7 FINK, RANK, AND W\;IGGINS

methane in the atmosphere. The results of our abund-ance determinations are given in Table IH.

Using the above calculated values of Aa for R(O) andP(2) and combining these values with the integratedabsorptances observed in the atmospheric measure-ments we obtain an equivalent path of 1.11 cm-atm(NTP). Our estimated error is 15%. The statisticalprobable error of our measurements is, of course, very

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

much smaller than the value quoted above. However,we believe our quoted error is realistic. The result wehave obtained for the methane abundance in the atmos-phere does not differ significantly from results previouslyobtained by others. However, we believe our result tobe more satisfactory since it was obtained by using arealistic effective temperature for the methane in theatmosphere.

VOLUME 54, NUMBER 4 APRIL 1964

Plasma-Interband Coupling in g' CuZnt

KEITH H. JOnNSON

Temnple University, Philadelphia, Peimzsvlvania 19122

AND

ROBERT J. ESPOSITO

Pitinan-Dautn institultefor Research, Frankford Arsenal, Philadelphia, Pennsylvania 19137(Received 29 October 1963)

Normal-incidence reflectivity data for f3' CuZn has been subjected to a Kramers-Kronig analysis for itsoptical constants in the spectral region 2 eV <hco•3 eV (4140! <X< 6200!) in order to explain the natureof an observed reflectivity edge at 5000 X. Extrapolations of the data into the infrared and ultraviolet weremade. The frequency dependence of the optical constants indicates a plasma resonance at 2.52 eV (4920 A),shifted and damped from its classically predicted value by the influence of interband transitions occurringat approximately the same energy. The spectral behavior of the optical constants in this region is not verysensitive to the exact nature of the extrapolation.

INTRODUCTION

THE alpha and beta phases of the copper-zincbrass alloy exhibit rather distinct color variations

as a function of zinc concentration and of temperature.These color variations have been the motivation forseveral investigations into their optical properties.'-3

Recently, Muldawer4 of Temple University has reportedon his measurements of the normal-incidence reflectivityof ordered fS' CuZn and other beta-brass type alloys as afunction of temperature.

Such data by themselves cannot be used conclusivelyto resolve the individual contributions to the opticalproperties of free-electron effects, interband transitions,and plasma oscillations. However, a knowledge of thecomplex dielectric constant

e:= 61-i2 (1)

and complex refractive index

=, - _ _ ( (2)

t Research supported by the Atomic Energy Commission, andthe U. S. Army Material Command.

' H. Lowery, H. Wilkinson, and D. L. Smare, Proc. Phys. Soc.(London) 49, 345 (1937).

2 N. F. Mott, Proc. Phys. Soc. (London) 49, 354 (1937).3J. A. Rayne, as reported in The Ferzi Surface, edited by

W. A. Harrison and M. B. Webb (John Wiley & Sons, Inc., NewYork, 1960), p. 269.

4L. Muldawver, Phys. Rev. 127, 1551 (1962).

and

where k is the extinction coefficient, can be applied todistinguish among these separate solid-state phenomena.The constants can be extracted from existing normal-incidence spectral reflectivity data by utilizing theKramers-Kronig dispersion relation

2orlnjrQ4J)a0 (.o) = 2w nJ X,,))I7r W 0 2-CO02

(3)

which connects the phase 0 at some arbitrary frequencywo with the modulus t r I of the complex reflectance

(4)

In Eq. (4), l r 12 is the quantity measured. The refractiveindex (2) can then be gotten from l r j and .0 through therelations

1- rl2II=

1+ jr1|-21rl coso

2 f r I sin+

1+r1 2- 21r coso

(5)

(6)

The dielectric constant (1) can be calculated in turnfrom it and le from the expressions

,El=1='-k 1(7)

andE2= 2nk. (8)

474 VOl. 54

r = (N - 1)/ (N+ 1) = I r I

PLASMA-INTERBAND COUPLING IN O' CuZn

We have applied the above K-K method to the datataken by Muldawer 4 on room temperature 13' CuZn inorder to obtain the spectral dependence of the opticalconstants in the immediate vicinity of a sharp re-flectivity edge at 5000 A. The relatively narrow limitsof the data have made it necessary to perform certainextrapolations into the infrared and ultraviolet. Never-theless, our confinement of the problem to a smallspectral region around 5000 A ensures that the opticalconstants obtained there are a reasonable basis for aninterpretation of the edge.

CALCULATION OF THE OPTICAL CONSTANTS

The data of Muldawer4 covers the wavelengthrange 3500 A<X< 7000 A, or the energy range 1.77eV< hco<3.5 4 eV. The error involved in applying theK-K relation (3) to such data, where infrared andultraviolet measurements are lacking, decreases in pro-portion to the remoteness of the chosen frequenciesowo from the limits of the data. As we are interested inthe validity of the optical constants near 5000 A, thiserror is kept at a minimum. Moreover, the error maybe significantly reduced by carrying out reasonableextrapolations of the reflectivity into the infrared andultraviolet.5-7

The Drude free-electron theory 8 has been shown toaccount for the optical properties of most metals inthe near infrared, except for a small contribution due tothe anomalous skin effect. This leads us to expect thatthe reflectivity will level off and remain constant in thisregion. Measurements of 1' CuZn recently extended byGoldman9 to the near infrared confirm this prediction.Therefore, we have extrapolated the 7000-A reflectivityvalue of Muldawer to infinite wavelength or zerofrequency.

Throughout the ultraviolet, the gross similarity of thespectral reflectivity among metals of a particular class,such as the noble metals'" and semiconductors," hasbeen attributed principally to absorption due to inter-band transitions. For the purpose of our uv extrapola-tion, we have assumed that the absorption spectrumof 1' CuZn is similar to that of its parent noble metalcopper. This is a reasonable assumption, for the grossrange of energy spacings between bands in the alloyshould agree roughly with that of the noble metalsbecause of the basic cubic symmetry and identical ioncores in both cases. There are, of course, detailed differ-ences between the band structures which the total uvabsorption spectrum does not readily reveal.'0

Thus, we have renormalized the 13' CuZn reflectivityresults of Muldawer 4 to 99% in the infrared andsmoothly joined it to fit the copper data of Ehrenreich

5 F. C. Jahoda, Phys. Rev. 107, 1261 (1957).6 H. R. Philipp and E. A. Taft, Phys. Rev. 113, 1002 (1959).7 B. Velicky, Czech. J. Phys. B11, 787 (1961).8 M. P. Givens, Solid State Phys. 6, 313 (1958).H. Goldman (private communication).

10 H. Ehrenreich and H. R. Philipp, Phys. Rev. 128, 1622 (1962).11 H. R. Philipp and H. Ehrenreich, Phys. Rev. 129, 1550 (1963).

80

60

40

20

IU 0 2

4 6 8 10 12 14 16 18 0c

ch oa 1eV )

FIG. 1. Spectral reflectivity profiles for Cu (Ehrenreichand Philipp'") and 1' CuZn (Muldawer4 ).

and Philipp'0 at 5.85 eV (2120 A). The fit between thelast measured point at 3.45 eV (3500 A) followed theprofile of the ternary alloy Au13Cu38Zn49, which isoptically similar to 1' CuZn in the visible.'2 The copperextrapolation was carried out to 18.8 eV (660 A); be-yond this point the reflectivity was assumed to have anaverage value of 1%. The phase angle and opticalconstants were calculated on a Remington Rand SolidState 90 computer which was programmed in FortranII, version 9000 language. The average cutoff re-flectivity (more correctly, the contribution to 4 in theinterval from 18.8 eV to infinity) is chosen by bracketingknown values of the optical constants in the visiblespectrum by successive approximation until the com-puter values agree with those measured by directtechniques. The d' CuZn cutoff reflectivity was chosento be 1% because this value yielded results consistentwith direct measurements of the optical constants ofcopper.

By choosing two uv extrapolations alternative to thatof copper described above, we have found that thefrequency dependence of the optical constants in theimmediate vicinity of the reflectivity edge (5000 A)is relatively insensitive to the detailed nature of theextrapolation. In the first extrapolation it was assumedthat the reflectivity decreases exponentially from itsmeasured value at 4.96 eV (2500 A) to a value of 1%at 12.0 eV (1032 A), beyond which it remains constantat 1% out to infinite energy. The second extrapolationwas a more extreme one; the reflectivity was assumedto remain constant at its 3.54 eV (3500 A) value out toinfinite energy.

DISCUSSION

The total reflectivity profile consisting of Muldawer's4

renormalized data plus the copper extrapolation isshown in Fig. 1 as a function of energy. In Fig. 2,

12 L. Muldawer (unpublished data).13 R. J. Esposito, F. Rothwarf, and J. N. Brown, "A Computer

Program for a Kramers-Kronig Transformation of the OpticalReflectivity," Frankford Arsenal M. R. Report.

| -- K CuZn

- Cu

I

100. -,-, ,, I,, I , I ,, ,, . . . . .

4r75April 1964

K. H. JOHNSON AND R. J. ESPOSITO

7.0

6.0

5.5

5.0

4.5

4.0-?0

FIG. 2. Spectral de-pendence of the absorp-tion coefficient of j3'CuZn for a uv copperextrapolation.

2.2 2.4 2.6 2.8 3.0

h c ( (eV)

the absorption coefficient 4irk/X is plotted as a functionof energy over the range 2 eV< ho<K3 eV, or betweenthe wavelengths 6200 and 4140 A. This coefficientexhibits a sharp minimum at 2.52 eV (4902 A) at thereflectivity edge. Such a minimum is associated withthe onset of interband transitions occurring at some highsymmetry point of the Brillouin zone.'"14 It is to thesetransitions that fi' CuZn probably owes its yellowishcolor at room temperature.2 4

If these transitions are taking place between theFermi surface and the next unoccupied valence band,the energy 2.52 eV is a first approximation to the magni-tude of the gap between the first and second zones. Onthe basis of a simple 1-OPW Harrison'5 type model forthe Fermi surface (assuming three valence electrons)the most probable symmetry point for these transitionsis that at the edge of the cubic zone in the [110] direc-tion (BSW symmetry point M). Here the Fermi spherehas just begun to penetrate the zone boundary. A first-order perturbation to the empty-lattice valence bandsalong this direction indicates the presence of a gap be-tween M states of proper symmetry, which can beadjusted to the observed value.

However, as in the noble metals, there is the un-specified interaction with the d bands, which mayhybridize with the valence bands so as to distort themappreciably from the empty-lattice situation. Moreover,optical transitions can occur between the d bands andFermi surface as well as from the Fermi surface to ahigher valence band. For example, the optical pro-perties of disordered fcc a CuZn, as an increasing amountof zinc is added, seem to support a rigid-band modelbased on transitions taking place between copperliked bands and the Fermi surface.3

A band structure calculation of j' CuZn has beeninitiated by one of us (K.J.) in order to determine fromfirst principles the magnitudes of energy gaps, theimportance of the d bands, and the identification of

14 M. Suffczynski, Phys. Rev. 117, 663 (1960).' WV. A. Harrison, Phys. Rev. 118, 1190 (1960).

optically observed interband transitions. This calcula-tion, together with extended optical measurements fromthese laboratories and completion of de Haas-vanAlphen studies of f' brass type alloys by Beck el al.16

should result in a fairly accurate physical model ofjS' CuZn.

In Fig. 3, the energy loss function

Im(1/e) = E2/ ('El'+ 62), (9)

E

Cr,

.I0I

- 2- 3- 4- 5- 6- 7- 8-l 9

-1o- II ;

-14

- 15

-162a0 2.2 2.4 26

hco Wev)

2.8 3.0

.40

35

FIG. 3. Spectral30 dependence of the

real part of the die-YL lectric constant and

25 + e the energy loss factora~~ for /3' CuZn, using

20 a uv copper extrap-olation.

'5

Jo

16A. Beck, J. P. Jan, W. B. Pearson, and I. M. Templeton,Phil. Mag. 8, 351 (1963).

17 H. Frdhlich and H. Pelzer, Proc. Phys. Soc. (London) A68,525 (1955).

18 P. Nozieres and D. Pines, Phys. Rev. 109, 762 (1958).19 P. Nozieres and D. Pines, Phys. Rev. 113, 1254 (1959).20 D. Bohm and D. Pines, Phys. Rev. 82, 625 (1951)."1 In thin film specimens it is possible to excite a surface type of

collective oscillation with electromagnetic radiation at the plasmafrequency obliquely incident and polarized in the plane of in-cidence. See R. A. Ferrell and E. A. Stern, Am. J. Phys. 30, 810(1962).

476

is plotted as a function of energy, again between thelimits of 2 and 3 eV. In the dielectric formulation of thecharacteristic energy loss problem, a sharp peak in thisquantity is associated with the excitation of longitudinalplasma oscillations by an externally introduced fastelectron traversing the metal.'7 -'9 Such a peak isevident in Fig. 3 at the energy 2.52 eV (4920 A), i.e.,at the same energy as that for the onset of interbandtransitions. The propagation of an electromagneticwave through a plasma constitutes a transverse plasmaoscillation.'9 -2 0 For a bulk specimen, at normal in-cidence and conventional intensities, there is no couplingbetween the transverse photons and the longitudinalplasmons of the characteristic energy loss problem.2 'However, to a high order of approximation, the dielec-tric constant e measured optically is identical to thatdescribing the energy loss experiments."7 ,9 Therefore,it appears that at the energy licw=2.52 eV there is aplasma resonance, where wp is the observed frequencyof the plasma oscillation.

The energy loss function (9) has also been plotted forthe two alternative uv extrapolations described above.

Vol. 54

PLASMA-INTERBAND COUPLING IN ,3' CuZn

.25

FIG. 4. Spectral de-pendence of the energyloss factor for t' CuZn,using, respectively, anexponential and con-stant extrapolation inthe uv.

,d +

No-

2.0 22 2.4 2.6 2.8 a0Th co (eV)

See Fig. 4. For both extrapolations the peaks are againobvious. For the exponential extrapolation the peakis at 2.49 eV; for the constant extrapolation it is at2.47 eV. These values are to be compared with the2.52-eV value above, obtained for the copper extrapola-tion. The agreement bears out the dominant influence ofthe sharp reflectivity edge on the spectral behavior ofthe optical constants in this region.

Another condition for a collective oscillation is thatEl pass through zero at hwo when E2 is small and roughlyconstant." 9 In Fig. 3 there are two zeros of El, one at2.60 eV and one at 2.73 eV, somewhat displaced fromthe 2.52-eV value for the peak in the energy loss factor.This displacement and a significantly increasing con-tribution by E2 near 2.52 eV suggests that the plasmaoscillations are considerably damped. This damping alsocontributes to the breadth of the ImE-E profile. Thepresence of two zeros of El close together is the result ofthe way the interband contribution to the measured El

adds to the free-electron contribution. This aspect of theargument is discussed below.

For an electron gas free of the perturbing interbandeffects of the periodic potential, the application ofplasma theory to the optical properties yields resultsequivalent to those of the classical Drude theory. In thiscase, the plasma frequency is given by

co,= (4,irne2/m)1, (10)

where m is the mass of a free electron, e its charge, and nthe concentration of conduction electrons. For 13' CuZnthe energy corresponding to (10) is 12.8 eV (X = 970 A),using the Hume-Rothery value for n. The shift of theplasmon energy to 2.52 eV, along with the damping ofthe oscillations, is a consequence of the presence ofinterband transitions at this energy. In other words,the positive interband contribution flb to the real partof the dielectric constant occurs at an energy for whichthe free-electron contribution elf is comparable in valueto Elb. The total measured real dielectric constant isgiven by

El= Elb+ Elf, (11)

which is greater than the free-electron value alone.Consequently, the zeros of El and peaks of Imc'- areshifted from their free-electron counterparts. It wouldbe valuable to perform characteristic energy loss ex-periments on 1' CuZn to check the existence of longi-tudinal oscillations at the optically predicted energy.

Thus, it is evident that the reflectivity edge observedby Muldawer 4 at 5000 A is the result of a superpositionof free-electron effects, interband transitions, andplasma oscillations. The behavior is quite similar tothat of the noble metal silver in the visible part of thespectrum.10 22 In copper, however, no plasma resonanceis observed in this region.' 0

ACKNOWLEDGMENTS

We wish to thank Dr. L. Muldawer and H.Goldman, of Temple University, and Dr. F. Rothwarf,of the Frankford Arsenal, for their helpful discussions ofthe experimental aspects of the problem, and for the useof their unpublished data. We gratefully acknowledgeseveral stimulating conversations with Dr. H. Amar ofTemple as to the theory of the Kramers-Kronig rela-tions; and we are indebted to Dr. H. Ehrenreich of theGeneral Electric Research Laboratories for sending tous the tabulated reflectivities taken by him and Dr.H. R. Philipp for copper and silver.

22 E. A. Taft and H. R. Philipp, Phys. Rev. 121, 1100 (1961).

477April 1964