Optical constants of ice from the ultraviolet to themicrowave
Stephen G. Warren
A compilation of the optical constants of ice Ih is made for temperatures within 60 K of the melting point.The imaginary part mim of the complex index of refraction m is obtained from measurements of spectral ab-sorption coefficient; the real part mre is computed to be consistent with mim by use of known dispersion rela-tions. The compilation of mim requires subjective interpolation in the near-ultraviolet and microwave, atemperature correction in the far-infrared, and a choice between two conflicting sources in the near-infrared.New measurements of the spectral absorption coefficient of pure ice are needed, at temperatures near themelting point, for 185-400-nm, 1.4-2.8-yrm, 3.5-4.3-,um, 33-600-,um, and 1-100-mm wavelengths.
1. Introduction
Theoretical calculations of absorption, transmission,reflection, emission, and scattering of electromagneticradiation in ice, and in ice-containing media such assnow and clouds, require knowledge of the laboratorymeasurements of the complex refractive index m of pureice as a function of wavelength. m is a complex func-tion, m(X) = mre (X) - imim (X), where X is the wave-length in vacuum, mre is the usual refractive indexwhich determines the phase speed, and mim is relatedto the absorption coefficient kabs, as kabs = 4rmim/X.(We will often refer to the imaginary part of the complexindex of refraction as the imaginary index of refraction,and similarly for the real part.) In the microwave andradiowave spectra it is more usual to report the complexrelative permittivity e = ' - ", or the dielectric con-stant e' and loss tangent, tan =-e/e'. In nonmagneticmaterials (such as ice) they are related to the complexrefractive index m which we report as M2= .
A. Earlier Reviews
The optical constants of ice have been reviewed byIrvine and Po]lack1 (IP) for the infrared, by Evans2 forthe microwave and radiowave regions, and by Ray3 andby Hobbs4 for the entire spectrum. The data recom-mended by IP have been superseded by better mea-surements everywhere except in the 1.4 -2 .8 -Am region.Ray fitted analytical formulas to the compilation of IPand to a few points in the microwave, but he ignored
The author is with University of Washington, Department of At-mospheric Sciences, Seattle, Washington 98195.
Evans's earlier review of the other available microwavedata. Hobbs's review is essentially complete up to 1972.In 1980, Wiscombe and Warren5 compiled the opticalconstants for the visible and near-infrared, but thatcompilation also has been superseded by new mea-surements. There are now many new measurementsavailable with which to prepare a better compilation ofm from the far-ultraviolet through the far-infrared.However, there has been little additional reliable workon microwave optical properties of ice since Evans's(1965) review, so for that part of the spectrum we relymostly on the references which he cited.
B. Crystal Forms of Ice
About ten different phases of ice have been discov-ered, but most of them can occur only at very highpressure and have only been observed in the laboratory.The three which exist at low pressures are ordinaryhexagonal ice Ih; the cubic form ice Ic which can remainstable at temperatures as high as -800 C but must beformed by condensation of vapor at lower temperatures;and amorphous (or vitreous) ice, which forms by con-densation of vapor at even lower temperatures. Theonly form ever observed to occur naturally on earth isice Ih. Temperatures in tropical cirrus clouds may becold enough to contain ice Ic, and Whalley6 has specu-lated that ice Ic could be responsible for a rare halo thatcannot be explained by hexagonal crystals. However,even if that speculation is correct, it does not complicateour specification of the optical constants: in the spec-tral regions where optical properties of both Ih and Ichave been measured (ultraviolet and far-infrared, asdiscussed below) they are practically identical. Thehigh-pressure forms of ice also do not exist naturally onearth. Even under the deepest parts of the Antarcticice sheet, the pressures are insufficient to form ice II orice III. This review is therefore restricted to the optical
properties of ice Ih, but the results can probably be usedfor Ic as well.
The ice crystal is very slightly birefringent, so that therefractive index varies with orientation of the crystaland polarization of the light. However, the birefrin-gence is so slight that we may safely ignore it. This hasbeen shown by measurements of m on single crystalsboth in the ultraviolet7 and in the far-infrared.8 Thuswe are able to use measurements made either on singlecrystals or on polycrystalline ice (bulk ice made up ofcrystals whose axes do not coincide), and we need notreport two different sets of optical constants. [Thereasons for near-isotropy of the ice crystal to radiationare explained by Johari 9 (p. 631).]
C. Features of the Ice Spectrum
The causes of the variation of m with wavelength forice have been reviewed by Johari. 9 Ice exhibits strongabsorption in the ultraviolet (UV) at X < 170 nm, dueto electronic transitions. Ice is very transparent in thevisible, mim reaching its minimum value at X = 460 nm.There are weak absorption bands in the near-infraredat X > 1.4 um, which have not been assigned to partic-ular vibrational transitions. The very strong band atA 3 m is due to vibrational modes involvingstretching of the 0-H covalent bond. A band at 12 umis due to rotational oscillation (libration). Up to thiswavelength the spectrum of ice shows similarities to thespectra of liquid water and water vapor. The strongfar-infrared absorption bands at 45 and 60 Am, however,are due to lattice vibrations which have no counterpartin water vapor, and the spectrum of liquid water is alsoquite different in this region. The absorptivity againdrops to low values in the microwave region, and re-mains small until the dielectric relaxation peak is ap-proached at a frequency of -3 kHz (wavelength 100km). The dielectric relaxation of liquid water, bycontrast, occurs at A\ 20 mm.3 The dielectric prop-erties of ice and water are therefore dramatically dif-ferent over -7 orders of magnitude in frequency in themicrowave and radiowave regions; water is highly re-flective, but ice is quite transparent.
D. Measurements: Purposes and Methods
This review is written with geophysical applicationsin mind, so emphasis is placed on obtaining the opticalconstants for temperatures T between 00 and -60'C.In this high-temperature region, the temperature de-pendence of the optical constants is poorly known.However, based on suggestive evidence from near-ul-traviolet and far-infrared measurements discussedbelow, the temperature dependence for X < 100 Am islikely to be small within 60 K of the melting point. Atlonger wavelengths, mim becomes increasingly sensitiveto temperature as X increases. We report a tempera-ture-dependent m for A > 167 Am, tabulating it at fourtemperatures: -1, -5°, -20°, -60'C. For A < 167,m we attempt a compilation only for one temperature,T = -7°C, which is the temperature of Schaaf andWilliams's10 infrared measurements, but we expect it
to be valid for most temperatures that would be foundon earth.
Unfortunately for geophysical applications, much ofthe spectrum of ice has been measured only at tem-peratures much colder than any terrestrial tempera-tures, and we must attempt to adjust these results tovalues appropriate for warmer temperatures. Many ofthe measurements we use were made not for the purposeof geophysical application, but rather to understand thebehavior of the water molecule within the ice lattice.The spectra are usually easier to interpret at lowertemperatures. Studies of the electronic structure in theultraviolet7 were thus made at 80 K; studies of the lat-tice vibrations in the far-infrared 1 were made at 100 K.Measurements at temperatures near 0C were made inthe visible, near-infrared, middle-infrared, and rad-iowave regions. No reliable measurements of m areavailable at any temperature for the near-ultravioletbetween 185 and 400 Am and in the microwave between1.25 and 32 mm. In these regions we must obtain theabsorption coefficient by a subjective interpolation.
In spectral regions where mim << mre, mim is deter-mined by attenuation of a beam of light through a blockof clear bubble-free ice. Blocks as long as 2.8 m wereneeded by Grenfell and Perovich12 to obtain sufficientattenuation for accurate measurement in the visiblewavelengths where ice is very transparent. Films asthin as 50 Am are needed to obtain sufficient trans-mission at 1-2-Mm wavelength. In the far-infrared,Bertie et al."1 used some ice films that were <1 Am inthickness. In the latter cases the uncertainty in mea-surement of sample thickness limits the accuracy ofmim .
The real index mre is obtained by reflectance mea-surements from a plane-shave ice block. The Fresnelformulas give mre unambiguously if mim is small. Thereal index mre is thus known reliably, and indepen-dently of mim, in the regions of weak absorption (visibleand microwave). I
In regions of large imaginary part (mim > 0.01 mre),the Fresnel reflectivity is no longer dominated by mrebut also contains a measurable contribution due to mim.A relationship between mre and mim can be obtained byusing dispersion analysis.1 3 Reflectance measurementscan thus be used to determine both mre and mim, but forthis method some prior information is also needed aboutmre or mim outside the region studied.
The real index is of the order unity at all wavelengthsshort of the dielectric relaxation; the imaginary index,by contrast, varies over the range from 10-9 to 100. Thereal and imaginary indices are related by the Kram-ers-Kronig equation:
2X2 r mim(X)dXMr(XO + PJX(X2 X2) (1)
where P indicates the Cauchy principal value of theintegral. To compute mre at a particular wavelength,one needs mim at all wavelengths. An analogousequation computes mim by an integral over mre.However, mim cannot be obtained accurately by thismethod in spectral regions where it is many orders of
magnitude smaller than mre, because mre is known ac-curately to at most 4 decimal places. By contrast, mrecan be obtained reliably from Eq. (1).
E. Compilation Procedure
We follow the approach that Hale and Querry14 usedin their compilation of m for liquid water. We reviewmeasurements of mim, displaying them in figures in theregions of disagreement. We compile our best estimateof mim (X) over the entire spectrum of interest. We thencompute mre(X) from Eq. (1), making an assumptionabout the behavior of mim at wavelengths shorter thanthose measured ( < 45 nm) in order to obtain the cor-rect value of mre where it is known in the visible spec-trum. Before using Eq. (1) to obtain mre (X), however,it is necessary to correct the far-infrared measurementsof mi () from 100 K to 266 K. This correction isconstrained as follows.
The real index is known independently of mim in re-gions where mim << mre. It is -1.31 in the visible' 5 and1.78 in the microwave.16 The difference of 0.47 betweenthese values can be attributed to an integral of mim be-tween these two wavelengths, in another Kramers-Kronig relation,
mre(VI) mre(V2) =I 2f2V2k.a0() dv, (2)
where v = 1/N. Since these two values of mre are accu-rately known, we use them to constrain our correctionin the far-infrared from measurements of mim at 100 Kto our estimate for 266 K. As explained below, thedominant contributions to the integral in Eq. (2) comefrom the 4 0 -10 0-Am region. Our first estimate of thetemperature correction gives kabs(V) which is slightlyinconsistent with Eq. (2); we then adjust the far-infraredspectrum until Eq. (2) is satisfied, before computingmre(X) from Eq. (1).
We start our compilation of m at X = 45 nm becausethere are no quantitative measurements at shorterwavelengths. We terminate it at X = 8.6 m. The di-electric properties at longer radio wavelengths havebeen adequately reviewed by others, including Johari,9Evans, 2 Jiracek,1 7 and Hasted.18
II. Ultraviolet, 44-185 nm
A. Sources of Data
The following references are listed more or less inorder of increasing wavelength:
Seki et al.7 (1981) grew a single crystal of hexagonal
ice and cooled it to T = 80 K for measurement of thereflectivity from 44 to 207 nm, using polarized syn-chrotron radiation as the light source. Kramers-Kroniganalysis was used to obtain ' and e". Usable values ofe" were obtained only for 44-160 nm, because e" be-comes too small beyond 160 nm to be constrained by theKramers-Kronig analysis. The corresponding valuesof mim are plotted here in Fig. 1. The spectrum wasmeasured for light polarized parallel and perpendicularto the c axis of the crystal, with identical results towithin the experimental error of 3%. Amorphous icewas also studied. Its spectrum was somewhat
smoother; the peaks at 70 and 140 nm were bothweaker.
Daniels'9 (1971) reported energy-loss measurementsfrom fast electrons for ice (probably amorphous) con-densed from vapor at T = 78 K, obtaining significantabsorption values for 8-28 eV (150-44 nm), with 0.4-eVspectral resolution. He obtained the ' and e" byKramers-Kronig analysis. The corresponding mim isplotted here in Fig. 1.
Onaka and Takahashi20 (1968) condensed ice attemperatures between 83 and 203 K and measured theUV absorption spectrum from 7 to 10 eV (177-124-nmwavelength). The peak at 144 nm (shown in Fig. 1 hereas obtained by other investigators) was found only fortemperatures in the range where cubic ice would beexpected. However, Daniels19 found this peak foramorphous ice, and Seki et al. 7 found it for bothamorphous and hexagonal ice. This discrepancy hasnot been explained.
Otto and Lynch2 ' (1970) measured the electronenergy-loss spectrum in the UV (5-35 eV) for ice frozenfrom liquid water and cooled to -10°C. It was thusprobably polycrystalline hexagonal ice. These mea-surements were unfortunately only qualitative, but theydiffer notably from those made at lower temperatures:Otto and Lynch found only a shoulder at 144 nm insteadof a peak. Daniels19 speculated that Otto and Lynch'sfailure to obtain the peak was due to poor spectral res-olution, but we cannot rule out the possibility that thespectrum depends on temperature near the meltingpoint.
0.5-
Seki et al.E / ~~~~~~~~~,hexagonalE / 80Ko~no
Z 0.4
o I/aophu , .0.3 -o
0.2 /Z amorphous078
0.1 A'nde r sn
0 _ i50 100 150 200
WAVELENGTH (nm)
Fig. 1. Imaginary refractive index of ice in the ultraviolet. Themeasurements of Seki et al.
7 are the ones used in our compilation.They are for a hexagonal single crystal at 80 K, both polarizations.
Dressler and Schnepp22 (1960) deposited ice fromthe vapor phase at temperatures of 175 K and 77 K andassumed the ice forms to be hexagonal and amorphous,respectively. Transmission measurements were madeto obtain the absorption coefficient from 140 to 170 nm.Scattering of light by' cracks or bubbles (which wouldreduce the transmission and lead to an excessive esti-mate of absorption coefficient) was shown to be negli-gible by measurements outside the absorption band atlonger wavelength. The film thickness was estimatedby measuring the amount of vapor deposited. Uncer-tainty in this quantity leads to an uncertainty in ab-sorption coefficient. The absorption coefficient foramorphous ice at 77 K was found identical to that forhexagonal ice at 175 K over the entire spectral rangestudied.
Browell and Anderson 2 3 (1975) grew ice by depo-sition from the vapor phase. They measured the re-flectance at two angles as functions of ice thickness.The period of the interference fringes was used to obtainmre for 161-320-nm wavelength at deposition temper-atures 77 K (amorphous) and 155 K (hexagonal). Theyalso obtained an upper limit for the absorption coeffi-cient at one wavelength only (147.5 nm) which they usedto correct Dressler and Schnepp's 2 2 values at all wave-lengths 130-170 nm, as shown in Figs. 1 and 2.
Shibaguchi et al.2 4 (1977, their Fig. 6) measured the
absorption spectrum of hexagonal H20-ice for 173-178-nm wavelength. (They also measured the 120-160-nm region, but only for D20-ice.) The absorptioncoefficient varies by a factor of -20 over this wavelengthregion. The corresponding mim is plotted in Fig. 2 forthe two extreme temperatures they used.
Minton2 5 (1971) measured the molar extinctioncoefficient for samples of polycrystalline hexagonal ice
10°
EE 0- Seki et al.z 80KA0
Dressler and Of Schnepp P pointer et al.
UJ lo-, ~ ~ ~~~ iquidwaeO f 0- Browel I and waterLI Anderson l \
x \ _ Shiboguchi2 10' ~~~~~~~~etoal. -
0< Minton: lo-, 223- 263 K C
l o -'140 150 160 170 180 190 200
WAVELENGTH (nm)
Fig. 2. Imaginary refractive index of ice and water in the ultraviolet.Our compilation uses Seki et al. 7 to point A, and Minton2 5 from pointB to point C, joining point A to point B by a straight line. Thestraight line coincides with the 253 K values of Shibaguchi et al.2 4
The dashed line extending from point C to longer wavelengths is ourchosen interpolation where no data are available.
1-10 mm thick at 181-185-nm wavelength. The ab-sorption coefficient varies by a factor of -20 over thiswavelength interval. The samples were frozen fromliquid water, then polished; clear samples of measuredthickness were chosen. Because of the possibility ofsome slight scattering of light by the ice, Minton re-garded his values as upper limits. The molar extinctioncoefficient was independent of temperature over therange from -10 to -40OC; the corresponding mim isplotted in Fig. 2.
Painter et al. 2 6 (1969) reviewed measurements ofmim for liquid water in the near-UV (their Fig. 9).Measurements from a variety of authors were in sub-stantial agreement. We have drawn a single smoothcurve through their Fig. 9 from 160 to 200 nm and in-cluded it in our Fig. 2 here, because we will use it in ourargument for the interpolation of the ice mim.
B. Compilation of Imaginary Index
The only available measurements of light absorptionby hexagonal ice in the ultraviolet short of 130 nm arethose of Seki et al. 7 They are plotted together withDaniels's' 9 measurements in Fig. 1. The values of Sekiet al. are preferred to those of Daniels because Daniels'sspectral resolution was poorer, his measurements usedelectrons instead of photons, and his ice was probablyamorphous rather than hexagonal.
Seki et al. 's measurements were taken at 80 K. Thespectrum may be somewhat different at higher tem-perature, but probably not by much, based on the smalltemperature dependence from 83 to 160 K of the UVspectra of D2 0-ice shown in Fig. 4 of Shibaguchi et al.
2 4
Dressler and Schnepp 2 2 also found no difference be-tween amorphous ice (77 K) and hexagonal ice (160 K)in their absorption at 140-170 nm. However, all thisevidence for weak temperature dependence comes frommeasurements well below the melting point. Thequalitative spectrum of Otto and Lynch2l near themelting point shows a shoulder instead of a peak at 144nm. There is thus a need for ultraviolet measurementsnear the melting point to resolve this discrepancy.
Based on the above discussion, we choose the valuesof Seki et al.7 for 45-161 nm. Beyond 161 nm, the ab-sorption is too small for mim to be obtained accuratelyby their reflection measurements. There are short gapsbetween available measurements (Fig. 2) in the 161-181-nm region. We choose to join points A and B inFig. 2 by a straight line. This amounts to assuming thatlnmim varies linearly with wavelength. The parallelbehavior of mim of liquid water (also plotted in Fig. 2)serves to justify this. The straight line connects Min-ton's25 measurements near the melting point (point B)with Seki et al.'s measurements at 80 K (point A) andpasses directly through measurements of Shibaguchiet al. 2 4 close to the melting point. In making this in-terpolation we choose to ignore Dressler and Schnepp's2 2
(DS) data. Browell and Anderson 2 3 (BA) called theminto question as being probably too high and tried toadjust them downward by a dubious ad hoc procedure,based on their own estimated upper limit for mim at 147nm. We choose to ignore the measurements of both DS
Fig. 3. Imaginary refractive index of ice and water in the near-ul-traviolet. Our compilation uses the dashed line to interpolate be-tween Minton 2 5 at 185-nm and Grenfell and Perovich' 2 at 400-nm
wavelength.
and BA, but we note that our interpolation passes closeto both of their results for 160-170 nm.
III. Visible and Near-Visible, 185-1400 nm
A. Sources of Data
Hale and Querry1 4 (1973) reviewed the opticalconstants of liquid water at 250 C from 200-nm to200-Mm wavelength. In Fig. 3 we have plotted theirchosen values for 200-400 nm because we will use themas a guide for interpolation in this spectral region wheremeasurements on ice are lacking.
Sauberer27 (1950) measured absorption of lightthrough blocks of lake ice 140-500 mm long, for 313-1100-nm wavelength. Although these blocks were ap-parently very clean and bubble-free, they were not longenough to attenuate the beam by more than 2% in theregion of minimum absorption in the visible.
Grenfell and Perovich12 (1981) measured absorp-tion of light through polycrystalline hexagonal ice at-4°C. To cover the variation of mim over the 400-1400-nm spectral range, three samples of differentthickness were used: 2.8 m for 400-850-nm, 200 mmfor 700-1100-nm, and 8.2 mm for 1000-1400-nmwavelength. The ice was demonstrated to be almostcompletely free of bubbles. These measurements su-persede those of Sauberer.27 The wavelength resolu-tion is given by Grenfell28: the full bandwidth at half-maximum is -20 nm at X = 400 nm and 30 nm at X =1400 nm.
B. Compilation of Imaginary Index
There is a gap in measurements between 185 nm(Minton2 5 ) and 313 nm (Sauberer2 7 ) where it is difficultto guess the actual behavior of mim. In fact, we choosenot to use Sauberer's measurements at all because theydisagree by a factor of 2 with the more accurate mea-surements of Grenfell and Perovich12 (GP) where theyoverlap at 400 nm. We therefore have to interpolate
between Minton's value at 185 nm and GP's value at 400nm. This interpolation is shown as a dashed line in Fig.3. Here again, the behavior of liquid water (also plottedin Fig. 3) serves as a guide and as evidence that noelectronic absorption features occur in the region ofinterpolation.
Essentially, we are taking Minton's measurementsat 185 nm as evidence that mim of ice is less than thatof liquid water over most of the spectral region that hasnever been measured for ice. We put high priority onthe measurement of the absorption coefficient of ice inthis spectral region, because of its geophysical impor-tance. The solar spectrum contains a large amount ofenergy in the 200-400-nm region, and most of the sun-light at 300-400 nm reaches the earth's surface, whereit is important in the energy budgetg of natural snowand ice surfaces.
For the 400-1400-nm region, we use the measure-ments of GP. They supersede the values of Luck29
which had been recommended by Irvine and Pollack'for 1100-1400 nm.
IV. Near-Infrared, 1.4-2.8 Am
A. Sources of Data
In the 1.4-2.8-,um spectral region ice has an absorp-tion coefficient that is too small to affect reflectionmeasurements, yet it is large enough that very thinsamples (a few hundred Aum) must be used to obtainmeasurable transmission.
Reding30 (1951) placed liquid water between twoAgCI windows, froze it, and cooled it to T = -78°C. Itwas therefore polycrystalline and probably not bub-ble-free. Two samples were measured, but the 48-iumthick sample was unusable because of excessive scat-tering of light by cracks and bubbles. For the usefulsample, the separation was 250 m when the liquid was
10
E
z01-H 10-2
oIL-
0103
z
Of
z0D00
20 2.5 30
WAVELENGTH (m)
Fig. 4. Imaginary refractive index of ice in the near-infrared. Ourcompilation uses Ockman's 3 2 0.1-mm sample for 1.45-1.61 ,um and1.89-2.11 ,m, Reding's 3 0 values (frequency shifted as described inthe text) for 1.61-1.89,um and 2.11-2.62,um. Straight lines are usedto join from measurements of Grenfell and Perovich1 2 at 1.4 m tothose of Ockman at 1.45 m, and from those of Reding at 2.62 ,m to
poured in, but expansion during freezing may have in-creased the separation between the plates. The ordi-nate on Reding's Fig. 16 is labeled with both percenttransmission and log(I/Io), but the two scales are mu-tually inconsistent. Values of mim derived using bothscales are plotted here in Fig. 4. They differ by -20%.Reding (personal communication, 1983) does not knowwhich is correct, but favors the percent-transmissionscale, which agrees with the reproduction of his spec-trum in Fig. 3 of the paper by Hornig et al. 3 1 In theirreview article, Irvine and Pollack' apparently assumedthe opposite.
No mention was made of a correction for reflectionat the interfaces. A neglect of this correction wouldcause mim to be underestimated, because less reflectionoccurs at an ice-AgCl interface than at an air-AgCl in-terface. By contrast, the other sources of error (scat-tering by bubbles, and increased separation of windowsduring freezing) would cause mim to be overestimated.These errors may be much larger than the 20% uncer-tainty due to Reding's mislabeling of the axes.
Ockman3 2 (1957) grew single crystals of hexagonalice on a glass window and measured their spectra at-30'C and-178°C for 1.4-2.5 jim. There was almostno difference in the absorption for light polarized par-allel and perpendicular to the c axis; we take the averageof the two. There was, however, a substantial effect oftemperature. As the temperature drops from -30'Cto -1780 C, the spectrum is shifted to longer wave-lengths and has more structure and sharper peaks. Asubsidiary minimum and maximum found by Ockmanaround 1.65 gum only at the colder temperature has alsobeen observed in the reflectance spectra of laboratoryfrosts33'34 to disappear at higher temperature.
Ockman measured his sample thickness more accu-rately than did Reding. For 1.4-2.5 gim, Ockman usedthree samples of thickness 102, 287, and 358 gim (un-certainty in thickness 3-5%) but only the spectrum ofthe thinnest sample was plotted in Ockman's 3 2 Fig. 17and in Fig. 5 of his subsequent paper.3 5 This is unfor-tunate, because (as IP pointed out) absorption mea-surements become inaccurate when transmission ex-ceeds -80%, since any random experimental errors, aswell as a systematic neglect of reflection at the ice-window interfaces, lead to larger errors in mim when thetransmission is large. This is because kab,/kabsl -I6T/TI/IlnTI, where 6 kabs/kabs is the relative error inkabs, and T/T is the relative error in transmission T,so for T > 0.8, 6kabs/kabsl > 5T/TI. However,Ockman (personal communication, 1983) was able toprovide us a graph of the absorption spectrum of thethickest crystal (358 m) which had been omitted fromhis thesis. We have converted those values to mim andploted them in Fig. 4. The exaggeration of mim in re-gions of large transmission is evident in Fig. 4, especiallyat 1.35 ,um where the measurements overlap those of athicker sample used at lower wavelength. Only thethickest (10-mm) sample gives accurate values of mimhere. We thus suspect that the values of mim we derivefrom Ockman's transmissions may be excessive.
Ockman's plots of kabs were obtained simply by usingthe relation I = Io exp(-kabsd), where I is the measuredintensity, Io that of the blank cell, and d is the samplethickness. He was only interested in obtaining thestrengths of the band maxima, where the errors inbackground level do not have severe consequences. Inprinciple, a correction for the reflection at interfacesshould be applied to his measurements, if we couldknow how to do it. The blank had two air-glass inter-faces. The sample had instead two ice-glass interfaces,if the ice was in contact with the glass. If not, therewould have been two glass-air interfaces and two air-iceinterfaces. Ockman thinks that the former situationis likely for the high-temperature (-29°C) results weare using, and that the ice became separated from theglass only upon cooling to -178°C for the low-temper-ature measurements. But if ice were in contact withglass, the reflection losses in the sample would be lessthan in the blank, because mre of glass is closer to thatof ice than to that of air. So the fact that substantialdifferences were observed among samples in the weaklyabsorbing parts of the spectrum means that there musthave been losses from cracks or from unexpected air-iceinterfaces. These cracks and interfaces can reduce thetransmission by 15%, judging by comparison of the0.1-mm sample with the 1-mm sample at 1.35 m(compare Fig. 5 and 6 of Ockman3 5 ). The losses due toscattering or unexpected interfaces thus more thanoffset the difference in known interface reflections be-tween sample and blank, so we choose not to correct forthose reflections. The obvious presence of scatteringin these measurements suggests that we should alwaystake the lowest values of mim, whenever two samplesdisagree. It appears (Fig. 4) that the thicker (358-gm)sample is more reliable.
B. Compilation of Imaginary Index
Our compilation in this region will be uncertain toabout a factor of 2. This is a region that needs to beremeasured carefully at temperatures near 0°C with theaim of establishing quantitative values rather than justthe location of peaks and shoulders.
The rationale for our choices is that(a) transmission measurements give inaccurate
values of mim when transmission exceeds -80%;(b) the spectrum shifts to shorter wavelength as the
temperature is raised; and(c) Ockman's3 2 measurements were apparently made
more carefully than Reding's.3 0
Irvine and Pollack' used Reding's data everywhereexcept in his data-gap between 1.97 and 2.13 m, be-cause Reding had used a thicker sample than the oneOckman published. They estimated temperature cor-rections from Ockman's measurements at two temper-atures. Our approach is to use Ockman's data for the0.1-mm sample with no correction (i.e., to assume notemperature dependence between -30 0C and -7 0C)except in the regions where his transmission exceeded80%. Ockman's results are generally to be preferredbecause he made them at temperatures close to the
melting point, and knew his sample thicknesses moreaccurately.
We join the values of Grenfell and Perovich12 at 1.4jim to those of Ockman at 1.45 jm with a smooth curve(lnmim approximately linear in N). We use Ockman'svalues uncorrected from 1.45 to 1.61 m. From 1.61 to1.89 jim we use Reding's values, shifted to shorterwavelength by -0.05 jm and with the minimum at 1.85jm broadened as Irvine and Pollack' did, because theminimum should not be so sharp at higher temperature.We use Ockman's data from 1.89 to 2.11 m. From 2.11to 2.62 ,m we use Reding's data, shifted -0.03 m toshorter wavelength, based on the position of the mini-mum near 2.3 m. [Where we use Reding's values, wetake the log (I/bo) scale. Although the % T-scale seemsmore likely to be correct, we prefer to take the lower mimvalues to compensate partially for the scattering.] Fi-nally, we draw a straight line (assuming lnmim linear inN) from Reding's (temperature-shifted) value at 2.62 mto that of Schaaf and Williams'0 (-70 C) at 2.78 jim.This compilation is uncertain to about a factor of 2,judging from Fig. 4.
Ockman supplied the plot of the spectrum from thethicker (358-jim) sample after the compilation in thispaper had been completed. In the regions where we areusing Ockman's measurements, the differences betweensamples are judged not to be large enough to warrantredoing the compilation, in view of the large uncer-tainties in the measurements when mim is small. Thespectrum of the thicker sample (plotted in Fig. 4)suggests, however, that our compilation of mi isprobably as much as 15% too high from 1.44,m to 1.63jim, because we used the 0.10-mm sample.
This compilation is somewhat different from thatgiven by Wiscombe and Warren.5 They simply tookIrvine and Pollack's' recommendation and applied thesuggested temperature correction to -7°C.
V. Middle Infrared, 2.8-33 ,lm
Schaaf and Williams' 0 (1973) measured the re-flection spectrum of ice at -70 C formed by freezing ofliquid water, so it was undoubtedly polycrystalline.They used the Kramers-Kronig analysis to obtain mreand mim. This method does not give mim accuratelywhen mim << mre, so their values of mim are inaccuratefor < 2.78 jim (dashed line in Fig. 4), and also at3.5-4.3 m where the uncertainty is -50%. To do theKramers-Kronig analysis they needed values of mi inthe far-infrared ( > 33 m) which were available onlyat 100 K. They tried to do a temperature correction onthe far-infrared data to estimate values at -7°C, sothere is some uncertainty in their values of m becauseof this, especially at the long-wavelength end near 33 mwhere contributions to the Kramers-Kronig integralfrom outside the measured region become appreciable.Schaaf and Williams's reflectivity measurements canbe reinterpreted in the future if far-infrared absorptionspectra become available for ice near the meltingpoint.
We use Schaaf and Williams's mim values as given for2.8-33 m. Subsequent measurements 3 6 3 7 of the 3-jim
WAVELENGTH (m
250 200
WAVENUMBER (cm-')
Fig. 5. Solid lines, transmission measurements for three ice filmsof unknown thickness, made at the two temperatures 100 K and 168K (from Fig. 2 of Bertie and Whalley 3 8). Dashed lines, our subjective
guess of the spectrum at 266 K, using the solid lines as a guide.
band have been made only at much lower temperatures(<180 K).
VI. Far-Infrared, 33 jgm-1.25 mm
A. Sources of Data
Bertie and Whalley3 8 (1967) (BW) measuredtransmission through thin films of ice in the 360-50-cm-' wave number region (28-200-jim wavelength).Their samples of hexagonal ice were prepared by con-densing water vapor onto a polyethylene window at T= 173 K. Three samples of unknown thickness wereused, each at two temperatures 100 and 168 K. Thespectra, plotted in their Fig. 2, are reproduced here(with additions) as Fig. 5. There are substantial dif-ferences in the spectrum at the two temperatures. Thetemperatures were uncertain to L20 K (D. Klug, per-sonal communication, 1982). BW made no correctionfor reflection at the interfaces. Because the thicknesseswere not known, we use these data only to estimate atemperature correction, not to obtain absolute valuesof mim. BW (their Fig. 3) also showed that the spec-trum of ice Ic is identical to that of ice Ih in this spectralregion. They also showed that the two strong absorp-tion bands (at 229 and 163 cm-') are both due to latticevibrations. Amorphous ice, by contrast, lacks thestructure shown in Fig. 5 and has only a single broadpeak, as is also shown in Fig. 2 of Leger et al. 3 9 Becausethese absorption bands are due to lattice vibrations,they are completely absent in water vapor.
Bertie, Labb6 and Whalley" (1969) (BLW) mea-sured transmittance, at T = 100 K only, through thinfilms of ice Ih of a variety of thicknesses, using the sametechniques as had Bertie and Whalley.3 8 Several of thefilms were <1,jm thick. They reported values for theoptical constants which were based on an indirect esti-mate of sample thickness. The main aim of the workwas to determine the origin of the infrared polarizabilitywhich is responsible for the difference, mre, of 0.47between the microwave and visible refractive indices.By performing the Kramers-Kronig integral (2) overtheir observed absorbances, they computed the samplethickness which yielded the correct mre [Samplethickness enters in the computation of kabs(v).] This
Fig. 6. Imaginary refractive index of ice instrongly absorbing far-infrared region to thcrowave region. Our compilation for T =point B and follows the 200 K measurements
to point C.
sumably hexagonal polycrystalline ice). They mea-sured the absorption spectrum from 42 to 17 cm-' atboth 100 K and 200 K. These measurements are plot-ted in Fig. 6. They supersede the preliminary mea-surements of BLW1" where they overlap them, becauseBLW's 1-mm thick ice block was too thin for accurateabsorption measurements at v = 30 cm-'. The ab-sorption coefficient was expected (theoretically) to beC 200K 4Vproportional to v4 around v = 40 cm-', changing to adependence at longer wavelength. Within experi-mental error, the data (Fig. 1 of WL) support this.
Iuv lOOK Mishima, Kug, and Whalley8 (1983) (MKW) usedsingle crystals of ice Ih to extend the results of WL to
0.5 1.0 1.5 1.25-mm wavelength at four temperatures (80, 100, 150,200 K). The absorptivity for polarized light was inde-pendent of the orientation of the single crystal. The
the transition from the values of mim for 100 and 200 K are plotted in Fig. 6..e weakly absorbing mi- MKW found the absorption to continue its decrease-60'C joins point A to with approximately the expected v2 dependence.of Whalley and Labb640 Under this assumption, the extrapolation of mim for T
= 200 K to X > 1.25 mm agrees with microwave mea-surements cited below at N = 110 mm and T = 213 K.
sample thickness was further verified by comparing thederived value of kabs at X = 2 jim with that measuredearlier by Ockman.32 The thicknesses of the varioussamples were then obtained by scaling them in regionswhere their spectra overlapped.
These authors did not correct their absorption mea-surements for reflection at the interfaces. They as-sumed that the reduction in transmission was due en-tirely to absorption (after subtracting the spectrum ofthe empty cell), computed mim(X), and used a Kram-ers-Kronig relation to obtain mre (X). They recognizedthat this procedure leads to some error, the worst casebeing a neglect of -27% reflection at v = 229 cm-1. Asthey stated, this "undoubtedly means that the absorp-tivity and hence the reflectivity are overestimated byan uncertain amount in this region." They properlyshould have used an iterative procedure, computingreflection from m, then recomputing m, etc., untilachieving self-consistency. Because the uncertaintyin our procedure for correcting the 100 K data of BLWto 266 K is much larger than the error due to neglect ofreflection by BLW, we have not reinterpreted BLW'stransmission measurements for the present compila-tion.
BLW reported mre and mim for ice at T = 100 K in the4000-60-cm-1 wave number region (2.5-167-jimwavelength). They also obtained some "preliminaryand not very accurate" measurements of a 1-mm samplefor 60-30 cm-'. This long-wavelength end of theirresults is shown here as the dashed line in Fig. 6.
Their conclusion regarding /Žmre (microwave-visible)was that 74% of the difference was due to the transla-tional bands in the far-infrared; 15% to the hindered-rotation band at 12 jim, and 7% to the 3-jim OH-stretching band. We use this information in Sec. VIIIbelow as a guide to adjusting our temperature correctionin the far-infrared.
Whalley and Labb64 0 (1969) (WL) grew blocks of iceof unspecified thickness by freezing liquid water (pre-
B. Compilation of Imaginary Index
Our source of mim for 33-167 jim is BLW."1 Theirmeasurements at 100 K must be corrected to 266 K forour use, because no measurements are available near themelting point. Our procedure for estimating a tem-perature correction is somewhat arbitrary. We use therelative measurements of BW3 8 at 100 and 168 K (120K) to estimate the temperature dependence. We notethat the peaks become less sharp and shift to longerwavelength as the temperature increases. Using thesetwo plots as a guide, we draw a subjective guess of thespectrum at 266 K, assuming these processes of broad-ening and wavelength shift to continue linearly withtemperature (dashed line in Fig. 5). The temperaturecorrection is then derived as the ratio of the 266 K plotto the 100 K plot. The ratio does not agree amongsamples where the spectra of two samples overlap. Inthese overlap regions, we choose the sample that gavea temperature correction closest to 1.0. This correctionfactor is plotted in Fig. 7. It is also constrained to makeBLW's mim at 300 cm-1 agree, after correction, withthat measured by Schaaf and Williams10 at 266 K (Fig.8). The correction is applied to the mim of BLW; theresult is shown as the dashed line in Fig. 9. By thisprocedure, the 229-cm-1 peak at 100 K moves to 209cm-1, somewhat further than the 214 cm-' expectedfrom measurements of the peak position published inFig. 4 of Zimmermann and Pimentel.4 ' This is ourinitial guess of mim. In Sec. VIII below we will adjustthe peak position to 214 cm-' and then slightly adjustthe magnitude of mim in order to match the known Amre(microwave-visible), obtaining finally the dotted linein Fig. 9.
We assume that no temperature correction is neces-sary at v = 60 cm-' ( = 167 jm). This is suggested byFig. 6, where the 200 K and 100 K plots of WL40 are seento converge toward an extrapolated meeting point at 60cm-1 (point A) which also agrees with BLW's measured
Fig. 7. Initial temperature correction which was applied to the 100K data of Bertie et al.," to obtain values of mim for 266 K. It is de-rived from the 266 K and 100 K curves of Fig. 5. This temperature
correction was later refined during the Kramers-Kronig analysis.
WAVELENGTH (/m)E 20 25 30 35 40
Bertie, Lbbi and0 Whobey 100 K
Cr temperote - /corrected ,
X0 0 1 to 266 KILi
LLW /chaf and°°5~~~~~~~~teprtr -n /
W I I S _ .
0 00t26K-
whr2 cha n Williams's0dt utb ondt h tmea
0
0 the 1l AL.L assmpLo 250wu500 450 400 350 0 5
WAVENUMBER (cmi')
Fig. 8. Imaginary refractive index of ice near X = 33 gin. This iswhere Schaaf and Williams's'o data must be joined to the (tempera-
ture-corrected) data of Bertie et al.
mi X there. The interpolation between the tempera-ture-corrected mim at 100 cm' and the measured mimat 60 cm' (Fig. 9) ignores the wiggles seen in the datain this region, on the assumption that they would bebroadened and smoothed at higher temperature.
For N> 167 jim the temperature dependence of mimbecomes appreciable even near the melting point. Ourcompilation for T = -~60o C connects point A (BLW) inFig. 6 with point B (WL, 200 K). A smooth curve isdrawn through the data of WL to meet point C of MKWat T = 200 K. The mi. at higher temperatures for60-33 cm'1 are obtained by using the difference be-tween the 100 K and 200 K measurements, and extrap-olating with the assumption that mim is linear in tem-perature. This is suggested by the fact that over theentire 42-17-cm- 1 region, mim (200 K) _ mim (100 K)+ 0.006. Beyond 33 cm-' (300 m), the extrapolationinto the microwave region is described below.
EE
z0U-
LuItLU
0X
0z
z2
300 260 220 180 140 100 60
WAVENUMBER (cmb1)
Fig. 9. Imaginary refractive index of ice in the far-infrared. Solidline, data for 100 K from Bertie et al.1 Dashed line, values adjustedto -71C by the function shown in Fig. 7. Dotted line, refined ad-
justment to -70 C as described in Sec. VIII of the text.
VII. Microwave and Radiowave, 1 mm-8.6 m
A. Sources of Data
Most of these data are plotted in Fig. 10.Lamb4 2 (1946) measured loss tangent at X = 30 mm,
as a function of temperature from -1 0C to -50'C.(Lamb's value at -10 C is reproduced incorrectly, afactor of 3 too low, in Fig. 7 of Evans's review. 2 )
Lamb and Turney 4 3 (1949) grew ice from distilledwater. They corrected Lamb's42 value of e' but not histan at = 30 mm. They reported new measurementsof tanb at = 12.5 mm, for temperatures from 93 to 273K.
Cumming 4 4 (1952) measured loss tangent at X = 32mm, at temperatures from 0 to -18'C, with a precisionof better than 5%. Ice grown from three sources (dis-tilled water, tap water, and melted snow) all gave thesame loss tangent. Cumming confirmed his intru-ment's calibration by obtaining good agreement withthe known loss tangent of other materials.
Perry and Straiton4 5 (1973) obtained correctedvalues of e' and e" at - 28oC, X = 3.1 mm, after Gough4 6
pointed out that they had earlier47 reported a wrongvalue of e' (1.9 instead of 3.2).
WAVELENGTH (mm)Fig. 10. Imaginary index of refraction of ice in the microwave and radiowave regions, according to various investigators. For X > 2 mm,measurements were made only at the points marked by symbols; they are connected here by lines only for display purposes. An error bar
shows the uncertainty quoted by Perry and Straiton.45
Vant et al.48 (1974) estimated tan3 at X = 30 mm to-be "of the order of 20 X 10-4 ( 0C to -35°C)," givingmim 2 X 10-3, i.e., between Cumming's -1C and-5 0C values. They also reported a loss tangent of 0.001± 0.001 at -35°C which seems to apply to a frequency
of 35 GHz ( = 8.6 mm), although their discussion doesnot make this clear. The corresponding mim is 9 X10-4. This is not inconsistent with our compilation butis too uncertain to be of any use to us.
Westphal (unpublished) measured loss tangent ofnatural glacier ice from three locations, for temperaturesfrom -1C to -60'C and frequencies from 150 MHz to2.7 GHz (2-m-111-mm wavelength. The measure-ments were reviewed by Evans2 and tabulated by Jira-cek.17 (Jiracek pointed out that the earlier presentationof Westphal's data by Ragle et al.
4 9 was incorrect becauseof lack of a multiplier.) Westphal found that annealing
the ice at -10 0C for a few hours resulted in lower losses,so the results were reported for the annealed samples.Two of the three ice samples gave identical values oftant, which are plotted here in Fig. 10. They haddensities of 0.898 and 0.902, in comparison with pure icedensity of 0.917, so they did contain a small amount ofair. Westphal estimated that his measurements of tanbwere accurate to 20%.
Vickers 5 0 (1977) reported decibel-power-loss (dB),which can be converted to mim, for bubble-free lake ice,17 N A 300 mm, at -5°, -10°, and -15'C. Althoughthe data are in tolerable agreement with Cumming44
and von Hippel 5 l at X = 32 mm, they show a trend ofmim increasing with X (Fig. 10) which leads to grossdisagreement with Westphal at X > 110 mm. Thereason may be that the dielectric losses were barelydetectable by Vickers, and he may have been reportingconstant instrument noise which would lead to an ap-
parent increase of mim with N when dB is converted tomim.
von Hippel5 l (1945, p. 220) reported e' and tanb forice at -12'C, 30 mm < X < 3 km. Evans 2 pointed outthat von Hippel's E' was in error, rising from 3.2 at X =30 mm to 4.2 at X = 300 m, whereas it should remainconstant at e' = 3.2 in this spectral region. We deducemin (shown in Fig. 10) from the reported loss tangentby use of a constant e' = 3.2 instead of von Hippel's e,but since von Hippel's e was in error we suppose his tanbmay also have been in error.
Yoshino5 2 (1961) measured e and tanb in glacier ice,from -18° to -36°C, 100 mm NX - 200 m. They werereviewed and rejected by Evans2 in comparison withsmaller values of other authors because "any unforeseenerrors in the experimental technique may increase thelosses but they cannot in any circumstances be expectedto result in measured loss less than the true value." Thesame statement applies to the values from vonHippel.
Johari and Charette53 (1975) measured e and tanbat six temperatures in the range from -I to -250 C, at5 and 8.6-m wavelength.
Johari54 (1976) reported e and e' for ice at -50 C, 3K X < 600 m. The measurements of e' match those ofWestphal at X = 3 m.
B. Compilation of Imaginary Index
We use data from Mishima et al.,8 Cumming, 4 4
Westphal (as tabulated by Jiracek17), Johari andCharette,53 and Johari.54 All other measurements areignored, as discussed below.
1. 300 im-111 mm
For the -60oC values, we use the 200 K measure-ments of WL4 0 and MKW 8 as far as 1.25 mm, then in-terpolate (assuming lnmim proportional to lnN) toWestphal's17 value at -60°C at X = 111 mm. For thehigher temperatures (-1, -5, -20'C) we start with thevalues we obtained at X = 300 jim by extrapolation fromthe 100 K and 200 K data of WL4 0 as described in Sec.VI.B. We then interpolate from these values, throughCumming's4 4 values at X = 32 mm, to Westphal's mea-surements at X = 111 mm.
There are several other conflicting sets of measure-ments in this region. Let us consider them in turn.
Perry and Straiton 4 5 gave a value of mim at X = 3.2mm, T = -280 C which looks realistic, but we choose toignore it because of severe doubts about the accuracyof their experimental technique.46
Lamb 4 2 measured mim (T) at X = 30 mm; Lamb andTurney 4 3 at N = 12.5 mm. These are lower by a factorof -2.5 than Cumming's44 measurements at 32 mm. Itis difficult to choose between the results of Cummingand Lamb. Evans2 stated that errors can only lead tohigher reported mim, which means that Lamb shouldbe favored. However, Cumming's measurements seemto have been made more carefully (Johari, personalcommunication, 1981). Furthermore, because of thegeneral trend of mim with X here, it is very unlikely thatthe measurements of Lamb and Turney and of Lamb
can both be correct. Here we choose to ignore both ofthem, and to use Cumming's values. However, ontheoretical grounds there is no reason to expect any-thing other than the frequency-squared dependence ofkabs through this spectral region.8 Our interpolationfor -60'C agrees generally with the frequency-squareddependence, but at the higher temperatures our com-pilation deviates from the theoretical relation becausewe choose to believe the measurements of Cumming. 4 4
(Cumming made no measurements below -18'C.) Itis clear that more measurements are needed in thisspectral region, which is of great interest for microwaveremote sensing. Measurements in this low-loss spectralregion are difficult because transmission methods re-quire long samples, and cavity-resonance methods re-quire cavity dimensions as small as the wavelength (F.Ulaby, personal communication, 1983).
The measurements of Vickers50 and Yoshino52 areprobably too high, as discussed above. Cumming'svalues are in agreement with those of von Hippel5l(1954), but just at the one wavelength X = 30 mm. vonHippel measured mim to increase monotonically as increases from 30 mm to 3 km, with values as much asa factor of 7 higher than Westphal's. 17 Since vonHippel's mre(X) is in error at X > 30 mm (compare Fig.6 of Evans 2 ), we tend to doubt his mim values as well,and prefer those of Westphal.
Although there are no reliable measurements of mimbetween 1.25 and 32 mm, we can be sure that there is nostrong absorption band in the unmeasured region, be-cause the infrared absorption bands fully account forthe measured microwave mre. 11,16 This is also sub-stantiated by the qualitative observation of Championand Sievers55 that ice is "completely transparent" at X= 2.5 mm.
2. 111 mm-8.6 mFor 111 mm-2 m we use Westphal's data from Ward
Hunt Island Glacial Ice (Table II of Jiracek17). Theresults were the same in Tuto Tunnel, Greenland, butwere generally higher at Little America V. We followEvans's2 recommendation and use the lower values.
For 2.0-8.6 m we use Johari and Charette's53 data attemperatures -1, -5, and -20'C. For -60'C we ex-trapolate Westphal's plot.
Beyond 8.6 m, Johari 5 4 has measured mim out to 600m, but only at -5°C. The temperature dependencebecomes extreme in this region, so we stop our compi-lation at 8.6 m. The longer wavelengths have also beenadequately covered in other review articles.
VIII. Adjustment of Far-infrared AbsorptionSpectrum
We now refine our temperature correction to mim inthe far-infrared until we obtain the correct differencebetween visible and microwave real indices, using theKramers-Kronig relations.
A. Real-Index-of-Refraction Reference Points
The real index can be measured independently of theimaginary index if mim << mre. The real index is well
established in the two regions of the spectrum where theabsorption is very small: visible and microwave.
We use the refractive index reported at X = 589 nmby Merwin.1 5 Ice is very slightly birefringent; mre =1.3090 for the ordinary ray, and 1.3140 for the extraor-dinary ray. These values differ by only one part perthousand, and we simply average them to obtain mre =1.3097 for polycrystalline ice.
The real refractive index is constant over a large re-gion of the microwave and radiowave spectrum and isthe square root of the limiting high-frequency permit-tivity E-. Goughl6 has measured e' for ice Ih from T =2 K to 270 K at frequencies up to 1 MHz, extrapolatingto en,. His data were found to fit very well to a quadraticfunction of temperature:
e- = 3.093 + 0.72 X 10- 4 T + 0.11 X 10- 5 T2 .
This gives us mre = 1.7772, 1.7837, 1.7865, and 1.7872at our referencetemperatures -60, -20, -5, and -P C,respectively. We assume these to be valid at X = 200mm. This slight temperature dependence is due to thetemperature dependence of the far-infrared absorptionspectrum. For the Kramers-Kronig analysis we use thevalue for -7°C, mre = 1.7861. This gives us a differencebetween the two real indices, Amre = 0.4764.
B. Refinement of Temperature Correction
The temperature correction which was guessed byextrapolating from Bertie and Whalley's38 low-tem-perature measurements in Fig. 5 must be refined in twoways. First, our initial temperature correction gave apeak at v = 209 cm-'. Although values of mim have notbeen measured at high temperature, the position of thepeak is known as a function of temperature as given byZimmerman and Pimentel.41 They show the peakvalue at v = 214 cm- 1 for T = -7°C. Second, when weperform the Kramers-Kronig integral (2) to obtainAmre, we obtain Amre slightly (<1%) different from thecorrect measured value of 0.4764.
Our procedure is first to shift the absorption spec-trum to higher frequency so that the peak positioncomes to 214 cm- 1 , and second to raise mim until weobtain the correct Amre. These adjustments are bothapplied only for 100 v 300 cm-', with maximumadjustment at 200 cm-' and no adjustment at 100 or 300cm-'. The adjustment is further refined during thecomplete analysis of mre () described in Sec. IX below,resulting in the dotted line in Fig. 9.
IX. Computation of Real Index
We compute mre () from mim () using two differentmethods and obtain identical results at all wavelengthswith both methods. The first method is the Kramers-Kronig (KK) integral used by Downing and Williams5 6
and Bertie et al."1;
v'2mim(v') -vv'mm(v)Mre(v) 1 +2 f-V2m(p)-VMi(y d Inv'. (3)
This equation is equivalent to Eq. (1) but is more con-venient for numerical computations because its singu-larity is of type 0/0.
The second method is the subtractive-Kramers-Kronig (SKK) integral recommended by Ahrenkiel57
and by Bachrach and Brown,58 used by Hale andQuerry14 in their compilation of liquid water refractiveindex:
mre (Xo) = mre (i) + 2(X21 - \° ) P X mim () d 1n\
( - X2)(X2 -X2)(4)
The SKK method requires specification of a known(measured) value of mre at some wavelength Ni (we useN1 = 0.5893 m) and is rather insensitive to mim atwavelengths far removed from 0. It is thus the pre-ferred method if mim is known only in a short spectralregion which is also the region of interest of mre. Forthe case of ice, however, we do have sufficient knowledgeof mim over the entire spectrum to allow us to do the KKanalysis almost as easily as the SKK analysis.
The KK integral (3) is done numerically (except atthe singularity) using trapezoidal integration for 100,000points equally spaced in logy. The results for mre differat most in the 5th decimal place if only 30,000 points areused. For each frequency v, the integral over the sin-gularity at v' = v is done analytically between the twogrid points v' = v - s and v' = v + t, with mim approxi-mated as linear in v between these two points: mima + b. This integral is2 fv+t v'(a + bv') - vmim(v) dv'r v-s (v' 2 - v2)
The singularity is of type 0/0, and the contribution of(5) to (3) is always negligible with the fine grid weuse.
The integral in (3) runs from zero to infinity, so wemust postulate mim outside the measured range. Ourcomputed mre turns out to be very insensitive to howthis assumption is made. For > 8.6 m we use mim asgiven by Johari5 4 for 8.6 m < X 600 m, and by Ray3 for
> 600 m. For < 45 nm we extrapolate mim from 45nm linearly in to mim = 0 at = 33 nm. A soft-x-rayabsorption band is postulated in the 2.33-2.40-nm re-gion, peaking at 2.36 nm, corresponding to the emissionspectrum measured qualitatively by Gilberg et al.
5 9 forice Ih and Ic. This also agrees more or less with thepeak postulated for liquid water in Hale and Querry's14
SKK analysis. We are unaware of any other absorptionband between 2.4- and 45-nm wavelength. We firstadjust the magnitude of this postulated x-ray band untilwe obtain the correct observed value of mre in the visi-ble, and then proceed to compute mre at all wavelengths.As long as its amplitude is adjusted to obtain the correctvisible mre, the exact shape and location of this peak hasno effect on mre ( > 45 nm) unless it is moved to con-siderably longer wavelength. If it is moved up to 24 nm,the ultraviolet mre is altered by up to 0.06 at the short-wave end (45 nm), with negligible difference for X > 200nm. (The SKK analysis, which does not require anassumption about the x-ray band, gives the same mre at45 nm as does the KK analysis assuming the x-ray band
at 2.36 nm.) The computed mre(X) is shown in Fig.11.
Seki et al.7 performed a KK analysis to interpret theirultraviolet reflectance measurements, obtaining bothmre and mim. They did not state what assumption theymade for the soft-x-ray band. When we do the KKanalysis with the above assumptions, we obtain slightlydifferent values of mre than they obtained, as shown inFig. 11(a). This suggests that whatever Seki et al. as-sumed was equivalent to assuming a band centered at8 nm instead of at 2.4 nm. The difference shown in Fig.11(a) can thus be taken as the uncertainty in the ultra-violet mre due to lack of knowledge of mim for X < 45nm.
The same comparison can be done with Schaaf andWilliams's 1 0 middle-infrared mre, since they also madereflectance measurements and performed a KK analysisto interpret them. They had to assume a temperaturecorrection for BLW's far-infrared measurements, as wehave done, in order to obtain values of mim outside theirmeasured region. They did not state how they did thistemperature correction. Comparison of their mre withours shows excellent agreement. The differences aresmaller than the width of the line in Fig. 11(b). Ap-parently we have independently arrived at the samefar-infrared temperature correction that Schaaf andWilliams used.
We also test the sensitivity of our results to the ter-mination point of our integration at long wavelength.Differences in mre appear only at the longwave end ofour compilation ( = 8.6 m), in the 5th decimal place,if the integration in (3) is terminated at X < 10 km. Ouruse of X = 1000 km for the termination is therefore quiteadequate.
The computation of mre uses the -5 0 C values of mimin the microwave. Our mre is forced to agree withGough's1 6 value at X = 200 mm. However, our calcu-lation of mre at longer wavelengths disagrees slightly(difference = 0.0015) with the measurement of Johariand Charette 5 3 at X = 5 m, as shown in Fig. 12. This isalso about the level of uncertainty of the measurement(Fig. 4 of Gough16 ; Fig. 2 of Johari and Charette 5 3 ).The compilation for other temperatures is forced to passthrough the values corresponding to the real part of thepermittivity, e', measured by Gough,16 and to parallelthe behavior of the -7° compilation at longer wave-lengths, resulting in a lack of agreement with values ofJohari and Charette.
The real index in the microwave depends slightly ontemperature, as we show in Figs. 11(c) and 12, but weobtain this temperature dependence from measure-ments rather than computing it. This is because thetemperature dependence of the microwave mre is dueto the temperature dependence of mim not in the mi-crowave but in the far-infrared 9' 1 6 where mim (T) has notbeen measured near the melting point.
The fractional uncertainty in mre is generally muchsmaller than that in mim. The real index may be takenas accurate to within the width of the line in Figs.11(a)-(c) for 200-nm-25-jim wavelength and for X > 300,um. The uncertainty in the far-ultraviolet, N < 200 nm,
z0
t-
U-
X
Li
U-0XLii0zj
Li
1.'
I 1
10-1 100 10 lo02 lo, 104 105
WAVELENGTH (mm)
Fig. 12. Real index of refraction at microwave frequencies. Thecompilation is based on the Kramers-Kronig analysis at T = -7°C.The compilation for other temperatures is forced to pass through thevalues corresponding to the real part of the permittivity, e', measuredby Gough,'16 and to parallel the behavior of the -7° compilation atlonger wavelengths, resulting in a lack of agreement with values ofJohari and Charette. These curves are the same as those in Fig. 11(c)
displayed here on an expanded scale.
may be taken as the difference between the dashed andsolid lines in Fig. 11(a). It is at worst -0.05 at X = 45nm. The only other spectral region where the uncer-tainty in mre is much larger than the width of the linein Figs. 11(a)-(c) is in the far-infrared, -25-300-jimwavelength. This is because the uncertainty in mimbetween 33 and 167 jm (Fig. 9) will significantly affectmre in a somewhat expanded region. New measure-ments near the melting point in the far-infrared wouldvery likely alter our compilation of both mre and mim inthat region.
The complete compilation of mre and mim is shownin Fig. 11 and tabulated in Tables I and II.
X. Summary and Recommendation for NeededMeasurements
A compilation of the complex refractive index of icefrom the ultraviolet to the far-infrared has been madefor temperature T = -7°C, and recommended for useat temperatures between -60°C and 0C. A temper-ature-dependent refractive index is compiled for themicrowave. Because of the uncertainties discussedabove, these compilations may be considerably in errorin some spectral regions. New measurements of theabsorption coefficient of pure ice within 60 K of themelting point are needed in five wavelength regions:
Table I. Real (me) and imaginary (mlm) parts of the complex index ofrefraction of ice Ih at -7C, from 45-nm to 167-Am wavelength (X). Data
sources are discussed in the text. These values are graphed in Fig. 11.Wavelengths were chosen for the tables in order adequately to resolve thevariations in both real and imaginary index. For intermediate wavelengths
not given in the table one should interpolate me linearly in logX andlgm,,, linearly in logX. Table I is on the next three pages.
A. Ultraviolet, 185-400 nmOnly liquid-water data are now available, and they
cannot be used for ice because ice absorption is verylikely to be considerably weaker than that of water inthis spectral region. The absorption coefficient hereis so small that it can be taken as zero in many appli-cations. However, for some purposes, such as radiativetransfer calculations for optically thick media,5 thecorrect nonzero values are required.
B. Near-infrared, 1.4-2.8 jimThe measurements now available (Fig. 4) are in
conflict and are of questionable accuracy because ofinappropriate or inadequately known sample thicknessand the possibility of significant light scattering in thesamples used.
C. Middle-Infrared, 3.5-4.3 jimThe imaginary index is too small here to be obtained
accurately by the reflection measurements of Schaafand Williams10; their values, while accurate elsewhere,are uncertain to ±50% in this narrow wavelength in-terval. Transmission measurements would be desir-able.
D. Far-Infrared, 33-600 jimMeasurements were made at T = 100 K, and a con-
siderable temperature dependence was shown between100 and 168 K. Measurements at higher temperaturesare not available. This spectral region is of use for somegeophysical applications (cooling to space by cirrus andstratospheric clouds), and these measurements are alsoneeded in order to do the dispersion analysis to obtainmre and mim in the middle-infrared, X < 33 m.
E. Microwave and Radiowave, 1.25 mm-2 m(1) There are no reliable measurements betweeen
1.25 and 12.5 mm.(2) Different sets of apparently reliable measure-
ments between 12.5 and 110 mm are in mutual conflict.The favored measurements are those of Cumming,4 4 butthese apparently disagree with the theoretical extrap-olation of Mishima et al.8
(3) For 110 mm-2 m it would be desirable to obtainadditional measurements to confirm or dispute ourchoice of Westphal's unpublished measurements, 1 7
since they conflict with other available measure-ments.
I thank Dennis Klug and Edward Whalley forprepublication results of their far-infrared measure-ments, Koichi Kobayashi for expanded graphs of hisultraviolet measurements, and Nathan Ockman forbeing able to locate an absorption spectrum plot thatwas not included in his thesis. John Bertie, Gyan Jo-hari, and Frederick Reding provided useful advice.Craig Bohren carefully reviewed both the first and finaldrafts of this paper. An anonymous reviewer also madehelpful suggestions. This work was supported by NSFgrants ATM-80-24641, ATM-82-06318, and ATM-82-15337. The computations were done at the NationalCenter for Atmospheric Research. This is ContributionNo. 693 from the Department of Atmospheric Sciences,University of Washington.
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(1980).6. E. Whalley, J. Phys. Chem. 87, 4174 (1983).7. M. Seki, K. Kobayashi, and J. Nakahara, J. Phys. Soc. Jpn. 50,
2643 (1981).8. 0. Mishima, D. D. Klug, and E. Whalley, J. Chem. Phys. 78,6399
(1983).9. G. P. Johari, Contemp. Phys. 22, 613 (1981).
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(1969).12. T. C. Grenfell and D. K. Perovich, J. Geophys. Res. 86, 7447
(1981).13. J. D. Neufeld and G. Andermann, J. Opt. Soc. Am. 62, 1156
(1972).14. G. M. Hale and M. R. Querry, Appl. Opt. 12, 555 (1973).15. H. E. Merwin, "Refractivity of Birefringent Crystals," in Inter-
national Critical Tables (McGraw-Hill, New York, 1930), Vol.7, pp.16-33.
16. S. R. Gough, Can. J. Chem. 50, 3046 (1972).17. G. R. Jiracek, "Radio Sounding of Antarctic Ice," in Research
Report 67-1, (Geophysical and Polar Research Center, U. Wis-consin, 1967).
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19. J. Daniels, Opt. Commun. 3, 240 (1971).20. R. Onaka and T. Takahashi, J. Phys. Soc. Jpn. 24, 548 (1968).21. A. Otto and M. J. Lynch, Aust. J. Phys. 23, 609 (1970).22. K. Dressler and 0. Schnepp, J. Chem. Phys. 33, 270 (1960).23. E. V. Browell and R. C. Anderson, J. Opt. Soc. Am. 65, 919
(1975).
References are continued on page 1225.
Table II. Real (rn,) and imaginary (mlm) parts of the complex index ofrefraction of ice h, from 167-,um to 8.6-m wavelength (X), for four
temperatures (T). These values are graphed in Figs. 11 (c) and (f). Forintermediate wavelengths not given in the table, one should interpolate m,.
linearly in logX, logm,,,, linearly in logX, m linearly in T, and logmimlinearly in T. Table I is on the next page.
24. T. Shibaguchi, H. Onuki, and R. Onaka, J. Phys. Soc. Jpn. 42, 152(1977).
25. A. P. Minton, J. Phys. Chem. 75, 1162 (1971).26. L. R. Painter, R. D. Birkhoff, and E. T. Arakawa, J. Chem. Phys.
51, 243 (1969).27. F. Sauberer, Wetter Leben 2, 193 (1950).28. T. C. Grenfell, J. Glaciol. 27, 476 (1981).29. W. Luck, Ber. Bunsenges. Phys. Chem. 67, 186 (1963).30. F. P. Reding, "The Vibrational Spectrum and Structure of Several
31. D. F. Hornig, H. F. White, and F. P. Reding, Spectrochim. Acta12, 338 (1958).
32. N. Ockman, "The Infrared-Spectra and Raman-Spectra of SingleCrystals of Ordinary Ice," Ph.D. Thesis, U. Michigan, Ann Arbor(1957).
33. R. N. Clark, J. Geophys. Res. 86, 3087 (1981).34. U. Fink and H. P. Larson, Icarus 24, 411 (1975).35. N. Ockman, Adv. Phys. 7, 199 (1958).36. M. S. Bergren, D. Schuh, M. G. Sceats, and S. A. Rice, J. Chem.
Phys. 69, 3477 (1978).37. S. Tsujimoto, A. Konishi, and T. Kunitomo, Cryogenics 22, 603
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Spectroscopy, (Macmillan, New York, 1962), Vol. 2, pp. 726-737.
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(1949).44. W. A. Cumming, J. Appl. Phys. 23, 768 (1952).45. J. W. Perry and A. W. Straiton, J. Appl. Phys. 44, 5180 (1973).46. S. R. Gough, J. Appl. Phys. 43, 4251 (1972).47. J. W. Perry and A. W. Straiton, J. Appl. Phys. 43, 731 (1972).48. M. R. Vant, R. B. Gray, R. 0. Ramseier, and V. Makios, J. Appl.
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(1964).50. R. S. Vickers, "Microwave Properties of Ice from the Great
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Books continued from page 1192
develop an electric field in the crystal). Unfortunately, almost noexperimental data are mentioned, either new or old; the authors areonly introducing the reader to what is possible. They get down tobusiness in the section devoted to intermolecular potentials, thesubject to which Califano and his colleagues have contributed most.Specific examples are drawn from the literature, using the opticallattice modes of a-N 2, ammonia, ethylenediamine, naphthalene, andethylene. In recent years the contributions of the Florence grouphave chiefly had to do with anharmonic interactions in molecularcrystals, and the final section of this chapter is devoted to this inter-esting topic. Anharmonic interactions in monatomic and ionic solidsform a subject with much history (it was the training ground of manyof the century's theoretical physicists, e.g., Born, Goeppert-Mayer),and while its general principles have been known for a long time, few,applications to molecular crystals have been made. Here, too, thereader is better advised to turn to the book by Califano, Schettino,and Neto. However, in the present article, Schettino and Califanomake quite clear the reason, on the average, third-order terms in acrystal's effective potential vanish. It is unfortunate that Califano'smost recent work on this subject-phonon linewidths (and thereforelifetimes)-could not be included in this review.
It is quite clear that the doing of infrared spectroscopy without amodern Fourier transform instrument is foolish besides being de-classe. Peter Griffiths gives us an almost up-to-date list of speci-fications to take along to market (a slightly more current version-alsoby Griffiths-appeared in the 21 Oct. 1983 issue of Science). Grif-fiths describes the history of FTIR, the design of several of the com-mercial instruments, optical layouts, low-cost interferometers, anddata systems. The smart FTIR shopper is well-advised to read thisarticle carefully before writing his RFQ.
The last chapter of this volume, by C. G. Cureton and D. M.Goodall, is devoted to the several kinds of vibrational photochemistrybeing done today. These largely fall into three categories: singlephoton electronic excitation (followed either by energy transfer toanother electronic state or to a high-lying vibrational level of theground state); single photon vibrational excitation (followed by energytransfer into the reaction coordinate by V-V relaxation); and multi-photon vibrational excitation leading to dissociation either from anexcited electronic or vibrational state. The authors have had thewisdom of beginning with a long section devoted to "concepts," whichconcentrates on some simple but necessary reminders about pumpand population relaxation rates, how multiphoton absorption isthought to work, energy distributions in molecules, RRKM theory,relaxation, and selectivity. This is followed by a section on multi-photon vibrational chemistry which includes a discussion of variousdetection techniques, including spontaneous vibrational and elec-tronic fluorescence as well as laser-induced fluorescence. This sectionis a cornucopia of dissocation reactions-food for thought for thosewho contemplate study of unstable species by any technique. Thissection closes with subsections on reactions of nascent photofragmentsand on isotope-selective photochemistry. Only after all of this issingle photon photochemistry discussed. The chapter ends with a13-page table summarizing three years' worth of published work(1979-1981).
Taken altogether, one can only conclude that vibrational spec-troscopy was alive and well as of 1983. One final thought: It maybe an accident of the editor's choices of chapter subjects, but onecannot help but be impressed with the contributions of G. C. Pimenteland his students that are to be found throughout this volume, bothcurrent and past. In order of the chapters in this volume, matrixisolation, the oriented gas model, and single photon vibrational ex-citation are but a few examples of those contributions.
