superconvergence relations and sum rules for reflection spectroscopy

Vol. 71, No. 8/August 1981/J. Opt. Soc. Am. 935

Superconvergence relations and sum rules for reflectionspectroscopy

D. Y. Smith*

Argonne National Laboratory, Argonne, Illinois 60439, and Max-Planck-Institut fur Festkorperforschung andPhysikalisches Institut der Universitat Stuttgart, 7 Stuttgart, West Germany

Corinne A. Manoguet

Argonne National Laboratory, Argonne, Illinois 60439, and Mount Holyoke College, South Hadley,Massachusetts 01075

Received August 15, 1980; revised manuscript received March 31, 1981

Superconvergence relations for the complex reflectivity F(w), its amplitude r(w)), and its phase 0(w) are considered.It is shown that infinite families of sum rules can be systematically generated for various moments of Re 1m(w) andIm F-m(W); these rules are analogous to similar sum rules for the refractive index. They provide self-consistencychecks on ellipsometric and interferometric measurements in which both phase and amplitude information are ob-tained experimentally. For the more common situation in which only the reflectivity amplitude is measured, itis customary to decouple amplitude and phase by considering the complex function In F() =In r(w) + iO(o.) Sim-ple superconvergence rules do not hold for this quantity because of the divergence of In r(w) at high frequencieswhere r(w) approaches zero. However, this difficulty can be partially overcome by treating the logarithmic deriva-tive of In F or by limiting consideration to finite energy intervals. The resulting rules are applicable to nonmetalsand include a superconvergence relation for dr/dw and finite-energy reflectance-conservation and phase f-sumrules. The derivative rule is applicable to reflectance and modulated-reflectance studies, whereas the finite-energyrules provide direct self-consistency checks on infrared spectra. Examples of the application of these rules to thereststrahl spectra of an ionic solid are discussed.

1. INTRODUCTION

Constraints on the optical properties of matter in the form ofsum rules and dispersion relations have received considerableattention in the last few years. The area of greatest activityhas been sum rules for the optical constants'- 9 and their ap-plication to specific materials.10' 1 ' The reflectivity has hadless attention. Although a number of sum-rule constraintsare known for magneto reflectivity,'12" 3 differential reflec-tance,14 and functions of the reflectivity amplitude and phaseincluding the ellipsometric function,'5" 6 simple supercon-vergence relations do not hold for the reflectivity amplitude,2

the quantity most commonly measured in reflectance studies.However, the existence of magneto- and differential-reflec-tance rules suggests that specific reflectance sum rules mayhold for other specialized experimental situations.

In the present paper a number of instances in which re-flectivity sum rules can be derived are considered. The largestclass of rules involves both amplitude and phase. This classis discussed in Section 2 and Appendix A. Although the rulesin this group provide self-consistency checks on ellipsometricand interferometric measurements, they do not supply thedesired direct check on reflectance data. Consideration ofthe dispersion relations for reflectivity in Section 3 shows thatthere are strict limitations on sum rules for the reflectivityamplitude arising from the fact that the amplitude is notuniquely determined by the phase. When this was taken intoaccount, two classes of rules involving the reflectivity ampli-tude alone were found. These include a sum rule for the re-

flectance of insulators and its derivative (Section 4) and fi-nite-energy sum rules for the reflectance (Section 5). Al-though they are subject to various limitations, these rulesshould be of particular value in infrared and Fourier-trans-form spectroscopy. For example, the finite-energy reflec-tance-conservation rule of Section 5 may be applied directlyto the reflectance spectra of infrared-active lattice modes inpolar materials to test measurements for self-consistency.Further, a finite-energy f-sum rule obtains for the phase,making it possible to calculate oscillator strengths directlywithout intermediate calculations of the refractive indexand/or dielectric function. An application of these finite-energy sum rules to the reststrahl modes of NaCl is given asan example. Use of the reflectance-conservation rule to setan upper limit on the efficiency of solar collectors employingtransparent heat mirrors is discussed in a separate publica-tion.17

2. SUM RULES FOR THE COMPLEXREFLECTIVITY

Sum rules for the normal reflectivity may be written in anal-ogy with known rules for the refractive index. Intuitively thisis suggested by the close connection between the complexreflectivity at normal incidence F(w) and the complex re-fractive index N(w) = n(w) + iK(W),

F(co) = r(co)ei0(w) = N(c) - IN(W) + 1

0030-3941/81/080935-13$00.50 © 1981 Optical Society of America

(1)

D. Y. Smith and C. A. Manogue

936 J. Opt. Soc. Am./Vol. 71, No. 8/August 1981

which holds at the interface of an isotropic medium withvacuum.

In the present paper detailed proofs of sum rules for thefunctions Pm, wm[f(w)]m, and W

2(m-l)[P(w)]m are given in

Appendix A. The resulting rules all involve rm(w)sin[mO(cw)]or rm(w)cos[mO(w)] in various combinations with powers ofw. Since these sum rules mix phase and amplitude, they areof limited interest for most current experiments, and only afew of the simplest of them will be discussed in the body of thispaper.

Application of the superconvergence theorem'8 to the dis-persion relations connecting the real and imaginary parts offm(w) yields two groups of sum rules (see Appendix A). Thefirst group is

f3 rm(c)cos[m0(c)Idc = 0 (m = 1, 2,3.. .). (2)

For m = 1 this becomes fSOr(co)cos 0(w)dw = 0, which is theanalog of the rule that the average refractive index is unity,i.e., the inertial sum rule.4

The second group is

Jo -wr-()sin[m0(co)jdW = i 8(m = 1)

(m > 2)* (3)

The case for m = 1 in this group is the analog of the f-sum rulefor the imaginary part of the refractive index. 4

To aid in visualizing these rules, the reflectivity amplitudeand phase for a Lorentz oscillator are plotted in Fig. 1 and thereal and imaginary parts of Fm(c), m = 1 and 2, are given inFig. 2. For the case of m = 1, Re fm(w) = r(w)cos 0(X) has adispersive behavior with equal areas above and below the coaxis, in agreement with Eq. (2). Further, form = 1, Im Fm(w)= r(w)sin 0(cw). This is always positive and is qualitativelysimilar to the original absorption band of the oscillator butis considerably broader and displaced somewhat towardhigher energies. The area under this curve, when weightedby energy, is proportional to the f sum, in accord with Eq.(3).

For higher powers both the real and imaginary parts offm(w), m > 2, oscillate as functions of w and average to zero

0.5

0 5 i 1'5 20 25 °0

Fig. 1. Amplitude and phase of the reflectivity for a Lorentz oscil-lator in vacuum with resonant frequency w0 = 10, plasma frequencyw = 10, plasma frequency w, = 10, and damping constant y = 1.The imaginary part of the dielectric function e2(W) normalized to unityat the peak is shown by the dashed curve for comparison.

13.5

125

2.0

(a)

* 5 l0 15 20 25 30

b)Fig. 2. The real and imaginary parts of Fm (w) for a Lorentz oscillator(a) with m = 1 and (b) with m = 2. The oscillator parameters are thesame as those for Fig. 1.

with the appropriate weighting, as given by Eqs. (2) and(3).

Other rules of this general class can be found by consideringfunctions such as cm[F(w)]m and W

2(m.-)[F(W)]m. The results

for the first two of these are given in Appendix A. Briefly, therules are similar to Eqs. (2) and (3): all involve rm(w)sin[m 0(co)] and/or rm(co)cos[m0(c)] in various combinations.This mixture of reflectivity amplitude and phase makes therules useful as self-consistency checks on ellipsometric andinterferometric measurements in which both r(w) and 0(w)are measured experimentally, but they cannot be used forreflectance studies in which only r(co) is measured.

3. SOME GENERAL RESTRICTIONS ONREFLECTANCE SUM RULES

The standard approach to achieve relations that do not mixamplitude and phase is to separate r(w) and 0(w) by consid-ering the logarithm of the reflectivity

ln F(w) = ln r(w) + iO(w). (4)

However, this function diverges logarithmically at infinitybecause lim<,,- ln r(w) = O[ln(w)].19 The function is con-sequently not square integrable over the real axis, and simpledispersion relations and sum rules do not apply to In r(w) and0(w). In particular, the dispersion relation connecting am-plitude and phase takes the form of a subtracted dispersionrelation2 0:



In r(co)-ln r(v)

= 2P f w('0(&w') ( U P2) dw' ;0 rO/ (@(s- C02 V/2 - P2)

whereas the phase is related to the amplitude by

0(w) - 2 In r(w2-') d '7r 0 W/)2 - CtZ2

(5)

(6)

These relations apply to both insulators and conductors, butwith the restriction that both v is 0 and c d 0 in the case ofconductors. Although the first of these, Eq. (5), appears tobe simply the difference between two Kramers-Kronig inte-grals, it is not. Since Iim<n, 0_ D(U) = 7r, the terms that appearto be individual Kramers-Kronig integrals would be loga-rithmically divergent; only their difference is convergent.

The first of these relations indicates that only the ratior(w)/r(v) is determined by 0(w). Furthermore, 0(w) is also notcompletely determined by a knowledge of the amplitude r(c)because of the possible presence of a Blaschke product in thereflectivity.2 ' However, in Eq. (6) this indeterminacy hasbeen eliminated on the physical grounds that lim<,,- 0(w) =7r; this guarantees that O(o,) is given by the canonical phaseshift alone.21' 22

Physically the reason for the indeterminacy in r(w) is thatthe reflectivity amplitude is not uniquely determined fromO(w) by causality.2 0 The same phase is associated with asystem having reflectivity amplitude r(w) as with systemswhose amplitudes are multiples of r(w) and Cr(w), where Cis a real constant such that Cr(w) < 1 for all w. This followsdirectly from Eq. (6), using the fact that

dc' =0.W/O t2 - W,2 =°

This indeterminacy restricts the possible sum rules, sincealthough two physically possible systems having reflectivityamplitudes r(w) and Cr(w) have the same phase, they neces-sarily differ in electron density. [This follows from Eq. (A3),which shows that lime-o,,- i-() .-' wU2 / 2 so that the electrondensities for the two systems differ by a factor of C]. Clearly,then, there can be no f-sum rule for 0(w) or for a function of0(w) alone since 0(w) is independent of particle number.

The same argument can also be used to dispose of the pos-sibility of f-sum rules for In r(w) of the form

S h(w)ln r(w)dw = NP (does not hold), (7)

where h(w) is a weighting function and N is the electrondensity. Applying this proposed relation to r(w) and Cr(w)and subtracting yields

ln C 3' h(w)dw = (CP - 1)NP, (8)

where the fact that the particle densities of the two systemsdiffer by a factor of C has been used. This is, however, acontradiction since the left-hand side of Eq. (8) is independentof the electron density NV, whereas the right-hand side is not.Hence no such rules exist.

Similarly, a simple inertial sum rule for In r(w) of theform

5 g(w)ln r(w)dw = i7 (does not hold), (9)

where wq is a universal constant and g(co) is a weighting func-tion with nonzero values for finite 2 3 w, can be ruled out. Tosee this, observe that if Eq. (9) holds for r(w) it also holds forCr(w). However, subtracting the two assumed expressionsyields

S g(co)dw = 0.0o

(10)

Thus g(w) defines equal areas above and below the g(co) = 0axis. Further, if the integral in Eq. 10 is to converge, g(w)must fall off faster than iow [this conclusion also follows fromEq. (9)]. Now for finite w, r(w) can be chosen with a largedegree of arbitrariness by picking the appropriate systemsubject to the constraint that 0 ' r(w) < 1, which guaranteesthat in r(w) < 0. [For example, ln r(w) for a free-electron gasis approximately zero up to the plasma frequency, which maybe chosen at will by picking the electron density.] Thus onecan choose r(w) to sample more or less of g(w) in Eq. (9). Theintegral therefore depends on the particular system chosen,contrary to the original premise.

From these considerations it is evident that since reflec-tance spectra differing by a constant multiplicative factor havethe same phase, there are severe limitations on the possibleform of sum rules, and most simple sum-rule forms involvingr(w) and 0(w) separately are eliminated.

Several strategems for circumventing these limitations havebeen found. The first24 involves considering normalizedfunctions, such as r(w)/r(0) or r-'(w)dr(w)/dw; this is con-sidered in Section 4. A second approach2 5 is to develop fi-nite-energy sum rules in which a portion of the reflectivityspectrum is considered to be superimposed on a constantbackground rb, which eliminates the indeterminancy of r(w)by requiring that r(w) approach rb at high frequencies. Thisis discussed in Section 5. A third strategem involves takingthe ratio of the reflectivity of two materials. The latter hasbeen discussed by Furuya et al. 14 and will not be consideredfurther here.

4. NORMALIZED-REFLECTANCE SUM RULES

The effects of the indeterminacy in the amplitude of F(w) canbe circumvented by normalizing F(w) to form a function thatis independent of the constant C. Two possible functionsare

r(w) = In (CO) = ln[r(w)/r(0)] + i0(w) (11)

and

FA ()= r'(w)A(w) = I = + i0'(w).

1(w) r(w)(12)

As will become apparent, these two functions are very closelyrelated, and both lead to formally equivalent rules. We firstconsider the second function A(w), since it is of a form notpreviously discussed in the literature.

From the properties of F(w) given in Appendix A it followsthat F'(w)/f(co) is analytic and bounded in the upper half planeand along the real axis except for an co 1 2 singularity in thecase of metals at X = 0. The high-and low-frequency limitsof F'(co)/F(w) follow directly from the limits of the refractiveindex and Eq. (1). In particular,



. '(w) 2

This is equivalent to the two conditions

l r'M@ 2

and19

lim 0'(O ) = o(cWr').

For present purposes it is sufficient to assume' 9 that

lim 0(U) = 0(w-1 In-a w),C.-m

a> 1.

This imposes no additional restriction since in actual physicalsystems O'(X) decreases even more rapidly. For example, inthe classical Drude-Lorentz models, lime,-- 0'(X) =O(W-2).

The low-frequency limit of A(co) for insulators is

i'(co) . dO(w)W-0 P(M) dw =o

2 dK(W) 1=L I ' (17)n2 (0) - 1 dco J"

where n(0) is the static refractive index. The second formarises from the small K(@) expansion of Eq. (1) for F(w) ap-propriate to insulators at small X where K(X) vanishes in thelimit w -b 0. In the case of metals the ratio has an CO-1/

2 sin-gularity at X = 0,


its detailed structure at low energies. This fact has already(13) been noted in Ref. 23, but Eq. (21) shows it far more clearly.

[The observed W- 2 high frequency of r(w) is a consequence ofthe similar asymptotic form of e(w) - 1. This in turn follows

(14) from the fact that the (classical) law of motion is a second-order differential equation. In fact, for what it is worth, itmay be easily shown23 that if Newton's second law were ayth-order differential equation, the limiting value of 0(W)

(15) would be 0(X) = y7r/2.]In the case of Eq. (20) the integrand is not superconvergent,

so that this dispersion relation does not yield a sum rule in thehigh-frequency limit. However, turning to the low-frequency

(16) limits, Eq. (20) gives the rule

d01 = -- Io-= 2 S ,-- r(w) dodwI 7r o r(co)

(insulators).

(22)

Dispersion relations can be written in a similar manner forthe function r(w)/w, where r(w) is defined by Eq. (11). Theseare closely related to Eqs. (5) and (6) but apply only to insu-lators because for metals r(w)/ow has a w-1/ 2 singularity at theorigin so that the function is not square integrable. The in-tegrands of these dispersion relations are not superconvergentso they give no high-frequency-limit sum rules, but the low-frequency limit yields

dO = -=_ 2 c 2InJr( I ddw So 7rjo r(0)

(insulators).

lim F(w) = (I( + 0) /2 (18),w-'o FMw [87ra(0)' 1/2

where (0) is the dc conductivity.The function F'(w)/F(w) is square integrable along a line in

the upper half plane parallel, to the real axis for both metalsand insulators. For insulators it is also square integrablealong the real axis. For insulators F'(w)Ii(w) thereforesatisfies the condition of the Titchmarsh theorem,2 6 and thereal and imaginary parts are related by the Plemelj formulas.Further, r'(w)/r(w) and 0'(X) are odd and even, respectively,so that the Plemelj formulas can be written as

r' (c) 2w '(w')= - P dIo

r(w) ir jo W '2 - w2 dw

and

0'(c) = - 2 P d'c'w'[r'(co)/r(w')J dr fo W/2 - ct2

(insulators) (19)

(insulators).

Sum rules then follow by taking the high- and low-frequencylimits in the usual way.2 In particular, the high-frequencylimit of Eq. (19) [employing Eq. (All) of Ref. 2] yields

0O) f 0'(w)dw =- lim | r(w) (21)o 2 I [ r(w)

where 0(0) = 0 has been used. Evaluating the right-hand sideof Eq. (21) with Eq. (14) gives 0(X) = 7r. This is, of course, awell-known result; what is noteworthy is that Eq. (21) showsthat 0(cO) depends solely on the asymptotic form of r(w) andis independent of both the magnitude of the reflectance and

The close connection of this result to Eq. (22) can be seen byintegrating the right-hand side of Eq. (23) by parts; this showsthat the integrals are identical.

Combining these results and Eq. (17) yields the principalresult of this section,

S - r'(W,) = W-21n Inr(w) ° r(0)

r dO r7 dK- - = -- -

2 dw o n2(0) - 1 dwl=o' (24)

which holds for insulators.From the form of the reflectance spectra it is clear that there

will be considerable cancellation in the two integrals in Eq.(24). This is illustrated in Fig. 3 for a single Lorentz oscillator.It will be seen that the cancellation is virtually complete sothat the right-hand side of the equation is small. This largedegree of cancellation is to be expected in a good insulatorsince K(co) is zero at w 0 0 and remains small at low frequen-cies. The primary contributors to K(w) at low frequencies ininsulators are optic phonon modes so that the largest valuesof [dK(w)/ddw] 1,,=o are to be expected in strongly polar mate-rials. NaCl has been investigated as a representative exampleof such a system (see Section 5 below). It is found that in thereststrahl region the positive contribution of the derivativefunction -'r'(w)./r(wo) exceeds the negative contribution byonly 10%. The resulting value of h-'dK(w)/dw,,=o is ap-proximately 14 eV'1. Extrapolation of the absorption spec-trum measured27 in the range from 70-166 Aim to zero fre-quency leads to a value of h'-dK(w)/dwj,=o in the range of10-18 eV-'. The two estimates are in good agreement con-


(a)

15-10

(b)Fig. 3. The integrands w&'[r'(w)/r(w)] and W-2 In [r(w)/r(0)] of Eq.(24) for a Lorentz oscillator in free space. For the first integral thelarge cancellation between positive and negative parts of the functionis apparent. The same cancellation occurs for the function w)-2ln[r(c)/r(O)], but the convergence of the integral is much slower. Theoscillator parameters are the same as those for Fig. 1.

sidering uncertainties in the measurements and the extrap-olation.

The derivative form of the normalized reflectance sum rule,Eq. (22), is of particular interest in connection with reflectancemodulation spectroscopy. 28

,29 In this manner of experiment

the quantity AR/R, where R = r2, is measured for some os-cillatory external perturbation or wavelength modulation. Inthe particularly simple case of modulation of the frequencyof the incident radiation by Awo, AR/R is given by

AR(w) r(W) AxR(M) r(w)

(25)

Equation (24) then becomes

AW-1 j W/-1 AR(cGo) dg -7 dO ]Jo R (W') do

2,7r dK

n2 (0) - 1 dwo (26)

In the context of this expression, the occurrence of positiveand negative values of AR with similar probabilities com-monly encountered in modulation spectra29 is a consequenceof small values of dK/dwc,,=o.

5. FINITE-ENERGY SUM RULES

The divergence in In r(w) at infinity, which precludes largeclasses of superconvergence relations for the reflectance, can

be successfully circumvented in many systems by consideringapproximate dispersion relations and sum rules for a finite-energy interval. This approach applies to systems with ab-sorptions lying in two or more widely separated regions.Examples include polar insulators and wide-band-gap semi-conductors. In these materials the interband electronictransitions occur at energies much higher than those for latticevibrations or free carriers, which generally lie far in the in-frared. Further, the two absorption regions are separated bya wide region of transparency in which the dielectric functionis real and is, to a good approximation, dispersionless.

In these systems the high-energy absorption may be viewedas providing a background medium with a frequency-inde-pendent dielectric function Eb, in which the low-energy pro-cesses take place. If there were no low-energy absorption, thenormal reflectivity would be nearly constant with the value

rb = (Eb 1 2

-1)/(eb1/

2+ 1) (27)

over the range 0 < w < co. Here (t is chosen to be an energy inthe region of transparency that is high compared with theenergies of the low-energy absorption but small comparedwith those of the high-energy absorption. At frequencies w< co the dielectric function is a linear combination of thecontribution from the low-energy absorption and the constantEb. In the following it will be shown that the reflectivity as-sociated with the low-energy processes may be viewed as su-perimposed on the dispersionless background rb id 0 and thatdispersion relations and sum rules may be written for quan-tities such as ln[f(w)/rb] and P'(co)/F(w) in the finite-energyinterval 0 < w < to. This procedure is well known in high-energy physics and has recently been applied 8 to derive fi-nite-energy f-sum rules for valence electrons.

Finite-Energy Sum Rules for In f(w)Restricting consideration to the region 0 < co < A, the dielec-tric function in the limit as w approaches co from below maybe expanded as

lim E(W) (b(C) - )COP 2/w

2 + ... Eb - wOpJ

2 /W 2 + **,

(28)

where w1,, is the plasma frequency associated with the infraredabsorption. The corresponding normal reflectivity is, fromEq. (1),

lim F(O) = rb I1- Ph _+ . . .w Ail- nb (nb 2- 1) W2 II

(29)

where nb = Eb1/2, so that the logarithm has the form

CO*2 1lim ln (w) = lnrb - 2 .... (30)

o-)b(flnb -n1) (02

Finite-energy dispersion-relation and sum rules may then bederived by employing a Cauchy-theorem integration abouta semicircular path of radius a, in the upper half plane. 30

Notice that Eq. (30) and hence the finite-energy sum rulesare not valid for Eb = nb 2 = 1, the vacuum background case,or more generally when nb is very small. The steps leadingto Eq. (30) require that the low- and high-energy absorptionsbe sufficiently widely separated that the dispersion associatedwith the low-energy processes be negligible at energies of theorder of hab. That is, copi 2 /a,2 << 1< Eb(a,) ~- Eb



An alternative procedure is to idealize the problem to onein which Eb is a constant to infinite frequencies. The functionln[F(w)/rb] then satisfies the conditions of the Titchmarshtheorem,2 6 and dispersion relations analogous to the Kram-ers-Kronig relations for the complex refractive index apply.In the actual finite-energy problem the integration can betaken only up to Xo before dispersion in rb becomes significant.This deviation of the physical system from the mathematicalmodel may be accounted for either by extrapolating the ex-perimental measurements as though the background wereactually constant or by truncating the integral at co. In thelatter case an estimate of the error introduced by the finite-energy approximation can be made by evaluating the integralfrom co to infinity using the asymptotic form for a harmonic-oscillator or free-electron model fitted to the low-energy ab-sorption. The results, including the largest correction terms,are

lnlr(w)/rbl ~-P ('W'(C' do)'+ R

and

Z2w P 2 lnIr(o')/rbI IU W CL? .P

7 Jo W'2 -c2

e03

where

R= 237rnb (nb 2 - 1)

(31)

(32)

(33)

In Eqs. (31)-(33) wpj is the plasma frequency and v is theaverage damping constant for a classical model fitted to thelow-energy absorption. In all cases in which this formulationis applicable one expects cpj and v to be much less than Xo andnb to be of the order of 1.5 or more, so the correction termsshould be negligible in practice.

Superconvergence RelationsSum rules for the reflectance over a finite interval follow fromthe high-frequency limits of these dispersion relations in theusual way.2 The results are

ln Ir(co)/rbIdco 0, (34)

or equivalently

z-1l | In r(w)dw ~ ln rb,

).2w0(w) dw - 7""

o 2nb (nb 2 -1)

(35)

(36)

These rules are exact in the limit of co - - for Eb a constantnot equal to unity. For finite Ct the first-order correctionterms to these expressions involve the ratios (op,i/Ct) 2 and(v/Ct), respectively. The correction terms are given in Ap-pendix B.

The first rule, Eqs. (34) and (35), is a finite-energy reflec-tivity-conservation rule: Whatever the low-energy processmay be, ln r(w) must average to ln rb over the range 0 to Ct.That is, for every region of reflectance greater than rb theremust be a corresponding region with reflectance less than rbintroduced by the low-energy process. This is a formalstatement of the common observation that in the infrared

spectroscopy of nonmetals a region of high reflectance is fol-lowed by a region of very low reflectance. A classic exampleis the low-reflectance region lying to the short-wavelength sideof the reststrahl region in ionic crystals; see Fig. 4. This re-flectivity-conservation rule provides a direct check on theself-consistency of infrared reflectance measurements on in-sulators and wide-band-gap semiconductors.

The second rule, Eq. (36), is a phase f-sum rule. The factorsdepending on nb in the denominator indicate that the greaterthe polarizability of the background, the less the effect of thelow-energy processes on the phase of the reflectivity. Theutility of this rule lies in the possibility of calculating the os-cillator strength (as measured by wp,1 2) immediately from thephase rather than having to make an intermediate calculationof K or e2 to apply the conventional f-sum rule. This shouldprove directly applicable to far-infrared optical studies inwhich advances such as asymmetric (or dispersive) Fourierspectroscopy3 l-3 3 now make possible the measurement ofreflectivity phase in addition to the amplitude.

As an example of these rules, the reflectance spectrum ofNaCl in the region of the infrared active lattice modes is shownin Figs. 4 and 5; the data are those of Geick.2 7 Figure 4 givesthe conventional R versus X plot, whereas in Fig. 5 ln R versus

X (Jm)

Fig. 4. Conventional R versus X plot of reflectance for crystallineNaCl at room temperature in the region of the reststrahl absorption.After Geick, Ref. 27.

001

0.001o

0.0000 I 2 3 4 5 6 7 8 9 10 I 1 12 13 14

ENERGY (102 '-Cr')

Fig. 5. The reflectance of NaCI replotted as In R versus energy. Thispresentation of the data provides an example of the finite-energy sumrule for In r(.), Eq. (35). Note the equality of areas traced by thecurve above and below the background reflectance line at Rb = 0.0433,the reflectance in the visible.


I

.,

II


1.0 -

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

ENERGY (102 cml')

Fig. 6. The phase of the reflectivity for crystalline NaCl in the regionof the reststrahl absorption. After Geick, Ref. 27.

w is given. At high frequencies the reflectance approachesthe reflectance in the visible, Rb = rb

2 = 0.0433. This is evi-

dent in both figures. However, the second shows clearly thatthe reflectance associated with the lattice modes is superim-posed upon a background rb and that it consists of two parts,a low-energy region with r > rb and a higher-energy regionwith r < rb. A comparison of the areas shows that the areaabove rb is equal to that below rb to within experimentalerror.

The phase of the reflectivity for the NaCl lattice modes isshown in Fig. 6. Note that at high energies the phase ap-proaches zero. This occurs in the region of transparencywhere the dielectric function is real and greater than unity.If it were not for the background dielectric function Eb > 1(originating here from virtual electronic interband transi-tions), the phase would approach wr as co - -, and the integralin the phase f-sum rule, Eq. (38), would not converge.

A numerical integration of co 0(w) yields the same numericalvalue for w,,i as a more complicated analysis involving cal-culation of n and k and then e2 followed by an integration ofthe f-sum rule for E2(W).

The NaCl data27 used here as an illustration resulted froma rather careful study, and it is not surprising that the rulesare satisfied by the measured data. In general, data are notalways this good and are often fragmentary. The rules thenprovide a criterion of selecting the most reliable data or themost likely extrapolation or interpolation.

Dispersion relations and sum rules for higher powers ofIn F(w)/rb can be written in analogy with those for the complexrefractive index. The majority of these mix lnlr(w)/rbI and0(w) so that they are not of direct applicability to the reflec-tance spectra. However, the modulus conservation rule

O Flnlr(w)/rb 12 dw-o 02(4)d,

dicted by Eq. (38) is 1.97, which is in good agreement with theexperimental value27 of 2.008.

This expression takes a particularly simple form when thematerial contains free carriers, in which case r(0) = 1:

2 j"In rb - - l w10(w)dw

7r 0(free carriers only).

(39)

At w = 0 Eq. (32) yields the trivial result 0(0) = 0. However,for insulators the reflectivity is analytic at the origin, and aseries expansion yields an expression analogous to Eq. (23) butfor the finite interval

dO 2o r(wo)I- :~- -Jw 2 In Id

dwo" 0 r(0)(insulators).

(40)

In deriving this a subtraction of lnlr(W)/rb I So (C0'2-C02)-ldw

- 0 has been performed to avoid the principal value singularityat w = w'. As might be anticipated from Eq. (24) this isequivalent to an expression relating d0/dwj0=o to the loga-rithmic derivative of r(w), which is derived below. The tworelations will be illustrated for our exemplary substance, NaCl,in the following subsection.

Sum Rules for the Logarithmic DerivativeThe technique of seeking dispersion relations and sum rulesfor the logarithmic derivative employed in Section 4 for infi-nite range problems can also be applied to finite-energy in-tervals. In the case of a nonzero background reflectance thefunction P'(WI)/P() converges to zero much faster than in thepreceding application, and it is easily seen that

lim F'(w) = 2w<,i 1+, ,(w) nb (nb 2 - 1) W3 (41)

As before, F'(w)/F(w) is analytic in the upper half plane andsatisfies the conditions of the Tichmarsch theorem2 6 for in-sulating systems, and dispersion relations similar to Eqs. (19)

90

80

70

60

50

40

30(37)

provides a check on the accuracy of the calculations of 0(o)from lnIr(c)/rbl by using Eq. (32).

Static-Limit Sum RulesIn the limit of w = 0 the dispersion relations for In r(w), Eq.(31), yield an expression for the static reflectance

lnIr(O)/rbl -Jf 1 -10(w)dw. (38)

The integrand c'-10(w) is given in Fig. 7 for the reflectivityphase in the restrahl region of NaCl. The ratio r(O)/rb pre-

20

10

3 4 5 6ENERGY (102_cm1)

Fig. 7. The integrand of the static-reflectance sum rule w-0(w) forcrystalline NaCl. Note that the factor w 1 emphasizes an unphysicaloscillation in the published data below 1 X 102 cm-'. For a passivesystems 0 > 0, but roundoff errors in the integration routine used byGeick introduced cancellation errors at small w, where 0(U) is givenby the difference of large numbers. The probable course of the curveis shown by the dashed curve.


E


10 1 2 3 4 5 6 7 8 9 10ENERGY (10

2cm

1)

Fig. 8. The integrand of the partial-energy sum rule for the firstmoment of r'(W)/r(co), Eq. (45), evaluated for the measured reflectancespectra of crystalline NaCl in the reststrahl region. The net areaunder the curve is zero to within the accuracy of the experimentaldata. Note that the first moment of r'(w)/r(co) is equivalent tof Sln[r(co)/rb]dw by partial integration but that the integrand of thelatter, which is shown in Fig. 5, is much simpler than the presentfunction.

and (20) hold for the finite-energy interval. In the case ofconductors, f'(co)/F(cv) has an c- 1/2 singularity at co = 0, so thefunction does not satisfy the Tichmarsh-theorem 26 conditions,but because of the more rapid falloff in the present case[compare Eq. (41) with Eq. (13)] the function cor'(cv)/r(c)does; by using this function one can easily show that dispersionrelations similar to Eqs. (19) and (20) also hold for metalsprovided that co 5d 0.

The resulting finite-energy dispersion relations are exactin the limit of X) - - for the idealized problem in which Eb isa constant at all frequencies. Limiting consideration to theactual finite-energy interval over which Eb is constantyields

and

r'(cv) 2- W O'cv) d

r(c) 7r o Cv' 2 - W2

dMM - a2oP W (a' 2 ',dw 7r o / --a

This at first appears new, but it may be shown to be an alter-native form of the average logarithmic-reflectance sum rule,Eq. (34), by integrating by parts. As a comparison of the tworules, the integrand of Eq. (45) is plotted in Fig. 8 for crys-talline NaCl. This is to be compared with In r(w) for NaClplotted in Fig. 5. It will be seen that the present integral isconsiderably more complex but has the advantage of rapidconvergence.

Static-Limit Sum RulesSum rules also follow for insulators by taking low-energy limitsof the dispersion relations. In particular, Eq. (43) yields

dO =-- _ (ol dwdcv =o 7r Jo r(w)

(insulators).

(46)

As might be expected, this equation may be shown to beidentical with Eq. (40) by partial integration. In form Eqs.(40) and (46) for the finite-energy interval are the same as Eqs.(22) and (23), respectively, for the infinite range. However,the functional form of the integrand is quite different becauseof the nonzero background reflectance over the finite-energyinterval. This may be seen by a comparison of the integrandsshown in Fig. 9 for a single Lorentz oscillator in a dielectricmedium with the same integrands for an oscillator in freespace shown in Fig. 3. An example of the integrands as en-

-01

-1 -02

-03

-04

-OS

-06(42)

(43)1.0X1

where the correction terms from the region beyond Xv are givenin Appendix B but should be negligible in practice.

Superconvergence RelationsSum rules then follow from the high-frequency limits of Eqs.(42) and (43) or by integrating the functions F'(c)/?(cv) andcor'(c)/r(c) about a semicircular contour of radius ct, in theupper half plane. The first result is trivial,

0'( c)dc = 0(6) - 0, (44)

and is simply a statement of the fact that the phase is zero inthe intermediate region between the low- and high-energyabsorptions where the material is transparent.

The second sum rule is

IScv) Ir(cv O (45)

-I1

(a)

,,,2 Injq,-

FOR A LORENTZ OSCILLATOR

o :2 yb2

Sp :2 ,b :2

2 3 4 5 6 7 R 9 10

(b)

Fig.9. The integrands co-l [r'()/r(w)] and -2 lnIr(o)/r(O)I of Eqs.(40) and (46) for a Lorentz oscillator in a dielectric medium with eb= 2. These functions are to be compared with the correspondingfunctions for an oscillator in free space given in Fig. 3. The oscillatorparameters are the same for both figures, but Eb = 2 in the presentcase, whereas in Fig. 3 eb = 1. Note that for Eb # 1 the integralsconverge much faster than in the free-space case.

.A I,0, .-- .


.


Z\, 0.8 a08606402 04

02

-0E2 1-04(-0'6-08 -- go0-1.2-- 14 -

- .6

*21.82 3 4 5' 6 7' 8' 9 10

ENERGY (102 Cm,)(a)

2.01.81.614-1.2-

I10r j2 If r (40 foa NOCe

0604

-02-

-04--0.6-08-10-12-14--1.6-1,8-2.0

I 2 3 4 5 6 7 8 9 1 0ENERGY 110

2cm

14

(b)

Fig. 10. A comparison of the integrals cw'[r'(w~)/r(co)] and W-21njr(w)/r(O)jI of Eqs. (40) and (46) for crystalline NaCl in the reststrahlregion. As in the case of the other finite-energy-reflectance sum rules,Eqs. (35) and (45) (Figs. 5 and 8, respectively), the integrand involvingthe logarithm is the simpler function.

countered in practice is given in Fig. 10 for crystalline NaClin the vicinity of the reststrahl absorption. As was mentionedin more detail in connection with Eq. (24), there is a largedegree of cancellation in the integrals, and the sum rule pre-dicts the observed value of dK/dwIw,=o to within experimentalerror.

6. SUMMARY

As a synopsis of the present study, the principal sum rulesobtained are listed below, together with a brief description.Equation numbers refer to the main text.

Sum Rules for the Complex ReflectivityThese rules apply to both metals and insulators and involvepowers of the complex reflectivity with various weightingfunctions. Since the real and imaginary parts of F(w) are in-volved, the reflectivity phase and amplitude are mixed. Someexamples are

5 rm(w)cos[mO(w)]dco = 0 (2)

J' wrm(w)sin[m0(w)]do = 80

(m = 1)

(m > 2)(3)

Related rules with various weighting functions are consideredin Appendix A.

Sum Rules for the Normalized ReflectanceFor insulators a sum rule for the derivative of the phase in thestatic limit holds for the infinite energy interval

5 co-l[r'(w)/r(w)]dwo = 5 w-2 In I r (co)/r (0) I dwo

or dO2 dw w=O

ir dK=n 2 (0 - 1dwn 2(0) - 1 dcw 1 (24)

where n (0) is the static index of refraction. In most instancesdK/dw I,,=o is small so that the cancellation within the integralsis almost complete.

Finite-Energy Sum RulesFinite-energy-interval sum rules apply to systems havingwell-separated absorptions. The absorption in the intervalin question must, to a good approximation, be describable astaking place in a background medium with a constant, realdielectric function Eb = nb 2. Then the following hold:

Logarithmic Background-Reflectance Rule

JO )[r'(w)/r(c)]d = J5 lnr(w)/rbjdw 0. (34)(45)

Here rb = (nb - 1)/(nb + 1) is the reflectance of the back-ground medium.

Phase f-Sum Rule

o 0(2)dw bnp2-1o 2nb (nb 2 -1) (36)

Static-Limit Phase-Derivative Rule for Insulators over aFinite Interval

f COw'[r'(.)/r(c)]dco = 5 Cr 2 InnIr(w)/r(O) Id

7r dO2d - - -2 dcol=o

7r dKn

2() - 1

n2(0)- I dwl=o (40)(46)

The first of these rules, Eq. (34), provides a direct check onreflectance measurements, as illustrated for the NaCl rest-strahl in Fig. 5. The second relation, Eq. (36), eliminates theneed for intermediate calculations of E2(W) in determining theoscillator strength from reflectivity data. The third relation,Eqs. (40) and (46), is the finite-energy equivalent of Eq. (24)and gives a second direct check on reflectance spectra pro-vided that the right-hand side of Eqs. (40) and (46) is known.In many instances the latter is negligible compared with theabsolute values of the areas involved in the integrals on theleft-hand side.



APPENDIX A: SUM RULES FOR THEREFLECTIVITY FUNCTION

The normal reflectivity at a vacuum interface with an iso-tropic medium is given in terms of the complex refractiveindex N(w) by

F?) (w) -= (Al)N(w) + 1

Since N(w) has an analytic continuation that is regular in theupper half plane, I+, standard arguments 22' 34 3 5 show that

(1) Equation (Al) defines an analytic continuation F(Q)of the reflectivity from the real axis into I+ that is regular andbounded by unity. In insulators the region of analyticity in-cludes the real axis, whereas for conductors there is a branchpoint at Q = 0. The latter arises from the square-root sin-gularity in the refractive index and corresponds to the familiarHagen-Rubens behavior 36 (Ir(Q) 12

- 1 K 12/2) observed inmetals.

(2) In both insulators and metals F(Q) is square integrableover the real axis or along any line in I+ parallel to the realaxis. This follows directly from Eq. (Al) and the square in-tegrability in I+ of N(9) - 1 for insulators and of c[N(Q) -

1] for metals.(3) The reality of fields implies the crossing relation

F*(Q) = rF-Q*) (A2)

for the nonmagnetic case. For a generalization to the mag-netic case see Refs. 12 and 37.

(4) The high-frequency limit of the reflectivity on the realaxis is

lim F(W) =-- wp2/W2. (A3)- 4

Here the plasma frequency is given by

COP 2= 4-r.Ve2/m, (A4)

where jV is the electron density. This asymptotic behaviorand the boundedness of F(w) guarantee that powers of thereflectivity Fm (w) for m > 1 are square integrable along thereal axis. Further, applications of the Phragmen-Lindelbftheorem38 to w 2

p(W), which is bounded on the real axis andanalytic in 1+, leads to the conclusion that Eq. (A3) holdsuniformly as Q - - in I+.

Many of these properties are similar to those of N() - 1,and a large number of dispersion relations and sum rules forthe reflectivity may be written in analogy with the rules forN(w) - 1. As examples we consider rules associated with thecombinations Fm(w), wm[F(w)]m, and W2(m-l)[F(w)]m. Manyother rules for powers of F(w) involving various moments orother weighting functions also hold and can be proved bysimilar methods. However, they contain no new physics anddo not appear to be of practical interest.

Sum Rules and Dispersion Relations for i m(w)To discuss dispersion relations for F(w) it is convenient39 tointroduce the Fourier transform

R(r) = (271- 1 2 J F(w)eiWTdw.

Since F(M) has no singularities on the real axis for either in-sulators or metals, this formulation holds for both types ofmaterials [cf. the treatment of N(w) - 1 in Ref. 4]. The re-sponse function R(T) represents the amplitude at time 'r of thereflected wave at the surface of a sample resulting from adelta-function excitation incident upon the surface at r = 0.For the case of normal incidence discussed here, causalityrequires that

R(r) = 0, T < 0.

Thus F(co) is a square-integrable function the Fourier trans-form of which vanishes for r <0. These are just the condi-tions for the Titchmarsh theorem2 6 to hold. Dispersionrelations therefore follow in the usual manner2'4 for F(W). Forhigher powers of the reflectivity consider the transform

Rm(T) = (27r)-1/2 f -m(w)e-i-rdw. (A6)

Application of the Faltung theorem leads to the recursionrelation

Rm(T) = (27r)-1/2 f R(s)Rm-i(T - s)ds. (A7)

Then by induction the causality of R (r) guarantees that Rm (T)= 0, r < 0 for all m > 1. Since Pm (W) is a square-integrablefunction, the conditions of the Titchmarsh theorem2 6 againhold, leading to the generalized reflectivity dispersion rela-tions

rm(o)cos[mO(w)] = 2 p f -A'rm(WI)sin[mO(c')]7 Jo c

2 -, 2

(A8)

and

= 2w Prf rm(w')sin[mO(W')]r nm P O C0/2 - W2

(A9)

Sum rules follow directly from these dispersion relationsby taking the high- and low-frequency limits. The low-fre-quency limit of Eq. (A8) yields an expression for powers of thestatic reflectivity

rm(O) = 2 J'wlrm(c)sin[mO(c)]ddw.at JO

(A10)

The high-frequency limit yields two groups of sum rules thatfollow directly from the asymptotic form, Eq. (A4), and thesuperconvergence theorem.' 8 Equating powers of co- 2 in thehigh-frequency limit of Eq. (A8) yields a generalized f-sumrule for Im Fm(c),

JO wsr-(w)sin[m9(wo)]dw = 80

(m = 1)

(m > 2)

Similarly, since there are no terms in co- 1 in the high-fre-quency expansion of Im Fm (w), Eq. (A9) yields

J rm(w)cos[m0(w)]dw = 0 (m = 1, 2, 3.. .). (A12)


(A5)


Sum Rules and Dispersion Relations for [wzfo)Jm, m 3 1A second group of sum rules for the reflectivity may be writtenby considering powers of coF(w). From the discussion of F(cv)it follows that [wf(c)]m is also square integrable over the realaxis and that its analytic continuation is holomorphic in I+with an asymptotic behavior Q-m as Q - -. This asymptoticbehavior in I+ and the boundedness of P(Q) guarantee that[wN(w)]m is square integrable along any line parallel to the realaxis in the upper half plane. The conditions of the Titch-marsh theorem26 are therefore satisfied, and dispersion rela-tions connecting Re[kF(w)]m and Im[wF(w)]m immediatelyfollow. 40 As a result of the parity of wm these relations takedifferent forms for m even and m odd. The results are

wmrm(w)cos[mO(w)]

2w fW(')mrm(w')sin[m0(co')] d1'

7r fo WDE - ,

(m odd) (A13a)

2 P .(W(.') m+lrm(w')sin[m0(w')] dco'

5r ' ,,/2 -W d2

(m even) (A13b)and

wmrm(wo)sin[m0(c)]

2 Pj(w')m+lrr(w')cos[m(co')] dw'= -- W/2 - W2

(m odd) (A14a)

2co r jX- (w) mrm (w')cos[m6(cI)] dcoor Jo 0,/2 - ,J2

(m even). (A14b)

Taking the limit as X - 0 of the first of these equations, Eq.(A13), yields the rules

f w0m- 2rm(w)sin[mO(W)]dw

taken using the superconvergence theorem in the usual way.2The results are, for Eq. (A13),

S cmr-(c)sin[m0(w)jdw

(m = 1)or= P2o

=O

and

f 'W)m+lrm (co)sin[m0(wo)]dwo

ad- for EA43CO P

and, for Eq. (A14),

S (0m+lrm(c0)cos[mO(c)]dw = 0

(A17a)(m = 3,5,7, ... )

(m = 2)(A17b)

(m = 4,6,8 .... )

(m =3,5,7...)

(A18a)

and

I cmrm(w)cos[m0(c)1dw = 0 (m = 2,4,6.. .).

(A18b)

Sum Rules and Dispersion Relations for W2(m-l[44)Im, m' I

An argument similar to that for [wF(w)]m may be made for thefunction 2 (ml)[F(w)]m. The only difference is that

lim c 2 (m-l)[F(w)]m = (. ) cop2m/n 2Co- ~1 4/

(A19)

2 r(0)O!:

(o

(m = 1)

(m = 3,5,7...)

and

Y ,m-lrm(-,)sin[m0(cw)]d° = 0 (m = 2, 4, 6...

(A15b)

The low-frequency limit of the second, Eq. (A14), yields

5 wm-lrm(c)cos[m(w)]ddw = 0 (m = 1, 3,5 .. )

(A16a)

and

S Om- 2rm(w)cos[m0(co)]dw = 0 (m = 2, 4, 6...

(A16b)

The high-frequency limits of the dispersion relations may be

both on the real axis Io and in the upper half plane I+. Thisasymptotic behavior guarantees square integrability over I0

(A15a) and along any line parallel to Io in I+. The Titchmarsh the-orem2 6 may then be applied to get the dispersion relations

C02 (m-l)rm(w)cos[m0(co)]

2 p uCO/ 2mrlrm(Wo')sin[m0(co')] dI '

7r JO C02

- C,2

and

w2 (m-l)rm(co)sin[m0(w)]

= -- P f w'/2(m-1)rm.(w)cos[mO(c')] I

7r J W/2 - ,2

Taking the limits of w -> 0 yields

S W2m- 3rm(co)sin[m0(co)]dw = {i20o

(A20)

(A21)

(m = 1)

(m ' 2)

(A22)

and



, w2m-4rm(co)cos[mO(w)Idw = 0 (m 2 2).

The high-frequency limit of the integrals may be taken byusing the superconvergence theorem,1 8 and comparison withEq. (A19) yields the generalized f- and inertial sum rules

S w2m-lrm(w)sin[m0(w)Idco = ( 2) +1 2m

E 22m+1 P~

(A24)

and

Ew C2(m-1)rm(co)cos[m0(co)]dco = 0. (A25)

APPENDIX B: CORRECTIONS FOR FINITE-ENERGY INTERVALS

Corrections for limitation of sum rules and dispersion-relationintegrals to the range 0 < w < X may be estimated by evalu-ating the integrals from X) to - for the optical constants of aharmonic-oscillator or free-electron model fitted to the low-energy absorption. As was shown in the discussion of Eqs.(31)-(33), these corrections are generally small and in manysituations can be neglected.

Retaining only the largest terms in the asymptotic expan-sions yields for the finite-energy sum rules for In F(w)

and

E In r(W)dcw = ln rb + ( 2 1) (1

0w(w)dw = b. 2.- 1)(7rw)

(Bi)

(B2)

Here the plasma frequency and damping constant for the fitto the low-energy absorption are denoted by ppj and v, re-spectively.

Similarly, the finite-energy dispersion relations for thelogarithmic derivative are

r'(w) 2W O '( ') do) - 6 vC0Wp, 2

r(c) or J /2 -... 2 5-rnb(nb 2

- 1) c)5

(B3)

and

dO(co)dw 7rP f C/2 - WJ2 d

4 Wp1 2

37rnb (nb 2 - 1) C.3(B4)

Note added in proof: After this manuscript was prepared,two relevant publications came to the authors' attention. Thefirst, by King,41 gives derivations of special cases of the rulesinvolving r(w) and 0(w) discussed in Section 2 and AppendixA of the present paper. The second, by Eldridge and Staal,42

gives more-recent values of the optional constants for NaClobtained by using an asymmetric Fourier-transform spec-trometer. These new data are of higher resolution than thoseof Geick used here; however, the general features of both setsof data are the same, as are the conclusions drawn regardingthe applicability of the sum rules.

Thanks are due to H. Bilz and L. Genzel for helpful dis-cussions and for bringing to the authors' attention the thesisof R. Geick. The aid of G. Graham in preparing numericalexamples and a number of figures is greatly appreciated. D.Y. Smith wishes to express particular thanks to H. Pick for hisprofessional interest and for his hospitality in making thefacilities of the Physikalisches Institut (Teil 2) der UniversitaitStuttgart available.

This work was supported in part by the U.S. Departmentof Energy. In addition, the financial support of the DeutscherAkademischer Austauschdienst, the Deutsche Forschungs-gemeinschaft, and the Max-Planck Geselschaft provided toD. Y. Smith is gratefully acknowledged.

* Permanent address, Solid State Science Division, ArgonneNational Laboratory, Argonne, Illinois 60439.

t Present address, Relativity Center, University of Texasat Austin, Austin, Texas 78712.

REFERENCES

1. W. M. Saslow, "Two classes of Kramers-Kronig sum rules," Phys.Lett. 33A, 157-158 (1970).

2. M. Altarelli et al., "Superconvergence and sum rules for the op-tical constants," Phys. Rev. B 6, 4502-4509 (1972).

3. A. Villani and A. H. Zimerman, "Superconvergent sum rules forthe optical constants," Phys. Rev. B 8, 3914-3916 (1973); "Gen-eralized f-sum rules for the optical constants," Phys. Lett. 44A,295-297 (1973).

4. M. Altarelli and D. Y. Smith, "Superconvergence and sum rulesfor the optical constants: physical meaning, comparison withexperiment, and generalization," Phys. Rev. B 9, 1290-1298(1974).

5. F. W. King, "Sum rules for the optical constants," J. Math. Phys.17, 1509-1514 (1976).

6. K. Furuya, A. H. Zimerman, and A. Villani, "Sum rules for theconductivity of superconducting films," Phys. Rev. B 13,1357-1358 (1976).

7. D. Y. Smith, "Superconvergence and sum rules for the opticalconstants: natural and magneto-optical activity, Phys. Rev. B13, 5303-5315 (1976).

8. D. Y. Smith and E. Shiles, "Finite-energy f-sum rules for valenceelectrons," Phys. Rev. B 17, 4689-4694 (1978).

9. D. Y. Smith, "Dispersion theory and moments relations in mag-neto-optics," in Theoretical Aspects and New Developments inMagneto-Optics, J. Devreese, ed. (Plenum, New York, 1981).

10. E. Shiles and D. Y. Smith, "Self-consistency analysis of opticaldata: aluminum," Bull. Am. Phys. Soc. 22,92 (1977); "The op-tical properties of silicon from the infrared to the x-ray region,"Bull. Am. Phys. Soc. 23,226 (1978); "Optical properties of semi-conductors: gallium arsenide," Bull. Am. Phys. Soc. 24, 335(1979).

11. E. Shiles et al., "Self-consistency and sum-rule tests in theKramers-Kronig analysis of optical data: applications to alu-minum," Phys. Rev. B 22, 1612-1628 (1980).

12. D. Y. Smith, "Dispersion relations and sum rules for magneto-reflectivity," J. Opt. Soc. Am. 66, 547-554 (1976).

13. K. Furuya, A. H. Zimerman, and A. Villani, "Superconvergentsum rules for the Voigt effect," J. Phys. C 9, 4329-4333 (1976).

14. K. Furuya, A. Villani, and A. H. Zimerman, "Superconvergentsum rules for the normal reflectivity," J. Phys. C 10, 3189-3198(1977).

15. T. Inagaki, A. Ueda, and H. Kuwata, "Sum rules for the complexreflection amplitudes of photons," Phys. Lett. 66A, 329-331(1978).

16. T. Inagaki, H. Kuwata, and A. Ueda, "Phase-shift sum rules forthe complex polarization amplitude of photon reflection," Phys.Rev. B 19, 2400-2403 (1979).

17. D. Y. Smith, "Limits of efficiency of solar-thermal energy con-version: greenhouse collectors and heat mirrors," Bull. Am. Phys.Soc. 24, 273 (1979).

18. For an introduction to this theorem see the appendix of Ref. 2.



Details are given in V. de Alfaro et al., "Sum rules for strong in-teractions," Phys. Lett. 21, 576-579 (1966); G. Frye and R. L.Warnock, "Analysis of partial wave dispersion relations," Phys.Rev. 130, 478-494 (1963), especially Appendix D.

19. The notation yo = o6/) as x X means that p(x)/4(x) - 0 as x-a- , and d = OP) as x - means that there exist positiveconstants A and xo such that I (x) I < A I 4(x) I, x 3 x0.

20. D. Y. Smith, "Dispersion relations for complex reflectivities," J.Opt. Soc. Am. 67, 570-571 (1977).

21. J. S. Toll, "Causality and the dispersion relation: logical foun-dations," Phys. Rev. 104, 1760-1770 (1956).

22. F. Stern, "Elementary theory of the optical properties of solids,"in Solid State Physics, F. Seitz and D. Turnbull, eds. (Academic,New York, 1963), Vol. 15, pp. 299-408, especially page 335.

23. The restriction that g(co) be nonzero for finite co is noteworthybecause of its physical significance. The point is that in the limitof high frequencies r(w) has a universal behavior

lime,, In r(a) = -2 In X

for all systems, independently of their composition. Funda-mentally this behavior is determined by the high-frequency formof the dielectric function ?(U): If lim e(U) = 1 + A/wY, lime asin r(w) = -y in w. The argument against Eq. (9) given in the textdoes not hold for a function g(co) that samples only the universalhigh-frequency portion of ln r(w) and vanishes for finitefrequencies. Such a weighting function is t/(W2 - p2), in the limitof large <. This function is negligible for co << gbut is appreciablefor X -; it also satisfies Eq. (10) as required. In particular, thisweighting function occurs in the high-frequency limit of Eq. (6)for the canonical phase shift, which in the limit of high frequenciesis or for all systems:

l 2 f - In r(co) dGus 7r JO C0 - t2

This is an expression of the form of Eq. (9) but is essentially trivialsince line W 9(w) = 7r is known from physical considerations andthe integral only provides a test of the known asymptotic behaviorof r(w). Mathematically, the fact that the high-frequency limitof the canonical phase shift is independent of the details of thereflectance at finite frequencies may be shown by dividing theabove integral into two parts: one for the asymptotic region fromCU to - in which r(w) has a power-law falloff r(cc) z-. ay (y = 2 inour Newtonian world!) and the second for the region 0 < X < Lc.In the limit of -X the second integral vanishes while the firstapproaches y7r/2. See Formula 864.3 in H. B. Dwight, Table ofIntegrals, rev. ed. (McMillan, New York, 1953).

24. D. Y. Smith, "Reflectivity sum rules," Bull. Am. Phys. Soc. 21,366 (1976). Note that the letters In inadvertently appear as atypographical error in the integral given in this abstract. Theexpression was intended to be Eq. 24 of the present work.

25. D. Y. Smith and C. A. Manogue, "Finite-energy sum rules forinfrared reflection spectroscopy," Bull. Am. Phys. Soc. 23, 342(1978).

26. E. C. Titchmarsh, Introduction to the Theory of Fourier Inte-grals, 2nd ed. (Oxford U. Press, London, 1962), pp. 119-128. Foran introduction, see H. M. Nussenveig, Causality and DispersionRelations (Academic, New York, 1972).

27. R. Geick, Zur Dispersion des NaCI in Bereich seiner UltrarotenEigenschwingung, Thesis, UniversitAt Frankfurt (1961); Z. Phys.166, 122-147 (1962).

28. G. Bonfiglioli and P. Brovetto, "Improved optical spectroscopytechnique," Phys. Lett. 5, 248-251 (1963); "Principles of self-

modulation derivative optical spectroscopy," Appl. Opt. 3,1417-1424 (1964).

29. B. 0. Seraphin, "Modulated reflectance," in Optical Propertiesof Solids, F. Abels, ed. (North-Holland, Amsterdam, 1972), pp.163-276.

30. For details, see C. Ferro Fontan, N. M. Queen, and G. Violini,"Dispersion sum rules for strong and electromagnetic interac-tions," Riv. Nuovo Cimento 2, 357-497 (1972), especially Sec.2.

31. E. E. Bell, "Measurements of the far-infrared optical propertiesof solids with a Michelson interferometer used in the asymmetricmode: Part I, Mathematical formulation," Infrared Phys. 6,57-74 (1966); E. E. Russell and E. E. Bell, "Measurements of thefar-infrared optical properties of solids with a Michelson inter-ferometer used in the asymmetric mode: Part II, The vacuuminterferometer," Infrared Phys. 6, 75-84 (1966). See also J. E.Chamberlain, J. E. Gibbs, and H. A. Gebbie, "The determinationof refractive index spectra by Fourier spectrometry," InfraredPhys. 9, 185-209 (1969).

32. J. Gast and L. Genzel, "An amplitude Fourier spectrometer forinfrared solid state spectroscopy," Opt. Commun. 8, 26-30(1973).

33. T. J. Parker, W. G. Chambers, and J. F. Angress, "Dispersivereflection spectroscopy in the far infrared by division of the fieldof view in a Michelson interferometer," Infrared Phys. 14,207-215(1974).

34. B. Velickf, "Dispersion relations for complex reflectivity," Czech.J. Phys. B 11, 541-543 (1961).

35. F. C. Jahoda, "Reflectivity of barium oxide single crystals," Thesis(Cornell University, Ithaca, N.Y. 1957) (unpublished).

36. J. M. Ziman, Principles of the Theory of Solids, 2nd ed. (Cam-bridge U. Press, Cambridge, England, 1972), Chap. 8.

37. D. Y. Smith, "Comments on the dispersion relations for thecomplex refractive index of circularly and elliptically polarizedlight," J. Opt. Soc. Am. 66, 454-460 (1976).

38. E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford U.Press, London, 1968), Sec. 5.61; see also H. M. Nussenzveig,Causality and Dispersion Relations (Academic, New York, 1972),Theorem 7.5.1.

39. An alternative proof not involving R(T) can be made by showingthat f(X) is square integrable along any line parallel to the realaxis in I+. This follows from the boundedness of t(9) in I+ andits It-2 asymptotic behavior as 9 - . Application of theTitchmarsh theorem then yields the dispersion relations.

40. This can also be proved by observing that differentiation of Eq.(A6) with respect to i- shows that the Fourier transform ofcwf(w) is just iR'QT) [the integral Sft'wF(w)e-1dw converges sinceat large co the integrand approaches co-le-@", a form encounteredin the definition of Si and Ci]. Since R(r) is zero for r <0, R'T)is also zero in this interval, leading to the conclusion that ccr(W)is a causal transform. Higher powers of wi'(co) may be similarlyshown to be causal transforms by repeated application of theFaltung theorem in analogy with the proof of Eq. (A7). Appli-cation of the Titchmarsh theorem yields the desired dispersionrelations.

41. F. W. King, "Dispersion relations and sum rules for the normalreflectance of conductors and insulators," J. Chem. Phys. 71, 4726(1979).

42. J. E. Eldridge and P. R. Staal, "Far-infrared dispersive-reflectionmeasurements on NaCl, compared with calculations based oncubic and quartic anharmonicity. I. Room temperature," Phys.Rev. B 16, 4608 (1977).


superconvergence relations and sum rules for reflection spectroscopy

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