25 FEB. 1974 .

PHI LIAi,·~~~:L~,b:JRESEARCHREPORTS

SUPPLEMENTS

PHILIPS RESEARCH LABORATORIES:hillps Res. Repts Suppl. 1973 No. 5led In the Netherlands

© N.V. Philips' Gloeilampenfabrieken, Eindhoven. Netherlands. 1973.Articles or illustrations reproduced, in whole or in part, must be

accompanied by full acknowledgement of the source:PHILlPS RESEARCH REPORTS SUPPLEMENTS

RADIATIVE TRANSFER BETWEENCLOSELY SPACED BODIES *)

BY

c. M. HARGREAVES

*) Thesis, University of Leiden, September 1973.Promotor: Prof. Dr H. B. G. Casimir.Co-promotor: Prof. D. Polder.

Philips Res. Repts. SuppI. 1973, No. 5

AbstractThe transport of radiant energy between dissipative media is modified-when it occurs in a space comparable to or smaller than the dominantwavelengths. After a review ofradiation-density size effects in isothermalenclosures, reasons are given for expecting size effects in non-isothermalenclosures. Calculations show that the radiative transfer by travellingwaves between two surfaces decreases as the surfaces approach oneanother; there is, however, also enegy transfer via the evanescent fieldand at very small spacings this increases as the inverse fourth power ofthe distance and soon completely dominates. The theory of Rytov andthe more general theory of Polder and Van Hove are described briefly.Proximity effects should begin to set in at distances le « hclk/I', inagreement with the deductions found earlier. Rytov's theory can beapplied only to dielectrics of high conductivity. For surfaces separatedby a large distance, only the Polder-Van Hove theory is consistent withthe Stefan-Boltzrnann law. At very small spacings the Z-4 dependence ofthe energy transfer is found. Next, the experimental method used tomeasure the radiative-transfer proximity effect is described. The follow-ing details are discussed: the emitting surfaces and the difficulties arisingfrom their small spacings (I'>J 1 fLm); thermallosses, spacing control andmeasurement, temperature regulation and measurement of the radiantflux. Finally, the results are given of measurements on chromium sur-faces near room temperature and down to 145K. The experiments atroom temperature confirm the Polder-Van Hove theory quantitativelyas regards the distance le at which the proximity effect sets in. There isalso substantial agreement with regard to the rate at which the radiativetransfer changes with distance. A shallow minimum calculated from thetheory, however, was not observed. The temperature variation of le isin agreement with the theory.

CONTENTS

1. PHYSICAL BASIS OF THE PHENOMENA1.1. Introduction. . . . . . . . . . . . . .1.2. The density of radiation in small cavities. .1.3. Radiative transfer between closely spaced bodies

2. THEORY OF RADIATION PROXIMITY EFFECTS.2.1. The theory of Rytov . . . . . . .2.2. The theory of Polder and Van Hove2.3. Comparison of the theories . . . .

3. APPARATUS AND METHOD OF MEASUREMENT3.1. The method of measurement .3.2. The emitting surfaces . . . . . . . . . . .3.3. Thermal fluxes and losses . . . . . . . . .3.4. Control and measurement of surface spacing .3.5. Temperature measurement and control3.6. Measurement of the mean radiant flux

4. MEASUREMENTS . . . . . . . . . .4.1. Measurements near room temperature.4.2. Measurements at low temperatures4.3. Discussion.

Appendix AAppendix B

References .

1.. 1

210

20202429

32323540485254

58606466

7477

79

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1. PHYSICAL BASIS OF THE PHENOMENA

1.1. Introduction

The intensive study ofthermal radiation at the turn ofthe century culminatedin Planck's quantum hypothesis and began a revolution in thought that hasdominated physics ever since. In these and subsequent investigations, the equi-librium density of radiation in a cavity was derived with due attention to thesize of the cavity but, once it had been established that size and shape playedno part when the dimensions were everywhere large compared to the wave-lengths, this aspect of the problem received no further attention. In this dis-sertation we examine another size effect related to this old problem, viz. theradiative transfer between closely spaced bodies.Size effects are inherent in any natural phenomenon with which a charac-

teristic length is associated. Observable effects may be expected whenever thephysical dimensions of the material system approach the value of the charac-teristic length. Examples are so abundant that one can say that a great deal ofphysics is concerned with the study ofnature's size effects. A few examples are:interferometry, Knudsen streaming in gases at low pressures, the onset of tur-bulence in fluid flow, the enhanced mechanical strength of small crystals("whiskers"), where wavelength, mean free path, vortex diameter and dis-location spacing are respectively the relevant characteristic lengths.In the case of thermal radiation, the characteristic length is, of course, a

wavelength, for example the wavelength of the maximum in the spectrum ofblack-body radiation, i.e. the radiation within a cavity in equilibrium with thewalls at a given temperature. If we examine the radiation inside a cavity whosedimensions are of the same order as or smaller than this wavelength, it wouldnot be surprising if we found a radiation density different from that given bythe Stefan-Boltzmann law and a spectral distribution differing from Planck'slaw. Although such effects were implicit in the various derivations of Planck'sformula, they received only passing remarks in a few early papers on the black-body problem and were nowhere discussed at any length. It is only quite recentlythat these questions have been treated in detail.In a non-isothermal system, where energy fiuxes are involved, we mayalso

expect size effects when the dimensions are of the same order as the dominantwavelengths. In particular, the radiative energy transfer may be expected todiffer from that given by the well-established laws for macro systems with dimen-sions much larger than the wavelength. The nature of these radiation transportsize-effects and their experimental demonstration form the subject of this dis-sertation.The first reference to a size effect in the transport of radiation is apparently

that of Rytov, whose work was published in 1953. He considered how the

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radiation from a half-space of a general lossy dielectric is modified by theproximity of a nearly perfectly conducting plane surface. The present work wasinitiated by quite other considerations, however, and especially by Sauer's cal-culation of the temperature corrections to the retarded Londen-Van der Waalsforce between conductors. Rytov's approach is more comprehensive but thetheory he gives is incomplete and somewhat restrictive; it has recently beensuperseded by a more general treatment by Polder and Van Hove.In the following sections of this chapter these various considerations are used

as a framework to discuss the physical basis of the radiation size effects in-vestigated here.

1.2. The density of radiation in small cavities

Planck's radiation formula 1),

8 'Tt '1'2 hvUv dv = -- dv

c3 exp (hvfkT)- 1(1)

expresses the spectral distribution ofthe energy density in a cavity in equilibriumwith the walls at temperature T.

The formula is valid only for a cavity large compared with the wavelengths,as Planck was aware. However he did not investigate this question and referencesto it in his papers are sparse and difficult to find. Early remarks by Planck,antedating even the Planck formula, refer to the difficulty of accurately defininga black body:

"Es ist wohl nicht überfiüssig, hier zu bemerken, daB es bis jetzt noch nichtgelungen ist, eine exacte Definition eines "vollkommen schwarzen" Körpers,welche sich auf beliebig lange Wellen anwenden läBt, aufzufinden".

"Die Schwierigkeit liegt darin, daB im allgemeinen FalIe die auf einenKörper auffallendende Wellen nicht unabhängig sind vom Körper selbst.Letzteres wird zwar gewöhnlich stillschweigend als zutreffend angenommen,es ist aber nur dann richtig, wenn die Wellenlängen verschwindend klein sindgegen die Krümmungsradien aller Körperoberfiächen" 2).Planck arrived initially 3) at his radiation formula by an interpolation be-

tween the Wien formula valid at high frequencies and the Rayleigh result validat low frequencies. Planck himself used the phrase "eine glücklich errateneInterpolationsformel" 3.4).Planck subsequently found 5) a broader physical basis for his result starting

from Boltzmann's statistical definition of entropy. Both here and in the initialapproach, Planck postulated that the radiation field at the frequency v andwithin the interval dv was in equilibrium with a set oflinear Hertzian oscillatorshaving frequencies within the same interval. In deriving the relation between

8 n v2u.dv = -- U.dv,

c3(2)

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the energy density u. of the radiation in this interval and the average energy U.of the oscillators,

Planck used an expression for the damping force on an oscillating charge (theradiative reaction) R = 2e2xi3c3, which is derived from the total rate of emis-sion of energy from a Hertzian dipole U = 2e2xj3c3, obtained by integratingthe Poynting vector over all directions. However, this expression is based onthe electric and magnetic fields at a large distance from the dipole, r» À,

where terms proportional to r-1 are dominant (e.g. ref. 6, p. 15). At smalldistances, I' ~ À, where terms in 1'-2 and r=? dominate, the same value of Uwould normally be calculated but in the presence of a boundary near r ~ À(such as a cavity wall) the field is distorted and therefore the value of Uwouldbe different. The radiation reaction R would then also be different and hencethe spectral density u. is no longer given by the expression (2)used by Planck.

The conditions necessary for the validity of (2) were no doubt recognized byPlanck but he does not explicitly give the condition r »,1. He states 7) (ex-pression (2) was obtained before 1900 and first used in an alternative deriva-tion of Wien's radiation formula):

"In einem von beliebigen elektromagnetischen Wellen durchzogenen Vacuumbefinde sich ein geradliniger elektrischer Resonator, dessen Eigenperiode einerim Verhältnis zu seinen Lineardimensionen grol3enWellenlänge entspricht unddessen Schwingungen nur duren Ausstrahlung von Energie in den umgebendenRaum, nicht durcb galvanischen Leitungswiderstand oder andere in seinem Innernwirksame, Energie consumierende Vorgänge gedämpft werden."Criticisms 8) ofPlanck's work led to other derivations ofthe Planck formula.

Debye's derivation 9) (1910), for example, made use of Rayleigh's asymptoticformula 10) for the number of possible modes of vibration in a cavity per unitvolume and per unit frequency interval, 8 n v2jc3• (Corrected by Jeans 11.12),Rayleigh's expression is equivalent to (2) found by Planck.)In Debye's derivation, as in the derivation of the Rayleigh-Jeans formula,

the lowest possible mode is necessarily limited by the cavity dimensions andthe question of (non)-excitation of modes of stilI lower frequencies simply didnot arise. Taking the asymptotic formula (2) to be correct, there can be no effectof the cavity dimensions. Consequently there is no discussion of any restrietionon the cavity size in the papers of Rayleigh, Jeans or Debye.In 1911Planck devised another route 13) to his radiation formula in which it

was supposed that only the emission from the oscillators took place discretely,the absorption being a continuous process. This is the theory given in the secondedition (1913) of his book 14). In this book there are two references to the size

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of the cavity. In the Introduction, sec. 2, p. 2, Planck states: "Throughout thefollowing discussion it will be assumed that the linear dimensions of all partsof space considered, as well as the radii of curvature of all surfaces under con-sideration, are large compared with the wave lengths ofthe rays considered".

In the last section of Part IV ofthe book, sec. 169, Planck gives a demonstra-tion of the Rayleigh-Jeans formula. Here the two conditions for the validityof (2) are stated explicitly:

2/-'JI» 1,c

2/-d'JI» 1c

(3)

However no reference is made to Planck's own formula in this connection andthere is no discussion of what happens when (3) are not fulfilled.

It is all the more curious that Planck did no investigate this question furtherbecause in thefourth edition ofhis book 15) he discusses the analogous problemoflattice vibrations in small crystals at low temperatures. Thesameproblemhadbeen treated independently by Schaefer 16). Effectively, the lattice vibrationsof long wavelengths that would otherwise contribute to the crystal energy arecut off in small crystals; this leads at low temperatures to effects that are inprinciple observable, for example a deviation from the Debye law for thespecific heat. These effects were first demonstrated *) only last year by Novotny,Meincke and Watson 17); their results have been explained quantitatively byBaltes and Hilf 18) along the lines suggested by Schaefer and Planck. NeitherPlanck nor Schaefer mention the corresponding case of a cavity containingelectromagnetic radiation. (Size effects in the transport of radiation also havetheir accoustical analogy, see 19).)

This brief review of the early papers on black-body radiation is not at allcomplete but reference must at least be made to the derivation of Planck'sformula by Bose 20) and to the many papers by Einstein on the subject 8.21).Neither authors discuss the influence of cavity size on the radiation density.

One may speculate on the reasons why the problem attracted so little atten-tion. Itmay be that a kind ofpsychological barrier had been raised in the nine-teenth century by associating radiant heat too strongly with optics, therebyendowing it always with wavelengths of the same order as visible light, that isto say "very much smaller" than common laboratory objects (such as thermalcavities).

Another reason may have been the lack of technical relevance. Until 25 yearsago only the fairly near infra-red was open to study and, since cavities of modest

*) Novotny et al. measured the (largely vibrational) specific heat of very small supercon-ducting spherical lead grains and found an enhancement over the bulk value that wasless than that predicted by the surface-term correction to the Debye theory. This deficiencyis due to the size effect of Schaefer and Planck.

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size yielded no anomalies, there was no reason to think about sizeeffects. Withthe development of far infra-red and submiIIimetre techniques, however, effectsbegan to emerge that proved to be due to the finite size of the cavities. A thirdreason is that theoretical attacks on problems of this sort had to wait onimprovements in the mathematical techniques (see, e.g. p. 490 of 22)).The effect of the shape of the enclosure, however, was investigated in the

limit v -+ 00. Lorentz 23) had drawn attention to the problem in the Wolfskehllectures of 1910: "Hierbei entsteht das mathematische Problem, zu beweiseiidaB die Anzahl der genügend hohen Obertöne zwischen n und n + dn unab-hängig von den Gestalt der Hülle und nur Ihrem Volumen proportional ist"("genügend hohen Obertöne" means: only those modes whose wavelengths arevery much smaller than the cavity). That the number of modes is indeed depend-ent only on the volume was proved by Johanna Reudler 24), a pupil of Lorentz,for three special cases, viz. parallelopiped, sphere and cylinder. The generalproof for any shape of cavity was given shortly afterwards by Weyl 25), againunder the conditions À. «cavity dimensions.For a cavity not everywhere large compared to the wavelengths, the asymp-

totic formula 8 'It V2/C3 for the mode density (per unit volume and per unitfrequency) of the electromagnetic field will certainly be inaccurate; Weyl hadconjectured for the scalar problem a first order correction ± ('It Sv/2c2) whereS is the surface area of the cavity. However, in the case of the electromagneticfield this correction turns out to be zero as shown much later (see below).The problem of the effect of the cavity shape on the distribution of eigen-

values *), generally known as Weyl's problem, is only one aspect of the eigen-value problem. Here we are concerned, rather, with the summation of certainfunctions of the eigenvalues, functions that may depend strongly on the lowereigenvalues for which the asymptotic formulae are not valid. The eigenvalueproblem is of importance in many branches of physics and much attention hasbeen paid to it in acoustics, vibrations of membranes and solids and in quan-tum mechanics. Many references are given by Clark 27) and Baltes 22). Muchof this work has appeared only in the last two decades. In 1939-1945 therewas an explosive development of microwaves, and a great deal of work wasdone on the eigenmodes of EM waves in cavities and waveguides. However,microwave problems were usually concerned with specific resonances in limitedfrequency regions. Few of these papers were concerned with the whole spectrumof EM modes in a cavity.An important application was the study by Casimir 2B) in 1948 of the cut-

off of electromagnetic vibrations in a cavity, with reference to the zero-pointradiation. He thereby derived an expression for the retarded London- Van derWaals force between perfect conductors at the absolute zero of temperature.*) The inverse problem has been stated by Bers, see Kac 26), as: "Can one hear the shape

of a drum?"

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This is the first macroscopie calculation of dispersion forces. The energy of thezero-point field is, of course, infinite because there is no upper frequency limit *)and each mode of vibration has the energy t lu», Nevertheless a finite energyresults by taking the difference of the total energy in two cases: when a per-fectly conducting partition lies at a large distance from one wall of the cavityand where it lies at a small distance *). Differentiation with respect to the par-tition-wall separation yields the force (per unit area)

he 7(,2 1p=---

240 [4(4)

The same result was later obtained by Lifshitz 29) as a special case of a for-mula for the force between arbitrary lossy dielectrics.

In the case of temperature radiation, a discussion ofthe influence ofthe cavity• size was given by Bijl 30). He considered the radiation inside a small cavity atlow temperature and showed qualitatively that equilibrium properties suchas the energy density and spectral distribution must be different from those ina macro cavity.In equilibrium, the partition function (the total number of states) is

z = ~ exp( -edkT),1=0

where the el are the total energy of each state, given by

n, nWI being the energy of the mode of frequency WI' Summing over the modenumbers ru, the partition function is found to be

IZ=IT-----

WI 1- exp(-nwdkT)(5)

and the free energy of the radiation is

F = -kTln Z = +kT ~ In [1- exp(-hwdkT)].0)1

(6)

For a given situation (temperature T, volume V) we can thus find F by

*) For real conductors of finite thickness there will be an upper limit at that frequency atwhich the conductors become transparent. Since we are only concerned with the differencein the radiation energy density in the two positions, we can omit from the calculation allfrequencies above this limit. This ensures a finite result.

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summing over all modes. From F, various equilibrium properties such as theenergy density can be calculated. Such properties will be independent of thecavity size and shape when the temperature is high enough: the probability ofexcitation of a mode of frequency w is given by (exp [nwlkT] - 1)-1 so thatfor large T many high-frequency modes are excited, i.e. those of wavelengthmuch smaller than the cavity dimensions. The energy of a mode being pro-portional to its frequency, these high frequencies effectively determine theenergy density and other properties of the radiation. Successive modes are veryclose together and the summation for F can be replaced by an integral.The situation is quite different, however, when the temperature is low or the

cavity small. The factor (exp [nwlkT] - 1)-1 is largest for lu» «kT, so atlow temperatures it is the lower frequencies i» «kTln that are principallyexcited, those of longer wavelengths. Now the (conducting) walls of the cavityimpose restraints on the lowest possible frequencies: the wavelengths of themodes in a cubic cavity of side L,

The longest possible wavelengths are thus of order 2Lln, where n = 1, 2, ...etc. Adjacent modes, corresponding to n = 1, 2, ... etc., have frequenciesof order v = elJ..= n e/2L or w = n n ell», so that their separation D..wis oforder n cjL. In order that the summation in (6) may be replaced by an inte-gral we must have the relative separation D..wlw« 1, i.e.

n elL 1--=-«1,n n cjL n

n» 1,

or alternatively,n elL «w «tatn,

orT» n e nik L.

This condition is equivalent to (3), mentioned by Planck (ref. 14, Part IV,sec. 169) in discussing the Rayleigh-Jeans formula. For T lower or L smallerthan given by (8), the summation must be retained and the energy density andother properties will deviate from the usual values.In general the energy density will be less than that in a macro cavity butthe

result now also depends on the shape ofthe cavity. Very recently these questionshave been further studied by Case and Chiu 31), Balian and Bloch 32,33,34)

and Baltes, Muri and Kneubühl 22,35,36,37). If the mode density is written asan expansion with Do(v) = 8 n v2 Vle3 as the first term (Vis the cavity volume),

8n VD(v) = -- v2 + a v + b,

c3

(7)

(8)

(9)

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where a is a coefficient proportional to the surface area and b a coefficientproportional to linear dimensions, a common result of the above-namedauthors 31.33.35.36.37.22) is that, for the electromagnetic field, the first ordercorrection (ex: surface area) vanishes in the limit 'V -+ 00. The reason for thiswas shown by Baltes and Kneubühl 22.37.38). The TE and TM modes havedifferent boundary conditions: each give the same surface area correction butof opposite sign.The total energy has been calculated by the authors of 22.31.35.38); for

example, Case and Chiu 31) obtained for the total energy in a cubic cavity ofsideL

:n;2(kT)4 :n;(kT)2 kTE= L3_ L+-+

15(lie)3 4 lie 2

lie 3k+ 15,7--- 1,64- + 0 [exp(-ÀLT)]L3(4:n;)2L 2ÀL

(10)

where À in the remainder term is the constant 4:n;klhc = 54·9cm=! K.-1 (not0·549crn=! K-\ as erroneously calculated by Case and Chiu *». The firstterm represents the Stefan-BoItzmann law. The first order correction term(ex: T3 and to the surface area) is zero, as it was in (9).This formula, however, is certainly not valid in the limit LT -+ O. It is

easy to show that for LT < 0·11cm K the first five terms of the formulayield a negative value for the energy. Baltes 39) pointed out that the formulawould also imply a non-zero entropy at T = 0, contrary to the third law.Baltes 39) has calculated the correct Iow-temperature value of the energy by

actually summing over the eigenvalues of effectively all modes, including thelowest, with the aid of a computer:

E = ~ (gnn1!21i e :n;/L) [exp (n1/21i e n/kT L)- 1]-1n~2

where gn is the weight of the eigenrnode w = n1/2 c nlL with n = n12 ++ n22 + n33, n, = integers (with at least two n, non-zero). For the lowestresonance of the cube the total energy is

E = (3. 21!21i e n/L) exp(-21/21i e n/kT L)

and this value of the energy falls off monotonically to zero as LT -)0 0, aswould be expected. The curve of E vs LT can be joined smoothly to the Caseand Chiu result (which is correct to 1% for LT> 0·3 cm K).

*) Also the values of LT in (41, a, b, c) of 31) are a factor 100x too large.

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Earlier, the energy density of temperature radiation in a cavity had beencalculated by Sauer 40) and also by Schram 41) in extending Casimir's result(4) to finite temperatures. Sauer considered a cubic cavity with a movable per-fectly conducting partition (fig. 1) exactly as in 28). Here, however, the freeenergy of the system is calculated by summing not only the modes of the zero-point radiation but also those of the temperature radiation.

x

.,L

~ ~L~-l~ _

Fig. I. Cubic cavity with perfectly conducting walls and partition.

The resulting force per unit area is

p = _h_c_17,_2+ _Ii _c17,_2_8~ n2 In [1 _ exp (_-_n) ] + _1i_c_17,_6_8_424014 P i__; 81 45

(11)

n=l

where 8 = kT/1i c 17,.The first term represents the Casimir force, whilst the two other terms are

the temperature correction of Sauer *). The term Ii C 17,6 84/45 represents theconstant I-independent pressure of the temperature radiation in the macrocavity (right ofpartition in fig. 1) and this adds to the Casimir term. The centreterm in (11) represents the pressure of the residual temperature radiation in themicro cavity and therefore works in opposition to the Casimir force (for finite81, this term is always negative). Because this radiation pressure is always lessthan that in the macro cavity (where cut-off effects are negligible), the tem-perature correction to the Casimir force is always positive.

For our purpose we require only the pressure p(I, T) of the temperatureradiation in the micro cavity as a function of I. We therefore drop the firstterm (net pressure of zero-point radiation) and the third term (temperatureradiation in the macro cavity) and get

*) The temperature corrections calculated by Lifshitz et al. 42) appear to be wrong. Also thecorrection given by these authors for imperfect conduction of the cavity walls is wrong 43).The temperature correction has also been considered by Fierz 44). I

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00

. Ii cn2 e ~ [ (-n)]p(l, T) = [3 ~ n2 In 1- exp Bi . (12)

n=l

Sauer has given approximations for the sum in (12) for the two cases, el «1and 4 n2 el » 1.For el «1, the sum is approximately given by -exp (-Ilel) because by

far the largest term in the expansion of the logarithm is obtained for n = 1.We get (the negative sign of p(l, T) arising from the expansion is now irrele-vant and is dropped):

Ii c n2 ep(l T) = -- ---, [3 exp (Ilel) . (I3)

For 4 n2 el « 1, the sum in (12) is transformed by Sauer using the Poissonsummation formula, and we obtain

Ii c n2e [n4 '(3) 1p(l, T) = -[3- 45 (el? - (2n)2 + 240 el +

- {(2n el)2+ 2 el+ 2~2} exp ~-4 n2 el)] . (14)

For 1-+ 00, (14) reduces to p = /i c n6 ()4/45 = 8 n5 k4 T4/45 113c3, whichleads to the Stefan-Boltzmann expression for the energy density e when weput e = 3p.

1.3. Radiative transfer between closely spaced bodies

It has been shown that in a cavity formed of two parallel planes with aspacing small compared to the dominant wavelengths, 1« Ii c nikT (see (8)),the radiation pressure is less than that in a larger cavity.

Consider now a non-isothermal cavity, e.g. two parallel plates at tempera-tures TA, TB, separated by a distance I. There is now a flux of energy betweenthe plates. Although we now have a non-equilibrium situation, there is still awell-defined energy density in the cavity, now the sum of two parts, one de-pending on TA, the other on TB' When I:1T= TA - TB is small compared toTA the energy density may be taken to be that corresponding to the meantemperature. For small I, the radiation pressure and therefore the density ofthe a-component (fig. 1) of the radiation will again be less than for I large.Since the flux of radiation from A to B must be related in some way to the

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energy density, it was speculated that this flux, too, would be less than normalwhen 1«/i c nikT.This argument leads to the apparent paradox that the radiative transfer would

decrease as the plates approach one another. However, there is a more com-prehensive way of regarding radiative transfer: in addition to travelling wavesthere is also energy transfer by what may be called direct coupling between thecharges in the two bodies. The random thermal motion of a charge can berepresented by a spectrum, each frequency of which can be regarded as anoscillating dipole. For the dipoles of high frequency the energy is radiated astravelling waves and absorbed by corresponding dipoles in the other body.For the low-frequency dipoles, however, we can no longer speak of waves.For metal bodies this is the case when it > 21~ 2/i c nikT. For dielectricbodies the condition is less stringent. The charges of the one body are nowsubject to the near field of the dipoles of the other; the consequent movementand damping of these charges represents a transfer of energy between the bodies.This mode of energy transfer has also been called tunnelling 45). One can alsosay that the internal reflection of plane waves at the interface dissipative me-dium-vacuum is modified by the penetration of the evanescent field into theother body, causing a sensible absorption there when the spacing is smallenough. Since the evanescent field falls off rapidly with distance above the sur-face, it is to be expected that this mode of energy transfer will exceed the con-tribution by travelling waves when the bodies approach each other very closely.At intermediate distances, however, it is not possible to regard these two modesof energy transfer as distinct. It is also not evident a priori that they are exactlycomplementary. Indeed, we may conjecture that, in principle, a minimum ispossible in the curve of energy transfer as a function of distance. Further dis-cussion of these matters is given in Chapter 2.As regards the energy transfer by travelling waves, a rough picture of the

behaviour to be expected can be given on the basis of Sauer's results (13), (14)for the thermal radiation pressure in a plane-parallel cavity (fig. I).We first consider an isothermal cavity with I «L, but 1 still large enough

for the radiation to be isotropic. All the standard expressions are then stillvalid. The total radiation density is given by the Stefan-Boltzmann law

4ae=-T4

,c

The radiation pressure is p = e/3.We now suppose the faces A, B to be at different temperatures TA' TB' If

D.T = TA - TB is small enough, (15) remains valid when we put T =

(15)

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(TA + TB)/2. There is now a flux of energy W from A to B:

W= a (TA4 - TB4) ~ 4 a T3 !lT (16)

or, if the surfaces each have the total emission coefficient e,

e I-Rw= 4a---T3 !lT= 4a--T3!lT

2- e 1+R(17)

where R = 1- e. The factor e/(2- e) = (1- R)/(I + R) arises from themultiple reflection and absorption at the surfaces.The above still refers to a macroscopie cavity. When the spacing I is now

decreased, the radiation pressure on the walls A, B decreases as shown bySauer. We now introduce the idea that the radiation transfer between the wallsalso decreases and does so proportionately to the pressure so that (17) becomes

1- R pel T)w= 4a--T3!lT '

1+R p(oo,T)(18)

where p(l, T) is the radiation pressure in the plane-parallel cavity of spacing Iat the mean temperature T, given by (13) and (14) and p( 00, T) is the pressurein a macroscopie cavity at the same temperature.Substituting for p( 00, T) = e/3 = 4 a T3/3c from (15) we get

1- R srW = 3c -- -p(/, T).

1+R T(19)

For BI« 1, p(/, T) is given by (13), so that (19) becomes

3n c k 1-R !lTW=-------

[3 1+ R exp (I/BI)(20)

For 4 n2 BI» 1, p(l, T) is given by (14), so that

3nck 1 - R [n4 1·202 1W = -[3- 1 + R 45 (BW - 4n2 + 240 BI +

- {(2n BW + 2 BI + 2~2}exp (-4n2 BI)] sr. (21)

For 1-+ 00, (21) should reduce to the usual expression (17) based on theStefan-Boltzmann law: we get

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1- R n4 e3w= 3--nck--tlT1+R 45

=-----tlT

which is (17), since Cl = 2n5 k4/15 h3 e2•

Formula (20) is valid for el « 1, say el ~ 0,2, or I ~ 0·2 /i e nikT. ForT R::j 300 K, this means I ~ 0·0005 cm = 5 !Lm.

For (21), el must be »1/4n2 R::j 0,025, say el ~ 0,2, i.e. I ~ 0·2 /i c nikT.For 300 K this gives I ~ 0·00048 cm, i.e. ~ 5 !Lm.

The part of the heat transfer W due to travelling waves only, calculated asa function of I using (20) and (21), is plotted in fig. 2. The value of the total

0·04,------------------,

t

0-01

°0~-2£-~4~-6~~8~-0~~~-'4~~~1 12 ~ 16 18- {(pm)

Fig. 2. Radiative transfer by travelling waves as a function of the spacing 1 between twoconducting surfaces at temperatures TA = 316 K, TB = 315 K, calculated from (20) and (21).The emission coefficients of the surfaces are assumed independent of the wavelength andequal in value, eA = en = 0'1. The distance at which the travelling-wave contribution dropsmost rapidly, 1* Ri 0·15/T, agrees well with the value found from the theory of Polder andVan Hove (see fig. 6 and ref. 47). The shape of the curve, however, is only roughly similar.

emission coefficient e (= 1- R) corresponds approximately to that of thechromium films used in the measurements and described in Chapters 3 and 4.The position 1*of the point of inflexion of the curve is obtained from (20) bydifferentiating twice and equating to zero. Only one of the solutions lies in therange ofvalidity et « 1, viz. 1*= 0'15/T,and this is close to the value 0'16/Tfound by Polder and Van Hove 47).

-14-

The calculation is naive in a number of respects. For example, at smalll it isno longer justifiable to assume that the emission characteristics of each surfaceremain unaffected by the proximity of the other surface. Furthermore, noaccount is taken of the variation of the reflectivity with wavelength, angle ofincidence and polarization. Nevertheless, the calculation gives a qualitativepicture of the energy transfer by travelling waves. It is given here primarilybecause it formed the starting point of this investigation.For a more exact account, a more sophisticated approach is necessary such

as that given by Rytov's theory 46) and the theory of Polder and Van Hove 47).These theories are discussed in the next chapter.When the surfaces are so closely spaced that the direct coupling between

charges becomes dominant, the way in which the energy flux depends on thespacing may be found by the following argument. We calculate the rate atwhich a single electric dipole (b) absorbs energy when situated in the near fieldof another oscillating dipole (a) and then integrate for the case of assembliesof such dipoles forming two parallel planes.The equation of motion of the dipole (b) in a field Eo sin t»t is

mx + yx + kx = q Eo" sin cot (22)

where q is the charge of (b) and Eo" is the x-component of Eo. It is assumedthat the damping coefficient y is small and that the absorbed energy is dissi-pated in some manner other than by re-radiation. With kjm = wo2 (woundamped resonant frequency), we find for the amplitude of vibration

so that

(23)

The damping force on the oscillating charge of (b) is R = -y x, so therate of absorption of energy is

d d- (R x) = -( -y x x) = -(y x x + Y x2).dt dt

The first term is zero when averaged over the cycle; the mean value of thesecond term y x2 thus gives the rate of absorption of energy by (b) at thefrequency w:

. Y w2 Y w2q2 Eo" 2

Wro•b = <yx2) = -2-IX/2 = 2 2 (2 2)2 + 2 2 2 (24)

m Wo - co y w

p sin Bsin [ro(t- rlc)] El .Eo= + =-smB

r3 2

(25)

-15 -

The exciting field Eo" is that due to the dipole (a) situated at a distance rfrom (b), fig. 3. For a Hertzian dipole of strength p, the near electric field atthe point r, B (where r «: clro) is (see e.g. ref. 6, p. 150)

2pcos B sin [ro(t- rlc)]Er = + = El cos B,,.3

where2p

El = - sin [w (t- "Ic)].,.3

We require to find <Eo,,), that is the average value of Eo", the componentof Eo along the direction x of oscillation of dipole (b). The averaging must bedone over all possible directions of p and of x. Referring to fig. 3 we have

E" = IEl cos u.

r

Fig. 3. Hertzian dipole of strength p at the point (a) and its field E at the point (r, 0) wherea dipole (b) is situated; x is the direction of oscillation ofthe dipole (b), Ex the field strengthin this direction.

-16 -

In the spherical triangle formed by the arcs u, v, IV, on the unit sphere ofcentre b we have the identity

cos u = cos v cos IV + sin v sin IV cos U

Er s,= - cos IV +- sin IV cos U,IEl IEl

so that

Ex = Er cos IV + Eo Sill IV cos U.

Inserting the values of Er and Eo from eq. (25),

Ex = El (cos fJ cos IV + t sin fJ sin IV cos U),

E/ = E12 (cos- fJ cos" IV + t sin? fJ sin2 IV cos? U ++ cos fJ sin e cos IV sin IV cos U).

In averaging this expression over all possible values of e, IV and U, the thirdterm containing cos U is zero. The averages in the first and second terms (theangles fJ, IV and U are mutually independent) are: (cos2 fJ) = t, (cos2 w) = t,(sin2 fJ) = t, (sin2 w) = tand (cos2 U) = t.

(26)

Hence the average value of Ex 2 is

and

(27)

The mean rate of absorption by dipole (b) of energy radiated by dipole (a)is then, from (24) and (27),

(28)

For the total rate of energy transfer this has to be integrated over the frequency., To perform this integration it is necessary to know how many dipoles thereare in the frequency interval dw at the frequency w. Our purpose, however,which is to find how the energy transfer varies as a function of the spacing, isnot materially served by such an integration: the behaviour is the same at allfrequencies for which w «cJr. We therefore write (28) in the form

1Ww•ab = F(y, w, q,p, m) -.

r6(29)

n2Ww,AB= n Ww,aB=f(y, w,q,p, m) 1

4• (31)

-17 -

Fig. 4. Assembly of dipoles such as (a) and (b) forming two parallel planes A and B.

We now imagine our two conducting surfaces to be represented by two planardistributions of the dipoles (a) and (b) as in fig. 4. The spacing I is assumed tobe very small compared to the linear dimensions of planes A and B. Supposethat there are n dipoles of frequency w per unit area in both surfaces. Then inthe annular elementary area at the angle 0( there are n r d« (I/sin a) 2n r cos a= 2n n 1'2 cot a de dipoles, all situated a distance r = I/sin a from the dipole(a). The total absorption of energy by plane B from the radiation of (a) aloneis therefore from (29)

1</22n n cos a sin" aWw,aB=F(y,w,q,p,m)! [4 d«

o

2n n 1</2= --F(y, w, q,p, m) I sin" a/41

~ 0

n=f(y,w,q,p,m)-.

14(30)

Each dipole (a) in plane A contributes such an amount to the total absorptionof plane B per unit area. Since there are n dipoles per unit area in plane A, thetotal energy flux from A to B per unit area at the frequency to is

This indicates that for r «c/O) = A/2n, where the near-field coupling isdominant, the energy flux between plane-parallel conducting surfaces should

-18 -

vary inversely with the fourth power of the surface spacing. Near room tem-perature where AmaxRj 10 {Lm, this implies that there should be an /-4 de-pendence at spacings «10/2n Rj 1·5 {Lm, say for l ~ t (Lm. The /-4 depend-ence at these spacings is confirmed by the theory of Polder and Van Hove 47)(see Chapter 2). .Another approach to the problem of the radiative transfer between closely

spaced bodies had been given earlier by Emslie 48). His calculation of thetravelling-wave component of the flux between two lossy dielectrics as a func-tion of the distance is based on interference between the electromagnetic wavesmultiply reflected between the two surfaces. Again, the simplifications of thecalculation are such that only a qualitative picture is obtained: for example,for dielectrics with complex refractive index given by n = 1,5, k = 10 (con-sidered as independent of A), a minimum is found at a spacirig Amax/3 and amaximum at Amax/20 (Amaxis the wavelength of the maximum in the black-bodycurve). Emslie also recognized that for very small spacings there must be energytransfer via the evanescent field.

Recently there have been further attempts to understand the problem of theradiative transfer between closely spaced bodies. A paper by Cravalcho, Tienand Caren 49), and a similar paper by Olivei S0) are concerned with the factthat the effective critical angle of a dielectric-vacuum-dielectric sandwich isincreased as the spacing is decreased. However, as it is assumed that the di-electrics are lossless, this work is not directly relevant to our problem.The radiation from a plane metal surface was considered by Boehm and

Tien 51) on the basis of a result of Fragstein 52) concerning the energy balancefor radiation (at normal incidence only) passing from one dissipative mediuminto another or into a loss-free dielectric. (See 53) for further references to thisproblem.) The effect on the emission of the presence of a second metal surfaceis then calculated as a change in the effective critical angle of the systemmetal-dielectric-metal as a function of the spacing. However, each of these calcula-tions excludes the other. The existence of a critical angle is irrelevant when onlynormally incident waves are considered. Quite apart from this, in a thermallyexcited medium all directions of propagation are involved, so that Fragstein'sconsiderations are inadequate, as pointed out by Rytov (see ref. 46, pp. 24-28).Domoto, Boehm and Tien53a) also performed some experiments to detect theproximity effect at very low temperature.

Caren 54) has calculated the energy density in a small conducting cavity asa function of one ormore dimensions in the cases of a cube, a long parallelopipedand a squat parallelopiped. In the latter case, geometrically equivalent to finiteparallel closely spaced surfaces, he finds that the total energy density increasesas the surface spacing decreases. Using this as a basis, Caren derives an ex-pression for the "heat absorbed by a cavity wall" but he does not make clearwhat he means by this phrase for an isothermal cavity.

- 19-

In a second paper SS), the same method is used again to obtain the energydensity in an isothermal cavity (closely spaced parallel planes as above). Sub-sequently the two walls are put at different temperatures and it is argued thateach wall emits and absorbs at the same rate as if each formed part of an iso-thermal cavity. The net heat flux is then taken to be the difference in the rateof emission (or absorption) of the two walls expressed in terms of the energydensity of isothermal cavities at temperatures equal to those of the walls. Theargument is rather similar to that used in our naive calculation above (p. 11).However Caren ignores the fact that only those components of the Poyntingvectors normal to the walls give contributions to the net heat flux. The netheat flux is therefore not proportional to the difference in total energy densityof the two isothermal cavities but only to the difference in that part of theradiation density corresponding to the radiation component normal to the twowalls.

-20-

2. THEORY OF RADIATION PROXIMITY EFFECTS

Rytov's work on the problem of the radiative transfer between closely spacedbodies 46) was published in 1953. Rytov's theory gives an expression for thespectral density of the energy transfer at the frequency al as a function of theseparation between the surfaces of two parallel infinite half-spaces. The theoryis restricted to the case of an arbitrary dissipative medium emitting to a nearlyperfect conductor.

The theory of Polder and Van Hove 47) is more general, since the propertiesof both media are quite arbitrary. The theory is basically similar to that ofRytov but differs considerably in treatment.Both theories are based on the fluctuation-dissipation theorem although, in

the form used by Rytov, this was merely a microscopie generalization of theNyquist-formula. The final form of both theories is such that they both requireto be evaluated numerically.We give here only a bare outline of these theories, followed by a comparison,

some numerical calculations and an extrapolation of the Polder-Van Hovetheory to large distances where agreement with the Stefan-Boltzmann law hasto be found.

2.1. The theory of Rytov

We consider a medium A at temperature TA occupying the half-space ofnegative z-values and bounded by the xy-plane at z = o. A second medium Bat temperature TB occupies the half-space z > I. The space between A and Bis vacuum. We first consider only the radiation originating in A.To Lorentz's form ofthe Maxwell equations for a general dissipative dielectric,

Rytov adds a source term, the fluctuating electric field K (as introduced byMrs. de Haas-Lorentz 56) in 1913, by analogy with the Langevin force 57) inthe theory of the Brownian movement). Magnetic losses are assumed to beabsent so that a corresponding magnetic source field does not arise. To thefields K generated by the thermal agitation of the medium, there correspondfluctuating currents - the sources of the thermal radiation in the medium.In order to calculate the energy flow at an arbitrary point it is necessary to

calculate there the time-averaged Poynting vector. This means calculating themean value of E xH, the product of the net fluctuation fields set up at the pointby all the fluctuating sources in the medium.The values of the fluctuation fields averaged over all sources, as well as their

time averages are zero. It wou1d therefore be necessary to calculate all localand instantaneous values of E and H due to each source and perform theaveraging on the product E XH and then add all the results, an impossiblycomplicated and lengthy operation.The averaging in time and over all the separate sources is facilitated by

- 21-

Fourier transformations. The time-variation of the field K is transformed tothe spectral representation in which each frequency component contributesseparately. The source field is characterized by its space-correlation in themedium. Since the separate thermal sources emit quite independently of oneanother, their correlation may be described by a c5-function. Rytov gives thefollowing expression for the correlation product of two sources separated bya distance ,.:

(Ka (r') s, (r") = C c5aP c5(r)

where C is a constant; c5aP = 1 for ex = {3 and otherwise 0 (ex and {3 are vectorcomponents of K); and c5(r)= 1 for r = 0, otherwise O.The strength of the source field is contained in the constant C, whose value

is found by an appeal to the known results of Kirchhoff's law and Planck'slaw (ref. 46, pp. 30-35); in another publication 58), C is derived from amicroscopie generalization of the Nyquist formula:

(1 1) (1),C=4:n: -+ Im --2 exp (hw/kT) - 1 8 - 1

4:n: ( 1 )-)- Im --exp (hw/kT) - 1 8 - 1 .

The constant term t represents the fluctuations set up by the zero-point radia-tion field. This is independent of temperature and gives identical contributionsto the radiative transfer between the two media; in the present problem it maytherefore be disregarded. The mean square value of the fluctuation field K istherefore dependent on the Planckian function and on the imaginary part ofthe dielectric constant. .

In fact, the geometry of the problem (media occupying half-spaces) makesit expedient to express also the spatial distribution of the source field K in termsof Fourier integrals. The calculation of the mean product E xH at some arbi-trary point therefore also involves integrations over spatial Fourier compo-nents.The Maxwell-Lorentz equations containing the source field K are solved

subject to the boundary conditions to yield values of E and H at an arbitrarypoint due to a single frequency component and a single spatial component ofthe source field K. The time-averaged Poynting vector at that point is thenobtained by integrating the product E xH over all frequencies and all spatialcomponents. In the problem of the radiation between media A, B, occupyingtwo half-spaces it is convenient to make the calculation for a point in thevacuum between the media. The time-averaged Poynting vector then representsthe energy transfer from A to B; the radiation from B to A is in the fust instance

-22-

neglected because, under the simplified boundary conditions used by Rytov(ref. 46, pp. 45, 73), the conductor B is assumed to be of high conductivityand therefore of low emissivity. However this reason is not adequate to thecase of the calculations given below where, for simplicity in the comparisonswith the Polder-Van Hove theory, we assume that the medium A is also ofhigh conductivity. As an alternative, we may (as Rytov states) first assume themedium B to be at such a low temperature that its own emission is negligible(while it is assumed to retain the same optical properties); later we calculatethe radiation B --+ A whilst the temperature of A is assumed to be very low.

Rytov's result, giving the Poynting vector for the spectral density of theradiation from A to B at the frequency w, is

(32)

1</2

(!1-2 cos" 8 sin? 8 + lal2 )~=f + X

1!1-cos 8 cos ~+ ia sin ~12 lie cos 8 sin ~+ a cos ~12o

X (a + a*) cos 8 sin 8 d8,

X (b + b*) cosh 1jJ sinh 1pd1p,

e Ii w3 1where Iow(A) = - -- ,

4n n2 e3 exp (liwlkT)- 1

T = temperature of medium A,aB = conductivity of medium B,11 = contribution due to travelling waves,12 = contribution due to evanescent field.

The quantities in the integrands refer to the medium A:

!1- = magnetic permeability --+ 1 for frequencies in the near infra-red,

= kl cos 8,= spacing A-B,= (e!1-- sin2 8)1/2,

I'YJ = klsinh1p,k = 2nl). = «[c,b = (e!1-- coshê 8)1/2.a

The result (32)is obtained by using at the surface B the simplified boundaryconditions referred to earlier. Essentially they are a statement of the fact that

-23 -

the magnetic field at the surface of a good conductor (highly developed skineffect) is substantially identical to that at the surface of a perfect conductor.A good conductor in this sense is one in which the imaginary part of the di-electric constant dominates i.e. e = e' - ie" R;! -ie", as is the case at lowenough frequencies when e" = 4n ao/w where ao is the conductivity (analternative statement of the Hagen-Rubens relation).In order to calculate, on the basis of Rytov's theory, the net radiative transfer

between two half-spaces A, B, both at considerable (i.e. not low) tempera-tures TA' TB' it is necessary to perform all the integrations twice, first puttingTB = 0 and then with TA = O.In the second case the medium B is the emitter,whilst A must now be the nearly perfect conductor in order that the simplifiedboundary conditions may be applied. The numerical calculations have there-fore been made on the basis of media A and B with identical properties andpurely imaginary dielectric constant obeying the Hagen-Rubens relation in thefrequency region of interest. For temperatures near 300 K, this implies fre-quencies up to '" 1014 S- \ i.e. wavelengths larger than '" 3 (Lm.Although realmetals do not obey the Hagen-Rubens relation down to quite such short wave-lengths, this need not be a deterrent to making the calculations on this basisfor the purpose of comparison with the Polder-Van Hove theory and with theradiative transfer at large distances calculated by standard methods.Figure 5 shows the results from Rytov's theory with e R::J -ie" = -i4n ao/w

Fig. 5. Radiative transfer between parallel surfaces, at temperatures TA = 316 K, T» =315 K, as a function of the spacing, calculated from Rytov's theory 46) for identical media,with e = e' - ie" Ri -ie" = -i4'TIXJo/w and ao = 7.1016 s-1; SI contribution due totravelling waves; S2 contribution due to the evanescent field; S total radiative transfer,SI + S2; Woo(G) radiative transfer at large distances calculated from Stefan-Boltzmann lawand assuming reflectivities depending only on the wavelength; Woo(I) radiative transfer atlarge distances taking into account also the angular dependence of the reflectivity.

w(mW/cm2]

t 0·10

0·08

\_

0·06

Wm(I)=0·0182Wm(G) = 0·0146

0·04

0·02

%~~*2~--4~~~6~L-~8~-{(pm)

-24-

and (10 = 7.1016 S-l. Initially the integrations were done after reduction of(32) to real quantities, on a program devised by A. K. de Jonge, who after-wards checked the calculations on a program written to accept complex quan-tities.

2.2. The theory of Polder and Van Hove

In the theory of Polder and Van Hove 47), the thermal agitation isregarded as acting directly on the free and bound charges whose randomfluctuations constitute the sources of the thermal radiation. The geometry isthe same as that considered above, two media occupying half-spaces, boundedby the planes z = 0, z = l.The problem is again to calculate the time-averaged Poynting vector at some

convenient point due to all the random thermal currents j in the medium. Asin Rytov's theory, the time-variation of each source is represented by its Fouriertransferm. The mean square strength of each frequency component is given bythe fluctuation-dissipation theorem 59.60) in the form

S" (X, m) ho:(ja (x., m)jb * (x2, m'» = c5(m- m') c5(x1- x2) c5ab,

2:n: exp (Iim/kT)- 1

where s"(x, m) is the dissipative part of the complex dielectric constant (afunction of m and in generalof position), c5(m- co') expresses the fact thatsources of different frequencies are uncorrelated, c5(xl- x2) that there is zerocorrelation between sources at different locations and c5ab(a and b are anytwo vector components) that there is no correlation between current compo-nents not parallel to one another.The spatial variations of the thermal currents in any plane z = z1 are

transformed with respect to the pairs x +t kx, y +t k.; the distribution oflocalized currents is thus represented as a pattern of periodic current wavesof all wavelengths in all directions in the xy-plane, characterized by the wavevector Ikl = (kx 2 + k/)l/2. The field at an arbitrary point x due to all thesources in all the planes is given by integration of the field

E (kx, ky, z, m) exp (i [mt- kxx- kyY])

with respect to kx, ky, z and m, where

E(kx, ky, z, m) = J dz e(kx, ky, z, z', m) j(kx, ky, z', m)

and the tensor e has to be found by solving Maxwell's equations subject tothe boundary conditions of the problem. The magnetic field H at the point xis found similarly. The time-averaged Poynting vector S, at the point x is thusobtained (equation (9) in the paper of Polder-Van Hove 47».

-25 -

The fields at x involved in Sz, are set up by the thermal currents. Theytherefore certainly have the same frequencies as the corresponding currentcomponents and they also have the same wave vectors k" in the x-directionas the source currents in the xy-plane. (We disregard k; for the moment.)The fields therefore contain the factor exp (iwt) exp (-ik"x) and they repre-sent plane damped electromagnetic waves in the medium, such as

E = Eo exp (iwt) exp (-ikxX) exp (± ikzz)

obeying the dispersion relation k/ + k/ = ew2je2, the imaginary part of eaccounting for the damping. The propagation vector in the z-direction is thenkz = (ew2je2 - k,,2)1/2 so that, in the space z > 0 above the medium A,where e = I (vacuum), kzv = (w2je2_ k,,2)1I2.

Consider now the field components of frequency w: for those spatial com-ponents with le; < «[c, kzv is real and there is propagation of travelling waveswith a component in the z-direction.For spatial components of larger wave vector, k" > «[c, however, kzv is

imaginary and there is no propagation in the z-direction; there is only anevanescent field that protrudes above the surface and drops off exponentiallywith distance. It is just these waves in the medium, which normally do' notcontribute to the energy emitted from the surface, that are responsible for theenhanced transfer of electromagnetic energy when the evanescent field is dis-turbed by the proximity of a second absorptive medium. For a given frequency,the above criterion separates the energy transmission into two categories: thatcarried by travelling waves (denoted wsln by Polder and Van Hove) and thatcarried by the exponentially damped evanescent field (denoted WexP). Thesituation is closely analogous to optical total internal reflection but the criticalangle is a complex quantity because of the dissipative nature of the medium.We now state the result of Polder and Van Hove giving the net differential

energy transfer between identical media A, B, with parallel surfaces separatedby a distance I:

w = wsln + wexp

I 00 èJ { hso }= (2n)2 f èJT· exp (/iwjkT)- I X

o

role 00

X {J k" (tilsIn + t.isln) dk" + J k" (tllexp + t.iexP) dk,,}dw (33)

o role

where

-26 -

(1- RII)2t sin - _

II - 11- Ril exp (i [PII- 2kzvl])IZ ,

(1- RJ)2(Lsln = -----------

11- R_t_exp (i [p_t_- 2kzvl])IZ ,

lekzv - kzl2

Ril =ekzv + kz '

Pil = 2 arg [(ekzv- kz) (ekzv + kz)*],

P_t_ = 2 arg [(kzv- kz) (kzv + kz)*].

PII/2 and P_t_/2are the phase jumps on reflection at each surface, 2kzJ is thephase angle associated with each traverse of kv to and fro across the gap I.

Further, with" = ikzv,1- cos XII

tllCXP = ,cosh [2" (1- <511)] - cos XII

1- cosX_t_t_t_cxp = ,

cosh [2" (1- h)] - cos X..L

ekIJll = I~"e + kz 12= 1- ekzv + kzl2,txe - kz - ekzv - kz

e><Ó.L = I~"+ kzl2 = I-kzv + kzl2,1" - kz -kzv - kz

XII = 2 arg [(-ir.e + kz)(-ir.e- kz)*] = 2 arg [(ekzv + kz) (ekzv- kz)*],

X..L = 2 arg [(-ir. + kz)(-i,,- kz)*] = 2 arg [(kzv + kz) (kzv- kz)*].

Polder and Van Hove have indicated the structure of their result by makingapproximate analytic integrations for the case of identical media with

e' «e" = 4:n:ao/w, ao = 7 . 1016 S-l.

Their curves, showing the various contributions to W as functions of I, arereproduced in fig. 6.Numerical calculations for the same dielectric constant and temperature have

been made with an existing program prepared by M. Van Hove. The resultsare plotted on a linear scale in fig. 7 to facilitate comparison with the Rytovresults.When the spacing between the surfaces is increased indefinitely, 1- co,

the net energy transfer should agree with that found from the Stefan-Boltz-mann law. For 1= co, it can be seen that tllCXP and t_t_cxp in (33) become zero.

w Ier(mW/cm2) 2

tla

la

-27 -

i04

10-4 10-3 10-2 10-' I la 102_____ ( (pm)

Fig. 6. The various contributions to the radiative transfer between parallel surfaces as func-tions of the spacing, calculated by Polder and Van Hove by approximate integrations of(33).The curves are taken from fig. 6 of the paper of Polder and Van Hove 47); the energy fluxis here expressed in milliwatts per square centimetre. The media have the same dielectricproperties and the same temperatures as for the Rytov calculations of fig. 5. The distance1* at which the gradient of the travelling wave contribution is a maximum is 1* = Q'163/T,see 47).

The transmission coefficients tilsIn and (Lsln remain finite but become independ-ent of I when averaged over the phase angle 1p:

Similarly; t_lsin = (I - RJ)/(l + R_l). These values are the same as those ob-tained simply by summing the energies absorbed at each surface as the radia-tion passes to and fro between them. This is just a restatement of the fact thatit is no longer necessary to add amplitudes, taking account of relative phases,when the interference peaks have becomes so close to each other as to formeffectively a continuum.

Restricting the argument to not-toa-high frequencies (w «wo), where theapproximation I el R:i I ie"l R::i 4n ao/w is valid, and writing Iiw/kT = x,Polder and Van Hove obtain from (33) and (34)

4 ( 1 )5/2 et) x9/2 eX k9/2 T7/2W =- - f dx----

et) 3 2n (eX_l)2 1i7/2c2 ao1/2

o

1

The integral has the value 55'2, giving

T7/2

Wet) = 93--.ao

(34)

(35)

(36)

- 28-

0·1

Woo ([)=O.0182\._

Woo (G) 0'014.5T

W(mW/cm2)

008

i '0'05

0·04

~w sin

002

Wexp

00 2 4 5 8 20la 12 14. 16-[(pm)

18

Fig. 7. Radiative transfer as a function of the spacing under the same conditions as fig. 6but integrated numerically from (33) and plotted linearly to facilitate comparison with fig. 5;W",,(G) and W",,(I) as in fig. 5.

This is in agreement with the Stefan-Boltzmann law provided that theeffective transmission coefficient terr is proportional to Tl/2/aol/2. We shallnow show that this is indeed the case.

We had e" = 2nk = 4n ao/w. For e' = n2 - k2 «e" = 2nk, n2 ~ k2 ~

~t 4n ao/w = 2n ao/w, which for metals in the infra-red is usually muchgreater than unity. Therefore we can estimate the reflectivity as

(n- 1)2 + P n2 - 2n+ n2 n - 1 2R~ ~ =---~1--

(n + 1)2 + P n2 + 2n + n2 n + I n

(the Hagen-Rubens relation) and the effective transmission coefficient becomes

terr = 1- R = ~(1- ~)-l~~~(~)1/21+ R n n 11 2n ao

The value of the effective transmission coefficient thus depends on determiningan effective wavelength corresponding to the temperature T. One way of doingthis is to take the wavelength corresponding to the maximum *) in the energy

*) It is to be noted that Wien's law describes only one (arbitrary) possibility, viz. the wave-length at which the energy per unit wavelength interval is a maximum, as noted by Czernyand Röder 61), Boerdijk 62) and others 63). Boerdijk shows that the wavelength at whichthe energy per unit frequency interval is a maximum is given by Ämox(v) T = 0'51; or, ifwe consider the energy per relative interval dÄjÄ or dv/v, .Ämox(rel) T = 0·37 in both cases.None of these Ämox deserves to be preferred above the others (the popularity of the firstis due only to its priority and possession of a name). In fact, none of the Ämox have anyspecial claim to be characteristic of T; for example, Czerny 64) showed that t of thetotal energy lies at wavelengths shorter than Ämol().)'

-29 -

vs wavelength curve of equilibrium radiation in a cavity at temperature T, asgiven by Wien's displacement law, Àmax T = hc/5k = 0·29. We then getterr = 3,2.105 T1!2/(JOl/2.The Stefan-Boltzmann law gives a radiative transfer of (J (TA 4 - TB4) terr

and for I:l.T/T «1 a differential transfer of 4(J T3 terr, i.e.

WAB = 4(J T3 x3·2. 105 Tl/2/(JOl/2T7/2

R3 73--.(Jo1/2

(37)

Exact agreement with (36) cannot be expected in view of the arbitrary natureof our choice of Àerr. The qualitative agreement between (36) and (37) showsthat the result of Polder and Van Hove at large distances is consistent with theStefan-Boltzmann law. The numerical calculations of sec. 2.3 will show thatthe agreement is in fact very close.

2.3. Comparison of the theories

The foregoing calculations using Rytov's theory (fig. 5) and using the theoryof Polder and Van Hove (fig. 7) can be compared directly. Both refer to sur-faces of dielectric constant e R3-i 4n (Jo/w, with (Jo= 7 . 1016 S-l, a ratherrough approximation to the dielectric properties of bulk chromium (see ref.47, fig. 8). Within this limitation the two theories differ in the following re-spects.(1) At large distances the Rytov result W<X)(Ry)is about twice that of Polder-

Van Hove W<X)(P-v.H).(2) The minimum of the Rytov curve is more pronounced than that of Polder-

Van Hove.The Polder-Van Hove theory is certainly correct at large distances, since it

agrees closely with standard calculations using Planck's law and a transmissioncoefficient based on Fresnel's formulae, whereas Rytov's result at large dis-tances is about twice this value. It has not been possible to localize in Rytov'swork where this discrepancy arises. Our experimental results (see chapter 4)will be compared only with the Polder-Van Hove theory.The standard calculation of the radiative transfer between two surfaces

separated by a large spacing is often simplified by assuming the reflectioncoefficient to be independent of the angle of incidence. In the present casehowever it proves necessary to take into account the angular dependence ofthe reflectivity.

For e R3 -ie" = -i 4n (Jo/wand (Jo = 7 . 1016 S-l, we have

n R3 k R3 (e"/2)1/2 »1,

~~-_._----- ._--~~~~~~~~~~~~~~~~~~~~~~~-

- 30-

and the reflectivity at normal incidence is

s; = [(n- 1)2 + k2]/[(n + 1)2 + k2] ~ (n- 1)/(n + 1).

The transmission coefficient for the net radiative transfer between identicalsurfaces, if it is assumed that the reflectivity has the same value Ro at all angles,is t = (1 - Ro)/(1 + Ro), see (17). Just as Ro, t is a function of co. It mayalso be written as a function of .il. For the above e" we calculated values oft(.il) that could be closely represented by

(38)

0·00806.il2 + 0·13105.il- 0·1008t(.il) = ----------

.il2_ 0·13105.il + 0·1008

To get the total radiative transfer between the two surfaces this transmissioncoefficient has to be included in the integration over .ilof the Planckian expres-sions

where Cl = 2n hc2 = 374. 10-7 erg cm2js and C2 = hcjk = 1·439 cm K. Theintegration is performed numerically, with TA = 316 and TB = 315 K (as inthe calculations of W(Ry) and W(P-v.H». The result is

WO()(G) = 0·0146 mW/cm2.

which is about 20% less than W",(P-v.H), see fig. 7.To take into account the angular dependence of the reflectivity it is necessary

to do a double integration (over Aand 8). For the reflectivities Ril (.il,8) andR.L (.il, 8) we use the approximations 65)

n2 + k2 - 2n cos 8 + cos- 8R.L = ,

n2 + k2 + 2n cos 8 + cos- 8

(39)

which are valid provided n2 + k2 » 1, a condition fulfilled here. The valuesof nand k are calculated from the same dielectric constant, e ~ -i 4n ao/co,ao = 7. 1016, as used above (figs 5, 6, 7).The transmission coefficient teA, 8)= (1- R)j(l + R) where R = t (Ril + R.L),

- 31-

0·6

0·4

40 60-8

Fig. 8. RJ., Ril' sin 0 cos 0 and the resulting transmission coefficient tOI.,O) as functionsof the angle of incidence 0, as calculated in Appendix A.

is now a function of 8 as well as of A (see fig. 8). As shown in Appendix A thenet radiative transfer is given by

11/2 co 1W co(I) = 2Cl J sin 8 cos 8 J teA, 8) - X

o 0 A5(40)

The double integration was performed numerically, with a program written byA. K. de Jonge, between the limits A = 3 !Lmto A = 30 !Lm(this range includesmore than 95 % of the integral).

For TA = 316 K, TB = 315 K, the result is

As may be seen from fig.7 the Polder-Van Hove result at large distances iswithin 5% of this value.

- 32-

3. APPARATUS AND METHOD OF MEASUREMENT

3.1. The method of measurement

From the speculations of Chapter 1 it had been deduced that a radiationproximity effect should become observable at distances «lien/kT. For roomtemperatures this implies distances of the order of a micron. For the wave-lengths involved in black-body radiation at T = 300 K (Ä'm"" ~ 10 !Lm),thepenetration depth in the case of metallic conductors is less than 0·1 !Lm66).The surfaces are therefore well defined and hence the distance between thesurfaces can in principle be determined with sufficient accuracy. At the timethat measurements were first contemplated, there was no theory for moregeneral dielectrics and it was not clear how the surface could be unambiguouslydefined for all wavelengths. For this reason it was decided to make measure-ments on metal surfaces. An advantage of conducting surfaces is, moreover,that it is possible to determine the spacing by measuring the capacitance.The apparatus is shown schematically in fig. 9. The one emitting surface A

is supported on insulating pillars within a guard enclosure C held at the sametemperature as A with the aid of a thermocouple which controls the currentto a heater winding on the guard enclosure. The other surface B is supportedon piezoceramic pillars which permit the adjustment of both the parallelismand the inter-surface spacing. The apparatus was mounted in a vessel evacuatedto about 10-5 torr to reduce gas conduction to an acceptable level. The guardenclosure ensures that, ofthe energy supplied to A via its heater winding, nearlyall is dissipated by radiation to surface B. Only a small fraction (some per cent)is dissipated via the supports of A and the connecting wires.Photographs of the apparatus are shown in figs 10 and 11.The measurements were carried out by holding the temperature of B constant

and recording the power that had to be supplied to A to keep the temperaturedifference TA - TB constant, for various values of the surface spacing. Athermocouple between A and B controlled the current to the heater of A soas to keep TA - TB constant. The mean power was determined by measuringthe integrated current over a period of time (some minutes for each measure-ment).The surface spacingwas determined from capacitance measurements. Ameans

of checking the parallelism was also essential. To do this, the surfaces weremade as thick metal films evaporated onto optical flats, the surface B consistingof three sectors on a pyrex glass substrate. In this way the parallelism could bechecked both capacitatively and from the interference colours (between thesectors). Direct contact or contact via a dust particle was betrayed by a risein loss angle of one of the capacitors and also by abnormal changes in inter-ference colour in response to the controls.

- 33-

WF'I ~~"~.!'I II ,I ,

-- I---TB- 1---- I---

"'£ 0 D...r=:J.Ir----'""-. l#;'~é~;# I~ ~/c,»: >/ ~ 4/d'_~.F ~ I' F

H4A t

~ ~~I--G '-f--oIJT

GeI--P f--o

10-'- =10: r- (--QC

H3Il"

j7)1

tIjTH, H2

• '2 I_-PX£(b)qA"1 ~_PX£(a)

•

"E

~ V~: ~</.

~.

~ It:,~~~

Fig. 9. Apparatus for measuring radiative transfer between closely spaced plane bodies. Aheated surface, B receiver surface, C thermal guard enclosure, D copper ring and flexibleconductors, E vacuum vessel immersed in temperature bath. F metal foil shims, G glassspacers, HIH2 heater for A, H3H4 heater for C, P 3 glass supporting tubes, Q fused quartzsupporting ring. TA leads to thermistor at A, TB leads to thermistor at B, LlTcopper-constantanthermocouple measuring TA - TB' (iT copper leads to Cu-Ge thermocouple measuringTA - Tc. PXE (0) are 3 piezo-electric ceramic tubes for adjustment of the spacing I betweenA and B; PXE (b) are 3 similar tubes for adjustment of the orientation of B. W is a glasswindow (at room temperature), through which optical interference between A and B canbe observed.

34 -

Fig. 10. Photograph of the apparatus. The surface of A and the rim of the guard enclosure Ccan be seen through the glass optical fiat placed on A to protect the surface during assembly.

Fig. 11. Photograph of the glass disk B, resting on an optical fiat of smaller diameter. Betweenthe sectors of B, a broad interference fringe can be faintly seen.

- 35-

For a rough estimate of the power flux to be measured, consider surfaces ofarea say, 5 ern", each having a total emission coefficient (over all wavelengths)of 0,1, and I:!..T= TA- TB= 20 K. The flux in the absence of proximity effectsis then of the order of milliwatts.

Although the method of measurement is in principle simple, the peculiardifficulties stemming from the microscopie scale of the experiments require thatthe apparatus and method be described in more-than-usual detail.

3.2. The emitting surfaces

In order to achieve the aim of measurements at spacings down to 1 fLm, thesurfaces clearly have to be of optical quality, flat to about 0·1 fLm and quitefree from small surface asperities. The latter requirement is not always easyto realize with evaporated films which, as noted above, had to be used to makeit possible to check the parallelism ofthe surfaces. The flatness requirement wasfairly easily realized by evaporating the films onto optically flat substratesprepared by standard optical techniques.

Further, it was desirable that the emitting surfaces should not have too higha reflectivity in the infra-red, for this would mean very small radiant fluxes withthe risk of large relative errors in the measurements. This requirement excludessuch optically well-behaved metals as Cu, Ag, Au, AI; they are also excluded,however, by the following requirement.It is essential that the surfaces be extremely hard, for the following reasons.

In order to measure the radiant flux without spurious conduction effects it isessential that the surfaces can be brought to within the working distance (1 fLm)with the total exclusion of all foreign particles of this size or greater. The onlyway to achieve this under normal laboratory conditions is first to bring thesurfaces into intimate contact and then to separate them. Now, it is (some-times) possible to bring two glass optical flats, when perfectly clean, into intimatecontact simply by placing them together; this, however, was never successfulwith metal surfaces. It was found that the only way of bringing the surfacestogether was, after careful cleaning, to slide the surfaces over each other witha gentle to-and-fro motion. Soft surfaces cannot withstand such treatment:even glass flats become scratched and damaged. However, foreign particles canbe worked out in this way and if the surfaces are hard the damage is restrictedto very shallow scratches.

Before the importance of hardness was realized, a number of softer metalssuch as Sn, Sb and Pt were investigated. Some bulk metals were also considered(the disk A, fig. 9, does not have to be transparent). Fig. 12 shows the infra-redreflectivities of a number of metals that were considered.

The testing of surfaces for small asperities was done in an auxiliary apparatus(fig. 13). On the polished surface of a bulk metal, asperities may occur due to the

-- 36--

100·r----------------------------------------------,

R(%)

9

10

.i 4

6

8(Sb)

o 2 6 8 TO 12- >. (um)

T4

Fig. 12. Infra-red refiectivities of various metals.1. Al, evaporated, aged 67).2. Rh, evaporated 68).3. Mo, polished bulk metal 69).4. Cr-Ni steel (18/8, N. 523, polishing quality), sample H. 18.5. Cr, polished bulk metal.6. Sn, polished bulk metal 69).7. Cr, evaporated on Pyrex glass, sample H. 21. This Cr film was evaporated at the same

time as the disk B' used in the provisional measurements.8. Sb, polished bulk metal 69).9. Pt, sputtered on glass.10. Cr, bulk metal, hand polished. Sample H. 29.

exposure of hard inclusions or crystal planes having a specially hard surface.In evaporated films, asperities are frequently due to the presence of foreign par-ticles during evaporation. Small asperities or local hillocks, even many micronsin height, are difficult to detect interferometrically when their lateral dimensionsare smaller than about a millimetre.The apparatus shown in fig. 13 is both an interferometer and an air-gap

capacitor. The flat (a) is of pyrex glass coated with a conducting transparentfilm of tin oxide. It is flat to better than half a fringe (Ä = 5890 À), i.e. toabout 0·1 (Lm.Moreover, owing to the method of preparation (pyrolysis ofSnCl4 at about 500°C, see for example 70,71)), these Sn02 surfaces are free ofasperities. The Sn02-quartz flat acts as a proof-surface against which anotherflat can be tested electrically for asperities. The flat to be tested is held by asuction clamp, and can be manipulated by screws to within a few microns ofthe proof flat.

Usually, however, after thorough cleaning of both surfaces, the flat to betested was first slid carefully onto the proof flat, and moved until the first orderinterference colours were almost uniform over the whole surface. After this,

- 37-

5

Fig. 13. Interferometer-capacitor for the detection ofsmall asperities on metal surfaces; a isa glass optical flat coated with an electrically conducting layer of tin oxide; b is the metalsurface to be tested; PXE denotes three piezoelectric ceramic pillars for fine adjustments.Parallel light enters from S and the interference field is viewed from E.

the suction clamp was placed on the back of the flat to be tested. With vacuumapplied to the clamp the screws were turned until the surfaces just or almostseparated. Further adjustment was done by means of piezo-ceramic pillars(Philips type PXE5 tubes; a radial field of 1000V/mm produces a longitudinalcontraction of about 0·02%, i.e. for tubes 50 mm long, ,...,10 {Lm""* ± 5 (Lm).With this fine control the flats were finally separated and oriented.The capacitor formed by the two flats is connected to a capacitance bridge.

After the flats have been set parallel (symmetrical disposition of interfer.encecolours) the capacitance value measured on the bridge gives directly the abso-lute spacing. The spacing is now gradually reduced, keeping the flats strictlyparallel. If contact is made via a conducting particle or asperity the bridgebecomes immediately unbalanceable. Less well-defined contacts (e.g. non-con-ducting or poorly conducting) are indicated by some change in the measuredcapacitance and usually a marked change in the loss angle.The apparatus described made it possible to narrow-down the choice of the

metal surfaces to be used for the measurements ofthe radiative transfer betweenvery closely spaced surfaces. Table I below indicates how various metals stoodup to the various criteria and why the ultimate choice fell on chromium.The chromium surfaces used for the measurements were evaporated films

about 1000A thick, on Pyrex glass for B (see figs 9 and 11) whilst for A (figs 9and 10) it was more convenient to use a conducting substrate, viz. germanium

- 38-

TABLE I

Comparison of some metal surfaces; + = satisfactory, 0 = unsatisfactory

metal preparation i.r. asperities hardness remarksreflec- (closest (estimatedtivity spacing from

achieved) scratch(IJ-m) damage)

Mo bulk, hand + 0·8 + sputtering of Mopolished on glass is possible

but bonding nottoo good

Cr-Ni bulk, hand + 0·7 + not evaporablesteel polished

Cr bulk, hand + > 10 + porous, intergran-polished ular cracks

Cr evaporated on + 0·5 + satisfactoryglass

Cr evaporated on + 0·74 + satisfactoryGe

Sn bulk, diamond + - 0 too softturned

Sb evaporated + - 0 too soft, high vap-our pressure

Rh electroplated, high - + satisfactory but re-hand polished flectivity rather

high

Pt sputtered on + - 0 too softglass

(highly doped to give good electrical conductivity). It is possible to produceoptical flats of high quality in germanium. Moreover, germanium has a some-what lower volume specific heat than most metals, offering some advantage inthe temperature regulation.The infra-red reflectivities (12to from the normal) of these chromium surfaces

measured on Hitachi infra-red spectrometers are shown in fig. 14. For À >40 IJ-mthe values are not very accurate because of the superimposed noise. The dif-

Fig. 14. Infra-red reflectivity of chromium surfaces used for radiative transfer measurements.Curve A: Cr on germanium substrate. Curve B: Cr on Pyrex glass substrate. (Thickness ofCr films ~ 1000 A). The vertical bars cover the extreme values of four measurements. Theuncertainty in the absolute values of RA and Rn is, however, about ± 1%, i.e. greater thanthe spread of some of the measured values. The intermediate curve is the mean of A and B.The mean curve is closely approximated by the formula

R(A) Ri 96.7- ~_ 100A A2

as can be seen from the points (+) so calculated.The curve B' represents the reflectivity of the Cr surface (disk B) used in the provisional

measurements. Its thickness is ~ 300 A, not 1000 A as reported in 79).

- 39-

95R(%)

I I

85

o

ferences between the reflectivity of the Cr film of disk A and that of B mustbe attributed to (a) the different substrates, (b) some difference in substratetemperature during evaporation, (c) film thickness (.-v 1000 A) not sufficientlygreater than penetration depth at all wavelengths.For the calculations of the radiative transfer from the theory we

have assumed that the media are identical, with the mean reflectivityR(A) = t [RA (A) + RiA)]. A complex dielectric constant corresponding tothis mean R(A) is then used in the theory of Polder and Van Hove (see sec. 4.1).The calculations are thereby simplified but a certain error is introduced. Themagnitude of the error may be estimated, as far as the radiative transfer atlarge distance is concerned, by comparing the values of the transmission coef-ficient

teA) __ l_R_(A_)1+ R(A)

-40 -

(as in (17)) with the values calculated from the expression

The error in teA) arising from use of the mean value R(A) instead of RA and RBranges from +4 % to +7 % for wavelengths from 5 to 40 !Lm.The mean error I

is about +6 %. The radiative transfer calculated from the theory has, therefore,to be reduced by 6% (see sec. 4·1).

Fig. 14 also shows the reflectivity of another chromium surface, this beingthe surface B' used in the provisional measurements. At the time that thereflectivity of this surface was measured, the reproducibility was rather poorowing to instrumental errors; this represents one of the major uncertainties inthe provisional measurements of the radiative transfer.Itwas later found that the choice of chromium for the emitting surfaces was

itself unfortunate from a theoretical point of view, owing to the complicatedoptical properties of this metal (see below, Chapter 4 and also 72.73.74)). Inview, however, of the imperative experimental demands of a hard conductingsurface of not too high reflection coefficient, capable of evaporation as a film,chromium seemed to be the most suitable choice.

3.3. Thermal fluxes and losses

Referring back to fig. 9, we mention here various relevant data. The disk A(germanium with a Cr layer, 1000À thick) has a diameter of 25 mm and anarea of 4·90 cm"; its thickness is 0·3 cm. The thermal guard enclosure is madeof molybdenum which has approximately the same coefficient of expansion (atroom temperature) as the three pyrex pillars; the latter are tubes of wall-thickness 0·1 mm. Disk A is mounted so as to lie '" 60 !Lmabove the rim ofthe guard enclosure. The heater windings both on A and on the guard enclosureare of manganin wire, glued by means of a silicone varnish which remainssomewhat plastic even after repeated heating to 200°C. The heater windingof A has a resistance of 19 Q, that of the guard enclosure 26 Q. All wires pass-ing through the guard enclosure are in thermal contact with the latter, althoughelectrically insulated from it.The assembly of the apparatus, in particular how the surfaces of A and B

are brought together, is of vital importance to the success of the measurements.This matter is dealt with in sec. 3.4 below.

Here we discuss the various thermal fluxes and losses in the apparatus.Fig. 15 shows the apparatus schematically and fig. 16 an electric equivalentcircuit. In order to make quantitative estimates, some further data concerningthe apparatus must be given (and some parameter values typical of the room-temperature measurements): see Table 11.

- 41-

TABLE TI

Apparatus data

tiT= TA- TB(JT = TA- TcT = <TA' TB) (mean value)

Data concerning radiation A-BDiameter of AArea of AWidth of annular spaceArea of annular spaceWidth of gap between Cr sectorsArea of gaps (glass)Radiating area (Cr-Cr) SCr

Total emissivity (all A) of Cr, eA= eB = eTransmission factor tAB (Cr-Cr)Radiating area Ct annular space-Cr) Ssp-Cr

Total emissivity (all A) ('" black body) esp

Transmission factor (annular space-Cr) tsp-crRadiating area (Cr-glass) SCr-g!

Total emissivity (all A) of glass eg!

Transmission factor (Cr-glass) tcr-g!Effective radiating area (A-B) S (see text, sec. 3.3a)

Data concerning thermal flux A-CEffective radiating area

Pyrex support tubes(3 x)

S'

(

lengthdiameterwall thicknessthermal conductivity

(

lengthwidththicknessthermal conductivity

Thermal conductivity of manganinThermal conductivity of constantanDiameter of BThickness of B

Ge crystal(Ge-Cu thermocouplefor (JT)

Effective radiating area of B (S")Thermal capacity of A, CA = (eVcp)GeThermal capacity of B, CB = (eVCp)pyrex

20K±0·1 K300 K

2·5 cm4·9 cm2

0·1 cm0·78 cm"0·13 cm0·5 cm"4·4 cm"0·10·0530·4 cm2

1·00·10·5 cm2

0·90·15·3 cm2

10 cm"1·5 cm0·25 cm0·01 cm0·01 W/cmK

1·5 cm0·05 cm0·01 cm0·74 W/cmK

0·24 W/cmK0·24 W/cmK5 cm1·2 cm40 cm"2·5 J/K42 J/K

-42-

(a) Radiationflux A-B

First we give an estimate of the flux to bemeasured. In the absence of proxim-ity effects or losses the flux would be:

W = a tAB (TA4- TB4) S~ 4a tAB ST3 ~Twatts,

where T = (TA' TB)' For S = 4·9 cm" (area of A) and tAB ~ 0·053 (Cr-Cr),T = 300 K and /::iT = 20 K, the flux is

WAB ~ 3·2mW.

This estimate is adequate for the purpose of judging the various experimentalerrors (see below, b-g) but for comparisons between theory and experiment(see sec. 4.1) two corrections are necessary.Firstly there is radiation from the annular space between A and C. This is

effectively a black body, which, in combination with Cr, has a transmissioncoefficient tSP-Cr ~ 0·1. Half this annular space is heated by the disk A andmust therefore be counted as part of the radiating surface. The second cor-rection comes from the gaps between the chromium sectors of disk B. Theglass exposed there has an emission coefficient about 9 X that of Cr, yieldingagain a transmission coefficient tCr-gl ~ 0·1.The additional emission from the annular space is independent of I and has

the fixed value WSP-Cr = 4a tSP-Cr Ssp_crT3 ~T. It must be added to thevalues W(l) calculated from the theory.The additional emission from the glass gaps may be conveniently taken into

account by calculating an effectieve radiating area S defined by

tABS= tAB SCr + tCr-Cl Sgl'

If both surfaces were completely of Cr, then S represents the area for whichthe net energy flux is the same as with the above combination of Cr andglass. (The value of tAB used here is only approximate but only a second ordererror is thereby introduced in the value of S.) From the data in Table n, theabove expression gives for the effective radiating area

S = 5·3 cm-.(b) Radiation loss A-C

The back of disk A is covered by an aluminium foil (e ~ 0'03) and theinterior of the guard enclosure is gold-plated (e ~ 0'03). The transmissioncoefficient is thus tAC ~ 0·015. The mean area S' of the radiating surfaces(back and sides of A, interior of C) is ~ 10 cm".

WAC ~ 4 a tAC S' T3 ~T ~ 10-5 W = 0·01 mW.(c) Conduction losses from A

Although small, the temperature difference TA - Tc = (JTwill cause finiteconductance of heat along the glass tubes, thermocouple wires (Cu, constantan,

(41)

-43 -

Ge), heater wires (manganin), and thermistor connections. We calculate thetotal thermal conductance G along these routes:

gglass (3 tubes)gcu (2 X 50 (lm diam. + 2 X 100 (lm diam.)gOe (thermocouple Ge-Cu for measurement of i5T)gmanganln (2x 180 (lm diam., 1= 2 cm)geonslanlan (1 X 100 [Lmdiam., 1=2 cm)total conductance G

0·17 mW/K0·41 mW/K0·23 mW/K0·061 mW/K0·01 mW/K0·9 mW/K

For i5T~O'IK, qeond.(AC) ~O'l mW.

(d) Gas conduction

All measurements were made under a pressure < 10-5 torr. At this pressurethe mean free path is of the order of a metre so that we are concerned withgas conduction in the Knudsen regime all round disk A and not only within'the gap AB (the question of the pressure between the disks AB in connection

i with outgassing is discussed in Chapter 4). The conduction of a Knudsen gasis independent of I but proportional to /:;,.T and to the pressure p, The con-duction between parallel plates is given by 75)

. _ ao y + 1 ( 2R )1/2 TA - TBq _ __ _- p erg/cm2 s,- 4 Y - 1 nM Tl/2

where y = cp/cv, R is in erg/mole degree, and the pressure p in dynejcm",M = molceular weight, and

which for equal areas SA = SB and equal accommodation coefficientsaA = aB = a, reduces to ao = a/(2- a). The accommodation coefficient, a,may take on values between 0 and 1. For a = 0, ao = a/2 and for a = 1,ao = a.

According to Corrucini 76) the above expression simplifies to

where p is in torr, when T is the temperature of the pressure gauge where pis measured. For our estimates at room temperature this formula is applicableand becomes for T = 300 K

-44-

We assume a = -t, a reasonable value for a surface carrying an absorbed gaslayer (see White 77)), so that ao = tI(2 - t) = t. Since most of the residualgas will be N2 we put M = 28 and y = 1'4, (y + 1) (y- 1) = 6. Thenqgas ~ 0·005 (TA - TB)Ptorr W/cm2

• For D..T= TA - TB = 20K, P = 10-5 torrand S = 5 cmê and for ~T= O'lK, P = 10-5 torr and S' ~ 10 cm",

tjgas ~ 0'OO5x 10-5 (20x 5 + 0·1 x 10) ~ 5.10-6 W.

The pressure was measured by means of a Penning gauge. Owing to thepumping action of this type of gauge, it may be that the pressure at the loca-tion of t~e surfaces is somewhat higher than that at the gauge, say 2 . 10- 5 torrinstead of 10- 5 torr at the gauge. The loss by gas conduction then becomesqgas ~ 10-5 W = 0·01 mW. Even ifthe pressure were to be 10-4 torr, the lossby gas conduction would still be only 0·05 mW, a small fraction of the totallosses.

(e) Edge effect

Assuming that the narrow gap (width I) at the rim of A (see fig. 9) is seenby A as a perfect absorber (eg = 1), whilst eA = eB ~ 0'1, we have

tAB = eA/(2- eA) ~ 0·053and

tA_gap = eAeg/(eA + eg- eAeg) = 0·1 X I/(H - 0'1) = 0,1,

i.e. '" 2 x tAB' However the disk B subtends an angle 2n at A, whilst the anglesubtended by the gap at A is very much smaller. It follows that the edge errorcan be estimated from the product of the relative areas and transmission factorsto be

tA_g 2n rl 41

For 1 ~ 10 [Lm and r = 1·25 cm,

4.10-3ij ~--- ~0·3%.edge -..::: 1.25

For w= 3 mW" qedge ~ 0·01 mW.

(f) Parts of heater wire not in contact with A

The total resistance of the heater of A is 19 Q and the total wire length is'" 100 cm. However about 3 cm at each end are not in direct contact withthe disk A and the heat in this 6 cm length is dissipated by conduction andradiation (most of which is not absorbed by A). Of the electrical energy dis-

-45 -

sipated in the 19 nwe must therefore deduct about 6% to get the true energysupplied to A. In the case of W= 3mWthis implies aloss ofqwlreendsRj 0·2mW.

(g) Thermal capacity effect

Apart from the various losses by conduction and radiation dealt with above(b-f) there is a further possible loss that can arise if the temperature of B is

I not held absolutely constant. Since ~T is kept constant by the feedback circuitcontrolling the heater of A, any variation of TB is followed by a similar variationin TA- If this variation is a steady drift in one direction it may give rise to asignificant error. Therefore the power corresponding to the heating-up or coolingoff of disk A has also to be added to or subtracted from qA, the input powerto A. Insufficient attention was paid to this point in the provisional measure-ments where the temperature drift of B was several degrees per hour. In thedefinitive measurements, however, it was possible to hold the drift r,< 0·35Klhour Rj 10-4 Kis.The specific heat of germanium is Cp ~ 0·31 lig at room temperature.

From the dimensions of disk A and its density (eGe= 5·4) we calculate itsthermal capacity, eVcp = 2·5 J/degree. The error due to temperature drift istherefore qdrlft= eVcp T::;;; 2·5 .10-4 W = 0·25 mW.

Total losses

(b) Radiation loss WAC(c) Conduction qAC(d) Gas conduction qcas(e) Edge effect qedge(f) Dissipation in wire ends qwlres(g) Thermal capacity effect qdrlft

Total (omitting (b) and (c))

± 0·01 mW± 0·1 mW

0·01 mW0·01 mW0·2 roW0·25 roW0·47 roW

The items (d, e, f) are losses ; (b) and (c) have the mean value ofzero; (g) may inprinciple be a loss or a gain but in practice the temperature drift was alwaysupwards, so that (g) too is a loss. The measured power qA therefore has to bereduced by the above amount to obtain the true radiative transfer between Aand B. The loss (f) depends on the measured power qA rather than on WAB,

as calculated above. Therefore this item will be larger as the other losses arehigher.

In the above example the totallosses are r- 15% of the flux, WAB, or ,...,13%of the measured power qA-Finally, in connection with the thermal capacityeffect (g), we have to con-

sider the dissipation of energy by disk B and its temperature stability, and alsothe time constants of A and B, which determine the time required to reachequilibrium.

-46-

All the radiant energy WAB has to be dissipated by B, partly by radiationto the ambient and partly by conduction. The temperature drop over B due to aflux of '" 3 mW is I.1TB R:! 0.020K. However, as shown below, the flux throughB will be about double this quantity at room temperature, for the guard en-closure C also radiates to B. tlTB is then still only ",0·05°K so that B maybe regarded as having a uniform temperature,The radiant flux WCB from the guard enclosure C to the disk B is estimated

as follows (figs 15 and 16). The space betweenA and C has an area Ssp R:! 0.8cm-. The area of the rim of C is Srlm R:! 2·8 cmê. The transmission factorbetween two surfaces with total emissivities el and e2 is t12 = ele2/(el + e

2+

- e1e2); for the space, esp = 1, whilst erlm R:! eB = 0·1, whence tsp -B R:! 0·1and trlm_B R:! 0·053. Hence WCB ~ 4 (J T3 tlT(Ssp tsp+ Srlm trim) ~ 2·7 mW,for T = 300 K and I.1T = 20K. There is also some radiant flux from the topof the quartz supporting ring Q (fig. 9); a rough estimate shows this to be

:0 r-

GlassCu wiresManganin

3wires'ÀB

. I"" JA P r-qA=1 'Ä BConstantan

Ge ön -~

...__

rCE rBE J(

W£;I ,, I

'OT

EFig. 15. Sketch of the apparatus showing temperature differences and thermal resistances.

LIT

Fig. 16. The apparatus represented as an electric equivalent circuit.

-47 -

certainly less than the flux from the gap and the rim of C. Taking it to be onehalf of the 2·7 mW just calculated, we get finally WeB ~ 4 mW. Adding thisto the flux WAB ~ 3 mW, the total power to be dissipated by B to the en-vironment E at room temperature is WBE Rj 3 + 4 = 7 mW.If this whole 7 mW is dissipated by radiation from B we can estimate the

temperature drop D>.TBE and thermal resistance rBE (see figs 15 and 16). Forradiation near room temperature, glass is effectively a black body so that

W BE = 7 mW = 4 a T3 D>.TBE S", D>.TBE Rj 0·3K,and

rBE = D>.TBEIWBE = 0.3/7.10-3 = 43 K/W.

Since tAB = D>.TABI(WAB + WeB) ~ 20/7. 10-3 Rj 3000, rBE «: tAB as requiredif the temperature of B is to be held constant by holding the ambienttemperature TE constant. In fact D>.TBE and rBE will be somewhat smaller thanestimated, owing to conduction via the flexible copper conductors D that con-tact the containing vessel (fig. 9). It is difficult however to estimate this con-tribution.From fig. 16, it can be seen that when any of the temperature levels is

changed a new equilibrium is established only when the thermal capacitancesCA and CB have been recharged to the new levels. As in an electric circuit,(D>.T)t = (D>.T)o e-t/er. In the case, for example, that the ambient temperatureTE changes, D>.TBEalso changes and since tBE «: rAB, the time constant involvedis effectively -r = CB r BE Rj 42x 43 ~ 1800 s Rj t hour.If the pre-set value of D>.Tis changed, it is primarily the level TA that changes,

TB remaining close to TE because rAB» tBE. However TB will change veryslightly (by a fraction of D>.TBE= 0·3 K, see above) and very slowly; the timeconstant CB tAB ~ 280,000 s Rj 80 hours. In view of this very large timeconstant, the drift observed experimentally (~0·35 K/hour) must be attrib-uted to changes in the ambient temperature, not to changes in the regulationof D>.T.For the measurements below room temperature the situation is less favour-

able owing to the radiant flux reaching B from the window W (fig. 9) at 300K.As a result, the equilibrium temperature of B is higher than that of the ambientbath.With the enclosure E immersed in a liquid nitrogen bath, TE ~ 80 K and

the temperature of B (measured) was about 130 K. Taking into account theradiating areas and the solid angles involved in the radiation from W (at'" 300 K) to B ('" 130 K, but the exact value has little effect on the flux fromWtoB, since Tw4» TB4) the value of WWB is ",25 mW. For D>.T= 28 K (thevalue in one series of measurements), WAB + WeB ~ 1 mW so that the totalflux reaching B is WWB + WAB + WeB ~ 26 mW. This is then the value of

-48 -

WBE, the power that has to be dissipated by B. If it were dissipated entirelyby radiation to E the value of TB should be given by

WBE = a (TB4_ TE4) 8",

26. 10-3 = 5·67. 10-12 (TB4_ 804) 40,TB Rj 111 K.

The measured value of TB under these conditions was >- 130K. The differencemust be attributed to the rough nature of the approximations used.

Under these conditions the effective thermal resistance 'BE is no longer verymuch smaller than the net resistance of the radiation paths to B (as it was atroom temperature) and the stability of TB is therefore less. Nevertheless it waspossible to hold the drift of TB to less than 0·5 K per hour.

3.4. Control and measurement of surface spacing

In designing the support and positioning system for the disks A and B thefollowing points had to be considered.(a) The surfaces A and B have to be first brought into intimate contact by a

sliding motion; afterwards they must be separable by means of the con-trols.

(b) The spacing should be variable from zero to about 10 (.Lm.

(c) The support system should represent a very large thermal resistance com-pared to that between B and the environment E.

(d) The thermal expansion of the supports of A should be as far as possibleidentical to that of the supports of B.

The system adopted is shown schematically in figs 9 and 17. PXE (a) repre-sents three piezoxide tubes (Philips type PXE 5, diameter 0·3 cm), by means ofwhich the spacing I is adjusted. Three further tubes marked PXE (b) allowadjustment of the orientation. The two sets of tubes are about the same length(= 10 cm) to minimise change of the spacing with temperature. Each tubechanges in length by ,....,12 (.Lm when a voltage of 250 V is applied across thewall (thickness ~0'3 mm).

The glass spacers (G, fig. 9) can be inserted from the side; each was individu-ally ground until its length was between 20 and 40 (.Lm too short when the glassdisk B rested in intimate contact with disk A (criterion: symmetrical dispositionof first order interference colours). The gap was then filled in each case by ashim (F) of metal foil chosen by trial and error until the surfaces A and Bwere just not separated (criterion: change in interference colours).

The most delicate and difficult preliminary to our measurements was theprocess of bringing A and B into intimate contact. After careful cleaning ofsurfaces A and B (lens tissue, triply distilled alcohol, camel-hair brush) theywere slid gently over each other and moved until the first order interference

-------_----------_--_ .._---- ---~

-49 -

02

b6

b}

c}

Fig. 17. (a) and (b). The pyrex disk B with its three sectors of evaporated chromium, 1,2,3.Cl> C2, C3 are the connections to these three sectors, cemented to three bevelled segmentsvisible in (b). PXE al, a2 and a3 represent the piezo-electric pillars to which the disk A isattached; these are primarily for the spacing adjustments. PXE b4, bS, b6 are the piezo-pillars on which B rests; these are used primarily to adjust the mutual orientation of A and B.The two dashed circles represent the quartz ring (Q in fig. 9). The innermost circle representsthe periphery of disk A.(c) Capacitative measurement of the absolute spacing between A and B.

colours (between the sectors) became roughly symmetrical. When the coloursare of first order the optical path difference is about zero or Ä (~0·6 [Lm);thismeans a physical spacing of about Ä/4 ~ 0·15 [Lm(there is a phase differenceof ~ 'Tt between the reflections at the two surfaces).Before the surface B was finally polished and coated with chromium, three

bevelled segments were ground on the rim. Wires cemented to these segmentswith Leitsilber paste provide contacts to the three sectors of the chromium filmon disk B (fig. 17). The three sectors form with disk A three capacitors whosetotal capacitance gives a measure of the absolute spacing between A and B.The equality of the three capacitances gives a check on the parallelism, addi-tional to the appearance of the interference colours in the gaps between thesectors.In calculating the spacing [in terms of the capacitance CA = Cl + C2 + C3

corrections have to be made for the gaps between the sectors 1, 2, 3 and forthe rim of the enclosure C. The capacitor CA has an area equal to that ofdisk A, less the area ofthe gaps between the sectors, giving a net area ",4,4cm"

---- --------~_._------~--~-----

- 50-

(see Table Il). The rim ofthe guard enclosure (diameter ~3'0 cm, width 0'3 cm)has an area ~n 3 . 0·3 ~ 2·83 cm" less the appropriate gap area of 3 x 0·13xxO'3 R:i 0·12 cm", giving a net area of ",2,7 cm". The rim, as noted earlier,lies 60 !lm below the level of the disk A. We calculate (all capacitances in pP)

EO Ac 2390Cc = = pP,

1+60!lm (1(llm)+60)

and CD R:i 560 pP, a :fixed capacitance in series with Cc, formed by threeinsulating bushings (fig. 17). However, since Cc ~ 40 pP, i.e. ~ CD/14 forall values of 1we may regard Cc as being connected directly to earth. Themeasured capacitance is therefore

3900 2390Cmeas R:i CA + Cc R:i -- + pP.

1(llm) 1(llm)+60

Since in our measurements 1~ 10 !lm, we neglect 1 in the second term andfind Cmm R:i 3900/1(llm) + 40 pP, and

3900 40/(llm)I=--+--!lm.

Cmeas Cmeas

Por 1= 1 !lm, the second term represents a correction of ",40/3900 R:i 1%for 1= 10 !lm, it is '" 400/3900 ~ 10%. In view of other errors and of the factthat we expect no rapidly varying effects at the larger distances we feel justifiedin neglecting the corrective term entirely.All distances reported in our results are therefore calculated on the basis of

39001=--.

C(pF)

(42)

The thermal resistance of the support system is very high. The PXE 5 materialhas a thermal conductivity of 0·0125 W/cm degr and the tubes have length10 cm, diameter 0·3 cm, wall thickness 0·03 cm. With three tubes in paralleland these in series with the other' three, the total thermal conductance is0.53.10-4 W/degr i.e. a thermal resistance of ",20000 K/W. This is somuch larger than rBE (",40 K/W, sec. 3.3) that it is unnecessary to calculatethe other thermal resistances in series with the piezoxide tubes: the tempera-ture of B is thus not affected by conduction via the support system. There is

- 51-

however transfer of heat to B by radiation from- the guard enclosure C and thequartz supporting ring Q. This causes no further measuring error (sec. 3.3) butit does have some effect on the equilibrium temperature reached by B.In the matter of the differential expansion of the various components of the

support system (see fig. 9) the Pyrex tubes P together with the Ge disk sufferabout the same expansion as the molybdenum enclosure C. (For !1T = 20K,M (Pyrex tubes) + tll (Ge disk) ~ 1+ 0·4 !Lm, whilst M (Mo enclosure)~ 1·6 !Lm).The two sets of piezoxide tubes likewise cancel (both undergo thesame small temperature change). This leaves substantially only the quartzring Q (length '" 4 cm) to be balanced against the glass spacers G (pyrex, length1·5cm) and a length of '" 2·0 cm in the base (pyrex). However there is a tem-perature gradient along Q which makes the comparison difficult. In practiceit was found that at room temperatures little expansion effect was noticeablebut that in experiments at temperatures well below room temperature the dif-ferential expansion changed the spacing and limited somewhat its adjustment.

104 10

5 5C(pF) tan fj tont:

t 2 2 tC

5Coo= 1130pF

2

102 '--_-L_'--J._J----L.LLLJ_ __ '---L----L-L.LL.L.LJ 100-1 2 5 1-0 2 5 10

- R (MIl)

Fig. 18. Variation of the measured capacitance C and the loss angle tan r5 (arbitrary units)of a ceramic capacitor as a function of the resistance shunted across it.

Fig. 18 shows how the capacitance and the loss angle of a capacitor vary withthe value of a resistance shunted across it. These curves (measured on a simpleceramic capacitor) served as a criterion against which the behaviour of thecapacitors 1, 2, 3 (fig. 17) could be judged: contact via a poorly conductingparticle shows up primarily as a change in the loss angle. There is little changein the measured capacitance value. For contact via a particle of high electrical

- 52-

conductivity, the loss angle rises rapidly and for R < 0·1 MQ the bridge ispractically unbalancable. The capacitance measurements therefore provide notonly the absolute spacing between A and B but also a valuable check on theexclusion of foreign particles.

3.5. Temperature measurement and control

The successful measurement of a small heat flux is contingent on a knowl-edge of the various losses and a strict control of the relevant temperatures.Estimates ofthe losses have been calculated above (sec. 3'3), where it was shownthat there are three temperature parameters to be held constant: the tempera-ture TB of the disk B, the temperature difference D..T = TA - TB and thetemperature difference öT = TA - Tc.The temperature TB was held constant by placing the vacuum enclosure

(E, fig. 9) in a temperature bath. Four different baths were used viz. waterat room temperature, melting ice, solid CO2 in alcohol and liquid nitrogen.A calibrated bead-thermistor (negative temperature coefficient) was used to

measure TB at the higher temperatures. A second such thermistor mountedon A (fig. 9) gives the value of TA- Each thermistor forms one arm of a bridgewhose parameters were calculated according to the method of Addink 78) togive an out-of-balance current that varies linearly with the temperature. Thedeviation from linearity for B is < 1% of the calibrated range (30 degrees).The out-of-balance currents of the two bridges are read by a millivolt recorderconnected alternately across the bridges A and B. Because of the linearity,the temperatures TA and TB can be read directly from the recorder chart. Thebridge voltage is about 30 mV and the power dissipated in the thermistors isless than 0·1 IJ-W.

Conventional mixed-oxide thermistors are unsuitable as thermometers at lowtemperatures (below '" -100 "C) because their resistance rises rapidly toinconveniently high values. For the measurements at low temperatures, there-fore, silicon carbide single crystals were used as the thermometers to measureTA and TB' A constant current of about 5 IJ-Awas fed through each crystaland the voltage over each was read by the recorder, switched alternately asbefore. No attempts was made at linearization. The crystals used were dopedwith Al (1·4. 1019 atoms/cm") and B (3·5. 1018 atoms/cm"). To obtain goodohmic contacts, the electrodes were Mo wires soldered with Au-Ta solder con-taining some AI. The power dissipated in the SiC thermistors is less thantIJ-W.It may be noted that silicon carbide crystals may be used as temperature

sensors down to far lower temperatures: as the temperature falls, the resistanceincreases far less rapidly than that of oxide thermistors and there is still a sensiblevariation below 20 K. Sintered SiC as used in commercial voltage-dependentresistors (surge and spark suppressors) can also be used down to below 20 K,

- 53-

but their behaviour is less reliable and the reproducibility inferior to that ofsingle crystals.For the regulation of ó'T, a copper-constantan thermocouple was used. In

the provisional measurements, however (room temperature only), the differencebetween the out-of-balance voltages of the bridges A and B was taken to be!1T = TA - TB and used with a simple on/off control to regulate the electricenergy qA fed to A. This system was neither sufficiently exact in !1T (bridges Aand B at a given temperature give similar but not identical voltages) nor suf-ficiently stable.

A much improved regulation was obtained with the thermocouple and usedfor the definitive measurements. The thermocouple measures ó'T directly andits voltage is fed to a millivolt potentiometer recorder. By means of a slave-slidewire, a voltage proportional to ó'T is applied to the input of an operationalamplifier whose output controls the current fed to the heater HIH2 of thedisk A. The circuit is shown in fig. 19. The helical potentiometer S can beadjusted by hand; it determines the value of !1T set up by the circuit.The temperature of the thermal guard enclosure (C, fig. 9) has to be held

very close to that of the disk A. As mentioned earlier (sec. 3.3c), ~T = TA- Tcwas in fact kept less than 0·1 °C. To measure ~T a thermocouple is again used,

, , i i i r:~8 mV IkD.

recorderA

Thermocouple(IlT)

SOOD.

IkD.

-12V___..__------------'

Fig. 19. Circuit for the regulation of!1T = TA - Tn. The thermocouple is copper-constantan.The recorder gives full-scale deflection for I mV. S is a helical potentiometer with which thevalue of ó'T can be set. DOA 40 is an operational amplifier. The NTN transistor BFY 50 de-livers the current for the heater HIH2 of disk A. This current also passes through a resist-ance R, the voltage across which determines the speed of the low-inertia motor M. Thenumber of revolutions of this motor in a given time yields the time integral of the current,from which the time-integrated power dissipated in HIH2 can be calculated.

- 54-

now a Cu-Ge couple giving a thermovoltage of ",200 fLV/K. This signal isapplied to the input of a chopped transistor amplifier having a very low drift(fig. 20). When this amplifier has been zeroed and care taken to reduce spuriouspick-up signals (twisted screened conductors to input, earthing at one place

Ge

Fig.20. Regulation of 5T = TA - Tc. The error signal 6v is derived from the Cu-Ge thermo-couple on the left. I = low-drift transistor amplifier ("Insist", type vnv 90, Philips); with1 kil across its output terminals, the voltage gain is 2000x. M = 0 - 48 mV, moving-coilmeter with regulating contacts. H3H4 is the heater winding on the guard enclosure C (fig. 9).

only), the output current is accurately proportional to the input voltage: foran input of 5 mV, the output current is 10 mA. For our purpose, a load of1 k.Q is connected to the output, giving (for 5 mV input) an output voltageiR = 10 volts, i.e. a voltage gain of (10/5) 10-3 = 2000 x. This means that for(JT = 1 K, Vlo = 200 fLV and Vout = 200 . 10- 6 • 2000 = 0·4 volt. The outputvoltage is applied to a moving coil millivoltmeter fitted with adjustable con-tacts which give an on/off regulation of the heating current through H3H4

(figs 9 and 20). This simple arrangement easily regulates (JT to within 0·1 K.The millivoltmeter gives a deflection of 1 division for 0·8 mW. During opera-

tion the meter swings about ± 12 divisions about the zero setting, i.e.± 12.0,8 = ± 10 mV. Since (JT = 1 K gives rise to 0·4 volt, this oscillationrepresents ± 10-2/0'4 ~ ± 0·025K, i.e. a peak-to-peak variation (JT f">::j 0·05K.In the calculations of thermal losses (sec. 3.3b, c, d) the value of (JT was put,conservatively, at 0·1 K.

3.6. Measurement of the mean radiant flux

The electric power qA dissipated in A, less the losses (sec. 3.3), is taken tobe a measure of the net radiant flux from A to B. In order to measure the smallpowers involved in our measurements « 1 mW at the lower temperatures)it is necessary to measure the average power qA over a period of time. Thecurrent varies slightly during this period and it is thus required to measurethe average value of Pr (r = resistance of HIH2' figs 9 and 19).The time average <i) of the current iet) can be conveniently obtained from

a low-inertia D.C. motor connected across a stable resistance that carries thecurrent: such a motor has a speed accurately proportional to the voltage across

- 55-

its terminals. The number of revolutions counted during a period of time ttherefore yields the average value of the fluctuating current. Provided that thefluctuation is not too large, we can take the square of the time-averaged cur-rent as an approximation to the average of the squared current: suppose thatthe root mean square fluctuation about the average value (i) is ± !:l.i.Then

tr r r(i]A) = - f i2dt= -{(i) +!:l.i}2 + -{(i)-!:l.i}2 =(i)2 r + (!:l.i)2r

t 2 2o

In our measurements M ~ 1 mA, whilst the smallest value of iwas for mostmeasurements> 10 mA. Hence the correction term is less than 1% and, inview of other errors, may be neglected.The number of revolutions of the motor can be read off from a counter

driven by the motor. In an interval dt, the motor count is dN = D vet) dtwhere D is a constant and vet) = iet) R is the voltage over the resistance R(see fig. 19). Hence

1 dNiet) = v(t)IR =--

DR dt

and the average current over the period t is

1ft 1 ftdN N(i) =- i(t)di= - -dt =-.

t DRt dt DRto 0

The average power dissipated in A is therefore

Fig. 21 shows the calibration curves of the motor when connected acrossR = 100 Q and R = 200 Q. The slopes of these lines yield the values of I/DR.A correction is necessary owing to the friction of motor and counter. A cer-tain threshold voltage Vo is required before the motor can rotate. The valueof this voltage is found by extrapolating the i vs Nit lines to Nit = O. Thevalue of (i) may now be written

1 N{i)=io+--

DR t

- 56-

where io is the threshold current through R corresponding to vo, the voltagefor which Nit = O.The average power is now given by

(1 N)2 ( u; N 1 N2)«JA) = io+-- r = io2+-- +---- r.

DR t DR t D2R2 t2

In fig. 21 it is seen that for R = 100 n, io = 0·5 mA and the slope l/DR =0·01212 A per Nit, so that

N N2)<tjA) = 10-6 (0'25 + 12·12- + 147 - r (W).t t2

For nearly all measurements the current <i) > 5 mA, so that Nit>DR(5. 10-3) = 5. 10-3/0'01212 = 0·4 and the first term within the bracketmay then be neglected compared to the second term (12'12 . 0·4 ~ 5) and thethird term (147 . 0·16 ~ 23'5).

35i (mA) i(mA}

t 30 t15

25

- Nit (counts/e)1~·5 ~2r- 2,.5~ _,3__ -n

40·

5

Fig. 21. Calibration curves ofthe integrating motor (M, fig. 20), for R = 100 nand R = 200 n.

------------------------------ ~0~----~~5~----~,------~f5--~0

- Nit (counts/s)

- 57-

Accordingly, the mean power is (I' = 19 n at room temperature):

(N N

2)tjA = 10-6 12·12- + 147- 19

t t2W

N (N)2= 0·230t + 2·79 t mW. (43)

The heater H1H2 is of manganin wire whose temperature coefficient ofresistance D.e.le between 295 K and 77 K is about 0·0003 per degree (seeref. 77, p. 379). At the lowest temperature at which measurements weremade, TA ~ 160 K. The heater H1H2 has a resistance at room temperatureof I' = 19 n. At 160 K its resistance is less by the amount D.r =(295 - 160) I' D.e/e = 135. 19. 0·0003 ~ 0·77 n, so that I' (160 K) ~ 18·2 n.As a result, the formula (43), based on r = 19 n, will be in error to theextent of 0·77/19 ~ 4%.For the measurements at intermediate temperatures (e.g. TA = 230 K),

D.r = (295 - 230) 19 . 0·0003 ~ 0·37 n, and I' (230 K) ~ 18·6 n, so that (43)is in error by only 0'37/19 ~ 2%.All the results are calculated on the basis of (43), except for some measure-

ments at the lowest temperatures where the current i is rather small. For thesemeasurements the integrating motor is connected across R = 200 n. The cor-responding expression for qA is then

tjA = 0·057 ~ + 0.686 (~)2 mW. (44)

- 58-

4. MEASUREMENTS

Measurements have been made at four temperature levels,with the apparatusimmersed in the following temperature baths: water at room temperature(293 K), melting ice (273 K), solid CO2-alcohol (193K) and liquid nitrogen(77 K).A detailed comparison with the theory is possible only for the measurements

near room temperature, since the dielectric properties of chromium at low tem-peratures are not known; the infra-red reflectivity of the chromium films wasmeasured only at room temperature.Table III below gives a summary of the measurements.

TABLE IIISummary of measurements

temperature TA TB T tlT We., (mW) Woo remarksbath (meas.) (meas.)

(S = 4·9 cm") corrected

in air, 293 K 323 306 315 17 14 - fig. 22provisional

results

water, 293 313 295 304 17·6 4·4 3·8 fig. 23

ice, 273 321 285 303 35·8 9·9 9·3 fig. 26ice, 273 300 282 291 18·2 4·4 3·8

CO2-alcohol,193 256 214 235 41·8 4·96 - fig. 28

" 231 209 220 22 2·15 - fig. 27

" 227 204 216 22·6 2·20 - fig.28

liq. N2 77 188 138 163 56 3·0 - fig. 29"

160 132 146 28·4 0·85 -

Figure 22 gives first some provisional results 79) measured at room tempera-ture *). They are believed to be the first experimental demonstration of theradiative proximity effect. These results were obtained with the same emitting*) In the earlier publication 79) the axes of the figure were wrongly shifted: at the position

of the y-axis the value of I should read 1 (.I.m.

- 59-

W(mW)

1

25

20

vv

15 v

v

:~:3 4 5o 2

Fig. 22. Provisional measurements 79) of the radiative transfer W between parallel metalsurfaces as a function of the spacing I. Mean temperature T = 315 K, !::iT = 17 K.

~-l~I(fJm)

surface A as in the later measurements but the surface B was different, in thatthe glass was coated with a much thinner film of chromium. lts infra-redreflectivity was measured up to Ä = 25 !Lm (curve B', fig. 14) but the absoluteaccuracy of these measurements is unknown. From the reflectivities of A andB' at Ä = 10 !Lm (RA ~ 0,9, RB' Rj O'Sl) we may calculate a transmissioncoefficient tAB (cf. (17)) that is roughly typical of radiation characterized byT= 300 K:

(1 - RA) (1 - RB)tAB = ~ 0·07.

(i.= 101'm) 1- RA + 1- RB- (1- RA) (1- RB)

For the temperatures of these measurements, TA = 323 K and TB = 306 K,i.e. !::iT = 17 K and T = 315 K, and an effective area S= 5·3 ern", the radia-tive transfer (at large distances) should be

- 60-

The measured value of W<Xl is nearly 3X this rough value. The poor agree-ment is due mainly to the following:(a) Errors in R(A), curve B' of fig. 14.(b) Poor regulation of D..T (sec. 3'5).(c) Thermal capacity effect due to temperature drift (sec. 3·3g).Nevertheless, these provisional results show the existence of a radiation

proximity effect which, for room-temperature radiation, begins to appear at adistance of about 2t !Lm.The theory of Polder and Van Hove 47) agrees wellwith this value (see e.g. fig. 7).

4.1. Measurements near room temperature

Figure 23 shows the first definitive measurements at room temperature. Theincrease in the radiative transfer begins at about 2t !Lmas found earlier. Thespread is much less than that in the provisional measurements, a reflection ofthe improved experimental conditions. The reproducibility over longer periodsis also much better.The direct correlation between the radiative transfer Wand the surface

spacing I is illustrated in fig. 24, which shows the history of the measurements

ur------r---------------------------------------------~W

T'"10

8

6

2 W",(I)

%~----~----~2---~3.----'+-----~5----~6~---*7--~8-((IJm)

Fig. 23. Measurements near room temperature (T = 304 K, D..T = 17'6).1: measurements;2: measurements corrected for losses (sec. 3.3); 3: calculated values, for surfaces with iden-tical dielectric constant e = e' - ie", e' = 1- 4:n:Gowo/(w02 + (2), ie" = i4:n:Gow02/w(w02 + (2), and Go = 1'6.1016 S-I, Wo = 1015 S-1 (see text); 4: calculated values, as3 but with Go = 1.3.1016 s-1, Wo = 1015 5-1•

- 61-

over one day. Each change in 1 is accompanied by a corresponding changein W. Even the small drifts in 1 are recognizable in the values of W.The measured values plotted in fig. 23 still haveto be correctedforthevarious

losses discussed in sec. 3.3. It was estimated there for measurements at roomtemperature and W ~ 3 mW, that the totallosses would be ",0,5 mW. How-ever two of these losses, the edge effect (sec. 3.3e) and the dissipation in the wireends (sec. 3.3f) are dependent on the measured power ij (= W + losses). The 'edge effect is so small that its variation can be neglected. The loss in the wireends ijwlres should, however, be put as 0·06 ij (mW) instead of the :fixedamount0·2 mW. The totallosses then become (0·3+ 0·06 ij) mW. In fig. 23 this totalloss has been subtracted from the measurements, yielding finally the measuredradiative transfer from A to B. It is with this curve that the theory has to becompared.

24.0 300 360 4.20 480 54.0 600-t(min)

Fig. 24. History of measurements of radiative transfer Wand surface spacing lover oneday (values correspond to the points 0 in fig. 23).

2·0I

(pm)r5

1·0

15_-ï

- 1111111

: _-jlï ï I..._-- r--..... I

--I I IL _J I~------.J I

W(mW)

t 10

5

60 120 180

To compute the radiative transfer from the theory of Polder and Van Hovewe require values of the complex dielectric constant of the emitting surfaces.Assuming a dielectric constant of Drudian form,

4n ao Wo 4n ao w0213 = 13' - ie" = 1- - i ,(45)

w02 + w2 W (w02 + (2)

Polder and Van Hove 47) approximated to the dielectric constant of chromiummeasured by Lenham and Treherne 72) using the following parameter values:

(45a)

-- 62-

The radiative transfer calculated on this basis differs from our measurements:the computed values are less than half the measured values, even for 1- 00.However, it is known that the properties of chromium films differ considerablyfrom those of the bulk metal 73.74.80.81).

In order to find the complex dielectric constant appropriate to our chromiumfilms, we searched by trial and error for values of ao and Wo that gave the bestfit to the mean value R(J.)of the measured reflectivities of our films (fig. 14).Fig. 25 shows some of the approximations to R(J.)obtained for various valuesof ao and Wo, calculated from (45) and

n = {e' /2 + t (e'2 + e"2)1/2 p/2, k = e"/2n,

(n- 1)2 + k2R(J.)-----

(n + 1)2 + pFairly good approximations to the measured R(J.)are given by the values

and by (46)

Compared with the bulk values, the lower conductivity and the higher col-lision frequency of the film are not implausible.

Using the above values, the radiative transfer per cm" has been calculatedfrom the Polder-Van Hove formula (33) for the temperatures TA = 304 K,TB = 303 K. For the purpose of comparison, these results are reduced to the

R _-----~~~-~=~------====~~(%) »> - _---------- _---0-3 R(>')

t95 ...--r:::-----~ ---- ....--~

, / ..............-:-V""'"I "",' _..

I /0::.'"I />

90 ;1/)1

85

8%~------~W~----~2~O-------3~0~-----~-~~0---À-r.-p-m-)~50~--~

Fig. 25. Drudian approximations to the mean infra-red reflectivity R(Ä) of the chromiumfilms A and B (see fig. l4). (1) 0'0= 3'5. 1016s-1, Wo = 1·41 . 1014 S-l; (2)0'0=2.1016s-1,wo=2.1014s-1; (3) 0'0=1·6.1016S-1, wo=101ss-1; (4)0'0=1'3.1016 s-1, Wo = 1015 S-1; (5) 0'0 = 7.1016 s-1, Wo ->- 00.

- 63-

effective surface area S = 5·3 cm- (see sec. 3.3a), and the temperature dif-ference (,:::,.r = 17·6) of the measurements by multiplying by 5·3x 0·94x 17·6.(The factor 0·94 represents the -6% correction for the R(Il.) averaging error,see sec. 3.2.) The constant correction Wsp= 40-fsp-crSsp-Cr T3 LlT(see sec. 3.3a)has the value 0-45mW. This has to be added to the calculated values of W.The calculated results so obtained are plotted as the dashed curves in fig. 23.

The agreement with the measured curve is not unsatisfactory.The value of W",(I) calculated according to (40) is also indicated in fig. 23.

20'r---------~-----------------------------------

Fig. 26. Further measurements near room temperature. Curve 1, TRi 303 K, LlT = 35·8 K;2, corrected measurements; 3, calculated from Polder-Van Hove theory. Curve 4, TRi 291 K,LlT = 18·2 K; 5, corrected curve; 6, calculated from Polder-Van Hove theory.

W(mW)

118

2

16

14.

I\\\\\\\\\\\\\\\,,

-,-,

" <, __ 3 _---------

12

8

10+

6

\\\\\\, 4.

-,',,_~ __--------...:5~-----I

________L---4.

2

%~------~------2~------*3--~--~4.------~5~----~6---[rpm)

-64-

The agreement with the Polder-Van Hove theory and with the measurementsis satisfactory.The measurements disclose no trace of a minimum in the W vs 1curve. We

come back to this point in the discussion below (sec. 4.3).Figure 26 shows further measurements near room temperature, corrected

for losses by subtracting (0·4+ 0·06 q) mW as for fig. 23. The theoreticalcurves, reduced to the conditions of measurement as described above, are alsogiven. The agreement between theory and experiment is reasonably good. Thereis no trace of a minimum.

4.2. Measurements at low temperatures

Figure 27 shows measurements with the apparatus immersed in a bath ofsolid CO2-alcohol. A noteworthy difference from the room temperaturemeasurements is that the increase of the radiative transfer here begins at asomewhat larger distance, viz. about 3t [Lm.

u:~----------------------------------------------------~

7 8-l(pm)

Fig. 27. Measurements at T Rj 220 K, dT = 22 K. (No corrections applied.) The dashedcurve is calculated from the theory of Polder and Van Hove.

10

B

6

2

\,,\\\. ,-, .....

............. 'J.

.......---- 2 _

2 63 4. 5

The level of Weo is rather higher than might be expected. Compared withWeo at room temperature (figs 23 and 26), the level of Weo at 220 K cor-responds to Weo ex: T3 dT, whereas, owing to the shift towards longerwavelengths at lower temperatures, we should expect approximatelyW<Xl ex: T7/2 dTJao1/2 (see (37)) or, since ao ex: Tr+, W<Xl ex: T4 sr. It appears

- 65-

that at low temperatures the errors due to losses are relatively larger than atroom temperature.It is not the intention to compare the level of the low-temperature measure-

ments with the theory, since the actual behaviour of R(It) at low temperaturesis not known. Nevertheless, W has been calculated (fig. 27) in order to showthat the theory predicts the onset of the proximity effect at a larger distancethan at room temperature, in agreement with our measurements.

Figure 28 shows further measurements at nearly the same temperature. Theseresults confirm the features already noted in fig. 27.

Measurements at still lower temperatures (liq. N2 bath) are depicted infig. 29. The values of Wex> are again higher than to be expected from (37).However, we are concerned primarily with the proximity effect: at these lowesttemperatures, the increase in the radiative transfer begins at a yet larger dis-tance (> 4 (.Lm), as the previous results and the theory had led us to expect.

20,-------.----------------------------------------------,

%~----~----~2------*3------*7------5~----~6~----~7~----~8~-L(prrJ

W(mW)

118

16

12

10

8

6

2

Fig. 28. Further measurements at low temperature. Upper curve T F::5 235 K, !1T = 41·8 K.Lower curve T F::5 216 K, !1T = 22·6 K.

- 66-

T40

W(mW)

1T2

TO

8

6

4

2

2

%~----~~------~2------~3-------74o------~5~----~6---<-[{IJm)

Fig. 29. Measureme~ts at low temperature. Upper curve TI'::! 163 K, !1T = 56 K. Lowercurve TI'::! 146 K, !1T = 28·4 K.

4.3. Discussion

The measurements presented above show the existence of a proximity effectin the radiative transfer of energy between closely spaced bodies. Two features,in particular, are thought to be positive evidence that the measurements repre-sent a true proximity effect of the electromagnetic field and not some spuriouseffect such as conduction by residual gas. These features are: (a) The distanceat which the proximity effect first appears agrees very well with that predictedby the tIieory of Polder and Van Hove. (b) The fact that this distance becomesgreater as the temperature is lower, is again in accordance with the theory.

Curves of W = fel) calculated from the Polder-Van Hove theory for themean temperatures 315 K, 304 K, 220 K and 145K are shown in fig. 30: theonset of the proximity effect occurs at larger distances as the temperature islower. In these calculations, the dielectric constant is assumed to be independ-ent of temperature which is, in fact, not the case. However, the onset of theproximity effect appears to be little dependent as regards distance on the valueof the dielectric constant, as is shown by the curves of fig. 31.

4%~_L--2~~--+4::~~5;=~~8~=Z==I;O==~=±~==~~U'

-[(IJm)

Fig. 30. Calculated values of the radiative transfer per degree W = f(/) at different temper-ature levels. (I) T = 315 K, (2) T = 304 K, (3) T = 220 K, (4) T = 145K. The calculationsrefer to identical surfaces with Drude-like dielectric constants, with the parameter valuesGo = 1.3.1016 S-1, Wo = 1015 S-1.

Fig. 31. Calculations at a single temperature level (T = 315 K) showing the effect of theconductivity Go and the collision frequency Wo of the media on the radiative transfer W(/).(1) Go = 3·5. 1016 s-1, Wo = 1·41. 1014 S-1; (2) Go = 2 . 1016 S-1, Wo = 2 . 1014 S-1;

(3) Go = 1'6.10165-1, Wo = 1015 S-1; (4) Go = 1'3.1016 S-1, Wo = 1015 S-1.

0·04

0·02

0·08W(I)

(mW/cm2)

r 0·05

0·04

0·02

- 67-

2

3

4)(3)

2)'..

(1)

°0~~--~2--~--4~_L--~5--~--8~~--~m~~~1~2--~~U-[(pm)

51(10%)(pm)

I :2

1 0'4-

'1r"oe0-2

100

- 68-

150 200 300 350--l~T(K)

250

Fig. 32. Comparison between theory and experiment: the spacing I (10%) at which W is10% greater than Woo is plotted against T. (+ = theory, 0 = measurements.) I/I (10%) isalso plotted against T.

The features (a) and (b) mentioned above are illustrated quantitatively infig. 32. Here the spacing I (10%) at which the radiative transfer is 10% aboveits value Woo at large distances is plotted against the temperature, both formeasurements and for calculations. The reciprocal of I (10%) is also plottedagainst T. The agreement between the theory and the measurements is satis-factory.

The calculations of sec. 3.3d show that there could be no spurious effect due togas conduction even if the gas pressure between the surfaces were as high as10-4 torr. However, it might be argued that gas desorption from the surfacesmight well cause a pressure between the surfaces very much higher than thatin the surrounding space because of the high flow resistance of the radial ductformed by the surfaces, especially at the smallest spacings. We have thereforeestimated (Appendix B) the pressure between the disks as a function of theradius on the basis of desorption data given by Diels and Jaeckel V) andThibault, Roussel and Jourdan 83). From this result it is estimated that theenergy transfer due to gas conduction at room temperature could be as highas 1 mW for a spacing I = 1 (Lm. If this were so, it would imply a substantialerror in the room-temperature measurements. However, the results of thecalculation rest heavily on the actual desorption rate, whose value remainsuncertain in the absence of definite information on the composition of thegas adsorbed on the surfaces and the partial pressures of those gases in thevacuum space.

The desorption rate after long periods of pumping may be expected to be

-69 --

much lower than that used in the calculation since the surfaces were frequentlyseparated by distances up to '" 8 (Lmduring the course of the measurements.In support of the contention that gas conduction in fact makes little con-

tribution to the energy transfer, we have two experimental facts. The first comesfrom an experiment in which the pressure in the vessel was allowed to rise andmeasurements of the energy transfer made at various measured pressures be-tween 3 . 10-6 torr and 1·3 . 10-4 torr (Penning gauge). The results are shownin fig. 33. Although the agreement with the calculations of sec. 3.3d ispoor, these

5

6

4L_~m~_~5~--~--~~5~.m~_~5~--~--~~--~m~-4r-i_~L-~--~Pv (torr)

Fig. 33. Effect of presure P» on the rate of energy transfer Q between two parallel platesmaintained at a temperature difference of 18·2 K, at a mean temperature of ~ 300 K. Thespacing of the surfaces is ~ 2 (J.m.

measurements do show that for a measured pressure in the vacuum vessel ofless than 0·6 . 10-4 torr there is no extra contribution to the energy transferfrom gas conduction. During these measurements the spacing was about2t (Lm. As all our radiation measurements were made at pressures well below10-5 torr (usually >- 10-6 torr), the data of fig. 33 show that gas conductionplays no part in the measured energy transfer except possibly at the smallestspacings.The second experimental fact in support of our contention is that the meas-

urements at low temperatures, where the desorption rate is very much lower,show the same qualitative behaviour as those at room temperature. If theproximity effect observed at room temperature had been due to conductionby desorbed gas between the surfaces this could hardly be the cas~ at tempera-tures as low as 146K, especially since here the effect sets in at larger distances.

One discrepancy between the theory and the measurements remains to bediscussed: the absence of a minimum in the experimental curves. Although its

-70 -

depth as predicted by the theory is only slightly more than the spread of themeasurements, at least an indication of this feature might have been expected.However, the appearance of the minimum in the theory depends on the relativestrength and form of the contributions ws1n and WCXP (see figs 6 and 7). Forexample, a slight displacement of the contribution WCXP with respect to wsln

(fig. 6) would be sufficient to change the sum W = WCXP + ws1n to a curvethat increases monotonically with decreasing I. This situation could arise ifthe surfaces were not sufficiently parallel. On the other hand, it is not impos-sible that the predicted minimum depends on the particular model used andmight disappear if we took averages over locally varying surface properties(the chromium films may be inhomogeneous and imperfectly smooth on amicro scale).The discrepancy may be accounted for by uncertainties in the dielectric con-

stant. In the first place there are errors in the measurements of the infra-redreflectivity of our chromium films; an error of ~ 1% in R(A) (see fig. 14)implies, moreover, an error of ~ 10% in the emission coefficient. Secondly,we attempted (sec.4.1 and fig.25) to find a Drudian dielectric constant yieldingreflectivities that were a reasonable fit to the measured values of R(A) in theinfra-red. Fig. 25 shows that this attempt was not entirely successful. Thisagrees with the opinion of Lenham and Treherne who have made extensivemeasurements of the optical constants of Cr and other metals 72.84). Theystate that the optical properties of chromium cannot be described within thecontext of a Drudian dielectric constant. Barker and Ditzen berger also recognizethe non-Drudian behaviour of chromium. They show 74), using a method de-scribed by Verleur 85), how a good fit to the measured infra-red reflectivitycurves can be obtained by assuming ad hoc a number of Lorentzian oscillatorsof given strength, frequency and line-width in addition to the collisional damp-ing of Drude's theory. The unusual optical behaviour of chromium may berelated to its unique magnetic properties 86.87.74). In our case we have theadditional complication that the chromium films may well have properties dif-ferent from the bulk metal 80.81.73).

Finally, we have seen that the Cr films on different substrates have differentinfra-red reflectivities (fig. 14). This may be due to the effect of mismatchstresses on the optical constants or different substrate temperatures duringevaporation. For the longer wavelengths, the penetration depth may well becomparable with the thickness of our films (1000 A). For the shorter wave-lengths where Cr is still in the relaxation regime (w > wo), viz. A ;:5 20 (km,

(Jr ~ (mc2f4n Ne2)1/2 ~ (c2f4n ao WO)1/2.

With our values of ao and Wo we find (Jr ~ 200 A and for values of ao and Wo(bulk Cr) from the literature, (Jr ~ 450 A, both much smaller than the thick-ness of our films. However, at longer wavelengths (A > 20 (km)which certainly

-71-

still contribute to the energy transport for radiation near T= 300 K, the pene-tration depth begins to depend on the wavelength: (Je ~ (cAf4nhTo)1/2. For ex-ample, at A= 100 (Lmthe classical penetration depth for bulk Cr (Je ~ 1000Á.Barker and Ditzenberger 74) calculate for bulk Cr at 300 K, (Jr = 200 A forA < 25 (Lm,rising to (Je ~ 1700 A at A = 1000 (Lm.

For all the above reasons, the dielectric constant (45,46) used in our calcula-tions is only an approximate representation of the dielectric properties of thefilms. The minimum may therefore be even shallower than that c~lculated.

An interesting aspect of the radiation between closely spaced bodies is thefact that, where the evanescent field is dominant, the radiative transfer nolonger has the temperature dependence associated with normal radiation(Stefan-Boltzmann law). As can be seen from fig. 6, there is even a regionwhere the radiative transfer (disregarding any variation of 0'0) is independentof temperature. This is the case where wexp is the leading term and the energytransfer is inversely proportional to the fourth power ofthe distance. The theoryof Polder and Van Hove takes no explicit account of the temperature variationof the dielectric constants of the media. Therefore (see (36) or (30) in ref. 47),the theory gives a radiative transfer at large spacings (evanescent field negligible)proportional to T7/2• This is confirmed by the numerical calculations forI = 14 (Lm,plotted in fig. 34.

W(mW/cm2)

t 5 jq~/1//III/1/1/11/1n

1O-2i--'--->"__'__'1--'-/l-+ -'--'--'---'-----1

2 2Jlm

100 200 300 400-T(K)

I111I1/111

5 SJlm !I

I"II,Jlm1IS/lm

J

Fig. 34. The radiative transfer per cm2 and per degree as a function ofthe temperature level T,for various fixed spacings. Theory -+---+-; measurements -0 0--.

-72-

This figure also shows curves derived from the theory for 1= 5 [Lm, 2 [Lmand 1·5 [Lm. Although these curves can no longer be described by a simplepower law, it can be seen that the radiative transfer depends far less stronglyon the temperature at the lower temperatures and the smaller spacings. Forexample, between 300 K and 145K, the energy flux for 1= 1·5 [Lmis propor-tional roughly to T3/2. These features are reflected qualitatively in the measuredvalues, also plotted in fig. 34.As stated in sec. 4.2 we should expect, for the experimental values,

W CX) ex:T4 /).T. Fig. 34 shows that the slope of the experimental curves in-creases with 1 but that for 1= 5 [Lm, where Wex:T3.1, the evanescent field isapparently not yet negligible.Clearly, the transport of energy via the evanescent field falls off less rapidly

with temperature than is the case with normal radiative transfer by travellingwaves.

In conclusion, we review briefly the theoretical situation and the measure-ments of the radiation proximity effect. The theory of Polder and Van Hove,although complicated in detail, has the following pronounced features.(a) At large distances, the theory is consistent with the Stefan-Boltzmann law

(fig. 7, secs 2.2 and 2.3).(b) At short distances (1 [Lm-O·l [Lm for T ~ 300 K), the leading term is

ex:0"0-1 1-4 and the only temperature dependence is that of the con-ductivity (fig. 6).

(c) At intermediate distances there is a transition region where the net energyflux is determined by the relative contributions of the evanescent field andof travelling waves: a minimum can occur here (secs 1.3,2.2; figs5,6 and 7).

(d) At very small distances, 1< 0·1 [Lmwhen T ~ 300 K, there is a regionwhere the total heat transfer is again constant, independent of distance(fig. 6). Below about 10-3 [Lm(RI:! lOA) the theory is no longer valid(see 47)).

The measurements are substantially in agreement with (a) (fig. 23). Nomeasurements were made at distances < 1 [Lmbut the reduced temperaturedependence predicted by the theory (b) is already noticeable in the trend ofthe measurements for 1> 1 [Lm(fig. 34). The position of the transition region(c) arid the form ofthe measured curves in this region agree well with the theorybut no minimum was detected (figs 23, 27 and 32). The region 1< 0·1 [Lm(d)is at present not accessible to measurement.

Future work on the radiation proximity effect might be directed towards afurther search for the minimum in the transition region, measurements at spac-ings below this region, investigations with media of other dielectric propertiesand extension of the theory to Iow temperatures where the anomalous ~kineffect may have an important influence on the dielectric properties.

-73 -

Finally, a word about the practical importance of the results. In the designof cryostats and other low-temperature equipment where radiative losses, thoughsmall, are often dominant, the proximity effect should be taken into accountin determining the spacing of components and radiation shields. The proximityeffect is certainly important in multiple-foil insulations where it is partly re-sponsible for the limit on the number of foils per unit thickness that can use-fully be employed, a limit well known in the technology of cryogenic super-insulations 88.89) but hitherto attributed entirely to conduction. Even at roomtemperature and higher temperatures the proximity effect need not be negligible.Micron-scale engineering in electronic devices and other fields is no longer ararity and developments involving yet smaller dimensions may follow. Theradiation proximity effect may prove to be an unexpected additional factor inthe design of such devices.

-74-

APPENDIX A

Radiative transfer between non-black surfaces separated by a large distance(sec.2.3)

A large distance (I) in this context is any distance at which proximity effectsare negligible, i.e. I» hen/kT (sec. 1.3).

Consider first the radiation emitted per second from an area S of a blacksurface within the solid angle dw at the angle e and within the interval dÀ atthe wavelength À (fig. 35). This radiation rate is the same as that emitted from

dwo = 210 sin ede,

Fig. 35. Geometry of calculation of (40).

a hole of this area in a constant temperature enclosure. Inside the cavity theenergy density u). is uniform and the fraction of this travelling towards thehole, within the solid angle dw, is dw/4n. Hence the total amount of radia-tion within dw contained in the skew cylinder of length cdt (which crosses Sin the time dt) is

dwdW 00(,1, e) = U;. 410 Sc dt cos e

or, for unit area and per second,

edWoo(À, e) = u). - dwo cos e.

410

Since

u). edWoo(À, e) = -- sin e cos ede

2

1</2 00

Woo = ~f sin 0 cos 0 f [u;.(TA) - u),(TB)] dAdO.o 0

\

-75 -

and the total emission is

1</2 00

Woo = ~f sin 0 cos 0 f u), dAdO.o 0

For black surfaces at temperatures TA, TB' the net radiative transfer is then

For non-black surfaces, a transmission coefficient must be introduced (see (17)).As we wish to take account of the angular dependence of the reflectivity, thetransmission coefficient is a function of 0 as well as of A.:

1- R(A., 0)teA, 0) = 1+ R(A., 0) .

The net radiative transfer then becomes

1</2 00

W 00(1) = 2C1 J cos 0 sin 0 J t(A., 0) [u),(TA) - u),(TB)] dA.dOo o

where Cl = 2n he? = 374.10-7 (mW/cm2),

u), = A.- 5 [exp (e2/A.T) - 1]-\C2 = hcjk = 1·439 (cm K) --+ 14390 (for A. in [Lm).

As an example we take the data o, = 3'5.1016 s-\ Wo = 1·41.1014s-1usedby Polder and Van Hove 47) as an approximation to the properties of bulk Cr.The program CHRI for the double integration first calculates the real andimaginary parts of the dielectric constant e = e' - ie", e' = 1- 4n 0'0 wol(w02 + w2), e" = 4n 0'0 W02/(W02 + w2) co, then the real and imaginary partsof the refractive index N = 11 - ik from 112 - k2 = e' and 2 Ilk = e" andthen the reflectivities R" and Rl from the formulae 65)

(112 + k2 )cos2 0 - 211 cos 0 + 1Ril = -------------

(112 + k2) cos- 0 + 211 cos 0 + 1and

112 + Jç2 - 2n cos 0 + cos" 0

Rl= ,n2 + Jç2 + 2n cos 0 + cos? 0

-76 -

which are valid for n2 + k2» 1; this condition is fulfilled for good conductors.The program prints out RII(A.,e) and R.t(A., e), followed by

1- t (Ril + R.t)t(A., e) = -----

1+t(R" +RDas arrays for e = 0, 10, 20, ... , 90° and A.= 3, 4, 5, ... , 30 fLm.Figure 8 (sec.2.3) shows the variation of Ril' R.t and t(A., e) as functions

of e for the single wavelength A.= 10 I-Lm. It is just because of the sharp dropin R" and the corresponding rise in t(A., e) for e > 70° that calculations ne-glecting the angular dependence of the reflectivity are inadequate. However, thepresence of sin e cos 8 in the integral reduces somewhat the effect of thehigh t-values near e = 90°.! The eRRI program calculates next the integrand of the A.-integration,

IL(A.,8)= 2Cl t(A., 8) [u;.(TA)- u;.(TB)].

From the resulting array in A.,8, the machine integrates with respect to A.(from 3 I-Lm to 30 I-Lm) and prints out

30

IT = cos e sin e J IL(A.,e) dA.,3

tlte integrand with respect to e. The integration with respect to e is done byhand using Simpson's rule.

r(.

'")

-77-

APPENDIXB

Pressure pr~file and gas conduction between closely spaced disks in vacuum(sec.4.3)

The gas desorbed from the surfaces of the two disks (fig. 36) has to escapeinto the surrounding vacuum space via the radial duct formed by the disks.The resistance of a non-uniform duct in the Knudsen regime is given by theKnudsen formula

3 UdR = --d,. sjcm3

4v F2

where dR is the resistance of the elementary length dr of duct, having a cross-section F and a periphery U, and v is the arithmetic mean velocity. A narrowradial duct has, at the radius r, a cross-section 2%,.1 and periphery ",4% rand the resistance of an elementary length dr is

3. 103 drdR = -- - sjlitre.

4%[2 v r

Assuming the gas desorption to be uniform over the surfaces, the desorptionper annular element dr is 4%g r dr, where g is the desorption rate intorr litre/em-s. The total gas current i, at radius r is

i,= J 4:n:g r dr = 2%,.2 g torr litrejs.o

",

Fig. 36. Pressure profile due to desorption of gas.

~ 78-

r :

The pressure drop dp over the annular element at r is

3 . 103 dl'dp = -i(r) dR(r) = -2n 1'2g -" 4:rt [2 v r

3grdr=- - __ .103,

2[2 V

P2 3g. 103

fdp = P2 - Pr = - (r22 - r2) torr.Pr 4[2 V

Writing v = (8 RTfnM)1/2 ~ 4·7 . 104 cmfs for air, r2 = 1·25cm and addinga factor 108 to allow I to be expressed in fLm, we get

1.6.106 gPr = P2 + [2 (1'56- r2).

11

According to Diels and Jaeckel V) the rate of desorption g of metal sur-faces in vacuum at room temperature after 10 hours pumping is of order10-9 torr litre/cm" s. According to Thibault, Roussel and Jourdan 83) this figure

I:may be at least an order of magnitude less for clean surfaces and after pumping I

for 100 hours another order of magnitude less. IAs our surfaces were macroscopically very clean and our measurements were i

made only after days of vacuum pumping (many weeks in some cases), it is /11probable that the gas desorption was not higher than 10-10 torr litre/ern" s.

Nevertheless we take the worse case of g = 10- 9 torr litre/ern- s and calculate thepressure profile between the disks in the worst situation, viz. when the spacing Iis only 1 micron. With these data we get

Pr = P2 + 1·6 . 10-3 (1'56 - r2) torr ~ 1·6. 10-3 (1'56 - 1'2) torr

when P2 < 10-5 torr. Fig. 36 shows Pr plotted vs r.The gas conduction is given by (sec. 3.3d):

. y + 1 TA- TBq = 0·014 ao -- Pton Wfcm2

y - 1 Ml/2

=0.005(TA-TB)Ptorr Wfcm2,

for M = 28, ao = tand y = 1·4 (sec. 3.3d).For TA - TB = 20 K,

l'Ij

\:

q=O'lPtorr Wfcm2•

Over the whole area of the disks the conduction is

~ ~Q = 0·1 f2n rpr dr = 0.16.10-3 2n fr (1·56- r2) dl'= 0·6mW.

o o

-79 -

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