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Rosenkranz Shape 5mm Band 1975

Rosenkranz Shape 5mm Band 1975 - 498 IEEE ramsxcnons ON...

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Unformatted text preview: 498 IEEE ramsxcnons ON antennas AND PROPAGATION, VOL. ”-23, NO. 4, run 1975 Shape of the 5 mm Oxygen Band in the Atmosphere PHILIP W. ROSENKRANZ, MEMBER, [BEE Abstract—The problem of absorption of microwaves by molecular oxygen in the atmosphere is treated by means of a first-order approx- imation to the impact theory of overlapping spectral lines. By including only the coupling between adjacent rotational states in molecular col- lisions, we have devised a simple approximate method for computing the interference between lines from measurements on the resolved lines. The need for an empirically determined function describing the linewidtlt/ pressure ratio is eliminated. Comparisons with measurements at atmo- spheric pressures show that the first-order interference accounts for the low absorption at the band wings near 1 atm pressure. It also predicts the correct amount of symmetry between high and low frequency wings. Improvement over previous models for the pressure broadening is ob- tained at frequencies 2 55 GHz. This approach is not specific to oxygen and could be adapted to other similar molecules. I. INTRODUCTION HE ABSORPTION of millimeter waves by atmospheric T oxygen is of interest in many fields. Use of the strong absorption region near 5 mm for remote sensing of the atmospheric temperature profile was first proposed by Mocks and Lillcy [1].A successful remote sensing experi- ment employing a microwave spectrometer has recently been fiowu on the Nimbus 5 satellite [2]. Attenuation in the window regions of the spectrum due to the wings of the 5 mm band is of interest in the fields of radio astronomy and communications [3]. For hose and for other studies of microwave propagation in the atmosphere there is a need for expressions for the absorption coefficient due to oxygen as a function of frequency, pressure, and temperature. These expressions should be accurate but must not involve prohibitive amounts of computation. The theory of bands composed of overlapping lines has been developed by many authors, notably Baranger [4], K011) and Griem [5], and Gordon [6], [7]. Gordon’s formulationis useful for our purpose since it allows a semiclassical interpretation of the molecular collision process. Computation of the collision cross section matrix that determines interference (or blending) of the lines requires an integration over all possible trajectories however, and the result depends on temperature. Computations of the pressure broadened spectrum of oxygen have been carried out in [8]—[11]. The complexity of these computations has heretofore preventcd'application of the theory of overlap- ping lines to the practical problem of atmospheric absorp- tion, however. ‘ Manuscript received November 17, 1974. This work was supported by NASA Contract NASl-10693. The author is with the Department of Electrical Engineering and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Mass. 02139. Previously, the solution to this problem has been to sum lineshapcs derived for isolated lines. The ratio lincwidth/ pressure, which would be constant for an isolated line, was reduced with increasing pressure to fit measurements of total absorption, particularly in the band wings. This is the procedure followed in [I], [12], and [13], where the Van Vleck—Weisskopf [I4] lineshape was used, and in [15] where the Gross [16] or Zhevakin—Naumov [l7] lincshapc was used. The lincwidth correction at 1 atm amounted to a factor of 2 to 3, depending on the model. This procedure is not entirely satisfactory because the functional form chosen to make the fit is ad hoc; thus there is a risk of erroneous results in regions of frequency-pressure-tcrnperature space for which data is unavailable at the time the fitting is done. The procedure that ‘will be developed in this paper in volves two approximations. The first is a scheme for determining the largest of the matrix elements that govern line interference, from measurements of resolved line- widths. The general principles of detailed balance and con- servation of probability are utilized. This approximation should be valid when rotational states are weakly coupled by collisions. Secondly, to eliminate matrix operations the band shape is expanded to first order in pressure. 11. FORMULATION A. The Oxygen Spectrum Van Vleck [18] has described the basic characteristics of the spectrum of molecular oxygen ”’02 in its ground state. Only odd values of the rotational quantum number N are allowed. Coupling of electronic spin with rotational angular momentum forms a triplet of states with total angular momentum J = N — l, N, or N + 1. The level J = N is highest in energy. Selection rules permit two types of magnetic dipole transitions. Transitions between the state J = N and the state J = N i 1 give rise to resonant absorption at frequencies vNi, listed in Table I for N s 39. Nonrcsonant absorption results from interactions with the states J = N i l in which AJ = 0. In this paper we will be concerned with the pressure range 1 to 1000 mbar. In this range collisions between molecules are the dominant line broadening mechanism. The absorption coeflicient 12 may be expressed as y = CP[v/T]2F. (1) When temperature T is given in K, pressure P in mbar and frequency v in 61-12, the constant C is equal to 0.330 for y in Np/km or 1.434 for y in dB/km. The function F together with the factor v2 determines the band shape. Copyright ©1915 by The Institute of Ekctrical and Ehctronics Engineers, Inc. hinted in U.S.A. Annals No. 507AP006 nosenmuz: 5 so»: oxvosN BAND ( TABLE I ‘-. FREQUENCIES AND AMPLITUDE} or Oxroen LINES ‘2‘ Frequencies ' (GHz) Amplitudes -~, N vN+ 17,; d,* d...“ i 1 56.2648 118.7503 0.913 0.816 I 3 58.4466 62.4863 0.982 0.976 5 59.5910 60.3061 0.992 0.991 7 60.4348 59.1642 0.996 0.995 9 61.1506 58.3239 0.997 0.997 11 61.8002 57.6125 0.998 0.998 13 62.4112 56.9682 0.999 0.999 15 62.9980 56.3634 0.999 0.999 17 63.5685 55.7838 0.999 0.999 19 64.1278 55.2214 0.999 0.999 21 64.6789 54.671 1 0.999 0.999 23 65.2241 54.1300 1.000 1.000 25 65.7647 53.5957 1.000 1.000 27 66.3020 53.0668 1.000 1.000 29 66.8367 52.5422 1.000 1.000 31 67.3694 52.0212 1.000 1.000 33 67.9007 51.5030 1.000 1.000 35 68.4308 50.9873 1.000 1.000 37 68.9601 50.4736 1.000 1.000 39 69.4887 49.9618 1.000 1.000 Frequencies for N s 31 are from Wilheit and Barrett [34], [35]; for N > 31 from Liebe and Welch [36]. See also [38]. We will follow Gordon [7] and represent the oxygen spectrum by six branches of lines: the v”: branches, two branches at the symmetric negative frequencies -vNi, and two branches at zero frequency. The zero frequency “lines“ in conjunction with the factor v2 in (I) produce nonresonant absorption. Pressure broadening is described as a Markov random process; collisions change the mag- nitude and direction of the radiating molecule’s rotational angular momentum and thereby transfer intensity from one line to another in the same or a different branch. We use a first-order (in pressure) approximation to the impact theory of overlapping lines (see the Appendix): F = P 2 mt whkdkz + (V — Will’s i (V " V02 '1' (Pillar): (2) where yk = mil: 2 M_ 1::ka - v! (3) In this formulation 21er is the transition rate matrix for the Markov process of intensity exchange, of which mere will be said in the following; d, is the amplitude of the kth line, v, is its frequency, and (I); is the fractional population of the initial state associated with'the line. We have assumed here that there is no first-order frequency shift of the lines. The frequency and pressure dependence of F is written explicitly in (2), but the (I), w, and y are implicit functions of tem- perature. The summation over k must include lines in all six branches whose populations are significant. For our purposes N s 39 is more than adequate. The second term in the numerator of (2) contains the factor (v —- vh); consequently at very low pressure where the lines are narrow their shape is nearly Lorentzian with half- widths PW”, at those frequencies where the absorption is - 7/ /f/ y'} r [9], 4%”74/ , Mew"? greatest. In the valleys between lines and on the wings of the band the interference terms are significant even at low pressure. As the lines widen with increasing pressure the interference is important in determining the band shape. Since hv << Id" at microwave frequencies it is of no consequence whether for each line we use the upper state population and amplitude of the upper to lower transition, or vice versa [18]. We choose the former because it simplifies our calculations by making the populations associated with lines v],+ and VA," equal. Thus we have for the fractional opulation of the state J = N _(2N+I) _ z 499 (Div exp(—2.0635N(N + 1)/r) (4) where the partition function is given by 3 " Z = 0.725?" 1'3 $43.16;“; 5m? wt?” with T in K. The amplitude of the vN+ line is + _ N(2N+ 3) ”2 d" ' [(N + 1)(2N + 1)] (6) and the amplitude of_the vN' line is d _ = [(N + 1)(2N — 1)]1f2 " N(2N + 1) Factors that do not depend on N have been absorbed into the constant C in (1). Values of these amplitudes are also listed in Table I. They are all close to unity. Negative frequency lines have the same amplitude as their positive counterparts. We will not require individual amplitudes for the zero frequency lines. (5) (7) B. Collision Cross Sections ‘ The transition rate matrix is expressed by Gordon [7] in the form of a matrix of collision cross sections a where 2nPw = p66; p is number density and i7 is the average relative speed. Thus 13 Zack?" (k is Boltzmann’s constant) and a is defined by w: 6 (3) an = $J‘o (”an “ jk)>b db (9) I where b is the impact parameter for the collision and 6” is the Kroneker delta. Angle brackets denote an average over relative speed a. The scattering matrix elements SJ, give the probability of a unit amplitude in line k being transferred into line j by a particular collision. The elements depend implicitly on the asymptotes of the collision and thus on the intermolecular potential. Lines associated with the same value of N have equal widths 5' ing diagonal elements of S are equal. The interes e rea should refer to Gordon’s paper [7] to see the “matrix of integrands" (I — S). If the atmosphere were isothermal, one might do the computation implied by (9) once and for all, and consider 500 the problem solved. Since the atmosphere is not isothermal, we wish to find a simple method of determining the inter- ference coefficients (3) as a function of temperature to an approximation sufficient for practical calculations. C. Merging of Lines If a subset of lines are close enough together that they may be treated as being at the same frequency, their transition rate matrix elements may be merged by summing over j [7]. Clearly this is always true for the lines at zero frequency. At sufficiently high pressure or at frequencies well removed from the resonant region it becomes true for the resonant lines as well. Gordon [7] made use of these facts to develop a simplified theory for use outside the resonant region: we may define Wb = 2 “’1',” for VJ- = 0, Vt = 0 (10) ,- where the summation is taken over all zero frequency lines. If k is a line at positive or negative frequency, Gordon’s theory gives for its total coupling _to zero frequency lines, 2 w}, = ——w,,/2, for v1 = 0, v, at 0. (ll) 1' Likewise, for the total coupling from a positive frequency line to all negative frequency lines: 2 “Us = ‘Ws/2, J' These results (lO)-(12) are independent of Ni. When the zero frequency lines are merged the elements coupling them to the resonant lines at positive and negative frequencies are symmetric [7]. Then yk vanishes for 12,, = 0. The nonresonant term in (2) is therefore given by the Van Vleck—Weisskopf expression fort-j < 0, v, > 0. (12) 070wa v2 + (Hub)2 I The coefficient in (13) lies in the range 0.699 i 0.005 for T = 200 to 300 K. It differs from its classical value of % because the amplitudes of the zero frequency lines are slightly larger than unity. Another consequence of the Spectrum’s symmetry about zero frequency is that the y; for negative frequency lines are equal to —1 times their positive frequency counterparts. F = nonresonant (13) III. TRANSITION RATES .4. Weak Coupling Approximation With respect to molecular collisions one may distinguish two limiting cases [19]. l) Weakly coupled rotational states. The S matrix is dominated by its diagonal and near-diagonal elements. This case may result from a weak angular de- pendence of the intermolecular potential, among other causes. 2) Strongly coupled rotational states, in which the elements of S approach randomness. We shall develop approximations that are appropriate when rotational states are weakly coupled in the majority of collisions. Preliminary results of computations by Barrett and Lam [37] indicate that oxygen belongs to this IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, JULY 1975 class. Computations for CO by McGinnis [20] also show diagonal dominance in the collision cross section matrix. Whether or not these approximations are applicable to oxygen line broadening in air must finally be decided by comparison of results with experiment. The approximation has two parts. i) In most collisions the magnitude of rotational angular momentum either stays the same (AN = 0) or is shifted into an adjacent state (AN = :2). The probability of larger JAN | is small enough to be neglected. 2) The reorientation at of the direc- tion of rotational angular momentum is small in most collisions. The elements of S coupling between branches VN+ and 121.,“ are proportional to sin“ (or/2) cos at whereas the elements coupling between each positive frequency branch and the zero or negative frequency branches are proportional to sin1 or [7]. We consider the former to be negligible in comparison with the latter. For all but the lowest rotational states this approximation is a corollary of 1) unless the angular distribution of changes in angular momentum is very asymmetric. The preceding implies that all of the ofidiagonal elements of S are much smaller than the diagonal elements. This statement is not true“ of a, due to the presence of the Kroneker delta in (9). The implication remains for a- and for w however, that certain offdiagonal elements are much larger than others. B. Ofi’diagonal Elements In our approximation, for each positive frequency line (v),+ or vN‘) the associated column of w has the following nonzero elements: 1) the diagonal element, which we shall denote by'wN; 2) an element denoted by wNT, coupling the line to the N + 2 line in the same branch; 3) an element denoted by wN‘, coupling the line to the N — 2 line in the same branch; 4) an element equal to —wb/2, coupling the line to the line at zero frequency; and 5) elements the sum of which equals —w,,/2, coupling the line to lines at negative frequencies. This matrix is illustrated in Fig. I. If the diagonal elements of w (including w,,) and one of the near-diagonal elements wNT are known the others can be found by successively applying tw0 conditions which w must always satisfy regardless of the intermolecular potential [21]. First, due to conservation of probability and the presence of the Kroneker delta in (9), the elements in any column of 19 must sum to zero. When this condition is combined with the statements of the preceding paragraph, we have wN1= wk — wN — W“. (14) Second, the fact that the intermolecular potential is real implies that the product w J1(1),, is symmetric with respect to interchange of j and k [22]. Thus 1 (DH (DH—2 (15) T _ WN—z — WN This condition is also necessary in order to satisfy the principle of detailed balance [2]]. nosrmcruNz: 5 MM OXYGEN BAND SUM IN EACR COLUMN --Wt/2 O 0 V5------- ----"31 ”as Fig. 1. Schematic representation of to matrix. Elements associated with zero frequency lines have been merged. Four shaded sub- matrices are alike. We solve for the elements in descending order for the following reason. Equation (15) is an exact relation between elements but (14) involves the weak coupling assumption and is therefore only approximate. Each time (14) is applied an error is introduced. If (IJN_2 > (1)” this error is damped by application of (15). The population reaches a maximum at N = 7 to 9 at atmospheric temperatures, so the error is not damped for the last few instances in which (15) is used; however, the decrease in population from the maximum to (bl is not so great as to cause serious problems. Since we have included lines up to N = 39 we set W391 = o. (16) This arbitrary cutoff introduces a large relative error in the interference terms associated with lines 39+ and 39‘ but progressively smaller errors in lower N lines. Since (1139 is very small at atmospheric temperatures these errors are insignificant. There is no magic in the choice of maximum N = 39; reduction of the cutoff to 35 for example changes w”1 and lower elements by less than 1 percent. C. Diagonal Elements The only undetermined parameters in the theory are now W” and w,” line halfwidths per unit pressure for the resolved resonant and nonresonant absorption. Two approaches are possible at this point. One is to use laboratory measurements of W” and w. The broadening of 0; lines in air is not the same as in pure oxygen. The weak absorption in oxygen, especially when diluted by nitrogen, makes laboratory measurements difficult; measurements on air are relatively scarce for this reason. The other approach is to adjust the values of these parameters to fit atmospheric 501 TABLE II NEAR-DIAGONAL w ELEMENTS AND INTERFERENCE Coerrrcmn-rs AT 300 K m Elements Interference Coefficients Population (MHzfmbar) (mbar‘ 1) N (by WNT WNl yn” YM— 1 1.365 - 02 —0.563 0.0 4.515 —- 04 -2.145 — 05 3 2.965 — 02 —0.422 -0.258 4.945 — 04 —3.785 — 04 5 4.115 — 02 —0.376 —0.304 3.525 — 04 -3.925 — 04 7 4.695 —- 02 —0.350 —0.330 1.865 — 04 —-2.685 -— 04 9 4.705 — 02 *0.331 —0.349 3.305 -— 05 —1.135 — 04 11 4.265 — 02 —0.315 -—0.365 — 1.035 — 04 3.445 - 05 13 3.545 — 02 -0.301 -—0.379 —2.235 — 04 1.655 -— 04 15 2.725 — 02 —0.289 —0.39l —3.3ZE - 04 2.845 — 04 17 1.955 — 02 —-0.277 —0.403 —4.32E — 04 3.915 - 04 19 1.315 — 02 -0.266 —0.414 —5.265 - 04 4.935 — 04 21 8.185 — 03 —0.256 —0.424 —6.135 -— 04 5.845 — 04 23 4.815 — 03 -0.245 -0.435 —6.995 —- 04 6.76E -— 04 25 2.655 — 03 —0.236 —0.444 —7.745 - 04 7.555 — 04 27 1.385 — 03 -0.225 —0.455 —8.615 «- 04 8.475 — 04 29 6.735 - 04 —0.219 —0.461 —9.115 — 04 9.015 -— 04 31 3.105 - 04 —0.204 —0.476 —1.035 —- 03 1.035 — 03 33 1.355 — 04 —0.210 —-0.470 —9.875 — 04 9.865 — 04 35 5.515 - ’15 —0.167 —0.513 —1.37.E — 03 1.335 — 03 37 2.135 — 05 -—0.248 —0.432 —7.07E — 04 7.015 - 04 39 7.745 — 06 0.0 —0.680 -2.585 — 03 2.645 - 03 Entries in the second through fourth columns are the same for lines vg” and vN‘. For N > 31 entries in the third through sixth columns are not accurate due to the_cutoil' at N = 39. measurements. These measurements have the advantage of a long path length; but because typical temperature changes at a given pressure level are fractionally small, one cannot separate temperature dependence from pressure dependence of the absorption coefficient. Therefore we will take the first approach and use atmospheric measurements to test the result. For the resonant halfwidths We adopt W” = 1.16 - 10‘3(300/T)°‘85, GHz/mbar. (17) Equation (1 7) is equivalent to the expression used by Meeks and Lilley [1] at low pressure. It is based on laboratory measurements in both oxygen and air. Broadening of the lines by 02 — N2 collisions is found to be smaller by a factor 13 = 0.75 than broadening by 02 — O2 collisions; linewidths measured in pure oxygen are therefore corrected by a factor (0.21 + 0.75 - 0.78) = 0.80 for air. Equation (17) is also supported by measurements made on the 9+ line by Lenoir er a1. [23] using a balloon-borne radiometer. There is some experimental e...
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