal plane; so we confine ourselves to thiscase and we let {3denote the angle between the a.c. field h and the static field.
where Gij = à2Gj'öx(öXj. We have Gn = G3a, G~2= G44 and G12= -G14 =G23= -G34• It is convenient to use as variables LI'!9-±= (LI'!9-1±LI-&2)/2andLlfjl± = (Llfjll±Llfjl2)j2. One then gets .
A Lt'!9-+ + iz Ll9J+ - nLlfjl- = 0,'B LI-&- + izLl m- = - Mh cos {3.cos-&,
FERROMAGNETIC RESONANCE ABSORPTION IN BaF.IIO" 119
whereas the non-diagonal element n is given, by
n = - 4G12= 2nM2 sinif cosif sin2a. (15)
For n = 0, which is the case for walls parallel or perpendicular to the staticfield, (13) splits into two separate sets of equations formed by the elementsof the first and third rows and by those of the second and the fourth onesrespectively, which give resonance frequencies obeying z~= AC an§'>z~= BD. It is seen that the first resonance mode can only be excitedfor sinfJ =!= 0, i,e. only by the component of the a.c. magnetic field which isnormal to the static magnetic field. The second resonance inode can onlybe e~cited by the parallel component. It can be seen that an a.c. field inthe z-direction excites the same resonance mode as that in the x-direction.In that case the right-hand side of the first equation of (13) contains hz•
In the more general case of n =!= 0 the two resonance modes are mixedup. The resonance frequencies then obey
2_AC+BD ~(AC-BD)2 2 c~tZr- ± +nB .2 2For BaFe12019, n is comparatively small at the frequency used, so that
In (16) also we have labelled (wrly)2 with the subscripts 1 and Ij, sincethe amount of mixing is small. The intensity of the absorption is proportion-al to the susceptibility for different excitations of the same mode. Thea.c. component of the magnetization in the direction of h is equal toM ~Llif-cosif cosfJ-Llcp+ sinif sinfJ~and can be calculated from (13). Onethen finds for the susceptibility a fraction whose numerator for e.g. fJ= nj2is proportional to sin2 if ~A (BD - Z2)- n2B~. This indeed does not vanishat the parallel resonance frequency, but is equal to n2B2D sin2ifl(AC-BD).We have to comparethiswith thevaluefor fJ= O,i.e. cos2if~D(AC-z2)-n2qwhich is approximately cos2ifD(AC -BD). The ratio of the two suscep-.tibilities is most 0·04 in the cases investigated for a = ± n/4.
For H> HA + NM all magnetization vectors have arrived in the
120 J. SMIT and H. G. BELJERS
direction of H, and the Weiss-domain structure has disappeared. We mustthen use the second resonance condition of (10):
(Wr/y)j_ = H~H - HA + (4n - 3N)M~ H>HA+NM. (18)
For H = HA + NM the two frequencies of (16a) and (18) for the trans-verse field are the same and satisfy .4(wr/y)j_ = (HA + NM) (4n - 2N}M· H = HA + NM. (19)
Only for the longitudinal microwave field does the resonance frequencybecome zero as in the simple case without Weiss-domain structure (fig. I).The absorption peak then vanishes also, since it is generated by the 'com-ponent of the field perpendicular to the magnetization. For transverseexcitation the resonance frequency does not vanish owing to the fact thatfor this mode .1#1 and .1#2 have the same sign so that the resultant a,c.magnetization is finite in the z-direction. Demagnetizing fields are thenset up, resulting in the term (4n - 2N)M in (16a) and (19), so that Wrremains finite. It is impossible that a movement of the Bloch walls com-pensates for this a.c, magnetization, since the results of Wijn's 4) experi-ments show that these walls are immobile at the high frequency used(24000 Mc/s).Fig. 1 shows the various curves obtained for a= 0 and a= n/2. It is seen
. that at a low, constant.frequency, and varying H, only one absorption peakfor the parallel a.c. field is obtained. If the frequency is increased to thatof (19) one should expect both for the parallel and the transverse a.c, fieldsone absorption peak. For still higher frequencies the peak for the trans-verse a.c, field splits up into two peaks. All peaks for fields lower thanHA + NM are broadened by the Weiss-domain structure.
It is also possible to obtain these different types of spectra at a constantfrequency by varying the temperature, since the resonance frequency of(19) for N = 0 is proportional to y8nK and J( decreases appreciably withincreasing temperature for BaFe12019 1). It appeared that at room tempera-ture at 24 000 Mc/s we just have resonance in the minimum of the curveof fig. 1 for transverse excitation. Accordingly one should expect this secondpeak to disappear at low temperatures and to split into two peaks atelevated temperatures. This has been actually observed as will be reportedin section 5.
4. Measuring technique
In order to prevent dimensional effects, the magnetic crystal should besmall compared to the internal wavelength. Therefore a thin sheet ofBaFe12019 with lateral dimensions of about 1 mm was glued to a reflectingpiston and the specimen was ground to 40 (.L thickness. X-ray diffraction
FERROMAGNETIC RESONANCE ABSORPTION IN B,nFe"O".----~--------------showed that it was a proper single crystal. The hexagonal axis was normalto the plate. A piece of a !" by i" wave guide between the piston and awall with a rectangular hole, i.e. an inductive iris, formed a resonantcavity which can be tuned to 24 000 Mcfs by the adjustment screw of thepiston (see fig. 3). The cavity can be exposed to a static magnetic field,which can be rotated in a plane normal to the crystal axis.
83669
Fig. 3. Diagram of the microwave apparatus, FTl and FT2 are magic Tees with crystaldetectors FDl and FD2' Eu is the reference cavity and FB is the balanced d.c. amplifierfeeding back to the klystron K. C is a directional coupler, D2 the local oscillator,DM the mixing crystal, DB the intermediate-frequency amplifier and DR the ammeter.PI is a tuning piston, and with piston P2 adaptation can be regulated.
The cavity is coupled to a wave-guide system for the 1·25-cm region,so that it resembles a measurement equipment for paramagnetic resonance,frequently described in present-day literature 5). By the application ofa directional coupler C, equilibrium can be established with zero outputin the heterodyne detector if the resonant cavity is adapted to the guide.This can be carried out by tuning the cavity to resonance with piston Ptand adjusting the proper coupling with P2• Small resonance absorptionin the crystal gives a small decrease in Q, giving a reflected wave resultingin a voltage V proportional to the absorption i.e. to the imaginary part ofthe magnetic moment. This signal is measured with a superheterodynedevice, consisting of a local oscillator klystron DL> mixing crystal DM'intermediate-frequency amplifier DB, rectifier, and d.c, meter. Anothernecessary component. of the system is the frequency stabilization described.by Pound 6). This makes it easy to eliminate dispersion, because the cavity
121
122 . J. SMIT and H. G. BELJERS
can be tuned properly for each measurement by adjusting the frequencytill minimum output is reached.
The temperature of the cavity can he raised up to 200 oe by a heatingfilament and is measured by a' thermocouple. For measurements at thetemperature of liquid nitrogen the cavity is put into a Dewar flask. In thiscase a big Oerlikon electromagnet was used with an air gap of 70 mm.In other cases an: air gap of 14mm in a Boas magnet, producing a maximumfield of 20 000 Oe was sufficient. The magnetic field was calibrated by protonresonance and it appeared that hysteresis had to be taken into account.
5. Experimental results
Absorption measurements were made at five different temperatures asa function of the magnitude of the magnetic field. The direction of the fieldmakes an angle of 45° with the h.f. magnetic field, the latter being in thedirection of the longer dimension of the wave guide and tangential to thereflecting surface of the piston. In this position the components of the a.c.field parallel and perpendicular to the d.c, field are of equal magnitude.Figs 4a, b, c, d, and e show the absorption of the crystal specimen at about24 000 Mcls, for the same input energy level, for temperatures -196, 20,112, 155 and 200 oe. The ordinate represents the voltage V induced bythe reflected wave, in arbitrary units, but the scale is the same at all tem-peratures. The low-field peak is only excited by a parallel a.c. field and thehigh-field peaks by a transverse a.c, field, as is shown in fig. 5 for roomtemperature and as was established also at the other temperatures. Thisconfirms the expectation of 'the-theory of section 3. .At low temperaturesonly the parallel-field peak is found, but at higher temperatures theperpendicular-field peak comes up and splits into two at the highest temper-ature. The mean values of H of the absorption peaks are listed in table I,together with the values of the measuring frequency, of 4:n;M and NM.
TABLE I
Values of the mean static field for absorption Hres, of 4:n;M and NM,and of the frequency f
T u.; f 4:n;M NM. (0C) (kOe) (Mc/s) (kgauss) (kOe)
FERROllIAGNETIC RESONANCE ABSORPTION IN BnFellO .. 123
o
T=-t96oe
'\a: I \LJ I\. -~
• T~20oCII
11\ i{ I
I
.b. \ Ir-..I 'v .,
<,.T=ft2°C I
I f\1\ I
£. ij \ \/ V \
T=155oe
A \
\ i\si. \
/ V
T=200oC
" ~. A '\1\ J
g_ \ I V \J IU \
30V.t 20
to
o
la
o40
Vf 30
20
la
o
f:20
. Ia
40V
f 30
20
la
o11 12 13 14 15 16 17 18 19 20
H- kOe 63670
Fig. 4. Absorption peaks as a function of the strength of the static fieldH at five differenttemperatures on the same (arbitrary) scale with the microwave field in the basal planeat 45° with the static fieldH,
124 J. SMIT and H. G. BELJERS
6. Evaluation of g and HA
T = 20°C. If we start with the room-temperature measurements, weassume that there we are just ill the minimum of the curve for transverseexcitation of fig. 1. This point is not sharp, since the sample is not anellipsoid but a disc, so that the demagnetization is not homogeneous. ThusHA + NM = 17·15 kOe at room temperature or HA = 17·0kOe. We nowapply equation (16b) with a = n/4 as a mean value for the low-fieldabsorption peak. One then finds Wr/Y = 8·65 kOe or g= 1·98.The minimumof Wr/Y for transverse excitation is, according to (19), equal to 8·78 kOe,so that the resonance absorption we actually find for transverse excitationexists only by virtue of the inhomogeneity of the demagnetization, Thepeak .height is therefore small.
T=20oC0
/i\III \I \
\I \ j_
J \ / <,/ -p<- r-- r-,
8V
.i6040
20
o12 14 15 16 17 18 19
H - kOe 83671
13
Fig. 5. Absorption peaks at 20 oe as a function of H both with the microwave magnetic:fieldparallel and perpendicular to the static field H.
T = 200°C. At this temperature the three peaks are well resolved.Subtraction of equation (16b) from (16a) gives HA + NM in terms of theresonance fields H" and Hl for a = n/4":
HA + NM= (2n2-N) 2M(Hl+Hm.
HI-H,,-L11 -L1"(20)
1.11 and L1"are small quantities which can he estimated at once withsufficient accuracy. One then finds HA + NM = 16·2 kOe, which issmaller than Hl (16·85 kOe). Obviously (Hl - HIJ) is too large. Apossible explanation of this discrepancy might he a disorientation of thecrystal with ·respect to the static magnetic field. The effect thereof has beeninvestigated in the Appendix. The result is that the two resonance peaks
FERRO~L\GNETIC RESONANCE ABSORPTION IN B.Fc"O"----------------------below saturation are not seriously affected and at most approach eachother slightly, this being contrary to the experimental finding. Only thehigh-field peak is shifted towards a lower value of the field. We' shallassume therefore that the effect of disorientation on the resonance condi-tions is negligihle, as we should expect from the accuracy of mounting,
The resonance peaks found are very broad, much broader than wouldbe expected from the Weiss-domain structure. It is possible therefore that. the two transverse resonance peaks are not well resolved, and in an:y casehave decreased their distance apart. This can apply quite well to thespectra at 112 and 155°C, but we shall assume that also at 200 °C the twotransverse resonance peaks are shifted over distances in the H scale whichare proportional to their difference with (HA + NM) and of oppositesign. Other plausihle assumptions do not give, however, substantiallydifferent results. We then obtain for HA + NM = 17·40 kOe, i.e., HA =17·3 kOe, g = 2·02 and Hl = 16·70 and 18·40 kOe, as compared with theexperimental values of 16·85 and 18·15 kOe.
T = 155°C. Also in this case (20) gives too Iowa value for HA +NM(15'8 kOe). For HA +.NM = 17·45 kOe or HA = 17·3 kOe one finds fromthe parallel excitation g = 2·01. The values of the fields for transverseresonance are then 16·90 and 18·25 kOe, as compared with the observedvalues of 17·15 and 17·90 kOe. The differences between the calculatedand the experimental values of the resonance fields are here larger than atT = 200°C, owing to the greater overlap of the peaks.T= 112°C. The best fit with the.experimental values can be obtained
by taking HA + NM = 17·45 kOe or HA = 17·3 kOe. From the value ofHres for the parallel excitation one then obtains wrlr = 8·57 kOe or g =2·00. For the transverse excitation the resonance fields are then found tobe 17·15 and 17·95 kOe.
T = -196°C. Since one cannot deduce both HA and g from a singleresonance peak, we assume g to he 2·00, and find then HA = 16·2 kOe.
TABLE IrCalculated values of the anisotropy field IfA = 2 KIM and the spectro-scopic splitting factor g
The mi~imum value of wr/Y for transverse excitationds, acoording to (19),equal to 10·1 kOe, which is well above that of the measuring frequency, sothat the second peak cannot occur.
The results obtained are collected in table 11.We do not consider the differences found for g at different temperatures
to be significant; they are within the experimental error. We conclude there-fore that the g-factor of BaFel2019 has the spin-only value of 2·00 with anaccuracy of ± 1 % at all temperatures.
7. Shape and intensities of the absorption lines
In the first place we see that the parallel resonance peaks are asymmetri-cal in the sense that their maximum is shifted to lower fields. This can heexplained by the occurrence of the term LI in (16b) which is maximum fora = n/4 and zero for a = 0 or n/2. For the transverse peak, LI is negligible.Secondly the width of the parallel-field resonance peaks decreases withincreasing temperature. This fact is understandable if the width is causedby the Weiss-domain structure in which case it should be proportional to M.
We shall now calculate the areas of the absorption peaks. For this weshall assume that the peaks are caused by separate single resonance" "phenomena occurring at wT(H). Consequently the non-diagonal elementsn in (13) and with them LI in (16a) and (16b) will he ignored ..
From one of the Kronig-Kramers relations 7) between the real and theimaginary parts of the susceptibility, i.e.
co •2 "()
I ( ) __ f WIX WI clX W - 2 2 WI'no WI - W
follows, if it is assumed that the integrand is only finite for values of WI
which are slightly different from that of WT:
co
dw2 j'n(w~-w2) X'(w) = d; X"(H) dH. (21)
If the damping is small, as has been assumed, X' is substantially independentof it, and can be calculated at once from (5) by adding to the right-handsides of these equations terms due to the microwave field, as in (13).If X refers to the ~-direction, one finds
(22)so that (21) becomes
co
S X~~(H)dH = nG"']1]~d(wr/y)2/dH~-1.o
(23)
FERROMAGNETIC RESONANCE ABSORPTION IN BnFc"O"
For the case that G is only due to a static field H, (23) reduces to nM/2.For the parallel excitation, G",,,, is, according to (13) and (14), given byD/sin2f), which is, for a = n/4, equal to M(HA + 2nM). The derivativeof (Wr/y)2 with respect to H is easily calculated from (16b) so that one gets,after multiplication by cos2f) in order to take into account that only thecomponent of the microwave field normal to the magnetization is activeand that only the component of the a.c. magnetization parallel to the a.c.field is detected:
co A
f. "(H)dH = nM ~H + NM _ HII ?XII 2 ~ H HA+NM~'
o 11
where HII denotes the field strength for parallel-field resonance.For the perpendicular excitation, G",,,, is given by A of (13) and (14) so
that we get for the first transverse resonance peak
whereas for the second one we get
co A
f ,"(H)dH= nM 2 Hl -H + (4n-3N)M.Xl 2 2Hl _ HA + (4n _ 3N)M (26)
HA+NM
We have calculated the factor of nM/2 for the various resonance peaks(table Ill). The field strengths used in (25) and (26) are the calculatedones.
TABLE III
Calculated line intensities I in units nM/2
T (0C) III
-196 0·2720 0·26112 0\25155 0·25200 0·25
0·250·260·26
0·380·370·34
The peak height for parallel excitation is practically constant. Accordingto table III the intensity is proportional to M, and this is consistent with
127
(24)
(25)
128 J. SMIT and H. G. BELJERS
a constant peak height, as follows from the discussion in the beginningof this section. The area of the peak at the lowest transverse field should,according to table Ill, be equal to that ofthe parallel-field peak.Experimen-tally it is somewhat larger; since the width is comparable, but the height isgreater. This is presumably connected with the shift of the resonance lines,which was discussed in the preceding section. The highest-field peak should,according to table Ill, have a larger area. The height is the same, but at200°C the width is certainly greater. The general agreement is therefo~esatisfactory.
8. Discussion
The above analysis was based upon the validity of the use of equation(2), i.e. no terms with higher power than the second. of sinif have beenconsidered. This has been established experimentally by Mr P. Jongenbur-ger of this laboratory, at temperatures below room temperature, with anaccuracy of about 1% by measuring the magnetization curve in the diffi-cult direction. This is contrary to what is found for other hexagonalcrystals 8) such as Co, MnBi and MnSb, where the second anisotropyconstant is appreciable and at least of the order of twenty or forty per centof the first anisotropy constant.We have seen that the g-factor of BaFe12019 has the spin-only value,
so that there is no contribution of orbital motion to the magnetic moment.Roughly this is what one would expect, since the magnetic moments arethose of the ferric ions, which have a 6S ground state. In this respect onecan compare BaFe12019 with MnFe204' for which substance g is alsoequal to 2·00. We are therefore forced to believe that spin-orbit interactionis inactive in BaFe12019. Mostly, however, the occurrence of crystallineanisotropy is ascribed to spin-orbit interaction, and this anisotropy isappreciable here. Another cause for crystalline anisotropy in uniaxialcrystals is dipole-dipole interaction, but then one can expect a term withsin2if only, the other ones being exactly zero. Since this agrees with ex-perimental finding, we regard it as highly probable that the magnetism inBaFe12019 originates from electron spins only, and that the anisotropy iscaused by dipole-dipole interaction. Preliminary calculations however,do not yet support the latter point ofview.
, .We wish to thank Mr L. van der Kint for his assistance during the
measurements.
Eindhoven, February 1955
FERROMAGNETIC RESONANCE ABSORPTION IN BnFellO ..
Appendix
For the case that there is a component of the field HI along the hexagonalaxis, we assume that the domain boundaries shift in such a way that themean static magnetic field inside the specimen along the easy directionremains zero, i.e.
Hl - (4n - 2N) ~(I + v) cos f}l - (1- v) cos f}2~M/2 = 0, (Al)
where (1 ± v)/2 are the total volumes of the two kinds of Weiss domains.The equilibrium orientation of the magnetic vectors is therefore unchangedand is given again by (12) so that (AI) yields
v = HI/(4n- 2N)M cos f}.
The free enthalpy now becomes instead of (ll):
G = (K/2) ~sin2f}dI + v) + sin2 f}2 (1- v)~ +- (HM/2) ~sinf}lsin epI(1 + v) + sin f}2 sin ep2(1- v)~ +- (H1M/2) ~COSf}1(1 +v) + cosf}2(I-vH ++ (M2/2) [(N/4Hsin f}lcos epI(1 +v) + sin f}2 cos ep2(I-v)~2 + (A3)+ (N/4) ~sinf}lsin epI(1 + v) + sin f}2 sin ep2(I-v )~2++ (n-N/2) ~COSf}1(1 +v) + cosf}2(I-vH2 ++ n~sinf}l cos (epl-a)-sin f}2cos (ep2-a)~2 (1-v2)].
The equations ofmotion, expressed in the variables t~Xl (1 + v) ± X3 (I-vHand t~X2(1+ v) ± X4 (I-v) ~yield the following determinant al equation for Zr:
A gl2v iZr + nv -ng2lV B 0 iZr =0-l,Zr 0 C 0-n -izr+ nv g43v D
in whichgl2 = (M cos f})2 (- 4n sin2a + N) ,g21= (M sin f})2 (4n - 2N) ,g43 = (M sin ff)2 (- 4n cos2a + N) •
The situation becomes most simple for N = 0, which is approximatelysatisfied for our sample (N/4n = 0·03). One then obtains
(Wr/y)2 = HA [HAcos2ff+2nM~I ± Ycos22ff+sin22il'sin2a(I-v2)O.For v = 1, i.e. just where the Weiss domains disappear, we arrive at the(a = O)-curves. The latter curves do not change for finite v. We obtaintherefore a picture which is very similar to that of fig. 1. For H = 0the resonance conditions are identical with those for v = O. For HI =1= 0the curves for a = Jt/2 converge more rapidly for increasing H to the
129
(A2)
130 J. SIlHT and H. G. BELJERS
curves for a = O. They meet them at the same value ofH, corresponding",:0 v = 1, both for longitudinal and for transverse excitation. The resonancefrequency for parallel :fieldis there still :finite.
Then, again, the transverse curve continues with a different slope. Thiscurve can be calculated for the case where the :fieldH makes an angle(n/2 - e) with the hexagonal axis in which e ~ 1. It is then found that(wr/y)2 is increased with respect to (18) by an amount of approximately
1) J. J. Went, G. W. Rathenau, E. W. Gorter and G. W. van Oosterhout, Philipstech. Rev. 13, 194-208, 1952.
2) D. Polder and J. Smit, Rev. mod. Phys. 25, 89-90, 1953.3) J. Ubbink, Physica 19, 9-25, 1953.4) H. P. J. Wijn, Physica 19, 555-565, 1953.6) E. E. Schneider and T. S. England, Physica 17, 221-233, 1951.6) Technique of microwave measurements, Radiation Laboratories Series, 11, McGraw-
Hill, New York, 1947, p. 58.7) R. Kronig, J. opt. Soc. Amer. 12, 547-557, 1926; H. A. Kramers, Atti Congr. Fis.
Como, 1927, p. 545.8) R. M. Bozorth, Ferromagnetism, D. van Nostrand Co., New York, 1951, pp. 568.
