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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 ﬁrstorder 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 ﬁrstorder 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 NASl10693. 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 frequencypressuretcrnperature 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 6112, 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 ﬁrstorder (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 ﬁrstorder 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 fortj < 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 neardiagonal
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 neardiagonal 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
NEARDIAGONAL w ELEMENTS AND INTERFERENCE Coerrrcmnrs
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 balloonborne radiometer. There is some experimental e...
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