GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS OF SPIN-WAVE MODES

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GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS OF SPIN-WAVE MODES

R. de Wames, T. Wolfram

To cite this version:

R. de Wames, T. Wolfram. GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS OF SPIN-WAVE MODES. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1090-C1-1095.

�10.1051/jphyscol:19711390�. �jpa-00214430�

GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS OF SPIN-WAVE MODES

R. E. de W A M E S a n d T. W O L F R A M Science Center, N o r t h American Rockwell Corporation

Thousand Oaks, California, 91360

Résumé. — Le spectre de modes déduit de l'approximation des ondes de spin extrapolés au cas limite des grandes longueurs d'onde est comparé aux solutions déduites de l'effet d'échange dipolaire dans le cas de couches minces d'épais- seur finie et de surface infinie ( = couche mince infinie). Des résultats de nombreuses expériences d'ondes de spin sont comparés avec le spectre de modes d'une couche mince infinie dans l'hypothèse d'une quantification géométrique. L'accord raisonnablement bon que l'on trouve suggère que la constante de propagation dans le plan de la couche mince est quan- tifiée par rapport aux dimensions finies de l'échantillon. De petites différences trouvées pour les modes d'ordre bas sem- blent indiquer qu'il existe quelque divergence par rapport à ce schéma simple de quantification.

Des solutions exactes obtenues dans le cas du cylindre elliptique aimanté dans la direction du grand axe sont comparées à celles correspondant au cas d'une plaque infinie aimantée dans une direction parallèle à la surface. On trouve que dans certaines conditions la constante de propagation est quantifiée géométriquement de façon simple. L'effet de la forme géométrique sur la fréquence des ondes de spin est discuté dans le cas du cylindre elliptique, circulaire, de la sphère, du disque et dans le cas de formes intermédiaires.

Abstract. — The mode spectrum derived from spin-wave approximation extrapolated to the long wavelength limit is compared with the dipole-exchange solutions for a film of finite thickness with an infinite surface — the infinite film.

Experimental results from a number of spin-wave experiments are compared with the mode spectrum of an infinite film with the assumption of geometrical quantization. The reasonably good agreement of such an analysis suggests that the propagation constant in the plane of the film is quantized according to the dimensions of the finite sample. Small discre- pancies for the low order modes suggest departures from the simple quantization scheme.

Exact solutions for the elliptical cylinder magnetized along the long axis are compared with those of the infinite plate magnetized in a direction parallel to the surface. It is found that under certain conditions the propagation constant is geometrically quantized in a simple manner. The effects of shape on spin-wave frequencies as a function of geometry are discussed for the elliptical cylinder, circular cylinder, sphere, disk and intermediate shapes.

I. Introduction. — Geometrical quantization in which the strongly excited exchange spin-wave modes of thin ferromagnetic metal films have an odd num- ber of half integral wavelengths across the thickness of the film has been known for some time [1, 2]. The excitation of these resonances by a uniform applied r. f.

field is frequently associated with the Kittel [2] spin pinning mechanism in which anisotropic interactions acting at the surfaces lead to an r. f. magnetization that vanishes at the surfaces.

In the case of the magnetostatic spin-wave modes in ferrite films the situation is quite different. The basic forces for these modes arise from the dipole-dipole interaction rather than the exchange interaction. The magnetostatic modes are solutions of Maxwell's magnetostatic equations rather than the Heisenberg exchange hamiltonian. The magnetostatic equations do not permit solutions in which the r. f. magnetization is pinned at the surfaces, and the eigenstates do not have an integral number of half wavelengths across the film. In order to employ the (pinned model) PM boundary conditions both dipolar and exchange interactions must be considered. The dipole-exchange equations have been solved [3, 4] employing the PM boundary conditions and it is found that the usual magnetostatic volume and surface modes as well as the uniform mode do not exist in this case. Instead, new modes, the magnetoexchange [3, 4] modes with dispersion quite different from the magnetostatic dispersion result. In this regard recent conclusions by Sparks [5] that the frequencies of the magnetostatic modes are essentially independent of the explicit pinning mechanism are incorrect.

Recent experiments on yttrium iron garnet films [6] have indicated that the propagation vector in the plane of the sample may be quantized in multiples of njL and ii/W where L and W are the sample length and width respectively. It is this type of simple quan- tization t h a t we wish to discuss in this paper. The question of the existence of modes with quantized propagation constants is entirely separate from ques- tions relating to explicit surface pinning models. In a finite sample the modes are always discrete and if t h e discrete frequencies lie on the frequency versus- wavevector curve of the infinite sample, then a one to one correspondence can be made and a set of quantized wavevectors can be deduced. In general, however, these quantized wavelengths are not simple multiples of the sample dimensions.

It is frequently supposed that the excitation of modes other than the uniform mode by a uniform r. f.

field is evidence for a pinning mechanism. In samples deviating from ellipsoidal geometry excitation of non- uniform modes is not forbidden by symmetry conside- rations. In addition the spin wave modes are admix- tures of exchange a n d magnetostatic waves [7] a n d the problem of excitation of these admixed modes has not been solved. Thus while pinning may occur, it is not essential for the explanation of the observation of multiple resonances in finite film samples.

In section II we analyze some recent experimental magnetostatic surface and bulk resonance data in terms of the infinite film results with t h e assumption of geometrical quantization for the in-plane propaga- tion vector k. Excellent agreement is found for the modes having large | k |. The modes having small | k |

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711390

GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS O F SPIN-WAVE MODES C 1

-

1091

show small but significant departures which indicate a possible breakdown in the validity of the simple quantization scheme.

The exact solutions for the magnetostatic surface spin-wave modes of an axially magnetized cylinder with elliptical cross section are discussed as examples in which a simple geometrical quantization occurs without pinning. In the limit in which the elliptical cross section approaches a planar geometry the solu- tions may be related to those of the thin film.

In section 111, shape effects on the frequencies of the elliptical cylinder, circular cylinder, sphere, disk, and intermediate shapes are discussed.

11. Dipole-Exchange Spin-Waves and Relation to Experiment. - The boundary value problem for any given geometry can be formulated but except for a few special shapes solution can be obtained only by numerical methods. For this reason, a number of ideal geometries have been investigated with the hope that the characterization of the resonant modes for these ideal cases will help in understanding the mode spectrum of the actual sample. Neglecting forces due t o exchange and electro-magnetic induction, Walker

[8]

solved the boundary value problem for samples of ellipsoidal shapes

;

similar calculations were made by Damon and Eshbach [9] for an infinite ferromagnetic slab magnetized in its plane. Both of the above inves- tigations revealed that the magnetostatic mode spec- trum for these geometries extended over a wider frequency range than the spectrum of the spin-wave approximation extrapolated to long wavelengths for which [lo]

w(q) =

y[(Ho - 4 nN, Mo + DM, q2) (H,

- -

4 EN, M, + DM, q2 + 4 nMo sin2

8 ) ] % .

(1) The above equation was obtained by taking the spin-wave excitations to be plane waves characterized by a three-dimensional propagation vector

q

and neglecting the effects due to boundary conditions at the sample surfaces. In eq. (1) D is the exchange constant,

8

is the angle between q and the applied dc field H,. N, is the demagnetization factor along the applied dc field and M o is the saturation magnetiza- tion which are taken to lie along the z-axis. As Ander- son and Suhl [lo] pointed out, eq. (1) has a shape effect because of the demagnetizing fields. This shape effect, however, is a shift in the resonance applied dc field and does not affect the dispersion. If we consider q as a variable, the spin-wave manifold for the infinite medium is bounded between

In figure l a the mode spectrum for this case is illustrated schematically. The dimensionless quantities are

Q =

014 nyMO,

^{Q H =}

Ho/4 nMO, and we have taken N,

=

0. For an infinite film with thickness S in the x-direction and applied field in the z-direction, a more appropriate graph is obtained if we regard the expression in eq. (1) to be a function of k

=

(k,, k,) and k, separately. Such a graph is illustrated in figure lb.

If k, is quantized according to the simple geome- trical scheme k,

=

nn/s (n

=

0, 1, 2 ...), then the

111'01 \

MAGENETOSTATIC

a

SURFACE BRANCH

( C l

1

^(dl

FIG. 1. - Characteristics of the spin-wave mode spectrum ; (a), as a function of the three dimensional propagation q ; (b), as a function of the propagation constant k = (k,, ke) for conti- nuous values of kz ; (c), as a function of kU for quantized values of kz ; ( d ) Solid lines are the levels discussed in Fig. l c ; the dotted lines are the levels deduced from magnetostatic theory ; (e) Illustrates the noncrossing of the levels discussed in ( d ) ; (f), Solid line with dots corresponds to various quantized values of k , for the magnetostatic surface state ; the straight

faint lines are exchange levels.

series of curves shown in figure l c are obtained for k,

=

0. On the other hand, Damon and Eshbach [9]

obtained from magnetostatic theory the dashed curves shown in figure Id. The dashed line at

(52;

+

corresponds to an infinite number of degenerate volume modes characterized by k,

=

nnlS, for k,

=

0.

If the exchange interaction is added, the degeneracies should be lifted and curves similar to the solid curves should result. The volume modes become the exchange levels. The curve labeled k,

=

ik, is the magnetostatic surface mode and we cannot guess from eq. (1) the correct behavior of this branch.

To treat these problems carefully, we have investi- gated the solutions of the dipole-exchange differential equation [7]. The results of this theory are illustrated schematically in figure le. The eigenstates are admix- tures of the surface and bulk states. The admixture is largest near crossover and they interact and repel each other. The n

= 0

(k,

=

0) exchange level does not exist and the magnetostatic type surface states (segments between the exchange levels) do not have quantized values of k, across the jilm.

Consider the film to have a finite surface length L in the y direction and width Win the z direction. Then the spectrum consists of discrete frequencies. If we assume that the in-plane propagation vector k

=

(k,, k,) is quantized in units n, n/L and n, n/ W, then the fre- quencies can be obtained from the infinite film results.

The discrete frequencies are illustrated schematically in

figure lf for a thick sample. In this case there are a

large number of exchange levels which are shown by light lines. For small values of 1 k 1 the interaction with these bulk levels is small and we do not show the repulsion of levels as the surface branch crosses the exchange levels. These magnetostatic modes can be excited in ferromagnetic resonance and other experi- ments.

In the usual resonance experiment the frequency Q is fixed and the applied field is varied. In figure 2 we

tan +s= ~l<1/2 = {n2/(1/2-

I i

&TG)~-I)+~/~

MAGNETO STATIC

^BULK

STATE DISPERSION SURFACE

FIG. 2. -Dispersion surface according to magnetostatic theory. k , and kz are the propagation constants parallel to the film. The ordinate is the resonant applied dc field for fixed

frequency Q.

show a more complete schematic of magnetostatic volume and surface frequency sheets as a function of both k, and k,. The discrete resonance frequencies are indicated by the dots. The various discrete frequencies may be labeled by (n,, n,) corresponding to

The lowest mode (1, 1) may lie on the magnetostatic volume or surface sheet depending upon the sample length and width and the frequency. If the resonance field

52,

is such that 522 ^> tan cp,, (1, 1) lies on the volume sheet while if 522 ^< tan cp,, it is on the surface sheet [9]. If n, is increased with n,

=

1, the points (n,, 1) follow a trace parallel to the k, axis and all or part of these points lie on the surface sheet. Volume modes are more likely for the set (I, n,).

These types of traces are shown schematically in figure 2. The very sharp distinction between the volume and surface sheet is removed by the exchange inter- action but the transition region is small when I k I is small. According to the expression for tan

cp,

in figure 2 the magnetostatic cutoff approaches

cp, =

n/2 when

52 +

0. The interaction with the exchange sheets has been omitted in figure 2 for simplicity.

In a recent paper [ l l ] Brundle and Freedman reported the experimental observations of magneto- static surface waves

;

similar data was reported by Bongianni et al. 1121 on thin films of Yttrium Iron Garnet (YIG). In their analysis Brundle and Freedman assumed a spatial quantization of (n,, 0). The compa- rison between theory and experiment is illustrated in figure 3. The solid dots are experimental resonant dc fields

;

the open triangles are the calculated points (n,, 0) based on the infinite film spectrum. If one normalizes the calculated points at a high k, value

2 3bo 5 0 4bo 4 : ~ 5b0

%

⁶⁶⁰

MAGNETIC FIELD H Oe

FIG. 3.

-

Experimental and theoretical positions of resonances in the Brundle-Freeman experiment. The solid dots are their experimental points, the open triangle refers to the assignment suggested by these authors, the open circles and the symbol

V are alternate assignments discussed in the text.

where boundary and shape effects are expected to be small, then the agreement is remarkably good. The absolute values of the applied field have been increased by -- 40 Oe to normalize the experimental value of 245 Oe for the (9, 0) level. Such small shifts can easily be accounted for by anisotropy fields. Hence, from the point of view of the resonant fields observed experimentally, such an assignment shows good agree- ment.

The basic problem, however, is that the first state (0,O) at (02 + QH)% is not an allowed value of k, or k, (Investigation of the spectrum of the elliptical cylinder in the limit the major axis a becomes very large illustrates this point.) If we start the assignment with n,

=

1 and keep n,

=

0, then the open circles are obtained which as shown in figure 3 do not give a good fit to the experimental data. In this case a shift of 36 Oe is necessary to match the tenth mode.

The first resonance should correspond to (I, 1)

k,

=

n/ W and k,

=

n/L. NOW since the subsequent

experimental fields reported by Brundle and Freedman

occur at lower dc applied fields, the resonances are

taken to correspond to (n,, 1) n,

=

1, 2 -... The calcu-

lations are represented by the symbol V and as can be

seen improve the agreement between theory and expe-

riment considerably with only small discrepancies for

the low order modes. The shift in the resonant field

is about 24 Oe for this assignment. According to this

scheme the first state (1, 1) observed by Brundle and

Freedman is a bulk state since (k,/k,),

=

2.02 while

tan8,

=

1.86 for

v =

3 G H z o r

SZ =

0.612, where we

have taken 4 nMO

=

1 750 Oe. The next resonant

state (2, 1) has (k,/k.J,

=

1,01 and consequently is a

surface state. All other resonances are also surface

modes. A quantization suggested by Sparks [13] based

upon spin pinning gives similar results except that due

to a numerical error in the calculated position of the

(1, 1) mode, the lowest Brundle and Freedman mode

could not be assigned. A similar quantization can be

carried out for the data reported in ref. [6] on parallel

resonance. A preliminary analysis was reported in an

earlier publication [7b] where we noted that the first

resonance according to the mode spectrum of figure 2

for the field aligned along the length of the sample

GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS O F SPIN-WAVE MODES C 1

-

1093

was a surface mode rather than a bulk state as stated in ref. [6].

In figure 4 we show the calculations for the case in which the dc applied field is along the length of the

t

³⁰⁰⁰ _POINT

cU

3360

-

FIG. 4. - Experimental and theoretical positions of ferroma- gnetic resonances. Points labeled B are for the applied dc field parallel to the width of the sample and the points labeled A are the field parallel to the length of the sample. In case A the theoretical values were shifted by 122 Oe while in case B the shift was 122 Oe in order to normalize to the points labeled

in the figure.

3300

sample, points labeled A. For this case (k,/k,'),

⁼

0.487 whereas tan q,

=

0.82, (Q

⁼

1.92)

;

hence, this mode is a surface state. The higher order modes having increasing values of n, are bulk states. Note that for this experimental condition the main series of observed resonances are at higher fields or lower frequencies rather than at lower fields or higher frequencies as in the Brundle and Freedman c-w experiments. Points denoted by B show the calculation for the dc field along the width of the film in this case (k,/k,),

=

2.05

;

hence, the first resonance is a bulk state. This disper- sion represents a different trace in figure 2 since the value of ky s

=

0.10 rather than 0.205 as in case ( A ) . All of the assignments in this figure are for odd integral values of n,. The abscissa of the figure indicates the values of k , S for these resonant modes. The first point to note is that the agreement between theory and experiment is reasonably good considering the simple quantization scheme used. The deviations occur for the low resonances where the experimental resonant fields are consistently lower for both types of experi- ment discussed here. At least part of these discrepan- cies may be due to a breakdown in the validity of the simple quantization scheme.

In the next section we discuss the spin-wave modes of the eIliptical cylinder as examples in which simple

1 I I

NORMALIZING/ -

- ^{POINT 0} -

m

quantization occurs without the imposition of explicit pinning mechanisms.

A comment needs to be made concerning the boun- dary conditions. The results discussed so far are based on boundary conditions which do not impose spin pinning. If the spins are assumed to be pinned at the film surfaces, then the usual magnetostatic volume and surface modes do not exist [3,

41.

111. General Discussion of Shape Effects. - In figure 5 the mode spectrum for a variety of ideal geometries are examined. This schematic diagram displays the mode spectrum as a function of shape and illustrates the need for further studies including dipolar and exchange interactions. First on the far right of the figure we have schematically illustrated the mode spectrum of an infinite plate magnetized in its plane. The solid line along k y s with a solid dot at the intercept is the magnetostatic surface branch, the dotted line with solid triangles for intercept are the exchange states which we know split the magnetostatic branch into segments. The variation of these branches as a function of k, for k,,

=

0 is also shown.

At the point denoted by the cylinder we have sketched the results or reference [14] for the infinite

A EXCXANGE STATES

BULK STATE

FIG. 5. -Schematic illustration of the mode spectrum for a number of ideal geometries. Note that ^{Q H} depends on the inter- nal field so that for these geometries the applied dc field has

different values to yield the same value of SZH.

cylinder deduced from the magnetostatic theory neglecting exchange forces. The states above (522 + ^QH)%

are surface states with various angular quantum num- bers. The states below (52; + QH)% represent an infi- nite degeneracy in the spectrum neglecting exchange for k,

=

0. In the adjacent portion of the figure on the right we have sketched the results for arbitrary values of b/a and k ,

=

0 for the elliptical cylinder [15]

deduced from magnetostatic theory. This portion of the figure shows how the levels at 52, + 9 for the circular cylinder go over into the infinite plate results.

The formula describing these modes [15] for arbitrary values of b/a is,

where 5 ,

=

3 In [(a + b)/(a

-

b)]. The above formula

reproduces the Damon Eshbach result ifpt, is replaced

FIG. 6.

-

Eigen-frequencies for the elliptical cylinder as a function of the axial ratio bja and the mode number p. The abscissa denoting thc various values of bla is a logarithmic scale while the quan- tized propagation constants kz, (p = 1, 2, ...) are linearly spaced. The solid line on the right hand of the figure is the infinite film result while the dots refer to the discrete modes of the elliptical cylinder

for a specific value of bla.

by

ky

b. Thus for

k, = 0

the discrete-values of

ky

are geometrically quantized and for small values of bla,

ky, N

pla if b is identified with the film thickness.

In figure 6 we illustrate the identification with the infinite film result and show how the integer p which labels the states for finite geometries can be related in a plane wave analysis to the propagation constant

ky.

The shaded lines on the left-hand portion of the figure indicate that this whole block is occupied by levels of larger values of

p.

Since

ky =

pC0/b we note that

ky

is shape dependent, i. e., only for bla

6

1 is

ky

solely determined by the length along its direction.

For bla 5 0.1,

ky

is determined to a good approxima- tion just p/a. We can conclude that the finite geometry gives a spatial quantization when

k, =

0. A study of the spectrum for the ellipse in the limit bla

6

1 but

k, #

0 reveals that this geometrical quantization ceases to be valid. In fact the frequency of the uniform state p

=

1 has a logarithmic functional dependence [15] on

k,

and cannot be reconciled with the infinite film calculations unless a frequency and field dependent quantization which is employed. When bla

^-t

1 these magnetostatic modes of the elliptical cylinder map into the surface states for the circular cylinder at

52,

+ 3 with p representing an angular quantum number. The effect of exchange on the levels of cylin- ders which are degenerate at (SZ; + a,)% in magneto- static theory have not been studied. In order to inves- tigate these states, one must employ the dipole exchange model. Such calculations are currently in progress. It is expected that the infinite degeneracy (for

k, =

0) at (52; + SZ,)% for the cylinder will be lifted by the exchange interaction so that these levels

will map smoothly into those of the plate. In the left- hand portion of figure 5, we have sketched the result for the ellipsoid deduced by Walker 181, neglecting exchange. As can be seen, none of the levels shown have an intercept at (52;

-k

52,)'/'. The evolution of these levels from the circular cylinder to the sphere to the disk need to be determined.

For the oblate spheroid which approaches the disk or the plate magnetized perpendicular to its surface the intercept of the frequencies is at

52,.

The dipole- exchange mode spectrum for the infinite plate magne- tized perpendicular to its surface is known [7b]. It is illustrated in the far left portion of the figure where

k l l

is the magnitude of the propagation constant parallel to the surface. The solid curve is the magneto- static branch and the dotted lines are the exchange levels with solid triangles for intercepts on the fre- quency axis. It would be interesting to study the mode spectrum for the oblate spheroid and see whether a spatial quantization can be deduced as was done for the elliptical cylinder as a

-,

co. A number of facts are still unresolved even for these simple geometries and the dependence of the spectrum on shape is by no means trivial. One final remark as to the excitation of these modes in a uniform r-f field is that for the elliptical cylinder when

k, = 0

only the p

=

1 level has nonvanishing intensity however, if

k, #

0 the odd integer, i. e.,

p = 3,

5 ..+ may couple into a uniform field because of the character of the Mathew functions which are admixed to form the eigenstates [IS].

IV. Discussion and Conclusions.

-

Some conclu-

sions from our studies are listed below.

GEOMETRICAL QUANTIZATION AND SHAPE EFFECTS O F SPIN-WAVE MODES C 1

-

¹⁰⁹⁵

I . Magnetostatic theory qualitatively describes the

behavior of the magnetostatic branch for k

,( lo4

cm-I even though this branch interacts with the exchange branches -provided the spins are not pinned on the large surfaces.

If the r. f. magnetization is pinned the usual magne- tostatic volume and surface branches d o not exist.

2. Comparison of theory and experiment indicates that the inplane propagation vector is approximately quantized geometrically in simple way. The quantiza- tion across the film does not correspond t o an integral number of half wavelengths. Quantization results from the finite size of the sample and does not necessarily imply any particular pinning model. Systematic devia- tions of experiment from theory occur for the lower order modes.

Investigation of the spectrum for the elliptical cylin- der shows that geometrical quantization occurs without explicit pinning mechanisms.

3. The evolution of the magnetostatic and exchange branches as the sample shape is varied is not comple- tely known. An infinite set of modes appear to be missing in figure 5 in the transition from the circular cylinder t o the prolate spheroid.

Since we cannot necessarily relate the strong exci- tation of only odd numbered modes in finite samples to spin pinning, little can be said with certainty regarding the state of the r. f. magnetization a t the sample boundaries. However, in the case of pinning a t the large surfaces of thin films it may be possible to distinguish between the pinned and unpinned condi- tions. As we mentioned in this case the usual magneto- static branches d o not exist. The magnetoexchange branches which occur for this pinned condition have

a dispersion which differs from that of the magneto- static branches [3, 41.

The pinned model predicts a linear dependence, rather than quadratic, of the exchange branch fre- quency on K the in-plane wave vector. Experiments to detect this linear dispersion in thin YIG films are being performed [16]. This linear dependence should not be confused with the Portis effect [I71 in which K = 0 and tire d~fferent exchange branches are separated linearly rather than quadratically.

Finally, we comment on the theory proposed by Sparks [5, 18,

191

which claims t o take into account shape effects. According to Sparks, the proof of the validity of his theory lies in the ((excellent agreement with a large amount of data)). Much of the claimed good agreement is apparently due to numerical errors.

The symbols in figure 2 of ref. [6] indicating, according to Sparks, the calculated positions of the resonances do not agree with his equations for the frequencies [20]. The inadequacy of the theory in accounting for mode intensities has been noted by Weber et al. [21].

The variational procedure described by Sparks [5]

depends upon the assumption of the circular precession approximation which unfortunately cannot be justified either experimentally or theoretically. Since the trial functions used are too crude t o even reproduce the infinite film results, they certainly cannot be employed for investigating the more subtle effects due to shape and finite size.

At this time there exists no adequate theoretical results describing the mode spectrum of finite samples other than those of ellipsoidal shapes. Accurate calcu- lations treating other shapes are necessary in order to better understand the resonance spectra of finite films.

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(M.). TITTMANN (B.

R.) and NEWKIRK

(C.).

- - .

^,,

phys. ~ e t l e r s ,

1968,

2 8 ~ , I

3

1

.

[201 A table of corrected results may be obtained from

B.

Tittmann. No erratum has been published at this time.

[21] WEUER (R.), TANNENWALD

(P. E.)

and BAJOUK (C.

H . ) , Appl. Phys. Letters,

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16,

35.