ON THE THEORY OF ELECTROMAGNETIC WAVES IN RESONANT CAVITIES Bound... · et la vibration- d'un simple circuit ... The average values of the magnetic and the electric ener ... THEORY

R 165 Philips Res. Rep. 6, 162-182, 1951

ON THE THEORY OF ELECTROMAGNETIC WAVESIN RESONANT CAVITIES *)

by H. B. G. CASIMIR 538.56:621.396.611.4

SummaryIn these lectures a survey is given of the theory of standing electro-magnetic waves in resonant cavities. The formal analogy betweenthe modes of vibration of a cavity, the modes of vibration of a net-work of discrete elements, and the vibration of a simple LC circuit isemphasized. Special attention is given to the theory of perturbations,and this theory is then applied to a number of examples. These includethe determination of the high-frequency properties of magneticmaterials by means of cavities into which small spheres of the mate-rial are introduced, and the coupling of two identical cavities by asmall hole in a dividing wall. In the last section some remarks aremade about the zero-point energy of empty space.

RésuméDans ces confërences, on donne un aperçu de la thëorie des ondesëlectromagnëtiques stationnairea dans des cavitës résonantes.On montre l'analogie formelle existant entre les modes de vibrationd'une cavitê, les modes de vibration d'un rêseau d'éléments distincts,et la vibration- d'un simple circuit LC. On aceorde une attentionparticulière à la théorie des perturbations, et cette théorie est ensuiteappliquée à un certain nombre d'exemples. Ceux-ci comportent ladétermination des propriëtës des matériaux magnétiques en hautefrêquence, au moyen de cavités dans lesquelles sont introduites depetites sphères du matériau, et le couplage de deux cavitês identiquesà l'aide d'une petite ouverture dans la paroi de séparation. Dans ladernière partie, on fait quelques remarques au sujet de l'énergie aupoint zéro dans Ie vide.

ZusammenfassungIn diesen Vorträgen wird ein Überhlick der Theorie der stehendenelektromagnetischen Wellen .in Hohlraumresonatoren gegeben.Die deutliche Analogie zwischen den Schwingungsarten eines Hohl-leiters, den Schwingungsarten eines Netzv. .zkes oder diskreterElemente und der Schwingung eines einfachen LC-Kreises wirdhervorgehoben. Besondere Aufmerksamkeit wird der Theorie derStörungen gewidmet und diese dann auf eine Anzahl Beispieleangewendet. Letztere schliellen die Bestimmung der Hochfrequenz-eigenschaften magnetischer Werkstoffe durch Hohlleiter, in welchekleine Kügelchen des Werkstoffes gebracht worden sind, sowie dieVerbindung zweier identischer Hohlleiter durch ein kleines Lochin einer Scheidewand ein. Der letzte Abschnitt enthält einige Be-merkungen über die Nullpunktenergie des leeren Raumes.

1. Introduetion

The theory of electromagnetic waves in resonant cavities is not in everyway a very modern subject. When Maxwell had formulated his equations

*) Course of lectures given on the invitation of the Consejo Superior de InvestigacionesCientificas, Instituto Nacional de Electr6nica, Madrid, November 1949. A Spanishversion of the paper has appeared in the annals of the Institufe.

THEORY 0 F ELECTROMAGNETIC WAVES IN RESONANT <;AVITIES 163

for the electromagnetic field and had shown that they possessed solutionsrepresenting electric waves it was a simple step, suggested by the acousticalanalogue, to study standing waves in closed cavities. As a matter of factsuch standing waves were discussed by Lord Rayleigh a long time ago.Yet it would last almost half a century until standing waves attainedpractical importance and cavities became useful instruments for theradio engineer. For many years' they belonged 'to the field of theoreticalspeculation and their study was one of the keystones of a great revolutionof physical science. The quantum theory of Planck, which led to the newquantum mechanics and to our revised attitude with resp'ect to the des-cription of nature, arose from a careful analysis of the problem of thermalequilibrium between radiation - described as a superposition of standingwaves - and matter. In spite of these spectacular successes many difficul-ties' remained to he solved and even to-day the problem of the interactionof the electromagnetic field and elementary particles constitutes a challengeto theory.

However, the notions of standing waves and resonant cavities havenow become daily tools for the electrical engineer who deals with theseone-time abstractions in the same matter-of-fact way as he deals withcurrents and voltages.In these lectures I shall try to give an outline of this theory. I shall

first point out the analogies and also the fundamental differences betweencavities and ordinary circuits. Next I shall show how in many cases theapplication of a simple perturhation method leads to valuable resultsespecially in the case of measuring-devices for measuring electric andmagnetic properties. In the final sections I shall come back to some ofthe problems of quantum electrodynamics.

2. The simple Le circuit

Let us first consider a simple Le circuit. Ifwe ignore damping, the energy,which is given by

remains constant and q will be given by

q = qoejw,l

with

The average values of the magnetic and the electric ener~y are equal:

tLi2= tq2fC = tW.During the oscillations electric energy is converted into magnetic energywhich is then converted back into electric energy, and so on.

164 H. B. G. CASIlIHR

If we introduce a damping resistance R, the equation for q becomes

L'q + Rq + q/ C= 0 ,

and so long as R/2L ~ Wo we have as an approximate solution

q ~ qoe-(RI2L)I+jW0t,

which can also be written as

withWil = R/2L.

We can derive this relation in another way. The average dissipation isgiven by the average of Ri2;hence

dW - R- R- = -Ri2 = --Li2 = -- W,dt L L

andW = Woe-(RIL)I = Woe-(RlwoL)wol.

Next we discuss the response of such a circuit to an impressed periodicvoltage. The corresponding differential equation is

L'q + Rq + q/ C = kejw1 ,

"with the steady-state solution

q= .L(w~ - (2) + jRw

Ifwe define the bandwidth Llw as the difference between those frequenciesta for which the amplitude of q is 1/{2 times the maximum value then, ifR ~ woL , one has

Llw = R/L.

The maximum amplitude, obtained for w ~ Wo is given by

Ikl' 1 woL .Iqresl ~ Rwo = woRC Cjkl =R Iqstatl ,

qstat = Ck

is the static response, which would be obtained at zero frequency. Equation(2.1) mayalso he formulated by saying that the impedance at resonanceis given by *)

(2.1)

in: which

(

WOL)-l woL -1 1Zres = R woL = (R) woc'

.) This is the impedance for Land C in series. In the case of parallel connection one hasZre. = QWoL.

THEORY OF ELECTROllfAGNETIC WAVES IN RESONANT CAVITIES 165

Let us define Q, the quality of the Le circuit, by

Q = woL/R.

We have seen that several important characteristics of the circuit aredetermined by this quantity:(a) The impedance at resonance

Zres = Q-1woL = Q-1/WOC;

(b) The amplitude at resonance

Iqresfqstatl = Q;(c) The bandwidth

.dw = wo/Q ;

'(d) The damping of spontaneous oscillations

W = Woe-w,tIQ.

3. More general networks

In the case of a more general network without damping the energy willhe a quadratic form of a number of charges qn and currents ÏJn, viz.

W = t ~Znm gn gm + t ~ Cnm qn qm •n,m n,m

Now according to a well-known mathematical theorem it is possible tointroduce new variables by means of the suhstitution

(3.1)

such that W is transformed into

W = t~vr2 + t~wr2 vr2 ,r r

and we have- .'"

The system behaves as a number of independent harmonic oscillators.If only one of these is excited so that v~) -=/::. 0 but v~) =0 for all n -=/::. p thenthe state described by

is called a .mode of vibration of the system.Damping is easily taken into accou:nt by calculating for each mode the

dissipation of energy *). Each mode of vibration has its' own Q and willdecay according to '

*) Strictly speaking the damping may slightly affect the transformation (3.1). It is per-missible to ignore this complication if the damping is small, i.e, if Q ~ 1.

166 H. B. G. CASUHR

and its bandwidth will be given by

Llwr = Wr/Qr .

The statements (a) and (b) of section 2 cannot immediately be generalized.In some cases the modes of vibration can be determined by symmetryconsiderations. Two examples will be discussed.For a combination of two identical Le systems the energy will have the

general form

and by the change of variables

qI = VI+ V2,

q2= VI-V2,(3.2) .

this is transformed into

so .that VI and V2 correspond to the two independent modes of vibration.In mode 1 the two systems are oscillating with the same amplitude and inphase; in mode 2 the amplitudes are equal but the phases opposite.

In fig. 1 is shown a cyclic arrangement of identical circuits. If thereare N systems the appropriate transformation is

(3.3)

Fig. 1. Cyclic arrangement of identical systems.

THEORY OF ELECTROMAGNETIC WAVES IN RESONANT CAVITIES 167

If N is even the case r = N/~ is of special interest. Then neighbouringcircuits are in counter phase. The frequencies of the various modes dependon the coupling but the transformation (3.3) is exactly valid, whateverthe coupling, so long as the symmetry is preserved.

4. Eddy currents

At high frequencieselectric currents are concentrated in very thin surfacelayers. The magnetic field will then be tangential to the surface and de-'crease according to

H· Hoe-(I+j)x/d,

where Ho is the value of the field at the surface, x the distance from thesurfacemeasured along the normal, and d the penetration depth givenby

,/ c2

d = V 2nawp,'

where p, is the permeability and a the conductivity. For copper at roomtemperature one has

d ~ 0·4tImicron

when .:t, the wavelength, is measured in centimetres.In a similar way the penetration law for the current is

• _. -(l+j)x/d~ - ~oe •

An interesting consequence of this skin effect is that currents can existon the inside of the wall of a closedspace, and this leads to a new typeof Le circuit.In fig. 2 is shown a "re-entrant" cavity. The shaded region can be re-

garded as a plane condenser and the current flowsalong the inner wall inthe way indicated by the arrows, so that one might say that it is flowingin a toroidal coil with only one turn. Another important exampleis thehole-and-slot system. In this case the self is formed by the currents on

66056

Fig. 2. Re-entrant cavity.

168 H. B. G, CASIllHR

the inner wall of the cylinder (for an infinitely long cylinder they give riseto a homogeneous field inside the cylinder] and the capacity is that of aplane condenser. We have as a rather rough approximation (l totallength,d width of gap, B depth of slot, 0 = nR2 area of bore):

C lB= 4nd'

4nL=-O

lc2 '

whence

LC 1BO, Ä l/BOc2 d 2n'= Vd'

The resonating system of the now famous multi-cavity magnetronconsists of eight such systems in a cyclic arrangement of the type discussedin section 3 and it is operated in the mode where adjoining cavities are incounter phase.

Fig. 3. Hole-and-slot system.

5. Resonant cavities

The theoretical considerations outlined in section 4 are only an approxi-mation: for many types of cavity they break down entirely because itis no longer possible to localize a self-inductance and a capacitance. Wehave then to use the complete Maxwell equations and to deduce formulaefor standing waves. These equations are

curlH =1 .-E,c

divE = 0,

1 .curl E = --H,

cdivH = 0,

with the boundary conditions

Etang = 0, Hnorm = 0.


These conditions are valid for perfectly conducting walls; the influence ofthe dissipation in the walls can he taken into account later on as a smallperturbation,, To find standing waves we put

leading tocurl H, = kEo,

curl E, = kHo'and hence to

LlHo+ k2Ho = 0,

LlEo+ k2 Eo = 0 .Further

div H, = 0,

div E, = 0,and

Ho,n= 0,

Eo,t = O.

Of course these equations are not all independent. If a solution Eo satis-fying (d), (f) and (h) has heen found then the field Ho derived by meansof (b) will automatically fulfil (a), (c) and the boundary condition (g).The total energy content of the state corresponding to the real part of

both Hand E is

where the integration has to be carried out over the whole of the cavity;it can he shown that

Since Hand E are in phase quadrature the magnetic energy reaches itsmaximu~ when the electric energy is zero and vice versa. In this respectthe state of affairs is analogous to the case of a simple Le circuit and opposedto that of travelling waves in free space where E and H .are in phase.If we call Ho,r, Eo,r the electric vectors corresponding to one definite

standing wave then the most general solution will he

H =j ~Vr Ho,r, E = ~Vr Eo,r ,with

(a)(b)

(c)

(d)

(e)

(f)

(g)

(h)

170 H. B. G. CASIMIR

These formulae are analogous to those of section 3. Again the system be-haves as a superposition of harmonic oscillators, but this time there is aninfinite number of such oscillators.

The energy dissipation for each of these oscillators can easily be worke~out so long as it is regarded as a. small perturbation so that we can applythe formula for skin effect taking for the tangential componen~ of H thevalue of Ho,t. Each Vr will decay according to

Vr ..:._ Vo r ej(wr' +jWr")1 = Vo r ej(wr' +iwr'/2Qr)l., 'The 'calculation gives

dW co -. d I I IHo,tl2dO"Tt =H ([ et2dn) do = - 2 HI IHol2dvww,

where the surface integral has to be taken over the entire boundary andwhere d is the penetration depth defined in section 4. For the quality we find

Q = 2 HI IBol2dv.dH 1Ho,tl2de

A very rough estimate of the order of magnitude of this quantity isobtained as follows. 'I'he field Hoand the tangential field Bo,t will be of thesame order. On the other hand the volume will be equal to the area multi-plied by a length D that is of the order of magnitude of the linear dimen-sions of the cavity. Therefore

(5.1)

Q ~Dld.

This shows that in the centimetric region high values of Q are easily ob-tained. In principle a cavity resonator leads to higher values than an Lecircuit and it is one of the attractive features of the centimetric region thatthere the resonant cavity is a practical proposition whereas for longerwaves the dimensions will soon become prohihitive.If a cavity is scaled down, that is, if all linear dimensions are reduced

by a factor lip, there corresponds to each solution Ho,r(x), Eo,r(x) a solu-tion Ho,r(Px), Eo,r(px), 'but its frequency will be higher by a factor p. Thevolume integral in eq.(5.l) is multiplied by a factor llpa, the surface inte-gral by llp~, and d is proportional to IIp'/·. Therefore Q is proportional toP'/I. For geo~etrically similar cavities the quality is proportional to thesquare-root of the linear dimensions and therefore to the square-root ofthe wavelength.

We mention two cases where explicit solutions are easily given. Fora cylindrical cavity .(of radius R) we shall only discuss solutions for whichthe only non-vanishing component of the electric field is Ez and where thisfield has moreover axial symmetry. We have .

THEORY OF ELECTROMAGNETIC WAVES IN RESONANT CAVI~IES 171

and, introducing polar coordinates,

E; +~ E~+ k2Ez' 0 ,r· .

. with the solution (of course to he finite for r = 0)

Ez = CJo(kr) .

The values of k are determined from the boundary condition

Ez(R) = CJo(kR) = 0,and this gives

k= 2'405 ,R

5'520k=--'- ,

R8'654

k=-'--,R

and so on.For a rectangular cavity with sides lx, ly, lz the complete set of modes

can be written down. If we put

7Ckz=-nz,

lz7C •

ky = ~ny,ly

(5.2)

.where nx, ny, nz are positive integers, and choose a vector e perpendicularto k,

(e. k) = 0,then a solution is

Ex = ex cos kxx sin kyy sin kzz ,

Ey = ey sin kxx cos kyy sin k~z ,Ez = ez sin kxx sin kyy cos kzz ,

corresponding to the frequency

OJ = c ik~ + k;+ k~.

There are two independent choices of e-vector for each k, unless one com-ponent of k is zero, when there is only one. (If k is parallel to one of thecoordinate axes no solution is possible.) The corresponding magnetic fieldis given by

1Hx = k (kyez - kzey) sin kxx cos kyy cos kzz ,

1 .H:y = k (kzex - kxez) cos kxx sin kyy cos kzz ,

1Hz = - (kxey - kyex) cos kxx cos kyy sin kzz •

. k .

(5.3)


6. Perturbation theory of resonant cavities l)It is often important to be able to calculate the change of the real and

imaginary parts of the resonant frequencies of a cavity when either thewalls are slightly modified or when a small object is introduced into thecavity. Suppose that this modification is confined to .a volume LI ~ V,bounded by a surface O'il' Let

with real Eo, Ho, be solutions for the unperturbed state. We write for thesolution in the perturbed state

leading to the equations

curl Ej = ok Ho+ koBl + okBl,curlHl = okEo+ koDl + okDl

(6.1)

(since inside LIboth the dieleciric constant and the permeability may bedifferent from unity we have to distinguish between Bl and Hl' and be-tween Dl and El)' We multiply the first equation by Ho, the second oneby Eo, add, rearrange, and integrate over space, Let us first integrate overV-Ll, that is, let us exclude that region where a change of conditions hastaken place. We find

ok IfI (IHoI2+IEoI2) dv = IfI -ko~(Ho·Bl) + (Eo·OlH dv +V-t.l V-J

+ I I I ~(Ho' curl El) + (Eo' curl Hl)( dv - ok I I I ~(Ho'Bl) + (Eo' Dl)( dv.V-J V-LI

In the integral on the left-hand side we mayalso integrate over the totalvolume: since the perturbation 'is supposed to be small ok·LlIV will bea quantity of higher order. Similarly the last integralon the right-handside can be neglected if we impose the condition that IBll,IDll shall neverbe much larger than IHol, IEol, and that they are small compared to IHol,IEol except in a region with a volume of the order of LI. This condition,which in most cases can easily be seen to be fulfilled, excludes certain casesof resonance.Because of the identity

div )[Ho X EJ + [EoX HJ( = -(Ho. curl El) '+

, + (El' curl Ho) - (Eo. curl Hl) + (Hl' curl Eo)we have


J J I HHo·curlEI) + (Eo·curlHI)dv =

= I I I ~(EI·curIHo) + (HI·curlEo)(dv- I I I div~[HoxEJ + [EoxHJ(dv =. = ko J J f ~(Eo. El) + (Ho· HI)( dv - J I HHoX El] + [Eo X HJ(n do ,

the surface integral being taken over the boundary of the cavity - whichgives no contribution because of the boundary conditions - and over 0"d.

Since outside 0"d we have El = Dl and HI = BI it follows that

if we designate by no the normal to O"d pointing away from LI, that is, theinside normal to the region V-Ll. This equation is important because itshows that the change of frequency is determined by the values of El' HIon a surface enclosing the perturbation so that entirely different agencieswill yet cause the same change of frequency so long as these values are thesame.

Next, let us integrate over the whole volume V.We make the additionalassumption that eq. (6.1) still holds, that is, that there are no currents norfree charges; of course this condition is not always fulfilled. But if it isfulfilled the integral of the divergence vanishes, and since El' Dl and HI' BIdiffer only inside LI there remains

éJk I I I (lHo12+ IEoI2)dv =- ko I I I ~(Ho·(Bl- Hl)) + (Eo· (EI-DI))( dv.Li

If Ho and Eo are with a sufficient degree of accuracy homogeneous insideLI, that is, if the dimensions of LI are small compared to the wavelength,then

éJk 4,:.0) (Ho. M) + (Eo· pH t) (Ho. M) + (Eo. P)~- ko = IIJ (IHoI2+ IEoI2)dv = ,2W '

where P and M are the total electric and magnetic moments. It should beemphasized, however, that the condition that Ho and Eo are homogeneousdoes not necessarily imply that El and Dl are homogeneous too. If a veryhigh # or e is found inside LI then El' Dl J?lay correspond to a much,shorter wavelength.

7. Applieations of perturbation tlieory

At first sight it might appear that we have, not made much progress.For we still have to find the values of El' HI. Fortunately approximatevalues for these can often be found by simple arguments. If for instancethe perturhation is a dielectric or magnetic sphere, small compared tothe wavelength (both inside and outside the sphere) then the electric andmagnetic moments are given by the well-known formulae

(6.3)


8-1 .p = -- R3E (0)

8 + 2 0 ,

where Eo(O) and Ho(O) are the values of Eo and Ho at the centre of thesphere; and eq. (6.3) leads at once to the desired result. For a metallicsphere of radius R small compared to the wavelength but large comparedto the penetration depth d we have

for such a sphere behaves, so far as the external field is concerned, like abody with e = 00 and p, = 0, and the argument based on eq. (6.2) showsthat again eq. (6.3) can be applied.

Next suppose that a small dent is made into the wall of a cavity. Insteadof this we imagine that a slab of matter with 8=00 and It = 0 is introducedinto the cavity. Then inside this slab B = 0, hence B1= -Ho' On the otherhand E = 0 (for D has to remain finite), hence El = -Eo. If the dent isnot only small but if also the angles with the original wall are small thenHl and Dl will be negligible, because of the continuity conditions for .thetangential component of H and the normal component of E. Therefore

~k If I (lHoI2-IEoI2) dV,1ko = I I I (IHoI2 + IEoI2) dv

(7.1)

This is a very useful formula for calculating the influence of small errorsin manufacture, accidental deformations or intentional displacements ofpistons or membranes on the resonance frequencies.

In our next example the application of perturbation theory is somewhatless straightforward. We consider two identical cavities coupled by a holein the dividing wall. Just as in the case of two identical Le circuits thereare two modes of the combined system, one in which the two halves arein phase (symmetric mode) and one in which they are in counter pbase(antisymmetrie mode) (fig. 4)·. In the symmetric mode the frequency isthe same as for the separated cavities: it is evident that the distributionof the field has not to be changed in order to fulfil the boundary conditionsapart from a slight inHuenée of the edge of the hole. In the antisymmetriecase, however, the E-lines have to curve away from the axis of symmetry'in the way shown in the figure in order to avoid "head-on collisions".

Q

Fig. 4. Electric-field patterns.660.58

.a = 4n (1 - ; ~ + ...).


Now let us consider a'third case in which there is only one cavity but wherea circular disc with e = 0 is inserted. It is easily seen that the lines of forceon the left-hand side satisfy exactly the conditions required in the antisym-metric case. Of course in order to obtain the complete field in the antisym-metric case the field on the right-hand side in fig. (4c) has to he reversed.This does not change the frequency. Now, so long as the hole is small.compared to the wavelength we can write at once

where N is the integralof IHol2 over one cavity. For a dielectric oblatespheroid of radius R, thickness 2h, dielectric constant e = 1 + 4n7G, andcoefficient of demagnetization a we have

4n y.IPllIEol = -R2h--.3 1+ ay.

In our case ,! = -1/4n, and if h<{;:.y.we can write

Hence

and

This formula has thus been derived with a minimum of computationaleffort.

We have not discussed the magnetic' field since on the axis this field iszero for the modes considered. In general, however, the field is not negli-gible and in order to satisfy the boundary conditions for the magnetic fieldwe have to give our disc also a f-t = 00.

We can now proceed as before bearing in mind that the magnetic fieldis now parallel to the disc and arrive finally at the result

8. Ferromagnetic resonance

The perturbation formulae derived above are often used for evaluating. the results of measurements on dielectric or magnetic properties of matter.

176 H. B. G. CASUllR

By means of these formulae it is possible to derive from the observed chang-es of the frequency the values of the electric and magnetic polarizations.As an example we quote the work of Beljers 2) on the magnetic propertiesof ferroxcuhe, a magnetic ferrite developed by the Philips company.Beljers used a cylindrical cavity in the mode where the wall ofthe cylinderis at the second zero of the Bessel function so that there is a cylindricalnodal surface at r = 0'435 R (fig. 5).

66059

Fig. 5. Cylindrical resonance cavity applied for gyromagnetic-resonance measurements").

At this place where the electric field is zero a ring of low-loss dielectricmaterial supporting a number of little spheres of ferroxcube can be insertedinto the cavity and we can now directly apply the formula

where Ho is the magnetic field at T = 0·435 R and M the magnetic momentwhich may he a complex quantity.

The cavity can be tuned over a limited range by displacing a cylindricalpiston. The influence of this displacement was found by direct calibration'but can also be calculated by means of eq. (7.1). The object of Beljers'swork was to determine the influence of an external static field (at rightangles to the high-frequency field) on the magnetic moment. A typicalcurve is reproduced in fig. 6. It will be seen that there is a very pronouncedresonance phenomenon. ,The physical explanation of this phenomenon isas follows. In an external magnetic field the magnetic moments of theelectrons carry out a precession with a frequency

ellWL=g-,

me


where ·eand m are the charge and the mass of the electron. In Larmor'sclassical formula g = 1, but g = 2 for a spinning electron without orbital'moment. Now if the frequency of a transverse field approaches.the frequen-cy WL or if, by changing the static field H the frequency WL is broughtclose to a.fixed frequency, then resonance absorption will occur. It may thusbe said that measurements of this type reveal very clearly the reality ofthe Larmor precession. It is very gratifying to see how this phenomenonformerly belonging to the more abstract field of atomic physics is nowwithin the reach of the technique of radio waves which is so mucb closerto our classical conceptions. The fact that the g-factors appear to be larger

!Jaf(11C/s)o

va

2,

, nI' ~: I

s I II ,!Jaf

V ~ I i 8I ,, ,7<

I \I \0 -r.II\.I

cl\ "-Cl r-,

0 I 4I \ ""I r--.. 3

\

d "V ,........ d....0. ~/ ... ..... ~

è!- --0 I iIt_ I

i"'--- ~ JV

\. I0 \ J

,

90

lJ1f(l1c~

4 o

J

V

60

50

2 oo20

V

o

400 500 Ixro3A/m

66060

Fig. 6. Gyromagnetic resonance 2) of five spheres consisting of "Ferroxcube 4" (diameter0·2mm) in a resonance cavity (fres= 9250 Mc/s). Measured detuning L1d + jL1d/2 for a

• variable, axially directed, magnetic field H.

co 200 300


than 2 indicates that orbital motion has an appreciable influence, andfurther accurate work of this kind will undoubtedly give valuable informs-tion about the mechanism of ferromagnetism.

9. The zero-point energy of empty space

In the introduction it was pointed out that the theory of standing wavesin cavities was already applied half a century ago to the problem of thermalequilibrium between matter and radiation and that it has led to the theoryof quanta. The basic idea of this application is to consider the radiationfield inside a cavity as a set of harmonic oscillators. If each of these oscil-lators would have the energy kT, the value holding in classical statistica,then the total energy content would be infinite, since there is an infinitenumber of oscillations. This difficulty was pointed out by Rayleigh andJeans; later it became known as the "ultra-violet catastrophe". Plancksolved the problem by assuming that an oscillator cannot have an arbitraryenergy content, but that the energy can only have the values 0, hv; 2hv, ... ,where h is a universal constant. Application of statistical mechanics thenleads to the result that the average energy of an oscillator is

lw .E - -;--;;--;;;---- e"v/kT -1'

which reduces to kT for kT~ hv but decreases very rapidly with increasing'11. The total energy content is given by

hvE = ~ iv/kT -1 '

where the sum has to be taken over all possible modes. It is easily seen thatthis sum is convergent. In problems of radiative equilibrium we are usuallydealing with waves of a wavelength much smaller than the dimensionsof the cavity and therefore with very high harmonics. In such cases thesum can be replaced by an integral

co hvE = f e'lvjkT -1 e(v)dv,

. 0

(9.1)

where e('JI)dvis the number of frequencies between '11 and '11 + dv, An ex-pression for e(v) can be derived from the value for the frequencies givenin section 5. We have

Y22 22 22Î2 2 2 nnx nny nnz2m = c 1 kx +ky + kz = C T + l; +.T·


The number of vibrations for an interval ~nx, dny, dnz is obviously2 dnx dny dnz (where the factor 2 is due to the possibility of two differente-vectors); the number of vibrations per dkxdkydkz is then

lxlylz V2 -- dkx dky dkz = 2 - dkx dky dkz•

n3 n3

Introducing polar coordinates in k-space and observing that only positivevalues of kx, ky, kz are possible, we find

n V' . V2 . - . - k2 dk = - k2dk

2 n3 n2

for the number of vibrations between k and k + dk and hence

8ne(v)dv = V - v2dv.

c3

For the radiation energy per frequency interval dv we thus find

8nV hv3E(v)dv = -- hikT dv ,c3 e v -1

which is Planck's famous formula.As is well known, a very similar theory has been worked out by Debye

for the specific heat of solids. Formula (9.1) applies also here but the. calculation of the distrihution of frequencies is more complicated and calls'for a thorough investigation of the modes of vibration of a crystallattice.However, Debye has shown that an approximate solution can be found bytreating the solid as an elastic continuum. In that case we have

e(v)dv = C v2dv,

where the constant C depends on the (transverse and longitudinal) velocityof sound. In order to obtain "the correct total number of vibrations, whichshould be 3N for a system containing N atoms, Debye introduces a maxi-mum frequency Vm determined by

t'V~C= 3N.

In modern quantum mechanics most of what has been said remains truewith one important exception: the energy levels of a harmonic oscillatorare no longer given by nhv but by (n + t)h'V, so that even in the loweststate the energy is not zero but! h», The existence of such a zero-pointenergy is intimately connected with Heisenberg's principle of uncertaintyaccording to which it is impossible that a particle is at rest at a definitepoint in space.


Let the energy of an oscillator he given hy

1 mw2E = _p2 + __ q2.

2m 2

The Heisenherg relation states that

LlpLlq F'::! h/2n,

where Llp, L1q are root-mean-square deviations of the momentum and thecoordinate. The lowest possible value of E will approximately he equalto the minimum value of .

_!_ h2

2(L1 )22m 4n2(L1q)2 + t mw q.

This minimum value is ohtained for

h(L1q)2 = __ ,2nmw

which givesEmin F'::! hv ,

How does this modification affect the theory of radiation and of specificheat? In the case of a solid we will have to add to the Dehye energy atotal zero-point energy

I'ms,= ~ ihv = ilt I vg(v)dv.o

Since the total number of possible frequencies is finite this expression willconverge. As a matter of fact the existence of this zero-point energy canto a certain extent he verified hy experiment and it is of great importancefor the theoretical interpretation of the difference in vapour pressure oftwo isotopes. Take for instance the case of Ne20 and Ne22• The fields offorce will to a very high degree of approximation he identical in Ne20 andNe22; therefore all frequencies will he proportional to MJ/" where M isthe atomic weight. Thus

This difference of 10% in zero-point energy leads to a difference in heatof evaporation and hence to a difference in vapour pressure. A closer analysisshows that this is the predominant factor, although there are other factorsthat should he taken into account 3).

But in the case of radiation the situation is different: here the expression~ lhv is wildly divergent and can have no direct physical meaning. We are


here touching upon one of the fundamental difficulties of modern theory.The standing electromagnetic waves in cavity resonators are almost asreal, tangible physical phenomena as the acoustical waves in solids; inthis latter case the existence of a zero-point energy is borne out by ex-periment, yet the application of this idea to radiation theory leads to im-possible results. Personally I do not believe that an entirely satisfactorysolution of this difficulty has been given. Has one to cut off at a certainlimiting frequency? Or has the zero-point energy to cancel other divergentexpressions arising from other types of field?However this may he, it is possible in some cases to obtain unambiguous

expressions for the difference in zero-point energy of empty space in differ-ent situations and to these differences we can give a physical meaning.From a mathematical point of view it may seem strange that anything. reasonable can he obtained by suhstracting two divergent series, but theprocedure can be made somewhat more rigorous in the following way.Let us introduce a convergence factor, for instance e-av and calculate

EI =~ ihve-av,(situation Ij

Ell =~ ihve-av.(situation lIJ

Iflim(E~-EII)~O

exists, then this limit will be interpreted. as an energy difference betweenthe situations I and n. The fundamental assumption in a reasoning ofthis kind is that the convergence factor is the same function of frequencyin situation I and in situation n. Of course this cannot he rigorously provedbut from a physical point of view it seems the only reasonable assumption.

The method outlined above can he applied to a number of problems,for instance, the theory of Van der Waals forces 5). Here I shall only discussone example, the attraction between two perfectly conducting plates 4).We take as situation I a cavity with a partition in the middle, as situationn the same cavity but with the partition at a distance a from one of thewalls. We have

EI = ~ ihvAe-avA + ~ ihvBe-avB = ~hvAe-aVA,A B A

The frequencies are given in each case by eqs (5.2) and (5.3). Now onefinds that if it were legitimate to replace the sums by integrals then

182 H. B. G. CASUliR

Further it can be shown that if a4;.l it is only for ~ that the discretenessc

of the sum has any influence at all. The obvious mathematical procedureis thus to replace the sums by integrals' throughout, but to apply the well-known Euler-Maclaurin formula for the difference between ~ and the

ccorresponding integral. The result of this calculation is

. n 1El - En = l2 -- he . -.

1440 a3

1 A D

1

a l-a

C 0

1

Fig. 7. C~vity with dividing wall in two different positions.

The interpretation of this result is that there exists an attractive forcethe value of which is given by

nIlF = he - - = 0·013 - dynefcm2 , (9.2)

480 a<1 a,}

where a", is the distance measured in microns. We may say that this forceis due to a zero-point pressure of the electroinagnetic waves.It is worth while to specify somewhat more closely the conditions under

which our expression fór the force should hold. We have assumed perfectlyconducting walls; on the other hand the main contribution to the forcearises from those waves for which the wavelength is of the order of magni-tude of the distance a. For these waves the assumption of perfectly con-ducting walls should be correct. In other words, our formula will be validwhen for waves with a wavelength of the order of the distance between thetwo plates, the penetration depth is small compared to that distance.For copper plates at a distance of Ill. this condition will clearly be fulfilled.Although the effect predicted hy eg. (9.2) is small it would not appear tobe outside the possibilities of experimental verification and this would bean interesting confirmation of our theoretical speculations,

Eindhoven, June 1950REFERENCES

1) J. Müller, Z. Hoèhfrequenztech. Elektronk. 54, 157-161, 1939.2) H. G. Beljers, Physica 14, 629-641, 1949.3) J. Haan tj es, Thesis, Leyden, 1936.4) H. B. G. Casimir, Proc. Kon. Ned. Akad. Wetenseh. Amsterdam 51, 793-795, 1948.5) H. B. G. Casimir, J. Chim. Phys. 46, 407-410, 194·9. .

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