Resonant Frequencies of the NosedIn CavityErnest Mayer Citation: Journal of Applied Physics 17, 1046 (1946); doi: 10.1063/1.1707674 View online: http://dx.doi.org/10.1063/1.1707674 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/17/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Resonant-frequency discharge in a multi-cell radio frequency cavity J. Appl. Phys. 116, 173301 (2014); 10.1063/1.4900994 Broadband electron spin resonance at low frequency without resonant cavity Rev. Sci. Instrum. 79, 046101 (2008); 10.1063/1.2901382 Acoustic resonant frequencies in an eccentric spherical cavity J. Acoust. Soc. Am. 64, 286 (1978); 10.1121/1.381974 Resonant Frequencies and Fields in a Cavity Containing a Magnetoplasma Dielectric J. Appl. Phys. 39, 5919 (1968); 10.1063/1.1656090 Effect of Long Waveguides on the Resonant Frequencies of Microwave Cavities Rev. Sci. Instrum. 34, 1441 (1963); 10.1063/1.1718269

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Resonant Frequencies of the Nosed-In Cavity

ERNEST MAYER*

Westinghouse Research Laboratories, East Pittsburgh, Pennsylvania

(Received June 13, 1946)

In this paper the resonant frequency of the nosed-in cavity is studied as a function of the cavity dimensions. Maxwell's differential equations and boundary conditions are converted into an integral equation which is solved approximately by the Ritz variational method. The size and shape of the cavity are fixed by specification of the dimensions (cf. Fig. 1) rl and '2, the inner and outer radii; el , the post length and tlI, the gap space. If the cavity is to resonate to the wave-length X, only three of its dimensions can be given independently; the fourth will be a function of the given three and the wave number k=211'/X. For- fixed ,tlr. the dependence of kEI on kEII is calculated with a precision of 1 percent.

I. INTRODUCTION

T HE present article is based almost entirely on a Westinghouse Research report. 1 A

number of mathematical details and proofs not given here and additional references will be found in the above-mentioned report.

Much interest has developed recently in the nosed-in type of cavity of large gap spacing shown in Fig. 1. In this article we obtain curves which relate the resonant frequency of the cavity to its dimensions. These dimensions' are: El, the post length; EII , the gap space; r1, the post radius; and r2, the outer radius of the cavity. As the cavity is a figure of revolution we use cylindrical polar coordinates (r, ip, z) with the z axis along the axis of symmetry and the z = 0 plane along the tip of the post. We are interested in the fundamental mode of resonance. In this mode the angular component of the electric field, E", is zero and the angular component of the magnetic field, H"" is independent of ip. Since the mode has angular symmetry, H", satisfies the partial differential equation

(1)

together with the boundary condition that, on all conducting surfaces,

a -(rR",) =0, an

(2)

* Now at The M. W. Kellogg Company, Jersey City, New Jersey.

1 T. Holstein and E. Mayer, Research Report SR-281, "Resonent frequencies of the nosed-in cavity."

1046

where a/an denotes differentiation in the direc­tion normal to the conducting surface, and k=2w/A is the wave number. H", can serve as a potential function for the other two non-vanish­ing components of the field:2

1 a ikEz=--rH""

r or

aH", ikEr=---.

az

(3)

(4)

For the cavity in Fig. 1 no explicit solution of (1) has been given. Only approximate methods are available for calculating the frequency of such a resonator. A practical procedure is to divide the interior of the cavity by a suitably chosen surface S into regions I and II, in each of which the variables of (1) can be separated. In these regions the solutions H,/ and H",II, respectively, can be set up so as to satisfy (2) at the conducting surfaces. However, owing to the fact that neither region is completely surrounded by conducting walls, each solution will contain an infinite num­ber of arbitrary constant~; furthermore, the wave

FIG. 1.

2 E. U. Condon, Rev. Mod. Phys. 14, 341 (1942).

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number k is undetermined. In order for Hi and H,p to represent the solution of (1) in the whole cavity, they must obey the following continuity conditions on the surface S:

(5)

(6)

where a/an denotes differentiation in the direc­tion normal to S. The imposition of these con­tinuity conditions determines not only the arbi­trary constants but also the wave number k.

This procedure has been applied by Hansen3

to the nosed-in resonator of small gap space. In his paper the surface S is chosen at r = rl. The interior is thereby divided into two regions in each of which H", can be expanded as a Fourier­Bessel summation, the amplitudes of the terms being determined by the condition that both summations yield the same value of the tan­gential electric field E. on the dividing surface S. This condition, which is equivalent to (6), to­gether with (5) can be used to determine the unknown amplitudes and the wave number k.

Hansen now assumes an analytic form E.(z) to approximate the component of the actual field on S. The assumed E.(z) is expanded into two Fourier series over the intervals Ell and EI+EII =Zo, corresponding to the regions into which Hansen's S divides the interior of the cavity. Substitution of the Fourier expansions into (3) permits evaluation of the amplitudes and therefore satisfies (6) by construction. The de­termination of the only remaining unknown, k, is now effected by a matching of H",I and H",1I at Zl, a particular value of z on the surface S employed by Hansen. With the value k thus found, say k l , the matching indicated in (5) is in general not satisfied at z~'Zl, because the analytic form of E.(z) is only an approximation. Had E. been the correct function, the k obtained from matching at a particular z would have guaranteed satisfaction of (5) everywhere on S. Hansen shows that kl is close to the correct k as it is not very sensitive to the error in Ez or to the choice of Zl. In practical calculations it turns out that his method begins to be inaccurate for resonators whose total height Zo exceeds 7r/2k.

A more general approach to the problem has

3 W. W. Hansen, J. App. Phys. 10, 1 (1939).

VOLUME 17, DEC~BER, 1946

1z -Z=o

11

FIG. 2.

been made by J. Schwinger. According to his method E. (or, more generally, the tangential electric field on S) is left undetermined. The amplitUdes of the Fourier terms are expressed by means of (3) and (6) as integrals of the unknown E •. Substitution in (5) leads to an integral equation in E. of which solutions exist onlv for a denumerably discrete set of k's. IIi applying such integral equations to wave guide problems Schwinger has shown how approximate solutions can be obtained by variational methods.

The frequency calculations presented in this paper extend beyond the range treated by Hansen, i.e., the method presented here is prac­tical for zO>1r/2k. These calculations are based on an integral equation of the form

f 1/;(r )G(r, r')dr' ~ 0, (7)

in which the kernel G(r, r') is a function of the cavity dimensions and the wave number k. The steps leading to (7) are stated below. The solution of the integral equation will be achieved by methods analogous to those employed by Schwinger.

II. THE INTEGRAL EQUATION

The derivation of the integral equation pro­ceeds as follows. We divide the cavity interior in to regions I and II by choosing* S along z = 0 as shown in Fig. 2. The solutions of (1) in both regions are of the form

H", =Zl(krr)e ikzz ,

where

and Zl(krr), the first-order cylindrical function, is a linear superposition of Bessel and Neumann

* This choice of S is merely a matter of m'lthematicaf convenience. The variational method applied here can be applied also when S is chosen at r=rl as in Hansen's paper (see reference 3).

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functions

Zl(krr) =aJ1(krr) +bN1(k,r) ,

a and b being constants of integrations. Using Htp as a potential from which E. and Er can be derived according to (3) and (4), we can now impose the boundary condition that the tan­gential electric field shall vanish on all conducting surfaces, and thereby arrive at relations between a and b and between k. and k from which simplified expressions for H,/ and HtpII can be written in closed form.

We start with region II, in which b=O since" the Neumann function is excluded because of its singularity at r =0. FUl:ther, the boundary condi­tion E.=O at r=r2leads to JO(krI~r2) =0. Hence, krII can have any of the values jn/r2, where jn is the n'th root of the zero-order Bessel function. In cavities of practical interest kr2 <jl = 2.405, and, therefore, all the kP's are imaginary with

_(kP)2= (jn/r2)2-k2= (k"II)2, k"IIreal. (8)

We can thus express H,/I as

H"P = L A,p cosh k"II(z+eII)J1(j"r/r2)' (9)

Here the phase of the cosh term has been ad­justed to make Er zero on the wall at z= -ElI.

In region I we find that in order for E. to vanish at r=rl and r=r2 we must have

ZO(k,Irl) =0 and ZO(k,Ir2) =0.

From the latter two conditions we can, in the first place, solve for alb so that we can write Zl as

Zl(kh) = -No(khl)Jl(k,Ir)

+ JO(k,Irl)N1(k,Ir) (10)

arid secondly we find that k,I must. satisfy

JO(k,Irl) N o(k,Ir2) - No(khl)Jo(k,Ir2) = O. (11)

Introducing the notation .I = kr1r2 we rewrite (11) :

J o(.\rt/r2)NoC.\) - NO(.\rt/r2) J o('\) = o. (11')

The solutions .I = .In of this equation are listed in Jahn}{e-Emde's4 Tables of Functions. As .In is always larger than the corresponding jn, we again have imaginary k.'s or

- (k})2= (.\n/r2)2-k 2 = (k"I)2, k"I real. (12)

In addition to the Zl'S listed in (10) the function l/r is a solution for the case kr = 0 since it satisfies (1) and its boundary conditions. Thus the complete solution of (1) in region I is

HtpI=AoI cos k(z-EI)/r

+ L AnI cosh k,,!(z- eI)Zl(.\"r/r2), (13)

where the dependence on z has been so adjusted that Er vanishes on the wall at z=eI.

We now define 1/;(r) by the relation

1/;(r)/r=iJHtp/iJz= -ikEr at z=O. (14)

This is the 1/; which appears in (7) as the unknown function. In solving (7) for the wave number k we shall obt.ain, as a by-product of the calcula­tions, the dependence of 1/; on r and hence the functional form of E r •

Following the procedure outlined above we solve for the amplitude An as integrals of the tan­gential field (in this case Er) on S. Substitution of (9) and (13) in (14) gives

1/;(r) --= L AnIIk"IIeIIJ1(jnr/r2),

r

1/;(r) =AoIk sin keI/r- L A,..Ik"I sinh k"IeIZ1('\nr/ r2). r

To solve for Amll we multiply in (15) by J1(jmr/r2)rdr and integrate over S obtaining

2kmII fro J1(jmr' /r2)1/;(r')dr'

rl Amll =-----------

(kmIIr2) 2 sinh kmII eII J I 2(jm)

(15)

(16)

(17)

4 E. Jahnke and F. Emde, Tables of Functions (Dover Publications, New York, 1943), p. 205. The tables from which rn can be obtained contain a number of errors. New and more accurate values were taken from tables by A. N. Lowan and A. Hillman, J. Math. Phys. 22 (Dec., 1943). These tables contain suitable information in the range rdrl:::::4. For r./rl>4 we calculated rn'S from asymptotic expansions quoted by Lowan and Hillman.

1048 JOURNAL OF ApPLIED PHYSICS

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An analogous integration of (16) with ZI(r mr /r2)rdr leads to

_2km1fT2 ZI(rmr' /r2)-.j;(r')dr' rt

A~= . (km1r2) 2 sinh km1eI[ZI2(r m) - (rdr2)2Z12(t mr /r2)]

(18)

The ZI'S in the denominator of (18) can be expressed in terms of J O'S5 so that AmI becomes

(18')

Finally to obtain AoI we integrate (16) with l/r·rdr and get

(19)

Having obtained the expressions for An we can formulate the boundary condition (5) as

(7')

where

G(r, r') cot keI 1 1 11'2 k"I coth k"Ielr"2Z1(r,,r/r2)ZI(rnr' /r2) -------- L -------------k log rdrl r r' 2 (k"Ir2)2[(Jo(t"rdr2)/Jo(rn))2-1]

m. THE EQUIVALENT VARIATIONAL PROBLEM

The solution of the integral Eq. (7') is achieved by the solution of the equivalent variational problem, which we shall now proceed to formulate. We define

U=j'J -.j;(r)G'(r, r')-.j;(r')drdr' ,

cot keI II -.j;(r) -.j;(r') . ----drdr'

k log rdrl r r'

(21)

where cot keI 1 1

G'(r, r') = . --G(r, r') k log rdrl r r'

and all the integrations extend, as in the subsequent expressions where the limits are omitted, from rl to r2. The two equations

U=l (22) and ou=o, (23),

where oU is the variation in U arising from an arbitrary variation o-.j; of -.j;, then constitute a varia­tionalproblem equivalent to (7').

6 G. N. Watson, TheOf'y of Bessel Functions (Cambridge, 1922), p. 76.

VOLUME 17, DECEMBER, 1946 1049

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To prove this equivalence we perform the variation

fJU

2 I fJ>/I(r)dr [

I GI(r, r')>/I(r')dr'

II cot kEI >/I(r) >/I(r') , ------ --drdr k log rdr1 r r'

II I cot hI >/I(rl)dr/]

>/I(r)G' (r, r')t/I(r')drdr' . k log r2/r1 rr'

II cot hI >/I(r) >/I(r') ------ --drdr'

. k log rdr1 r r'

=0.

Since fJ>/I is arbitrary, the bracketed quantity must vanish; or, in view of (21)

I I cot kEI >/I (r/)

G'(r, r')>/I(r')dr' - U --dr' =0 k log rdr1 rr'

(23')

which, for U=l, is identical with (7/).

IV. PROPERTIES OF THE SOLUTION

The theory of integral equations asserts that the solution of (23) (or its equivalent (23 /» is in general an infinite set of eigenfunctions t/ln which belong to corresponding terms of a set of eigenvalues Un of U. Owing to the particular form of G/(r, r') a number of special features of the solution, outlined below in (a), (b), and (c), can be proved.1 These features serve to guide us in the application of approximate methods for solving (23).

For specified values of EI, Ell, r1, r2, and kin the range

kr2<j1=2.405, kEI <1I"/2, (24)

the following theorems hold: (a) There exist only one eigenvalue (denoted

by Uo) and one corresponding eigenfunction (denoted by >/10) which satisfy (23).

(b) Equation (22) with U = Uo, that is Uo = 1, constitutes an equation among the five param­eters from which EI can be evaluated uniquely when the other four are specified. (The value of EI so obtained will be denoted by EoI.)

(c) The value of EI obtained from (22) with an arbitrary >/I cannot exceed EoI; i.e., EI:5EoI.

In view of (c) we may expect that6 if we find

8 R. Courant and D. Hilbert, Methoden aer Mathema­tischen Physik (Interscience Publishers, Inc., New York, 1943), Vol. 1, p. 146. .

1050

a sequence of >/I's such that the corresponding EI'S form a monotonically increasing series, the limit of this series is EOI.

V. THE RITZ VARIATIONAL METHOD

In this section we describe the Ritz procedure by which an approximate solution of (22) and (23) can be obtained. We select n functions >/Ii(r) each of which j:;an be made:to satisfy the bound­ary conditions on >/I(r) and set up

n

>/I(r) = L aJ-t/liCY). (25) i=l

Variations in >/I are now effected by variations in ai' With (25) substituted in (22) and (23) we can determine ai and solve for EI as a function of ElI, r1, !'2 and k.

In using (25) we find the following notation convenient:

Gii~ J J >/Ii(r)G'(r, rl)>/Ii(rl)drdr'=Gii,

cot kEI If >/Ii(r) >/IiCY' ) I ij = --~rdr'=Iji.

k log (r2/r1) r r'

With these formulas we convert (22) and (23) into

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and

respectively. The variation is performed on a; giving

As the oa/s are arbitrary, the bracketed term must vanish; or, after simplification,

n

1: a;(Gij-h) =0; j= 1,2· . ·n. i=l

This system of n linear equations for the na/s ·has a non-trivial solution if,· and only if, the determinant formed from the coefficients of the a/s vanishes;

Gll-Ill G12-I12

G21 - 121 G22 - 122 =0. (26)

This expression constitutes a transcendental equation from which it is possible to solve for EI

when the other four parameters a~e specified.

VI. THE CHOICE OF 'I"'S

In choosing suitable 1/;/s, we take into account the behavior of the correct solution 1/;0 and its derivatives on S. Thus as r = r2, we must have Ez-tO and hence from (14) and

a aE. div E=-rEr+r-=O,

ar az

it follows that

(27a)

(27b)

VOLUME 17, DECEMBER, 1946

I

1 .. t ...... l;- 4-=" .. . 1 ...

.. ' .. ~ .. .. 0 LO

p

1\ l-\ ..-

I\~

'.0 .. -

\ 1,\ \

FIG. 3.

FIG. 3 f-." • .!III "'._I.U1 f-

1\ .0

Finally, from the analogous two-dimensional case it is to be expected that

a2if;o/ ar2 > 0 on S. (27c)

Now, we seek a set of 1/;/s which can be combined according to (25} to make 1/; behave approxi­mately as 1/;0, A vital property of our 1/;;'s must be, however, that they permit the evaluation of the integrals for Gij and Iij. The question of this integrability is important enough to warrant the relaxation of some of the conditions in (27) in order that a practical set of 1/;;'s be found.

After a number of trials it was found that a family of 1/;/s which can be made to agree satis­factorily with conditions (27) and also permit evaluation of Gij- Ii; without undue difficulty consists of power series in r. The basic 1/;n's considered in this family are

1/;0= 1, a constant,

.-1/;q=al'+a3'[(r-r2)4- aq], a quartic 1/;n,

and

a superposition of parabolic and quartic 1/;n'S. The constants ap and a q are determined from

which will insure that all the I;/s vanish except

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2

0

FIG. 4 I Krl.I.9 Ke1<· co - t----,\

\\ , \~

1\\ l\ r~

"" r0: ~ k~-'- --- ~

..... t-:--Ih

/

2 3 4 5 6 7 8 9 1.0

~rt

FIG. 4.

I--

Iu. For 1/Ic the determinantal equation reduces to

Gu-Iu=O (26')

from which values of kEI as function of kEI! with fixed kr1 and kr2 have been calculated (Fig. 3). In view of more accurate* final curves, the value of kEl thus obtained is about 10 percent below kEi.

Considerable improvement" over the if;c curve is obtained by use of the parabolic if;p, which leads to the determinantal equation

I Gn - I u G121 = o. (26") G21 G22

Further increase in the calculated value of kEl is obtained by use of if; q. When the more general 1/IPH is e~ployed, there is no appreciable ~ncrease in kEl over the result with if;q, as shown in Fig. 3. The convergence of the Ritz process illustrated in Fig. 3 indicates that if;q is sufficiently good for

the calculation of characteristics of the nosed-in resonator, and therefore this function was em­ployed in preference to more general functions, with which the numerical work becomes exceed­ingly laborious without compensating gain in accuracy. The calculations with if;q are given in greater detail in the next section.

The experimental check on these curves was obtained at the Westinghouse Research Labora­tories. The resonator of total height kZ;) = k(El +EII )

= 2.5 having kEl and kEII corresponding to point P on Fig. 3 shows an agreement within 2.5 percent of the calculated value. Further meas­urements at the Bloomfield Division of Westing­house Electric Corporation led to similar agree­ment, the experimental values of kEl lying about 2.5 percent above the calculated curves. The disagreement is ascribed to the deviation of the actual cavity from that treated by the theory. Instead of the smooth top shown in Figs. 1 and 2 the actual cavity is provided with a corrugated diaphragm (to permit tuning).

The radial variation of if;p, if;q, and if;p+q is illustrated in Fig. 4. It is to be noted that (27b) is not satisfied by these if;'s, but this does not result in serious trouble because the infinity of if;o at r ~ r1 is integrable.

VII. SUMMARY OF PROCEDURE AND RESULTS

A practical set of if;n's consists of the two functions

where the constant a q , defined by

fro

1/Iqfir/r=0, n

IS

12 log rdr1- 25 +48(rdr2) -36(rdr2)2+ 16(rdr2)3-3(rdr2)4 aq=r241--------~·---------------------------------------

12 log r2/r1

From the definitions of Iu and Gii we are led to the following formulas for these terms:

log rdr1 Iu=r2 cot kEI,

kr2

* Since from (c) in Section IV '01 is the upper limit of the .1 that can be calculated from an arbitrary tit, the highest curve on Fig. 3 is the most accurate.

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In these formulas the E's, which are functions only of the ratio r1/r2 are defined by

+rdr2Jo(j"rdr2{ ( - j:2 +~) (rdr2) - (rl/r2)2+HrI/r2)3

where the Ho and HI are the zero and first-order Struve functions, respectively. Next we set up the determinantal equation

jGn-In G131 =0 Gal G33 '

(26"')

which connects the wave number k and the geometric parameters eI , eII , rl. and r2. If, now, k is assigned, this equation can be solved for fiI

when the other three parameters are specified provided that kr2 <jl = 2.405.

It is convenient to fix the value of rl/r2 and let kEII assume an arbitrary set of values corre­sponding to which kEI is determined by (26'''). Expanding the latter into

(28)

VOLUME 17, DECEMBER, 1946

(1-a q/r24) 1f" ]

+ -2H1(j"rl/r2)(1-3/J.2) , 4r1/r2

we note that kEI appears on the left in III and knIEI on the Tight in G13, which, however, is not very sensitive to variations in EI. We can thus first obtain kEI approximately from III = Gu , calculate from (12)

and substitute this in Gu _ It turns out that only coth klI eI need be calculated in this manner because for n> 1 coth kn1eI "", 1. Similarly, we have from (8) coth kIHEI! = coth keII[jN (kr2)2 -1 J! and for n> 1 coth k"IIEII "'" 1. * With these expres­sions for the coth's the summations in the right member of (28) are readily performed ,up to a

* In order for this approximation to hold k.1I must not be taken much smaller than unity_

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~~~~~-+~+-~F~~~~~~~

H-+++-+-l-+-++-+---I- ",,' U -+--+-i,-+-+--l-l

FIG. Sa.

I-+-+-+-+-l-!-+-+~-l-J~ FIG.S' 1-+-+-+-+---1 '"' ·1,5

J

• • c·

FIG. Sb.

I ,

.I

~ .. - -- -- -- -- -- --

1~"" FIll. So

-11 Ii SO

I I I I • •

• c·

FIG. Sc.

number of terms N which depends on the accu­racy desired. The number N increases as the ratio r1/r2 is taken smaller.

With kEII 2: 1 as the independent variable the full-lined curves in Figs. 5a-f were calculated giving kEI for the following fixed values of the other parameters: rt/r2=.1, .3, A, .5; kr2=1.1, 1.3, 1.5, 1.7, 1.9, 2.1. A precision of 1 percent was obtained with five terms in the summations deriving from region I and seven terms in the summations from region II.

By drawing on Fig. 5 the straight lines kzo = constant (shown only on Fig. Sf) and taking the intersections of the latter with the k(;I curves corresponding to the values of rl/r2 we obtain the points from which we plot Fig. 6a-f giving

1054

I,'

I FIG. 5d "',-1.7

" . \

"

.f ~-~- - -" -- =""" :~

I _J-..L...-:~ ,. J • ,,'

FIG. 5d.

I H-t-+-t--1H-+--+ :::: +-+-l-+++-+-l-+-l

FIG. Se.

, k ..... U

1\ .... l).-t-

J 1-<1\ 1-- - - f.-- -- -- -- -... ~ -- --1\ .. -- ~ ff ... ~r; ~~ r;.: -- ! ---~ .: ~\ ~~-

,(.- R;T" : ~~.- I-'I~

_'0 '

\:' ~ [\ I \~-[\ \ \j I I I

I • • FIG. Sf.

hI VS. rl/r2 for fixed total heights kzo• On these curves the ordinate and the slope at rt/r2 = 0 are obtained directly from (28) as limiting values for krl---tO are taken. We can write

where f(rl/r'J.) is a function which remains finite, and hence

f(rrJr2)

log rdr/

with the slopes becoming infinite as r2/rl/log r2/r1 ..

From Figs. 6a-f we can now read the ordinates

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4 .0

! tef.-'.I "'t -us

.$

" I.

1\. , '.

t-- ,!,I-l-t--I--t- lI.Ol-t-...

OI-l-..

'I, ~\" '~ '~ t:=:::: II:"

« -t- ur-r-t-r- '.0 .. '

r-----I--f-··l-H~I'_I'_I'_I_+_+_+____I

.• f-HH---t---t--t--+--+-+-+-+---t---i ..

J

0 .. .. . . .. .• " .. .. A .. .. FIG. 00 • FIG. 6b. FIG. 6c.

.0

I 0

, -1.7 ,

I "ft"'l.t Kt,"2J

.. i\

\ \

r-----~ ~ '0r----- I--

~ r- = nr - r-----

r- '.0 r-- t--.

. ..

.. l ~~

\1

0 I;~

~"'" ~ ~ ~ :0r- r-----...... t--!r----- r---

•

.. ' ",'

-~l :--~-l. -.. f· rf!---P= :--r-

r-!~ I K.-¥' \\\\\ -r-

i~ .. \ ~

l'\ ~ \ 1'--, ~ !:;:.:-. ... 1-I-

" l'---1"-f:::: E:::: r- t-: l::- ... " t- 3.0 r--I-

'. r- r-t- ... r-.. 2.0 r--t-t-- l-t., r-----

0 .I .. .. .. .. . . 0 • 1 .. .. .. .. .. 0 .. FIG.6d. FIG. 6e. FIG. 6f.

for other values of rl/r2 and obtain by interpola­tion curves for kEI vs. kell. On 5a-f the curves for rl/r2 = .2, indicated by the dotted lines were

plotted in this manner. A check was obtained on the interpolation by a direct calculation for kr2= 1.9.

VOLUME 17, DECEMBER, 1946 1055

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