Absorption by the Tails of the Oxygen Microwave Resonances at Atmospheric Pressures

A variety of methods axe available for calculating atmospheric absorption of microwaves and millimeter waves by oxygen resonances. We compare these methods for the absorption by the tails of the 60 GHz band, the line at 118.75 GHz, and the nonresonant oxygen absorption. For practical, atmospheric calculations the tails of the 60 GHz band consist of frequencies less than 48 GHz and greater than 71 GHz. Gordon's method is selected as being the simplest and most reliable for calculating the absorption by the tails of the 60 GHz band. The interference parameter of the 118.75 GHz line can be chosen such that its high-frequency tail approaches zero as frequency increases.


I. INTRODUCTION
T HE MAGNETIC DIPOLE transitions of the oxygen molecule produce a microwave absorption band at 60 GHz, a single resonance at 118.75 GHz, and a nonresonant absorption that is effective at frequencies below 60 GHz [l].These three parts of the oxygen absorption are really just one unified phenomenon.The absorption can be calculated by computer-intensive inversion of very large matrices [2], one such inversion being needed for each temperature, pressure, humidity, and frequency of interest.Atmospheric pressures are sufficiently small that several approximate methods are available.These approximate methods are based on the line widths being very small compared with line-center frequencies, first-order in-pressure perturbation theory, etc.Such approximate methods are much more computer efficient than the matrix inversion.The subject of this paper is a comparison of these approximate methods at frequencies well removed from line centers.This study is restricted to pressures and temperatures typical of the atmosphere.Only pressure broadening is considered, so pressures must exceed about atm.The models treat one or more of the three parts of the oxygen absorption as though they were separate phenomena.
Van Vleck [l] gives an approximate formula for both the nonresonant absorption as well as the absorption by the tails of the 60 GHz band; to do so he approximates the 60 GHz band as a single resonance.Van  where f is the electromagnetic frequency and y is the line width parameter, both in gigahertz, fo = 60 GHz, P d is the dry-air partial pressure in atmospheres, and T is Kelvin temperature.The third term in (1) is the nonresonant absorption.This result does not include interference effects between lines in the band [3].At atmospheric pressures y Q fo.
The frequency f must be many line widths removed from 60 GHz for (1) to be valid; it is therefore consistent to neglect the width in the denominator of the first two terms within the brackets in (1).If we neglect y2 relative to bothfi and (fof ) 2 .and use a common denominator for the three terms, (1) becomes where x = ( f/fo)2 and g = (y/fo)*.The formulas ( 1) and (2) should be used only outside the 60 GHz band; for most applications (1) and ( 2) apply to the ranges f 6 50 GHz andf 5 70 GHz.For f 2 8 GHz the last factor in (2) may be replaced by unity.Forf < 10.7 GHz the second-to-last factor may be replaced by unity to within 7 percent accuracy; the remaining expression in (2) is then the Debye formula for nonresonant absorption.Forf 2 100 GHz, (1) and (2) tend to an unphysical constant value; this value is three times the absorption in the 1 to 10 GHz range where (1) and (2) give nearly frequency-independent absorption for atmospheric pressures.
Gordon [3] notes that the assumptions leading to the Van Vleck-Weisskopf [4] line shape do not apply to the oxygen microwave spectrum.Gordon [3] reduces the matrix of cross sections to a 3 x 3 matrix by assuming that all the lines in the band have the same resonance frequency.The matrix inversion then gives the following simple analytic formula for the absorption outside of the band center: where as in (2) x = (f/f~)~ and g = ( ~/ f ~) ~.
For atmospheric pressures, we may neglect y2 relative to both fi and I f ;f I ; then Gordon's formula gives Excluding frequencies 50 to 70 GHz, the approximation (3b) formulas for calculating Si, yi, and from temperature, agrees with (3a) to within one part per thousand for atmopressure, and humidity as well as for the spectroscopic data spheric conditions.For the coefficient A we match (3b) to base.Equation ( 7) uses the Van Vleck-Weisskopf line shape Liebe's [5] formula (same as that by Rosenkranz [SI) for the as modified by Rosenkranz [6].The summation in (7) consists nonresonant absorption at low frequencies; this gives (in units of 41 lines in the 60 GHz band as well as the line at 118.75 dB/(km GHz)) GHz (six lines between 368 and 834 GHz are also available but 300 2 not used here).In addition to (7) we obtain the oxygen A=0.0113pd (--) .
(4) nonresonant absorption from Liebe's model: The numerical coefficients in (1) and (4) differ by 5 percent because (1) corresponds to the classical result of one-third of f 2 + y 2 ' the mean-squared dipole moment arising from diagonal matrix where A is given in (4).We see that ( 8) is identical to (5).
elements and two-thirds from off-diagonal elements, whereas Note that [5] includes an ad hoc factor [I + (f/60 GHz)~] -1 Rosenkranz [6]  senkranz [6] and Liebe et af.[8].Smith [2] obtains good agreement between measured and calculated dispersion in the which agrees with (3b) to within percent iff < GHz.This 60 GHz band.H~ the Rosenkranz theory to second is the classical Debye for nonresonant absorption and order in and thereby shou,s that the first-order theory is the same as the third term within brackets in (1).In the is accurate for pressures as great as atm.We use Smith,s frequency ranges GHz < < 50 GHz and ' 70 GHz' interference coefficients in (7) in place of Liebe's;all other (3b) reduces to parameters remain the same.This gives yet another calculawith an accuracy of better than 7 percent for atmospheric conditions.At sufficiently high frequencies, f %-60 GHz, then cyG a f -2 .As pointed out by Stankevich [7], this decrease proportional to f -2 in the high-frequency tail is contrary to the constant value produced by the Van Vleck-Weisskopf formula (1).Rosenkranz [6] gives a theory for atmospheric oxygen microwave absorption that uses line-by-line summation over the 60 GHz band and the 118.75 GHz line as well as including the nonresonant absorption.This theory introduces interference coefficients to first order in pressure, which describe the collisional transfer of energy between states.Rosenkranz [6] found good agreement between his theory and data for frequencies 50 to 70 GHz.Using dispersion spectroscopy, Liebe et af.[8] obtained a spectroscopic data base consisting of oxygen line strengths, widths, and interference coefficients for use in the Rosenkranz theory.Liebe [5], [9] gives a model of atmospheric absorption and refraction.Our line-by-line summations of the Rosenkranz theory use Liebe's computer programs.
For the contribution by the 60 GHz band and the 118.75 GHz line, the absorption coefficient in the theory by Rosenkranz is given (in dB/km) by where Si, fi, yi, and 6i are the line strength, resonant frequency, width parameter, and interference coefficient of the ith line, respectively.Our S; is Liebe's strength parameter multiplied by 0.1820 fi.The reader is referred to [5] for the tion of the tails of the oxygen band: The width parameter y in ( 2), (3b), and (8) determines the absorption at frequencies less than about 10 GHz for atmospheric pressures; it is the width parameter of the nonresonant absorption.For air, the measurements by Kaufman [ll] at 9.26 GHz give the value (in GHz) Measurements by Maryott and Birnbaum [ 121 give a value 18 percent less than Kaufman's.On the basis of fitting their data in the 60 GHz band to the Rosenkranz theory, Liebe et af.[8] give a value of y that is 6 percent greater than Kaufman's; Liebe's [5] where e is the water-vapor partial pressure in atmospheres.Radiometric data at frequencies 2.5,4.75,9.40,and 10.0 GHz are reported by Smoot et al. [13] Liebe I51 shows that these data support his value of y; these data also support Kaufman's value of y, but not the value by Maryott and Birnbaum.For our calculations we use Liebe's as in (9), with e = 0. oxygen absorption obtained by adding the nonresonant absorption to this sum.The two short-dashed curves differ from the Rosenkranz-Liebe theory only in that Smith's interference coefficients are used.The difference between the Smith and Rosenkranz calculations is greatest for lowest temperature, and the 30 percent difference at 30 GHz for -40°C shows that the difference is observable using atmospheric radiometry for extremely cold air.Gordon's theory agrees well with the Rosenkranz theory.Of course the theories are forced to agree at low frequency by using the same asymptote for the nonresonant absorption.Van Vleck's theory predicts the least absorption below 60 GHz.Measurements of oxygen absorption at 9.26 GHz by Kaufman [ l l ] agree significantly better with Gordon's theory than with Van Vleck's for pressures greater than a few atmospheres.In any case, Figs. 3 and 4 show that above 50 GHz Gordon's and Van Vleck's theories deviate from the Rosenkranz and Smith theories as the 60 GHz band is approached; this is expected since Gordon's and Van Vleck's theories apply only to the tails of the band.Figs. 5 and 6 (for 40°C and -40"C, respectively) show the tails of the 60 GHz band with the same curve types for the various theories as used in Figs.1-4; the resonance at 118.75 GHz is also shown.Note that the impact approximation for oxygen absorption fails for frequencies greater than about 300 GHz [ 2 ] , [3], and that atmospheric nitrogen-collision-induced absorption is important beyond 200 GHz [7].In interpreting Figs. 5 and 6 it should be kept in mind that absorption coefficients of dB/km are beyond present experimental capabilities.Also, the three parts of the oxygen absorption (the 60 GHz band, 118.75 GHz line, and nonresonant absorption) are shown separately to illucidate the behavior of the theories; however, only the sum of the three parts is meaningful.

GRAPHICAL COMPARISON
The lower solid curve for the 60 GHz.band is the sum of resonances in the 60 GHz band using the Rosenkranz-Liebe interference coefficients.It crosses zero and is negative at high frequencies; the frequency of zero crossing is dependent on temperature.The upper solid curve for the 60 GHz band is this sum of resonances with the frequency-independent nonresonant absorption added.Adding the nonresonant absorption is an approximate means of correcting for the negative absorption; in fact Liebe [9] made slight changes to interference coefficients to improve the correction.However, the nonresonant absorption has different temperature dependence than the high-frequency asymptote of the sum over resonances, so the correction only works at one temperature.Thus at -40°C in Fig. 6, the Rosenkranz-Liebe theory including the nonresonant absorption follows Gordon's theory which is the medium-dashed curve, whereas for 40°C in Fig. 5 , it is well above Gordon's theory and is approaching a positive constant because the nonresonant absorption is an overcorrection.In Figs. 5 and 6 the sum of resonances in the 60 GHz band using Smith's interference coefficients is in good agreement with Van Vleck's theory.These curves are approaching a positive constant at asymptotically high frequency; this is the unphysical high-frequency tail of the Van Vleck-Weisskopf line shape.
Three curves are given for the 1 18.75 GHz line in Figs. 5  and 6.The solid curve uses the Rosenkranz-Liebe interference coefficient; this curve tends to a negative constant value at high frequencies.The short-dashed curve uses Smith's interference coefficient; this curve approaches a positive constant value at high frequencies.The long-dashed curve uses 6; = - yi/fi for the 118.75 GHz line such that the high-frequency tail approaches zero from above as f -2 (see Section HI).The condition 6; = --yi/h for the 118.75 GHz line requires that Liebe's temperature exponent for 6 be changed from 0.9 to 0.8, that a slight humidity dependence be given to hi as it is to yi in ( 9) and that Liebe's interference parameter [5] as be set to -0.134/kPa, which is intermediate between the Rosenkranz-Liebe value of -0.44/kPa and Smith's value of -0.0583/ kPa.These changes have little effect except in the far tails.
The total oxygen absorption in the Rosenkranz-Liebe model is the sum of the values given by the upper solid curves in Figs. 5 and 6 and their absorption by the 118.75 GHz line.In the next section we give the value of the high-frequency asymptote of the 118.75 GHz line.This value shows that the total oxygen absorption at frequencies above about 200 GHz is only slightly less than the value in the upper solid curves in Figs. 5 and 6.Rosenkranz [14] cites atmospheric radiometric data at 90 GHz as supporting his model over Smith's; consequently these data also support Gordon's theory because Gordon's theory agrees with the Rosenkranz theory at 90 GHz.We obtain further support for the theories by Rosenkranz and Gordon from the dry-air zenith attenuation given by Kislyakov and Stankevich [15].For the Rosenkranz theory we include the addition of the nonresonant absorption to the high-frequency tail of the 60 GHz band.For Gordon's theory we include the absorption by the 118.75 GHz line using hi = -yi/fi for this one line.From the data by Kislyakov and Stankevich [15], the ratio of dry-air zenith attenuations at 73 GHz to those at 140 GHz is 11.This ratio is 11, 12, and 8 from the theories by Gordon, Rosenkranz, and Smith, respectively.Thus these data favor the Gordon and Rosenkranz theories over Smith's.

ALTERNATE LINE SHAPES
The comparison of Smith's theory with Van Vleck's in Figs. 5 and 6 suggests that the constant high-frequency asymptote and the disagreement with radiometric data are caused more by the constant asymptote of the Van Vleck-Weisskopf line shape than by Smith's interference coefficients.Gordon criticizes use of the Van Vleck-Weisskopf line shape for the oxygen band.For these reasons it is useful to consider other line shapes.In the following we consider the case f --+ 03 because it is evident from Figs. 5 and 6 that the behavior of .theline shape in this limit strongly affects the 2s; 6; calculated absorption at frequencies lower than those at which = ffGR-' the impact approximation is expected to fail.
By analogy with (7), the absorption coefficient for the full Lorentz line shape with interference coefficients is i i f i The second formula (1 1) is obtained from (10) by placing the two terms over their common denominator.With 6; = 0 and the last two terms in the denominator of (1 1) replaced by 4f2yf, (11) becomes the Gross line shape.Thus, as emphasized by Hill [16], there is very little difference between the Gross and full Lorentz line shapes for large y;/J.Therefore, by analogy with (1 l), the absorption coefficient for the Gross line shape is Equations (1 1) and (12) have the same asymptotic behavior for both f + fi and f Q fi provided that n / f i Q 1.When the terms in (7) are put over a common denominator, we get where d = 1 for CXFL but d = 3 for (YGR.One can see that there are differences between CYFL and (YGR of order ( ~; l f i ) ~ relative to unity in these limits.All three line shapes give.aconstant absorption as f -+ 03, but the constant is different for cyR than for CYFL and CYGR; this constant also depends on the choice of ai.
Table I presents values of the summations in (14)-( 17).Study of this table reveals the following.The first pair of columns shows that the Van Vleck-Weisskopf line shape with the Rosenkranz-Liebe interference coefficients gives more absorption on the low-frequency side of the 60 GHz band than with Smith's interference coefficients.This is also evident in Figs. 1 and 2. Use of the full Lorentz (or Gross) line shape gives substantially more absorption below 60 GHz than does the Van Vleck-Weisskopf shape with the same interference coefficients.In fact, the full Lorentz (or Gross) line shape with Smith's predicts more absorption below 60 GHz than does the Van Vleck-Weisskopf shape with the Rosenkranz-Liebe 6;.Below 60 GHz, the absorption calculated using the full Lorentz (or Gross) shape is less sensitive to variation of the interference coefficients than if the Van Vleck-Weisskopf shape is used.
Introducing interference coefficients into the full Lorentz or Gross line shapes therefore does not improve the highfrequency behavior.This problem would be alleviated if a new line shape were constructed in an ad hoc manner by replacing 6i in (1 1) and (12) with 26; ff/( f 2 + e).
The asymptotic behavior of these line shapes is examined by taking the limits f + 0 and f + 03.As f + 0, we have where b = 1 for CY& but b = 0 for CYGR.
As f + 03, we have The second pair of columns shows that in the highfrequency limit the Rosenkranz-Liebe 6; values produce negative absorption for any of the line shapes, whereas Smith's 6; values produce positive absorption from the Van Vleck-Weisskopf shape but negative absorption from the full Lorentz (or Gross) shape.In interpreting these results, note that slightly different 6; would be obtained from Liebe's dispersion data [8] if the full Lorentz dispersion line shape were used instead of Van Vleck-Weisskopf dispersion shape.
The third pair of columns in Table I shows the coefficient of the term proportional to f -z in (16) and (17).These coefficients of the f -2 term forf + 03 show great sensitivity to values of the 6; as well as to the line shape.
The 1 18.75 GHz absorption is plotted in Figs. 5 and 6 for the case 6; = -y;/fi.From (16) we see that this results in an approach to zero absorption from above as f -+ 03.Using a full Lorentz (or Gross) line shape, (17) shows that one must set 6; = 0 for the 118.75 GHz line to obtain zero absorption from this single line as f -+ 03.In reality the 60 GHz band and 1 18.75 GHz line are two aspects of a single phenomenon, so one should only require that the sum of both the band and this line give nonnegative absorption, which tends to zero as f + VI.CONCLUSION

03.
Gordon's theory agrees with Rosenkrantz's for frequencies less than roughly 1 0 0 GHz, and they are both favored over  Smith's on the basis of radiometric data.Further atmospheric radiometric data from very cold conditions at frequencies near 30 and 90 GHz would be useful.The data at 33 and 91 GHz presented by Smoot et al. [I31 have the potential for testing these theories.Gordon's theory has physical behavior in the high-frequency limit, whereas the line-by-line summations do not.Gordon's theory is favored over Van Vleck's on the basis of Kaufman's high-pressure laboratory data.Based on these considerations, as well as the simplicity of Gordon's formula relative to line-by-line summations, Gordon's theory is recommended for calculating atmospheric oxygen absorption for frequencies less than 48 GHz and greater than 71 GHz.For frequencies between 48 and 71 GHz, the line-by-line summation is needed, using the theories by Rosenkranz or Smith.There will then necessarily be a jump discontinuity in absorption coefficient at 48 and 71 GHz; if this is undesirable then the line-by-line summation should be extended over a greater range of frequencies.Using Gordon's theory, one must add the absorption by the 118.75 GHz line as though it were a separate phenomenon; an ad hoc method of doing so that preserves zero absorption as f + 03 is to use 6; = -?;/A for this one line.As noted previously, this choice of 6; for the 1 18.75 GHz line causes very little change in the dependence of yi on temperature, pressure, and humidity and corresponds to a value of 6; that is between those of Rosenkranz-Liebe and Smith.
It is noted that the high-frequency tails of the 60 GHz band and 118.75 GHz line are sensitive to the choice of line shape as well as being sensitive to the choice of interference coefficients.It is possible that the tails could be obtained using lineby-line summation if the proper combination of line shape and interference coefficients could be determined.

ACKNOWLEDGMENT
The author thanks Dr. H. J .Liebe for helpful discussions and 535 f 0.025) -He obtained this by fitting the data to Gordon's theory.

FigsFig. 1 .Fig. 2 .
Figs. 1-6 show the absorption coefficient due to oxygen (in dB/km) for dry air at 0.7 atm pressure and for temperatures -40°C and 40°C.Figs. 1 and 2 show the frequency range 0.2 to 30 GHz, and Figs. 3 and 4 show the range 20 to 55 GHz.The two solid curves are the Rosenkranz theory as calculated from Liebe's model; the lower curve is the sum over the resonances in the 60 GHz band and the upper curve is the total

Fig. 5 .Fig. 6 .
Fig. 5.The oxygen absorption at temperature 40°C and air pressure 0.7 atm for frequencies 65 to 250 GHz.For both the 60 GHz band and 118.75 GHz line.the Rosenkranz and Smith theories are given by the solid and shortdashed curves, respectively.The uppermost solid curve for the 60 GHz band is the Rosenkranz line-by-line summation with the nonresonant absorption added.The theories of the 60 GHz band by Gordon and Van Vleck are given by the medium-dashed and long-dashed curves, respectively.The long-dashed curve for the 118.75 GHz line uses the interference coefficient 6, = -*/,/A.