P. K. SISGH: Effect of Dimensional Resonance in Magneto-Kerr Effect 553

phys. stat. sol. (a) 91, 583 (1985)

Subject classification: 18.3 and 20; 22.8.2

Materials Division, National Physical Laboratory, New Delhi')

Effect of Dimensional Resonance and Damping Constant on Various Parameters Used in Magneto-Microwave Kerr Effect in Ferrites

BY P. K. SlNGII

Dedicated to Prof. Dr. Dr. h. c. Dr. E. h. P. GORLICH on the occasion of his 80th birthday

The effect of internal multiple reflections and damping constant on various parameters (viz. the permeability tensor components, effective permeabilities, propagation constants, reflection coefficients and the amplitude ratio (R), and phase difference (6) of the two orthogonal linearly polarised components of the reflected elliptically polarised wave) used in the analysis of Faraday configuration magneto-microwave Kerr effect (MMKE) in ferrimagnetic materials are studied theoretically. It is found that damping constant drastically affects the values of all the parameters and internal multiple reflections lead to the excitation of dimensional resonance phenomenon.

Der EinfluB von inneren Vielfachreflexionen und der Dampfungskonstante auf verschiedene Parameter (z. B. die Komponenten des Permeabilitatstensors, die effektiven Permeabilitaten, Ausbreitungskonstanten, Reflexionskoeffizienten sowie des Amplitudenverhaltnis (R) und der Phasendifferenz (6) der beiden orthogonalen, linear polarisierten Komponenten der reflektierten elliptisch polarisierten Welle), die bei der Analyse der Faraday-Konfiguration des Magneto- Mikrowellen-Kerreffekts (MMKE) in ferrimagnetischen Materialien benutzt werden, wird theore- tisch untersucht. Es wird gefunden, daB die Dampfungskonstante die Werte all dieser Parameter drastisch beeinflufit und daB die inneren Vielfachreflexionen zur Anregung von Dimensions- resonanzphanomenen fuhren.

1. Introduction When the wavelength of electromagnetic radiation inside a material is smaller than twice the relevant dimensions of the specimen, standing waves which can result in high losses may be excited. This phenomenon is known as dimensional resonance and was first observed by Brocknian et al. [l] in 1950 in manganese-zinc ferrite below microwave frequencies. In a core made of Ferroxcube I11 placed inside a coil, they observed large magnetic and dielectric losses a t a particular frequency. They also observed that as the core dimensions were decreased, the resonance effects were shifted towards higher frequencies. They attributed this effect to a cavity type of resonance which occurs due to the high dielectric constant and high value of perme- ability below microwave frequencies. Brockman, Dowling and Steneck [ 11 have also shown that for a sample 2 cm thick with E' = 105 and p = lo3, one can observe a similar type of losses a t 2.5 MHz. In this case the half wavelength (1,/2) inside the rnediuni given by

C 1,/2 = - (p&')-1/2 2f

becomes equal to the size of the specimen. Here, ,u and E' are the permeability and

l) Hillside Road, 110012 New Delhi, India.

584 P. K. SINGH

relative permittivity of the sample, respectively, f is the frequency of the electro- magnetic wave, and c the velocity of light.

The effect of dimensional resonance a t microwave frequencies was discussed by Fox et al. [2] and Rowen [3] in connection with a first study of Faraday rotation bandwidth. In 1956, Sensiper [4] found experimentally that the loss (db/inch) at microwave frequencies increases as the sample thickness is increased. He attributed these losses to dimensional resonance. Many other workers [5] have assigned the name “body resonance” to such type of losses and ascribed these resonances to multiple internal reflections which form standing waves in the sample, provided that the skin depth is large enough to permit such multiple internal reflections. The name “body resonance” has also been suggested for spin wave resonance. Spin wave reso- nance is a special variation of dimensional resonance. The basic difference from the usual “body resonance” is that the spin wave resonance (SWR) is observed only in relatively thin films such that exchange dominates. This type of resonance has led to many useful studies [6 to 81 of exchange constants, homogeneities of the films, etc. In contrast to above, the type of resonance which we propose to study in the present work, is observed in bulk samples and exchange does not play any role. Surprisingly, very little attention has been given so far to the systematic study of dimensional resonance in spite of the fact that large losses were expected due to i t a t microwave frequencies. Though many references [2 to 4, 9 to 111 give the preliminary idea of dimensional effects, workers have generally by-passed the effect of dimensional resonance in their experiments by choosing small dimensions of the sample. The dimensional resonance (or body resonance) in a metal-backed sample can be excited when the following well-known optical interference condition is satisfied [12] :

2k;d = nn ,

where n is an integer, d the thickness of the specimen, and k & are the real components of the propagation constants inside the material and are equal to 2n/1,, where 1, is the wavelength inside the sample. The dimensional resonance is excited when n is an integer. In the present study, we take account of the dimensional resonance effect by means of the multiple reflection formula given in the following section.

In this paper we have studied the effect of internal multiple reflections on various parameters used in the analysis of the magneto-microwave Kerr effect (MMKE) in ferrites [12 to 141 with a view to understand the phenomenon of dimensional resonance. This is due to the fact that the dimensional resonance phenomenon observed in Kerr effect measurements has been successfully used as a tool to study the microwave behaviour of partially magnetized ferrites, its permeability, and effective linewidth, etc. 112 to 141.

The analysis of MMKE [12 to 141 is, generally, done in terms of R and S, the ampli- tude ratio and the phase difference of the two orthogonal linearly polarized components of the reflected elliptically polarized wave from the magnetized sample. These two parameters can be measured directly with the help of a microwave interferometer [12 to 141 as a function of applied dc magnetic field ( H ) . Theoretically R and 6 are related to permeability tensor components through the propagation and reflection coefficients as discussed in Section 2. In the present paper we have studied (theoret- ically) the effect of internal multiple reflections and the damping parameter (01 ) on permeability tensor components (p and K ) , scalar permeabilities (,u&), propagation coefficients (k+), reflection (e+), and effective reflection coefficients (e+e) and R and 6 and the results are presentedin the form of graphs.

The calculations reported here are carried out for a ferrite sample having thickness 0.70 cm, diameter 3.02 cm, saturation magnetization 380 G, resistivity = lo7 Qcm,

Effect of Dimensional Resonance and Damping Constant in Magneto-Kerr Effect 585

dielectric constant 15.5, and tan 6 = 0.002 at 9.44 GHz. For a saniple of these speci- fications, the ferromagnetic resonance (FMR) is expected a t magnetic field H,,, = = 3574 Oe and the sample remains partially magnetised upto 260 Oe.

2. Theory of Herr Effect Let a linearly polarized electroniagnetic radiation, propagating in z-direction, be incident upon a ferrite sample with its polished flat sprface in the x-y plane. The linearly polarized wave can be resolved into two counter-rotating circularly polarized coniponents. The two components, positive and negative, have different propagation constants (or wave numbers) inside the sample magnetized in the propagation direc- tion of the microwaves. These can be expressed as [12, 151,

The scalar permeabilities ,u* are equal to ,u f K , where ,u and K are permeability tensor components and other symbols have their usual meanings. The two circularly polarized waves also have, in general, different reflection coefficients a t the ferrite surface which are given as [12, 131

where Ei, and E,+ represent the incident and reflected components of the electric field, respectively, and kn is the free wave propagation constant. It is well known that the reflection coefficients in the case of high resistivity materials cannot be accurately represented by the above expression because there is always an effect of multiple reflections. The modified expression for the effective reflection coefficients, taking such internal multiple reflections into account depends upon the thickness of the sample. The expression for a sample backed by a, metallic plate is given as r12, 161

The effective reflection coefficients (e+e) are, in general, different both in magnitude and phase. The total reflected wave has components E,, and E,, which are parallel and orthogonal, respectively, to the linearly polarized mode of the incident wave [17]. We can define X I and X , as

X l = (e+ - @-I/(@+ + e-) and x, = ( e + e - e-e . ) / (e+e + e - c ) 9 (6) where i = 1 and i = 2 represent the cases without and with multiple reflections, respectively. X , and X , are coinplex quantities and can be written as

Now, from the definition of R and 6 [12, 131,

and

X , = X i + jX;’ and X , = Xg + ?Xi. ( 7 )

R = lXil @a)

s = - ( n/2) + arg IXy/Xll . (8 b)

3. Results and Discussion 3.1 Permeability tensor components and scalar permeability

The analytical expressions for the components ,u and K of the permeability tensor were derived by Polder [18] and later modified [19] to include losses through the damping term and are given in [20]. The real parts of ,u and K have been plotted as

586 P. K. SINGH

a I

1 2 3 4 I 2 3 4 H IVCi ~ *

Fig. 1. a) Theoretical p' vs. H and b) K' vs. H curves for a = 0.001, 0.010, and 0.100

functions of magnetic field in Fig. 1 a and b, respectively. These curves are shown for three different values of OL to demonstrate the effect of damping on the tensor compo- nents. We see that the behaviour of p' and K' versus field is approximately the same. The value of p' starts from unity a t H = 0 while the H = 0 value of K' is negative, but both the parameters attain very large values near the ferromagnetic resonance field (Hres). The values of p' and K' become very large and negative near H,,, and exhibit a sharp increase at slightly higher fields to attain large positive values. In fact, in the absence of damping, i.e. if we neglect the losses, the two curves show a discontinuity a t resonance. When the loss term is included, the values of and K'

m=0010 i_/ 2

0 6 I I L; I f 1

1 4 i b a=OOIU

0050 2- -

I I 0 1 2 3 4 5

2- -

I I 0 1 2 3 4 5

become reduced a t resonance and their values decrease as the damping param- eter is increased.

Fig. 2 gives the theoretical variation of p" and K" with magnetic field. Both parameters give a sharp peak a t a field equal to H,,, for low values of a. The peaks broaden for higher values of the damping parameter. The values of p" and K" away from resonance are negligibly small.

Big. 2. a) Theoretical $' vs. H and b) K" vs. I$ curves for a = 0.010, 0.050, and 0.100

Effcct of Dimensional Resonance and Damping Constant in Magneto-Kerr Effect 587

Fig. 3. Theoretical variations of the real parts of effective permeability with dc magnetic field for different values of LI

In the Kerr effect in Faraday confi- guration, the scalar permeabilities p, and K , which are the sum and differ- ence of the diagonal and off diagonal coniponents of the permeability tensor, respectively, are complex and represent two senses of polarization. The theoretical variations of the real parts of p, and p- with magnetic field ( H ) are presented in Fig. 3. p i is initially approximately equal to unity and decreases with in- creasing field. Purtherniore, p i is large and negative near H = H,,,. After a

singularity it becomes positive and falls off with increasing magnetic field. This component represents a circularly polarized wave with rotation in the same sense as the Larrnor precession. The value of pL is greater than p’+ in the region H < H,,, and less than p; for H > H,,,. The value of p l decreases slowly and linearly with increasing H . This represents a circularly polarized wave with sense of rotation op- posite to the Larinor precession. The variation of the imaginary parts of pi with H is presented in Fig. 4. The value of p t is maximum a t resonance, similar to the be- haviour of K” or p”, whereas p: is nearly equal to zero. Thus, the damping parameter changes the values of effective permeabilities drastically.

It was shown in (3) that the propagation constants are directly proportional to the square root of the scalar permeabilities [15]. The other parameters which affect the propagation constants are the dielectric constant, the operational frequency, and

3.2 The propagatiom constants

Fig. 4. Theoretical variations of the imaginary part of ( p f K ) with H for different values of a

a

L=0501

0010

0.050 * I v 6 = 0700

050

001

1 I 2 3 4

P. K. SING11

Fig. 5 . a) Theoretical k; vs. H curves and b) theoretical variations of imaginary part of k, with dc mangetic field for different values of damping constant

the conductivity of the sample. As the two scalar permeabilities for the two sens- es of rotation have different values, the propagation constant will also be dif- ferent in these two cases. The variations of real part k', and imaginary part k'; of the propagation constant k , with H have been given in Fig. 5a and b, respectively. It is clear from the k'+ versus H variation that a t lower and higher fields (i.e. away from resonance) the value of k'+ does not change much whereas in the vicin- ity of FMR, i t changes drastically. A similar behaviour is also found for k:. The propagation constant for the anti- Larmor wave, for which FMR is not possible, does not show any drastic change with magnetic field or with damping parameter as can be seen from Fig. 6. In fact, the real part k' decreases almost linearly with magnetic field in the saturated state of magnet,isation and does not show any appreciable change with oc, whereas k l increases linearly with H and is not affected significantly by the damping parameter.

3.3 Heflectioit atid effective veflectioia coefficieitts

The reflection and effective reflection coefficients can be expressed by (4) and (51, respectively. In the calculation of the effective reflection coefficient, the effect of internal multiple reflections has been included through the multiple reflection formula. The variations of e', and e;e with magnetic field are shown in Fig. 7 . The value of p i is negative and its absolute value increases slowly with magnetic field. In other words in this region 0 < H < 2.5 kOe, de;ldH is almost constant. Between 3 and 4 kOe, the absolute value of e', increases sharply and then falls rapidly with increasing magnetic field. Furthermore, for small values of oc (viz. 01 = 0.002) becomes

Effect of Dimensional Resonance and Damping Constant in Magneto-Kerr Effect 599

A i4 I I I

I

H ikOcl ~

Fig. 6. Theoretical variations of real (k:) and imaginary (k:) parts of propagation constant k-with 11 for different values of a

positive and again it starts increasing with magnetic field. After 4 kOe the absolute value of PL starts increasing slowly in a similar fashion as seen earlier a t low fields. On the other hand, the value of e i e shows sharp variations both near ferromagnetic resonance and away from it. Moreover, eke takes both, negative and positive values. It is evident from the figure that two peaks exist, one on either side of the ferro- magnetic resonance field. In fact, these peaks are seen in the region where e'+ varies linearly with magnetic field. It is important to note that in the vicinityof ferromagnetic resonance, the values of e ; and e i e are almost the same (for example, see the curve for 01 = 0.002 in Pig. 7). The other important feature is that the peak height decreases with increasing values of 01 and the peak shifts a t lower fields. Two questions which come inimediately in the mind are (i) why are there peaks only in eke a t fields away from the ferromagnetic resonance which shifts with damping parameter ? (ii) why are the two values of e ; and almost equal in the vicinity of ferromagnetic resonance !

1 7 2 3 4 HlkOe! ~~ -

Pig. 7. Theoretical variations of real parts of reflection coefficients wit.hout multiple reflections e+ ( -x-x-) a.nd with multiple reflections eoe (- ) a.gainst H

590 P. K. SINGH

The answer of the first question is easy. It is known that in the expression of e + e the effect of multiple reflections is included which may give maxima in the versus H curves. This can be confirmed from the effective reflection coefficient formula in the following way. For the occurrence of maxima or minima in the e ; e versus II curves one has

de+e/dH = 0 , (9)

(de;e/dH) + (dp;e/dH) = 0 . (10) 1.e.

Differentiating ( 5 ) with respect to H, one gets

dk+ dH {$ ( 1 - exp ( - 4 j k + d ) ) + __ ( 2 j d ) exp ( - 2 j k . d ) ( 1 - p:)} = 0 . ( 1 1 )

On assunling e+ and k+ to be the real quantities, by the separation of real and imagi- nary parts in the above equation, we get

d k ; d H

sin ( 4 k ; d ) + ~ (2d ) (1 - e;”) cos ( 2 k ; d )

must be zero which means deh For a maximum or minimum in e;., __ dH

de ; d k ; 2 ~ ( 1 - cos (4k;d) ) + __ ( 2 4 ( 1 - @.I.) sin ( 2 k ; d ) = 0 dH d H

or dk;

~ 2 sinz ( 2 k ; d ) + ~ ( 2 d ) ( 1 - p:) sin ( 2 k ; d ) = 0 . d H d H

But, in general, e+, -, and ~ are not equal to zero. Hence,

de ;

I de; d k ; d H d H

sin ( 2 k ; d ) = 0 or

where n is an integer. As both ternis of ( 1 3 ) contain the sine function, they will be zero siniultaneously if the above condition is satisfied. The condition described by (14) is the well-known dimensional resonance condition. It is clear from the above analysis that maxima or minima will occur in e;. versus H curves when the dimen-

sional resonance condition is satisfied, i.e. when k’d equals -, n:, ,..., - . Strictly speaking, however, the values of p+ and k+ are never purely real, and their imaginary parts may be negligibly small. This is the case at fields away from ferro- magnetic resonance for small values of 01 (say OL = 0.001) . Due to the complex nature of p+ and k + , the real part of (12) will not be represented correctly by ( 1 3 ) for high values of the damping parameter. In fact, (13) will also contain an additional term C related to the imaginary parts of e+ and k + . In this case ( 1 3 ) can be written as

2k;d = n n , ( 1 4 )

n: 3 n nn: 2 2

d k ; ~ 2 sin2 ( 2 k ; d ) + ~ ( 2 4 ( 1 - Q:) sin ( 2 k ; d ) + C = 0 . d H dH de;

Effect of Dimensional Resonance and Damping Constant in Magneto-Kerr Effect 591

The additional term C will directly affect the positions of the maxima and minima. Equation (15) cannot be satisfied to zero a t the field where the dimensional resonance condition is satisfied. Therefore, the extrema of the dimensional resonance will be slightly shifted (from the positions where the above condition of maxima or minima is fulfilled). Furthermore, this discrepancy will increase with increasing 01 because the constant term C will be more and more important. This phenomenon can be noticed in Fig. 7 , from the peaks seen near 2.1 kOe. The dimensional resonance condition is satisfied a t 2.12 kOe and the maxima for 01 = 0.002 is also a t the same field because in this case e.:. and k'.; are very small (see Fig. 8 and 5 b) and e+ and k+ can be considered real. But, the shift in the position of maxima is noticed for high values of 01 and the discrepancy increases as n increases. This is due to the fact that k; and

The answer to the second question can be obtained from (5) for e k e . This equation can be written as

are not very small for high values of a.

A t dimensional resonance, i.e., when 2k; d = nn, this equation becomes

Near the ferromagnetic resonance region k& changes quite drastically, and hence, many dimensional resonance conditions will be satisfied in a small region. Therefore, we can say that e k e given (17) will also hold good in ferromagnetic resonance region. T t has been shown earlier that near FMR k: attains very large negative valnes. Therefore, the second ternis both in the numerator and the denomerator of (17) can he neglected in comparison to e+ and 1, respectively. Then

e k e = Q. . (188)

This indicates that near ferromagnetic resonance, the real parts of the two reflection coefficients p+e and e+ will be the same.

It is alos interesting to consider here the case far from ferromagnetic resonance. It has been shown earlier that lc: is either zero or negligibly small a t fields far from resonance, particularly for low values of n. In this case, (17) will reduce to the following if the dimensional resonance condition is also satisfied :

This means that a t dimensional resonance the value of the effective reflection coeffi- cient will be negative or positive depending upon the odd or even values of n in the condition given by (14). The value of e k e will tend towards unity for small values of a.

The theoretical variation of p; with H , demonstrated in Fig. 8, shows a peak near FMR whose height is sensitive to a. Furthermore, its value away from resonance remains negligibly small for higher values of a. In the Q:~ versus H variation presented in Fig. 8b, large variations are seen a t FMR as well as away from FMR. The important features which can be noticed f y n i this curve are: (i) many maxima are seen a t positive and negative values of p+. both at ferromagnetic resonance and away from FMR, (ii) the positions of t,he maxima start for higher values of 01,

592 P. K. SINCH

I I T Fig. 8. Theoretical a) e’; vs. H and b) e’;e - vs. H curves for different values of a

t c3-a

a=0050 -05-

-0g --0001 -

- -

: * Qr

0

(iii) a t the positions wher: gives a maxima, the value of Q+. is either zero or very small and vice versa, (iv) the values of e; and e>c are same near the FMR. The explanation of

HlkOel ~ the above features follows much the same lines :s given earlier.

For maxima or mininia in the e y e versus H curve, de+/dH must he zero. From

-04

- 08

0 I 2 3 4 5

(12).,

or dk; de;-

__ 2 sin (2k;d) cos (2k;d) + ~ ( 2 d ) (1 - e;”) cos (21c;d) = o . dH dH (19)

It has been mentioned earlier, however, that in general __ and - are not zero. dp; dk; dH d H

Therefore, for a maxima or minima the conditions

i.e. sin (2k;d) = 0 and cos (2k;d = 0 ,

2k;d = nn and 2k;d = (2n’ + 1 ) 2 / n

must be satisfied, where n and n‘ are integers. These conditions give extrema which will occur when k;d = n/4, 3n/4, 5n/4, etc.

From (21) , we can say that the first term of ( 1 9 ) will be zero when the dimensional resonance condition is satisfied in eL,, but in this case the second term of (19) will remain finite. The extrema in the e‘;e versus H curve will be observed only when the condition given by (21) is satisfied.

I n practice, we find that the extrema do not occur exactly a t the same field where the above-mentioned conditions are satisfied. These discrepancies are due to the facts that Q + and k , are not purely real quantities and that (19) will contain an addi- tional C term similar to the one discussed earlier. The same value of e‘; and e y e near FMR is quite expected from condition (18).

The theoretical variation of the real and imaginary parts of e - and ePe with H have been presented in Fig. 9. None of the four components, e l , e le , p y , and e‘Le, show

Effect of Dimensional Resonance and Damping Constant in Magneto-Kerr Effect

-

593

any significant change with damping and all are negative. Q - decreases with increasing H while Q: is negilgibly small. Both remain practically unaffected by changes in 01.

The real part of ede increases with mag- netic field whereas its imaginary part shows the opposite behaviour. Small 01- dependences are also seen. Such types of behaviour are quite expected because the parameters from which their values have been derived, do not change appreciably with H or with a. The other point which can be mentioned here is that Q- and e - e do not ex- hibit any maxima. This is due to the fact that k- does not change appreciably with H and, therefore, cannot be “tuned to” di- mensional resonance.

I I I 1 1

i, I 7 3 4

Big. 10. Theoretical X i vs. H curves for (x = = 0.002, 0.010, and 0.100

H ikUd -- - --

38 physica (a) 9112

594 P. K. SINGH

Fig. 11. Theoretical X y vs. H curv- es for different values of 01

H ikOei z

3.4 Amplitude ratio R and phase difference d

The theoretical variations of Xi and X;' with magnetic field are presented in Pig. 10 and 11, respectively in the case where internal multiple reflections are not considered. The real part Xi as well as the imaginary part X;' show changes in their values with magnetic field only near FMR and in this region their absolute values are also sensitive to the damping. At the fields away from resonance Xi is almost field independent while X ; is zero. It is to be noticed that for H < H,,,, X i is positive and for H > H,,,, X ; is negative. It is seen from (8a) that R is simply the absolute value of X,. Hence we get only one maximum a t FMR in the R versus H curve when the effects of multiple reflections are ignored. On the other hand, 6 is the ratio of the imaginary and real parts of XT. For H < H,,, Xf is almost zero and X i is positive, so that 6 will be - 4 2 . For H > H,,,, Xi' is again zero but X i is negative and finite. This gives a 6 of f 4 2 . Near FMR the value of Xi and X ; are finite and give a value of 6 between -nl2 and +n/2. A typical curve showing this trend is given in Fig. 12.

- IZl l" , 1 1 1

The theoretical curves of X i and X g versus magnetic field show many niaxi- ma and minima. Such a behaviour is quite expected because their parent parameters posses similar dependences. These two curves are shown in Fig. 13a and b. It is apparent from thesefigures that the value of Xi is negative for the entire region of magnetic field under study while X t takes both negative and positive values. A comparison of the two curves shows that a t the field where X i gives a niaximum (viz. at 2.2 kOe), Xi is nearly zero. Another notable fea-

Fig. 12. Theoretical R vs. H and 6 vs. H curv- es (without multiple reflections) for different values of o(

Effect of Dimensional Resonance and Damping Constant in Magneto-Kerr Effect 595

-08

-12-

-16-

- - The values of damping constant a t peaks have been calculated accurately by fitting of data to the theoretical curves which have been used to evaluate the effective linewidth (AH,ff). It has been consistently observed that the values of AHeffat dimen- sional resonance peaks which generally lie in the region outside the spin wave mani-

-

I I I I I fold, are an order of magnitude higher than

- - -

~

-

b - -

~ o , o li I

4. Conclusion

Theoretical R versus H and 6 versus H curves contain maxima and minima which occur in the regions outside and inside the spin wave manifold. The peaks are either

596 P. K. SIKGH: Effect of Dimensional Resonance in Magneto-Kerr Effect

5

Qz I 4

3

2

I

0 C I I I I I 1

Fig. 14. Theoretical variation of a) R and b) S (with multiple reflections) as a function of dc magnetic field for different values of a

due to the dimensional resonance or aue to the ferromagnetic resonance. The heights of the maxima are highly sensitive to the damping parameter. The experi- mental observation of such peaks would make i t possible to determine the damping parameter correctly by fitting the data to

4, =0010

'o I20

60 0 I00 I I 8 O l o b 0 7 2 3 4 ~ ( k u ~ ) pb the theoretical curves.

Ackn ouwledgemen ts

The author is extreiriely grateful to Dr. Shiva I'rasad for many stimulating discus- sions and to Mr. s. B. Singh and Prabhat Singh for useful suggestions.

References [l] F. G. BROCKMAN, P. H. DOWLING, and W. G. STENECK, Phys. Rev. 77, 85 (1950). [2] A. G. FOX, S. E. MILLER, and T. E. WEISS, Bell Syst. tech. J. 34,5 (1955). [3] J. H. ROWEN, Bell Byst. tech. J. 32, 1333 (1953). [4] S. SEXSIPER, Proc. IRE 44, 1322 (1956). [5] M. H . SEAVERY and P. E. TANNENWALD, Phys. Rev. Letters 1, 168 (1958) and J. appl. Phys.

[6] J. T. Yu, R. A. TURK, andP. E. WIGRN, Phys. Rev. B 11, 429 (1975). [7] W. S. AMENT and G. T. RADO, Phys. Rev. 97, 1558 (1955). [8] B. HOCKSTRA, R. P. VAN STAPLE, and J. M. ROBERTSON, J. appl. Phys. 48,382 (1977). [9] ALLEY, Bell Syst. tech. J. 32, 1155 (1953).

[lo] L. G . VAN UITERT, Proc. IRE 44, 1294 (1956). [ll] G. W. GORTER, Proc. IRE 43, 1945 (1955). [12] P. K. SINGH, S. PRASAD, R. M. MEHRA, P. KISHAN, S. B. SINGH, and G. P. SRIVASTAVA, J.

[I31 P. K. SINGH, S. PRASED, P. I<. SINGH, and S. B. SINQX, J. appl. Phys. 53,9180 (1982). [14] S. PRASAD, and It. M. MEHRA, 5. Phys. Chem. Solids 39,353 (1978). [15] B. LAX and K. 5. BUTTON, Microwave Ferrites and Ferrimagnetics, Me Graw-Hill Publ. Co.,

[I61 P. K. SINGH, Ph. D. Thesis, Delhi University, Delhi (India) 1981. [17] M. E. BROWDIN and R. J. VERNON, Phys. Rev. 14OA, 1390 (1965). [18] D. POLDER, Phys. Rev. 73, 1120 (1948) and Phil. Mag. 40, 99 (1949). [19] C. L. HOGEN, Proc. IRE 44, 1345 (1956). [20] B. LAX and K. J. BUTTON, see [15] (p. 154).

30, 2275 (1959).

Magnetism magnetic Mater. 21, 297 (1980).

New York 1962 (p. 299).

(Received May 17, 1985)