R 783 Philips Res. Repts 27, 7-27, 1972

CALCULATION OF THE STRAY FIELD OF A MAGNETICBUBBLE, WITH APPLICATION TO SOME BUBBLE

PROBLEMS

'by W. F. DRUYVESTEYN, D. L. A. TJADEN and J. W. F. DORLEIJN

AbstractThe stray field of an isolated bubble in a sheet of a uniaxial magneticmaterial has been calculated. Numerical results are presented in self-contained tables. Knowing this stray field we can determine the inter-action between two bubbles in different sheets, situated one above theother. It is shown that the character of the stray-field distribution leadsto the use of a spacing between the magnetic overlay and the bubblematerial. The repulsive force between two bubbles present in the samesheet has also been calculated. The normal component of the stray fieldof a hexagonallattice of bubbles is found to be much smaller than thenormal component of the stray field of an isolated bubble.

1. IntroduetionMagnetic bubbles are cylindrical domains which can occur in thin sheets of

uniaxial magnetic materials. In fig. 1 we have drawn such a magnetic bubblein a thin single-crystal sheet, situated in a magnetic field Ho. The axis of easymagnetization and the field Ho are both normal to the sheet; we shall definethis normal to the sheet as the a-direction. The static stability of such a singlebubble in an infinite plate 1), and in a finite plate 2) has been treated in litera-ture. The interaction with other bubbles present in the same sheet; or in othersheets, was not considered in refs 1 and 2. The interaction between bubblesarranged in a hexagonallattice in a single sheet has been calculated 3.4) andexperimentally verified 5). Itwas found that for a fixedvalue ofthe bias fieldHothe radius of a bubble in a lattice is smaller than the radius of an isolatedbubble, because in the sheet the a-component of the stray field of a bubble hasthe same direction as Ho. The presence of the surrounding bubbles increasesthe effective bias field. In the region outside the sheet just above a bubble, thez-component of the stray field is antiparallel to Ho. A bubble located in another

z

Fig. 1. Magnetic bubble in a thin sheet.\

8 W. F. DRUYVESTEYN. D. L. A. TJADEN and J. ·W. F. DORLEIJN

sheet directly above the given bubble is subjected to a total field which is lowerthan Ho.

The radii of these two bubbles can be found from the expressions for theenergy of the system formed by two plates. Differentiation of the energy withrespect to the coordinates of the bubbles gives the force on the wall of thebubbles. For a stable situation these forces should vanish. Alternatively onecan calculate the stray field of one bubble at the wall of the other bubble. Insec. 2.1 expressions are given for this stray field; in secs 2.2 and 2.3 the z-oom-ponents of the stray field of a hexagonal lattice of bubbles and of a cluster ofbubbles is given. The results of sec. 2.1 will be used in sec. 3. .

In sec. 3 the radii of two bubbles in line above each other are calculated.From the analysis of the stray field of a bubble it is found that in a bubbledevice, where a magnetic overlay is present for the transport of the bubble(T-bar circuit), one should use a spacing between the overlay and the bubblematerial (sec 3.1). In sec. 3.2 the repulsive force between two bubbles in thesame sheet is calculated and compared with the results for this repulsion whena dipole approximation is used.

2. Stray fields

2.1. Stray field of an isolated bubble

With respect to the scalar potential and the field outside the sheet the bubblecan be taken as a homogeneously magnetized cylinder of magnetization 2Ms.The potentialof a flat.circular disc of radius R and a constant charge densityhas been given by Bateman 6). From this we obtain for the potentialof thebubble stray field for z > ° (see fig. 1):

00 dkeper,z) = 411: Ms R J J1(kR) Jo(kr) e-kz (I - e-kt) -, (1)

o kwhere t is the thickness of the sheet, R the radius of the bubble and M, the(saturation) magnetization. The origin of the coordinate system (z = 0, r = 0)is taken at the upper surface of the sheet in the centre of the bubble. The fieldcomponents follow from eq. (1):

"bep - 00

Hr =- - = -4 11:M, R J J1(kR) J1(kr) e-kz (1- e-kt) dk, (2)"br. 0"bep 00

Hz -'- - - = -4 11:M, R J J1(kR) Jo(kr) e-kz (1- e-kt) dk. (3). . öz 0 .

According to Eason et al. 7) integrals of this type can be expressed in terms ofcomplete elliptic integrals of the first, second and third kind (Hr containselliptic integrals of the first and second kind, while in the expression for Hzintegrals of the third kind also occur). The expression for Hr in terms of elliptic

- ,

CALCULATION OF THE STRAY FIELD OF A MAGNETIC BUBBLE 9

integrals of the first and second kind was recently given by Goldstein andCopeland 8). Using the ALGOL procedures for !he elliptic integrals, aspublished by Bulirsch 9), the expressions for the external bubble field can beevaluated numerically. In appendix A we give expressions for the integralsoccurring in eqs (1)-(3) in terms of the function used by Bulirsch 9). Inappendix B some numerical results for the bubble stray field are given; theresults are presented in self-contained tables.

2.1.1. Comparison with dipole approximation

It seems relevant to compare the exact results for Hrl4nMs and Hxl4nMswith results obtained when a simple dipole approximation is used. For a dipole'of strength 2 M; n R2 t located at r = 0, z = -t12 the field components aregiven by

3 t R2 r (z + t12)2 [r2 + (z + t12)2 ]5/2

t R2 [2 (z + t12)2 - r2]

(4)

-----=------------------4 n Ms 2 [r2 + (z + t12)2 ]5/2

The same coordinate system as shown in fig. lis used. Obviously, eqs (4) and(5) are good approximations for large values of rl R and zl R. In table I wehave compared some results obtained from eqs (2) to (5)..

(5)

TABLE I

Comparison between bubble stray field and field of a dipole with strength2M; in: R2 t; Rit = 1

rit zit Hrl4nMs Hrl4nMs Hxl4nMs Hxl4nMseq. (2) eq. (4) (dipole) eq. (3) eq. (5) (dipole)

10 4 10 4 -0.323.10 4 -0.240.10 2 -0,707.10 0 -0'800 . 10+10·1 10-4 -0,326.10-1 -0'217.10+1 -0,708. 10-0 -0,710 . 10+11 10-4 -0·283 . 10+1 -0·429 . 10-0 ;-0·321. 10-0 +°.143.10-010 10-4 -0,755 . 10-4 -0,745. 10-4 +0·498. 10- 3 +0·494. 10- 310-4 0·1 -0·340. 10-4 -0,116.10-2 -0·640. 10-0 -0·463 . 10+10·1 0·1 -0,343.10-1 -0,108.10+1 -0'641.10-0 -0,426 . 10+11 0·1 -0.652.10-0 -0,417.10-0 -0·266. 10-0 +0·649. 10-110 0·1 -0'903 . 10-4 -0,892 . 10-4 +°.496.10-3 +0·492. 10- 310-4 1 -0,132.10-4 -0·296. 10-4 -0,187.10-0 ":::'0·296.10-00·1 1 -0,132.10-1 -0·293 . 10-1 -0,186.10-0 -0,292.10-01 1 -0,892 . 10-1 -0,118.10-0 -0'960 . 10-1 -0,919.10-110 . 1 -0,215.10-3 -0·213 . 10-3 +0·454. JO-3 +0·452. 10- 3

10 W. F. DRUYVESTEYN. D. L. A. TJADEN and J. W. F. DORLEIJN

2.2. The stray field .of a bubble lattice

Consider a rectangular bubble lattice as sketched in fig. 2. The nearest-neighbour distance in the x-direction is taken as D, in the y-direction as pD. 'A hexagonal lattice of bubbles corresponds to p = V3, a square lattice top = 1. In a way similar to that followed' in refs 3 and 4, the potential can beexpressed' as a double Fourier series, 'the coefficients of which are given inref. 4. The stray field then follows from the gradient of the potential. For thez-component of the stray field we find:

co co

Hz b'I k [2n( n y)]--=-' cos - mx+- X4 n M, ' P (m2 + n21p2)1/2 D ' P

-co -com nm+n=even

(6)

where k = 2 RID, kmn = (2nID) (m2 + n2Ip2)1/2, and z = ° corresponds tothe upper surface of the sheet. The x and y coordinates are indicated in fig. 2,

Fig. 2. Bubble lattice; the shaded regions are the bubbles.

TABLE 11

The z-component of the stray field of a hexagonal bubble lattice, Hzl4nMs'P,= V3; k = 0·4= 2RID; Dit = 12;Rit = 2,4; IX = 0°; r = x

zlt=» 1 3 . 6 12-rit

° -0·2375 ' , -0,618.10-1 -0 ·85.10-2 -0,21.10-3

1 -0·2266 -0,543.10-1 -0,76.10-2 -0,19.10-3

2 -0·1461 -0·345 . 10~1 -0,54.10-2 -0,14.10-3

3 +0'32 .10-2 -0,115.10-1 -0,25.10-2 -0,70.10-4

4 +0·399. 10-1 +0'50 .10-2 +°.21.10-3 +0·4 .10-6

5 +0'392'. 10-1 +°.131.10-1 +°.20.10-2 +°.52.10-4

6 +°.377.10-1 +°.153.10-1 +°.27.10-2 +°.71.10-4(

CALCULATION OF THE STRAY FIELD OF A MAGNETIC BUBBLE '11

TABLE III

The angular dependence of the stray field of a hexagonal bubble lattice. p = V3;k = 0,27; Dit = 12; Rit = 1,62; zit = 3

,

rit = 6, r = DI2 rit = 12, ~= D,

ct CO) Hz/4nMs ct CO) Hz/4nMs

0 0.75.10-2 0 -0·402. 10-115 0.77.10-2 15 -0·28 .10-230 0.79.10-2 30 0·78 .10-245 0.77.10-2 45 -0·28 .10-260 0.75.10-2 60 -0·402 . 10-1

,

r = (x2 + y2)1/2 and ct = arctan Cylx). In tables II and III we have' givensome results for p = V3; in table II, Hzl4nMs is given for ct = 0° and Rit =2·4. Table III gives the angular dependence for the case Rit = 1'62, Dit = 12(k = 0·27) and zit = 3 at rit = 6 and rit = 12, which corresponds to r = tDand r = D respectively.It seems relevant to compare the results for the bubble lattice with the results

for the isolated bubble. In some devices where different sheets are placed aboveeach other a low stray field is desirable. Depending upon the information storedthe stray field can be caused by one bubble, a cluster of bubbles or in anotherextreme case by a lattice of bubbles. The numerical results in appendix B showthat for a given value of z the z-component of the stray field of a bubble is forsmall r-values negative but for larger values positive. For a bubble lattice thepositive and negative values thus cancel out to some extent. If we take as anexample Rit = 0,2, rlR = 10-4, zjR = 5, then Hzl4nMs = -0·0145 for anisolated bubble (table IV-3), whereas for a bubble lattice with Dit = 1 wefind Hzl4nMs = -0,00047. As we shall see in sec. 2.3 the maximum stray fieldof a cluster of bubbles can be larger than that of an isolated bubble. If theinformation stored corresponds to a cluster of bubbles the "cross-talk" be-tween the two platelets will thus show a maximum.

2.3. The stray field of a cluster of bubbles

If we approximate the bubble field as a dipole field

!Iz t R2 [2 (z + t12)2 - r2]-- =- --------4 n Ms 2 [r2 + (z + t12)2]5/2

the change of sign 'of Hz occurs at z + tl2 = r!V2. This means that at a fixedvalue of z all bubbles within a circle with radius r = (z + t12) V2 give a nega-

12 W. F. DRUYVESTEYN, D. L. A. TJADEN and J. W. F. DORLEIJN

tive contribution to the local stray field Hz, while the bubbles outside this circlegive a positive contribution. The fact that a cluster of bubbles gives a largestray field will be illustrated in the following calculation. Consider áhexagonalbubble lattice with Rlt = 0,2, D]t = 1, k = 0,4, z/R = 5, r/R = 10-\ oe_.:.0°.Using the results presented in table IV-3 we found for all bubbles located withina radius r = 3t!V2 = 2·1D (1 bubble in the centre, 6 for distance D, DV3and 2D respectively) a total stray field Hz/4nMs = -0·038. For a bubble lat- 'tice we found Hz/4nMs = -0·00047 and for an isolated bubble -0·0145. Thestray field of such a cluster is thus 2·5 times that of one bubble.

(

3. The interaction between two bubbles situated one directly above the other

Knowing the stray field outside the platelet of a bubble the interaction be-tween two bubbles located in two sheets in line above each other can be cal-culated. Let us consider sheets of different materials and thicknesses (see fig. 3).V!e shall use the subscript 1 and 2 to indicate the different sheets, and the fol-lowing abbreviations:tI and t2 the thicknesses of the sheets,d the separation between the sheets,RI and R2 the radii of the bubbles,MI and M2 the (saturation) magnetizations,11 and 12 the materiallengths, defined as aw/4nM/,where aw is the wall surface energy per unit surface,

hl = Ho/4nMl and h2 = Ho/4nM2•

The radii of the bubbles are calculated from the condition oE/oR = 0, whichsimply means that the force on the wall of the bubble should vanish. Let usconsider first the case of an isolated bubble in one sheet (d -+ (0). The forceon the wall of bubble 1, divided by 8 n2M/t12 can be written:

where, Fl (00) denotes the (normalized) force for d -+ 00. The expression forOED/oRl in terms of elliptic integrals of the first and second kind is given byThiele 1). For the case of a finite separation d, the force Fl(d) can be written:

2tM21 + I t t22R, d..- z=o

1 tM,1 + I t t,

Fig. 3. Configuration with one bubble directly above another.

CALCULATION OF THE STRAY FIELD OF A MAGNETIC BUBBLE 13

where (h12> 4 nMI is the z-component of the stray field of bubble 2 averagedover the wall of bubble 1:

(9)

For bubble 2 a similar equation holds:2R2

F2(d) = Fioo) + - (h21>.t2

(10)

Another way of obtaining the force equations (8)-(10) is of course by firstwriting down the total energy *) of the system of two plates:

E= 2nRl aWl tI + 2nR2 aW2 t2 + 2HoMl nR/ tI + 2Ho M2 nR/ t2 +Rl

ED! + ED2-16n2 MI M2 R2 J rdr Xo

00 dkX f ll(kR2) lo(kr) [e-kd_ e-k(t2+d) - e-k(d+tl) + e-k(t2+d+tl)] k' (11)

o

where ED1 and ED2 are the demagnetizing energies ofisolated bubbles in a singlesheet 1) (d -+ (0) at radii RI and R2 and thicknesses tI and t2 respectively. Thelast term in eq. (11) is found by integrating the magnetic-charge density 2M1times the potentialof the stray field of bubble 2 over the upper and lowersurfaces of bubble 1. Differentiation of eq. (11) with respect to RI or R2gives the expressions for the forces F(d) of eqs (8)-(10). The stability

*) A symmetrical expression in Rl and R2 is obtained when the integration to rin eq. (11)is, performed. UsingXl Xlf Jo(lex) X dx = _ Jl(leXl) we find:o leE=2nRll1wl/l + 2nR2 CTw2 12+ 2Ho(nR12 11Mi +nR22 12M2) +

co+ EDl + ED2 _ 16 n2u,M2 Rl R2 f Jl(leR2) Jl(kRl) X

odk

X [e-kd_ e-k(t2+d)_ e-k(d+tl) + e-k(t2+d+t1)] _.

k2

14 W. F. DRUYVESTEYN. D. L. A. TJADEN and J. W. F. DORLEIJN.

of the bubbles is determined 'by the second derivatives of the energyrespect to R, and R2:

"ö2E-->0"öR/

with

.and (12)

The expressions for the second derivatives can easily be found. The numericalcalculation goes in the same way as the calculation of F(d). Stability againstelliptic deformations will not be discussed. For the two extreme cases of zerospacing (d = 0) of identical materials and infinite spacing (d -- (0) we canuse the results of Thiele 1). If the material quantities 1and M and the relevantdimensions tand d are given Rl and R2 can be calculated as a function of Hoby equating the right-hand sides of eqs (8) and (10) with zero. Knowing Rland R2' we can check the stability using eq. (12). Results for identical sheetsfor different values of dft are presented in figs 4 and 5. For identical sheetsRl = R2' so eqs (8) and (10) are also identical. The curves for -d = 0 andd __ 00 in figs 4 and 5 are reproduced from the results of ref. 1; the largestvalue of R corresponds to the run-out radius, the smallest value of R to thecollapse radius. For the intermediate values of d we have assumed that the run-out radius is somewhere between the two values for the two extreme cases. Ifsheets of different materials or sheets of different thicknesses are placed ontop of the other then R, =1= R2• If 12< 11 or M2 > Ml or t2 > tl (d = 0)then R2 > Rl' An example is given in fig. 6 where R, and R2 are plotted asfunctions of Ho/4nMl for different values of 12ft2• It can be seen that thedifference in radii, R2 ":_Rl' is small. This is related to the fact that the z-com-ponent of the stray field of a bubble is concentrated in the region r ~ R, andfalls off rapidly for r > R (see table IV).An interesting situation will occur if the differences between the two sheets

are large. Consider as an example two sheets of the same material but of dif-ferent thicknesses, t2ftl ...:....0·1. Under certain conditions, which will be speci-fied below, a bubble in plate 1 can have two different radii at the same magneticfield. One stable situation occurs when one bubble is present in plate 1 and nobubble in plate 2. Another situation with a bubble in plate 1 as well as in plate 2can also occur; the radius for the latter case can be 20% larger than for theformer case. This bistable state of the bubble 1 in plate 1 can .occur if we taketwo sheets of the same material with llftl = 0·4 and 12ft2 = 4, d = 0·1 tland Ho = 0·18x4nMl• For a single sheet with llt = 4 the bubble collapsefield is much smaller than 0·18 X 4nM 1 (Ho cOllapsef4nM1< 10- 3), while for1ft == 0-4 the collapse field is about 0·21 x4nM. A situation with a bubble inplate 2 and no bubble in plate 1 is thus not possible. If there is no bubble inplate 2 a bubble with Rlftl ~ 1·5 is stable in plate 1. From eqs (8)-(10) and(12) we found a situation with Rlftl ~ 1·84 and R2ftl ~ 1·8 to be a stableconfiguration.

CALCULATION OF THE STRAY FIELD OF A MAGNETIC BUBBLE 15

0·06'-----'----'--L.._--'---'-----'_---L;__-'----' _ __J

0-70 0·72 MI, 0·76 0·78 0·80 0-82 0·81, 0·86 0·88 0·90-Ho/l,rrM,

Fig. 4. Bubble radius normalized to the sheet thickness in identical sheets as a function ofmagnetic field; /l/tl = 0·01. The curves for the two extreme cases, d = 0 and d-+ 00 cor-respond to stable bubbles (Rcollapse < R < Rrun out).

0·11,

0'10

d parametert, ",t2 ",1I, ",12 =0·1,M, =M2

0'28

Fig. 5. Bubble radius normalized to the sheet thickness in identical sheets as a function ofmagnetic field; /l/tl = 0'4. The curves for the two extreme cases, d = 0 and d -+ 00 cor-respond to stable bubbles (Rcollapsc < R < Rrun out).

3.1. Spacing between magnetic overlay and bubble material

In a magnetic bubble device one often uses a magnetic overlay, such as aT-bar or Y-bar structure, to move the bubbles around 10.11). It is evident thata strong attractive force between the magnetic overIay (PermalIoy) and thebubble is very desirable. It has been found 11.12) that the overlay should notbe in direct contact with the bubble material, but that a small spacing in theorder of 0·05X 2R to 0·25X 2R 11.12) should be used. This result follows easilyfrom an analysis of the bubble stray field. Consider a simple modelwith aPermalloy structure magnetized up to saturation by an in plane magnetic field,

16 W. F. DRUYVESTEYN, D. L. A. TJADEN and J. W. F. DORLEIJN

ID

1·6

.RI+R2'2 parameter'1=0'4MI =M2tI =t2 =1d =0·1

3'2

L0·8

0·16 0·24 0·32 0·40 0·48-Ho/47rMI

Fig. 6. Radii of the bubbles for different sheets; dJtl = 0'1; 12 is used as a parameter.

resulting in a magnetic charge density at the Permalloy structure. The attractiveforce between the Permalloy and the bubble will then be proportional to ther-component of the bubble stray field. The results in table IV show that for afixed value of rfR, Hr has a maximum value for z =1= O. If we take as an examplerfR = 2·5 and R = t then Hr has a maximum for zf2R Ri 0·3.As far as the propagation velocity of the bubble is concerned we can deter-

mine the optimum spacing, z l' between the bubble material and the Permalloyas follows. Suppose we have a magnetic pole Q at a distance 1", z from thebubble. The force in radial direction, upon the bubble is F = Hr (1", z) Q.When the motion of the bubble through the platelet, under the influence of F,is assumed to be viscous, then the velocity V of the bubble is proportional to F.That means that the time for the bubble in a position (1', z) from a magneticpole to reach that pole,

r dr'TocJ-.hr(r',z)

o

Now it is clear that this integral will diverge at 1" = O. We therefore per-formed, the integration from 0·05 Runtil r. We found that the shortest timewill occur if there is some spacing between the Permalloy and the bubblematerial. If we take as an example rfR ~ 2·5 with R = t then the shortesttime is found if a spacing of 0·12X 2R is used. Without spacing this time isabout 11% larger. This example shows' clearly that the character of the stray-field distribution of a bubble makes the use of a spacing between the magneticoverlay and the bubble material advantageous.

CALCULATION OF THE STRAY FIELD OF A MAGNETIC BU1lBLE 17

3.2. The repulsive force between two bubbles

A method for obtaining the coercive force, He, of a bubble material usingthe repulsive force between two bubbles has been reported 10.13). In this ex-periment the minimum stable separation between two bubbles is measured. 'From a balance of forces between the wall pinning force, expressed in terms ofthe coercivity He, and the repelling force due to the stray fields of the twobubbles, the quantity He/4nMs can be determined. In ref. 10 the stray fieldof a bubble is approximated by a dipole field. In that case one obtains forthe driving force

3 n t2 R4F1 =4nM/----

1124(13)

where 112 is the separation between the centres of the bubbles. The wall coer-civity follows from FI using

4nM, 32 in: M/ R t

For the dipole approximation one thus obtains:

He 3 n t R3

---=----- (14)

---=--- (15)

It is clear that the dipole approximation is only correct when 112 is large.It seems therefore relevant to compare the result obtained in eq. (15) with amore exact expression using the stray fields as given in eqs (2) and (3). Forthe driving force we then obtain:

~ 00 dkFI = 32 n M/R2 f cos ct d« f JI(kR) Jo(kr)(l- e-kt) k' (16)

o 0

We have calculated eq. (16) using a numerical-integration procedure for theintegration with respect to ct. Using eqs (14) and (16) the value of He/4nMscan be obtained. In fig. 7 we have plotted Hc/4nMs as a function of IdR withR = t for the case that the exact expression for the stray field is used (eq. (16),drawn curve in fig. 7) and for the case eq. (15) is used (dashed curve). Figure 7indicates for which values of He/4nMs the dipole approximation is a useful.approximation.

4. Conclusions

Exact solutions have been presented for the bubble stray field. Due to thisstray field there is a coupling between bubble domains in different sheets. A

18 W. F. DRUYVESTEYN. D. L. A. TJADEN and J. W. F. DORLEIJN

2.10-3

""""""" R /, • ,

, '--bubble

""""~~~---dipole

-,

"" "',

"'"

10-3'-11-1-----'1-----+----'-"...0..""'-15 6-1/2>R

Fig. 7. The coercivity as ~ function of the minimum stable separation between the centresof two bubbles. The drawn curve refers to results obtained from the exact values of the strayfield, the dashed curve follows from a dipole approximation.

3

configuration of two domains in line above each other has been calculatedrigorously. A consideration of the stray fields of isolated domains, clusters ofbubbles and bubble lattices, gives results useful for studies on the cross-talkin certain bubble devices. In a worst-case consideration a cluster of bubblesis shown to have twice as large a stray field as an isolated bubble. From thecharacter of the stray field of a domain it is indicated that a spacing betweenbubble material and T-bar overlay structures may be desired in devices. Acomparison is made between the exact stray field of a bubble and the field ofa dipole. The range of validity of a dipole approximation depends on the ratioof bubble radius to sheet thickness. The results of a dipole approximation inthe calculation of the coercitive field are compared with an exact calculation.At low values of He, which corresponds to a large separation between thedomains in the sheet, the agreement is fairly good.

Appendix A

We shall use the notation

Eindhoven, October 1971

, "J12a cos? cp + b sin" cp dcpeel (k',p, a, b) = "cosê cp + p sin? cp (cos" cp + k'? sin" cp)1/2

, 0

due to Bulirsch, who published a fast Algol procedure for it 9). Then we find:

eo dk 2 1

J Jl(ak) Jo(bk) e-ek - = - [cel (k', 1, a + b, a- b) +k n (é + b2 + C2)1/2

o 'c2 C+ eel (k', q, a+ b, a- b)] - - [1 + sign (a- b)],

(a + b)2 2a '

CALCULATION OF THE STRAY FIELD OF A MAGNETIC BUBBLE 19

00 2 1'J J1(ak) J1(bk) e-ck dk = - cel (k', 1,-1,1),o n[(a+bY+c2]1/2.00 2 c'JJ (ak) J. (bk) e-ck dk = - - X

o 1 0 n Ca+ b)2 [(a + b)2 + c2]1/2

1+ sign(a- b)X cel (k', q, a+ b, a- b) + ,

2awhere

c >0, k'2 = (a- b)2 + c2 q2 = (a- b)2

(a + b)2 + c2' a + b

and

sign (x) = ~~( [x]

for x = °for x =1= 0.

Appendix B

Making use of the equations of appendix A we have evaluated numericallythe expressions for the bubble field outside the sheet. The results are repre-sented in tables IV-1-7. Each table refers to a particular value of the ratio Rit,which is the radius of the bubble divided by the thickness of the sheet. In thehorizontal direction is given the value rjR, which is the radius vector (incylindrical coordinates) normalized to the bubble radius. In the vertical direc-tion is given the value zl R, which is the z-vector (in cylindrical coordinates)normalized to the bubble radius. The point z = 0, r = ° coincides with thecentre of the domain at the upper surface of the sheet. The values of the fieldcomponents in the r- and z-directions, normalized to 4nMs are indicated byhr and hz respectively.

TABLE IV-l

, I R/t· .05 I~ r/R 0 .1 .2 .3 .5 .7 1 2 5 10

z/R0 ~: o· -.502(-1) -.102( 0) -.155( 0) -.278( 0) -.446( 0) - co -.139( 0) -.200(-1) -.457(-2)hz: -.999( 0) -.999( 0) -.999( 0) -.999( 0) -.999( 0) -.999( 0) -.499( 0) .123(-2) .114(-2) .894(-3).1 o -.494(-1) -.999(-1) -.153( 0) -.272( 0) -.428( 0) -.761( 0) -.138( 0) -.200(-1) -.458(-2)

-.899( 0) -.899( 0) -.896( O} -.892~ 0) -.876( 0) -.834( 0) -.429( 0) -.735(-2) .711(-3) .837(-3).2 o -.473(-1) -.954(-1) -.145( 0) -.254( 0) -.384( 0) -.546( 0) -.136( 0) -.200(-1) -.458(-2)

-.803( 0) -.801( 0) -.797( 0) -.789( 0) -.760( 0) -.693( 0) -.382( 0) -.156(-1) .284(-3) .780(-3)

.3o ·-.440(-1) -.886(-1~ -.134( 0) -.230( 0) -.332( 0) -.424( 0) -.132( 0) -.199(-1) -.458(-2)

-.711( 0) -.710( 0) -.704( 0 -.694( 0) -.657( 0) -.582( 0) -.343( 0) -.232(-1) -.140(-3) .724(-3).5 0 -.358(-1) -.715(-1) -.107( 0) -.177( 0) -.239( 0) -.282( 0) -.121( 0) -.197(-1) -.458(-2)

-.552( 0) -.549( 0) -.543( 0) -.532( 0) -.493( 0) -.427( 0) -.280( O} -.358(-1) -.971(-3) .611(-3)

.7o -.274(-1) -.546(-1) -.812(-1) -.131( 0) -.m( 0) ~.199( 0) -.loB( 0) -.194(-1) -.457(-2)

-.425{ 0) -.423( 0) -.418( 0) -.4oB( 0) -.376( 0) -.327(0) -.232( 0) -.444(-1) -.177(-2) .500(-3)1.0 o -.176(-1) -.349(-1) -.517(-1) -.820(-1) -.106( 0) -.125( 0) -.884(-1) -.188(-1) -.455(-2)

-.292( 0) -.29Q( 0) -.286( 0) -.280( 0) -.259( 0) -.230( 0) -.178( 0) -.507(-1} ~.289(-2) .336(-3)2.0 o -.446(-2) -.885(-2) -.131(-1) -.211{-1) -.280(-1) -.359(-1) -.409(-1) -.159(-1) -.437(-2)

-.105( 0) -.104( 0) -.103( 0) -.102( 0) -.981(-1) -.924(-1) -.817(-1) -.440(-1) -.564(-2) -.174(-3)

5.0 o -.374(-3~ -.746(-3) -.112(-2) -.184(-2) -.255(-2) -.354(-2) -.607(-2) -.684(-2) -.332(-2)-.186(-1) -.186(-1 -.186(-1) -.185(-1) -.184(-1) -.181(-1) -.176(-1) -.149(-1) -.634(-2) -.116(-2)

10.0 o -.474(-4) -.948(-4) -.142(-3) -.236(-3) -.329~-3) -.467(-3) -.893(-3) -.168(-2) -.160(-2)-.441(-2) -.441(-2) -.440(-2) -.440(-2) -.439(-2) -.437 -2) -.434(-2) -.413(-2) -.303(-2) -.130(-2)

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TABLE IV-2

1 R/t· .1 I~ r/R 0 .1 .2 .3 .5 .7 1 2 5 10

z/R0 hr: 0 -.501(-1) -.101( 0) -.155( 0) -.278( O~ -.445( 0) - co -.138( 0) -.185(-1) -.326(-2)

hz: -.995( 0) -.~5( 0) -.995( 0) -.995( 0) -.995( 0 -.995( 0) -.495( 0) .468(-2) .357(-2) .177(-2).1 o -.494(-1) -.998(-1) -.153(·0) -.271(0) -.428(0) -.760(0) -.137(0) -.186(-1) -.328(-2)

-.8g6( 0) -.895( 0) -.~93( 0) -.88B( 0) -.872( 0) -.831( 0) -.425( 0) -.397(-2) .310(-2) .171(-2) i

.2 o -.472(-1) -.953(-1) -.145( 0) -.254( 0) -.384( 0) -.545( 0) -.135( 0) -.186(~1) -.331(-2)i-.799( 0) -.798( 0) -.793( 0) -.786( 0) -.756( 0) -.69Q( 0) -.378( 0) -.123(-1) .263(-2) .165(-2)

.3 o -.440(-1) -.885(-1) -.134( 0) -.230( 0) -.332~ 0) -.424( 0) -.131( 0) -.185(-1) -.333(-2)-.708( 0) -.706( 0) -.701( 0) -.691( 0) -.653( 0) -.578 0) -.339( 0) -.200(-1) .217(-2) .159(-2)

.5 o -.357(-1) -.715(-1) -.107( 0) -.177( 0) -.239( 0) -.281( 0) -.121( 0) -:184(-1) ~.336{-2)-.548( 0) -.546( 0) -.54o( 0) -.528( 0) -.489( 0) -.423( 0) -.277( 0) -.327(-1) .127(-2) .147(-2)

.7 o -.274(-1) -.545~-1} -.811~-1} -.130( O} -.171( O} -.198( O} -.loB( 0) -.182(-1) -.339(-2)'-.422( O} -.420( 0) -.414 0) -.405 0) -.372( 0) -.324( 0) -.229( 0) -.415(-1) .397(-3) .135(-2)1.0 o -.176(-1) -.349(-1) -.516(-1) -.818(-1) -.106( 0) -.125( 0) -.878(-1) -.177(-1) -.343(-2)-.289( 0) -.287( 0) -.284( 0) -.277( 0) -.256( 0) -.227( 0) -.175( 0) -.479(-1) -.821(-3) .118(-2)-2.0 o -.443(-2) -.881(-2) -.13;(-1) -.210(-1) -.279(-1) -.357(-1) -.404(-1) -.149(-1) -.342(-2) -

-.102( 0) -.102( 0) -.101( 0) -.997(-1) -.957(-1) -.900(-1) -.793(-1) -.417(-1) -.388(-2) .622(-3)5.0 o -.362(-3~ -.723(-3) -.108(-2} -.179~-2) -.247~-2~ -.343(-2) -.585(-2) -.636~-2) -.273~-2)-'.172(-1) -.172(-1 -.172(-1) -.'-71(-1)-.169-1) -.167-1 -.162(-1) -.136(-1) -.520-2) -:.520-3)

10.0 o -.430~~) -.860~~) -.129(-3) -.214(-3~ -.299~-3~ -.423(-3) -.807(-3) -.149(-2) -.132(-2)-.372(-2) -.371 -2) -.371 -2) -.371(-2) -.370(-2 -.368 -2 -.365(-2) -.345(-2) -.243(-2) -.876(-3)

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TABLE IV-3

I R/t = .2 I~r/R 0 .1 .2 .3 .5 .7 1 2 5 10

z/R0 hr: 0' -.4ga~-1) -.101( 0) -.154( 0) -.276( 0) -.443( 0) _00 -.133( 0) -.133(-1) -.144(-2)

hz: -.gal( 0) -.981 0) -.981( 0) -.981( 0) -.981( 0) -.981( 0) -.482( 0) .157(-1) .710(-2) .180(-2).1

o -.491(-1) -.992(-1) -.152( 0) -.270( 0) -.425( 0) -.757( 0) -.132('0) -.135(-1) -.148(-2).-.882(0) -.881( 0) -.879( 0) -.875( 0) -.858( 0) -.817( 0) -.413( 0) .669(-2) '.660(-2) .176(~2)

.2 o -.469(-1) -.947(-1) -.144( 0) -.253( 0) -.382( 0) -.542( 0) -.130( 0) -.137(-1) -.152(-2)-.786( 0) -.785( 0) -.780( 0) -.773( 0) -.743( 0) -.677( 0) -.366( 0) -.201(-2) .611(-2) .173(-2)

.3 o -~437(-1) -.879(-1) -.133( 0) -.229( 0) -.330( 0) -.421( 0) -.127( 0) -.138~-1) -.156(-2)-.695( 0) -.693( 0) -.688( 0) -.678( 0) -.641( 0) -.566( 0) -.327( 0) -.101(-1) .562 -2) .169(-2)

.5 o -.355(-1) -.710~-1) -.106( 0) -.176( 0) -.237( 0) -.279( 0) -.117('0) -.,40(-1) -.164(-2)-.537( 0) -.535( 0) -.528 0) -.517( 0) -.478( 0) -.412( 0) -.266( 0) -.235(-1) .465(-2) .161(-2)

.7 o -.272(-1) -.541~-1) -.804(-1) -.129( 0) -.169( 0) -.196( 0) -.104( 0) -.140(-1) -.171(-2)-.411( 0) -.410( 0) -.404 0) -.394( 0) -.362( 0) -.313( 0) -.219( 0) -.329(-1) .370(-2) .153(-2)

1.0 o -.174~-1) -.345(-1) -.510(-1) -.809(-1) -.104( 0) -.123( 0) -.847(-1) -.139(-1) -.179(-2)-.279( 0) -.278 0) -.274( 0) -.267( 0) -.247( 0) -.218( 0) -.166( 0) -.402(-1) .237(-2) .140(-2)2.0 .0 -.432(-2) -.858(-2) -.127~-1) -.204(-1) -.271(-1) -.346(-1) -.385(-1) -.122(-1) -.198(-2)

-.955(-1) -.953(-1) -.945(-1) -.932 -1) -.892(-1) -.835(-1) -.729(-1) -.360(-1) -.111(-2) .978(-3)5.0 o -.328~-3~ -.654(-3) -.978(-3) -.161(-2) -.223(-2) -.309(-2) -.521(-2) -.522~-2) -.181(-2)

-.145(-1) -.144-1 -.144(-1) -.144(-1) -.142(-1) -.140(-1) -.135(-1) -.",(-1) -.353-2) -.302(-4)10.0 o -.345(-4) -.690(-4) -.103(-3) -.172(-3) -.240(-3) -.339(-3) -.643(-3) -.114(-2) -.912(-3)

-.275(-2) -.275(-2) -.275(-2) -.274(-2) -.273(-2) -.27:2(-2) -.269(-2) -.253(-2) -.167(-2) -.490(-3)

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TABLE IV-4

1 .R/t - .5 II~ r/R 0 .1 .2 .3 .5 .7 1 2 5 10'z/R0 hz.: o· -.4.57(-1~ -.927~-1~ -.14.2(0) -.257~ 0) -.4.18(0) - ex> -.980~-1) -.424(-2~ -.291(-3~

hZ: -.894.(0) -.895( 0 -.895 0 -.897( 0) -.901 0) -.907( 0) -.4.17(0) .450 -1) .660(-2 .952(-3.1 o -.4.55(-1~ -.921(-1) -.141( 0) -.253( 0) -.4.03(0) -.728( 0) -.1OO( 0) -.457(-2) -.318(-3)-.803( 0) -.803( 0 -.801( 0) -.798( 0) -.785( 0) -.749( 0) -.353( 0) .350(-1) .636(-2) .94.4(-3).2 o -.437(-1) -.884(-1) -.135(0) -.238(0) -.362(0) -.516(0) -.101(0~ -.489~-2) -.345(-3)-.714( 0) -.71·3(0) -.709( 0) -.703( 0) -.676( 0) -.614( 0) -.310( 0) .253(-1 .610 -2) .•934(-3).3 o -.4{)9(-1) -.823(-1) -.125( 0) -.215~ 0) -.312( 0) -.398( 0) -.994.(-1) -.517(-2) -.372(-3)

-.630( 0) -.628( 0) -.623( 0) -.614( 0) -.579 0) -.508( 0) -.276( 0) .163(-1) .583(-2) .923(-3).5 o -.332(-1) -~664(-1) -.996(-1) -.165( 0) -.222( 0) -.260( 0) -.933(-1) ~.568(-2) -.423(-3)-.481( 0) -.479( 0) -.473( 0) -.462( 0) -.425( 0) -.362( 0) -.222( 0) .916(-3) .527(-2) .900(-3).7 o -.254(-1) -.504(-1~ -.750(-1) -.121( 0) -.157( 0) -.181( 0) -.841(-1) -.609(-2) -.4.72(-3)-.364( 0) -.362( 0) -.357( 0 -.348( 0) -.317( 0) -.270( 0) -.180( 0) -.104(-1) .468(-2) .875(-3)

1.0 • 0 -.160(-1) -.318(-1) -.470(-1) -.74.3(-1) -.955(-1) -.111( 0) -.686(-1) -.652(-2) -.541(-3)-.242( 0) -.24.0( 0) -.237( 0) -.230( 0) -.211( 0) -.184( 0) -.134( 0) -.203(-1) .378(-2) .832(-3)2.0 o -.375(-2) -.74.4(-2) -.110(-1) -.176(-1) -.233(-1~ -.293(-1) -.304(-1) -.663(-2) -.722(-3)

-.757(-1) -.755(-1) -.747(-1) -.736(-1) -.699(-1) -.648(-1 -.552(-1) -.231(-1) .109(-2) .660(-3)5.0 o -.236(-3) -.4.70(-3) -.703(-3) -.116(-2) -.159(-2) -.220(-2) -.361(-2) -.311(-2) -.833(-3)-.937(-2) -.936(-2) -.934(-2) -.930(-2) -.917(-2) -.899(-2) -.862(-2) -.677(-2) -.161(-2) .131(-3)

10.0 o -.206~-4~ -.412(-4~ -.618~-4~ -.103(-3~ -.143~-3~ -.202~-3) -.380~-3) -.643(-3) -.455~-3)-.151(-2) -.151 -2 -.151{-2 -.151 -2 -.150{-2 -.149 -2 -.14.7-2) -.137 -2) -.84.4{-3) -.195 -3)

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z/R0 hr: 0 -.326(-1) -.666~-1)_-.104(0) -.196(0) -.340(0) -co -.505(-1~ -.123~-2) -.755~-4)

hz: -.707( 0) -.7~( 0) -.712 0) -.719( 0) -.739( 0) -.768( 0) -.321( 0) .519(-1 .393 -2) .498 -3) !

.1 o -.3~3(-1) -.699(-1) -.loB( 0) -.201( 0) -.337( 0) -.652( 0) -.560(-1) -.146(-2) -.903(-4)-.640( 0) -.~1( 0) -.~2( 0) -.~3( 0) -.64~( 0) -.627( 0) -.266( 0) .~0(-1) .385(-2) .496(-3)

.2 • 0 -.3~(-1) -.695(-1) -.107( 0) -.1~( 0) -.306( 0) -.~51( 0) -.596(-1) -.168(-2) -.105(-3)-.572( 0) -.572( 0) -.570( 0) -.567( 0) -.553( 0) -.506( 0) -.232( 0) .360(-1) .375(-2) .493(-3)

.3 o -.327(-1) -.662(-1) -.101( 0) -.178( 0) -.2~( 0) -.342( 0) -.615(-1) -.189(-2) -.120(-3)-.505( 0) -.504( 0) -.501( 0) -.495( 0) -.~70( 0) -.~12( 0) -.205( 0) .282(-1) .365(-2) .489(-3)

.5 o -.273(-1) -.5~7(-1) -.822(-1) -.137( 0) -.187( 0) -.218( 0) -.611(-1) -.227(-2) -.148(-3)-.385( 0) -.383( 0) -.378( 0) -.370( 0) -.340( 0) -.286( 0) -.163( 0) .142(-1)· .3~1(-2) .481(-3)

.7 o -.209(-1~ -.~17(-1) -.621(-1) -.100( 0) -.131( 0) -.149( 0) -.568(-1) -.261(-2) -.176(-3)-.288( 0) -.287( 0 -.282( 0) -.275( 0) -.2~9( 0) -.209( 0) -.131( 0) .343(-2) .313(-2) .472(-3)

1.0 o -.132(-1) -.261(-1) -.385(-1) -.6oB(-1) -.778(-1) -.892(-1) -.475(-1) -.301(-2) -.215(-3)-.187( 0) -.166( 0) -.183( 0) -.n8( 0) -.161( 0) -.n8( 0) -.960(-1) -.686(-2) .267(-2) .454(-3)

2.0 o . -.288(-2) -.572(-2) -.846(:-2) -.135(-1) -.lTI(-l) -.220(-1) -.211(-1) -.351(-2) -.326~-3)-.5~3(-1) -.5~1(-1) -.535(-1) -.525(-1) -.~95(-1) -.~~(-1) -.378(-1) -.135(-1) .111(72) .377 -3)

5.0 o -.155(-3) -.309(-3) -.~2(-3) -.760(-3) -.1~(-2) -.1~(-2) -.231(-2) -.181(-2) -.428(-3)-.581(-2) -.581(-2) -.579(-2) -.576(-2) -.568(-2) -.555(-2) -.530(-2) -.405(-2) -.798(-3) .101(-3)

10.0 o -.122(-4) -.243(-4) -.3~(-4) -.604(-4) -.842(-4) -.119(-3) -.223~-3) -.367(-3) -.245(-3)-.856(-3) -.856(-3) -.855(-3) -.85~(-3) -.850(-3) -.845(-3) -.833(-3) -.TIO -3) -.458(-3) -.946(-4)

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I R/t ..2 II~ r/R 0 .1 .2 .3·.5 .7 1 2 5 10

z/R0 ~: 0 -.144(-1} -.300(-1} -.482(-1} -.101( O} -.207( O} - 00 -.175(-1} -.319(-3} -.190(-4}hz: -.447( O} -.449( 0) -.456( 0) -.467( 0) -.506( 0) -.572( 0) -.218( 0) .370(-1) .206(-2)' .252(-3).1 o -.179(-1) -.371(-1~ -.590~-1) -.119( 0) -.226( 0) -.525( 0) -.230(-1) -.444(-3) -.266(-4)-.415( 0) -.416( 0) -.420( 0 -.427 0) -.447( 0) -.463( 0) -.174( 0) .332(-1) .204(-2) .251(-3).2 o -.198(-1) -~408(-1) -.641(-1) -.124( 0) -.214( 0) -.347( 0) -.273(-1) -.566(-3) -.342(-4)

-.377( O} -.37B( 0) -.380( 0) -.382( O} -.384( 0) -.367( 0) -.150( O} .288(-1) .201(-2) .250(-3).3 o -.203(-1) -.414(-1) -.642(-1) -.119~ 0) -.187( 0) -.255( 0) -.303(-1) -.685(-3) -.416(~)

-.337( 0) -.337( 0) -.337( 0) -.336( O} -.327 0) -.293( 0) -.131( 0) .241(-1) .197(-2) .249(-3).5 o -.181(-1~ -.366(-1) -.555(-1) -.950(-1) -.133( o~ -.157( 0) -.330(-1) -.908(-3) -.564(-4)-.260( 0) -.259( 0 -.256( 0) -.252( 0) -.233( 0) -.196( 0 -.103( 0) .149(-1) .187(-2) .246(-3).7 o -.144(-1} -.287(-1) -.429(-1) -.700(-1) -.922(-1) -.104( 0) -.323(-1) -.111~-2) -.708(-4)-.195( O} -.1g4( o) -.191( 0) -.186( 0) -~168( 0) -.14o( 0) -.822(-1) .718(-2) .175 -2) .242(-3) I

1.0 o -.912(-2) -.181(-1) -.267(-1) -.423~-1) -.540~-1) -.611(-1) -.281(-1) -.136(-2) -.917{-4)-.125( 0) -.124( 0) -.122( 0) -.11B( 0) -.107 0) -.900 -1) -.598(-1) -.650(-3) .154(-2) .235(~3)2.0 o -.191(-2) -.378(-2} -.558(-2) -.887(-2) -.116(-1) -:142(-1} -.128(-1) -.176(-2) -.151(-3)

-.340(-1) -.339(-1} -.335(-1) -.328(-1) -.308(-1) -.280(-1) -.228(-1) -.709(-2) .731(~3) .200(-3)5.0 o -.909(-4) -.181(-3) -.271(-3) -.445(-3} -.611(-3) -.838(-3) -.133(-2) -.975(-3) -.216(-3)

-.329(-2) -.329(-2) -.328(-2) -.326(-2) -.321(-2) -.313(-2) -.298(-2) -.223(-2) -.389(-3) .602(-4)10.0 0. -.665(-5) -.133(-4~ -.199(-4) "...330(-4)-.460(-4) -.650(-4·) -.121(-3) -.197(-3) -.127(-3)

-.458(-3) -.458(-3) -.458(-3 -.457(-3) -.455(-3) -.452(-3) -.446(-3) -.410(-3) -.239(-3) -.462(-4)

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TABLE IV-7

I Rit· 5 l~ r/R 0 .1 .2 .3 .5 .7 1 2 5 10

z/R0

~: 0 -.291~-2) -.61~(-2) -.101(-1) -.235(-1) -.613(-1) - 00 -.315~-2) -.517(-4) -.306(-5)hZ: -.196( 0) -.197 0) -.202( 0) -.209( 0) -.239( 0) -.306( 0) -.1n( 0) .168-1) .835(-3) .101(-3)

.1o -.5~1(-2) -.114(-1) -.185(-1~ -.~13(-1) -.954(-1) -.336( 0) -.607(-2) -.103(-3~ -.611(-5)

-.1oo( 0) -.189( 0) -.192( 0) -.198( 0 -.219( 0) -.253( 0) -.865(-1) .158(-1) .831(-3' .101(-3)

.2o -.722(-2) -.151(-1) -.242(-1) -.509(-1) -.102( 0) -.203( O~-.859(-2) -:154(-3) -.915(-5)

-.175( 0) -.n6( 0) -.n8( 0) -.182( 0) -.192( 0) -.198( 0) -.723(-1 .144(-1) .822(-3) .101(-3)

.3o -.825(-2) ~.170(-1) -.270(-1) -.53~(-1) -.936(-1) -.142( 0) -~106(-1) -.203(-3) -.122(-4)

-.160( 0) -.160(,0) -.161( 0) -.162( 0) -.1~( 0) -.155( 0) -.623(-1) .126(-1) .•8"(-3) .100(-3)

.5o -.833(-2) -0169(-1) -.260(-1) -.~3(-1) -.681(-1) -.830(-1) -.130(-1) -.299(-3) -.182(-4)

-.126( 0) -.126( 0) -.125( 0) -.124( 0) -.1n( 0) -.100( 0) -.483(-1) .860(-2) .780(-3) .995(-4)

.7o -.69'7(-2) -.140~-1) -.210(-1~ -.350(-1) -.471(-1) -.537(-1) -.135(-1) -.386(-3) -.241(-4)

-.955(-1) -.951(-1) -.938-1) -.916(-1 -.835(-1) -.693(-1) -.383(-1) .493(-2) .739(-3) .982(-4)

1.0o -.~55(-2) -.903(-2) -.134(-1) -.213(-1) -.273(-1) -.307(-1) -.123(-1) -.500(-3) -.326(-4)

-.6" (-1) -.608(-1) -.597(-1) -.579(-1) -.520(-1) -.~35(-1) -.277(-1) .920(-3) .660(-3) .956(-4)

2.0o -.926(-3) -. 1~(-2) -.271~-2) -.~30(-2) -.558(-2) -.680(-2) -.577(-2) -.700(-3) -.575(-4)

-.159(-1) -.159(-1) -.157(-1) -.153-1) -.1~3(-1) -.129(-1) -.104(-1) -.286(-2) .339(-3) .825(-4)

5.0o -.404(-4) -.805(-4) -.120~-3) -.198(-3) -.271(-3) -.371(-3) -.582(-3) -.409(-3) -.865(-4)

-.143(-2) -.142(-2) -.142(-2) -.141-2) -.139(-2) -.135(-2) -.128(-2) -.949(-3) -.152(-3) .266(-4)

10.0o -.281~-5) -.562(-5) -.~3~-5) -.140(-4) -.195(-4) -.275(-4) -.513(-4) -.823(-4) -.522(-4)

-.191(-3) -.191-3) -.191(-3) -.191-3) -.190(-3) -.189(-3) -.186(-3) -.171(-3) -.982(-4) -.181(-4)

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CALCULATION OF THE STRAY FIELD OF A MAGNETIC BUBBLE .27

REFERENCES1) A. A. Thiele, Bell Sys. tech. J. 48, 3287, 1969.2) R. Szymczak and R. Wadas, IEEE Trans. Magnetics 7, 361, 1971.

W. F. Druyvesteyn, R. Szymczak and R. Wadas, to be published.3) J. A. Cape 'and G. W. Lehman, Solid State Comm. 8, 1303, 1970 and J. appl.

Phys. 42, 5732, 1971.4) W. F. Druyvesteyn and J. W. F. Dorleijn, Philips Res. Repts 26, 11, 1971.5) J. W. F. Dorleijn, W. F. Druyvesteyn, F. A. de Jonge and H. M. W. Bo o ij, IEEE

Trans. Magnetics 7, 355, 1971.F. A. de Jonge and W. F. Druyvesteyn, Proceedings of the 17th M.M.M., Chicago1971. ..

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