THE UNIVERSITY OF MICHIGAN 7455-1-T TECHNICAL REPORT ECOM-01378-5 October 1966 Calculation of Effective Absorption Coefficient and Spectral Density of Volumetric Power Production by Spontaneous Emission Near the Center of the 183, 3GHz Line of Water Vapor. Technical Report One Contract No, DA 28-043 AMC-01378(E) Prepared by K. Nishina and M. L. Barasch The University of Michigan Department of Electrical Engineering Radiation Laboratory Ann Arbor For U. S. Army Electronics Research and Development Activity White Sands Missile Range New Mexico

THE UNIVERSITY OF MICHIGAN 7455-1-T TABLE OF CONTENTS ABSTRACT v I. INTRODUCTION 1 II. CONSTANTS OF A WATER MOLECULE 2 III. ROTATIONAL ENERGY LEVELS AND THE FREQUENCY OF THE RADIATION 6 IV. THE VALUE OF THE DIPOLE MOMENT MATRIX ELEMENT 12 V. CALCULATION OF EFFECTIVE ABSORPTION COEFFICIENT 24 5.1 Derivation of Formula 24 5.2 Calculations 28 5.2.1 Check of Numerical Calculations for t = 14. 5C. 28 5.2.2 A General Expression a(T,PW, Pt) and a Test Calculation for T = 1000 K. 32 VI. VOLUMETRIC POWER PRODUCTION OF SPONTANEOUS EMISSION 38 VII. LINE-SHAPE AND SPECTRAL DENSITY FOR VOLUMETRIC POWER PRODUCTION 42 VIII. POPULATION OF STATES - ADDITIONAL COMMENTS 44 IX. APPROXIMATIONS MADE AND POSSIBLE IMPROVEMENT ON THE PRESENT CALCULATIONS 46 X. SUMMARY OF NUMERICAL RESULTS 48 10.1 Expressions for S. 48 10. 2 The Effective Absorption Coefficient, a 49 ACKNOWLEDGMENTS 49 RE FERENCES 50 iii

THE UNIVERSITY OF MICHIGAN 7455-1-T ABSTRACT Expressions are obtained and presented for the effective absorption coefficient and the power per unit volume per unit frequency interval arising from spontaneous emission in a gas containing water molecules, both near the center of the line at 183. 3 GHz of water. Since the molecule is an asymmetric rotor, and since the constants of the molecule, to which the calculations are sensitive, seem not to have been determined unambiguously, much re-investigation of the notation, numerical work and interpolation to apply approximations derived from symmetric rotor theory, and checking of previous calculations to select a satisfactory set of molecular constants, has been necessary. Finally, expressions for the two quantities first mentioned here are presented, in a form which permits application to many problems of practical interest. For example, the radiative intensity emerging from such a gas could be predicted only after computations such as those presented here. ii. i,.1._. i i V i i ii

THE UNIVERSITY OF MICHIGAN 7455-1-T I INTRODUCTION A microwave radiation line at 183.3 GHz has been attributed to the 313 - 22 13 20 transition of water by King and Gordy (1954). They attributed the line to this transition on the basis of calculations made by King, et al, (1947). The latter publication calculates the frequency of this radiation and also the absorption coefficient associated with it. In this report an attempt will be made first to reproduce their calculations, both of the frequency (Chapter III) and of the absorption coefficient; the results are compared with their values. Because of the complicated notation in such calculations and the availability of various values for the constants involved such a checking procedure was felt necessary. After a good agreement is demonstrated, a general expression for the absorption coefficient will be given (see Eq. 5. 34)with temperature and pressure as variables. Another quantity calculated in this report is the source intensity, due to spontaneous emission (Chapter VI) resulting from the transition. Such a quantity, together with the absorption coefficient will furnish numerical data needed for radiative transfer calculations of the intensity emerging from a body of gas containing some water molecules. A general expression for the source intensity with temperature and pressure as variables will also be given. The investigation will be formulated and computations presented in such a manner that calculations similar to the ones presented here can be performed easily with different parameters. This is necessary because new and improved values of molecular constants are published frequently, and because values of temperature and pressure other than those assune d by King et al may be of interest. 1 - -.-.,. -.-I —

THE UNIVERSITY OF MICHIGAN 7455-1-T II CONSTANTS OF A WATER MOLECULE Before we embark on the calculations of various quantities, a discussion of values available for the constants describing water molecules seems in order. As for the rotational constants of the molecule, the values theauthors could find are listed in Table II-1. Those various sets of values agree with each other (if some of them are rounded off to the proper number of decimal places), except for the value of C furnished by Townes and Schawlow (1955, p. 639). The latter group quote Herzberg, but apparently miscopied that value for C. Their value of C is expressed in MHz, and after a conversion it gives -1 -1 C' = 9.96 cm, whereas Herzberg's value is C' = 9.28 cm. The discrepancy is serious for the purpose of checking the transition assignment, since this much difference could result in a significant discrepancy in the value of the transition frequency. As for the rest of the table, if we adopt the values for the lowest vibrational level in Herzberg, the values are consistent, although A' by Randall, et al, (1937, p. 163) gives 27.81 cm when rounded off, whileHerzberg gives 27.79 cm, but this difference is unimportant for practical purpose. Penner does not.mention how his values were obtained, incidentally. For the above reasons, it does not make any difference which set of values we use, except Townes and Schawlow's. Here in our calculation we will use the values by Randall, et al, (1937). This has the advantage that these. values were used by King, -et al, (1947) and Randall, et al, (1937) in similar calculations to the present one, therefore using these values makes the comparison of our results with these groups easy. In energy value calculationsKing et al (1947) carried many significant figures in order to obtain some accuracy in -----------------—.1.1 ----- 2' —---

THE UNIVERSITY OF MICHIGAN 7455-1-T c - 0co S E=, "^I?~; S~Co' COD 0) CM C\1 co ro Co o o X ^ -X- IC')O CD CD CD CD I) CI CC C Co 4-I ^. * * * 0 Ev I S I - " I I C) 0' 0)~~ ~ ~ ~ Q COa) CO O >(c M ~ i si ~ ~ ~ a C1 1 8Mi m t o o. Co I --— I - 0 " I o II CV1 O= ft NN N'! ~t co 0 ) Cl 0> 0 10 10 10 10 LOL LO n L Co Z.o N. 0 o E- 1 o o cD 0^ ----- ~ ~ ~ ~ ~ ~ ~ c 0 0 C-, r I;; nI r>, *' ~Cc O *^'g c c~ co r^' N 0^2' ~ co I H -- - ~ % %C >M 0l am r a) ~ ( o Co o oo 3

THE UNIVERSITY OF MICHIGAN 7455-1-T frequencies resulting from energy differences so at least for the purpose of checking the computational procedure and notation, the many digits in Randall, et al, (1937) are useful. Therefore we will use A' = 27.8055cm B' = 14.9997 cm (2.1) C' = 9.2793cm1 K = 0.43643 These values give the following parameters: b -B = -0.16400 (2.2) p 2A - B - C b A- B 0 2=A -0.56032 (2.3) o 2C - B -A b is a measure of deviation from a prolate symmetric top, and b from an oblate one (Townes and Schawlow, 1955). The axes of symmetry for the H20 molecule are shown in Fig. 2-1. IA, IB and IC are the moments of inertia around the a, b and c axis respectively, where IA < IB < IC, or A > B > C, for this molecule, in our notation. As for its dipole moment, the radiation of H20 is mainly associated with the electric dipole moment (Van Vleck, 1947a, p. 426) and this moment is parallel to the axis b (King, et al, 1947, p. 436). The latter statement can be justified by symmetry considerations. As for the magnitude of the moment, Townes and Schawlow (1955, p. 639) gives the value u = 1. 94 x 10 (esu cm) whereas Van Vleck (1947a, p. 427) uses u = 1.84 x 10 (esu- cm), which appears to have been used by King, et al, (1947) in their calculations. In the present report, the 4

THE UNIVERSITY OF MICHIGAN 7455-1-T latter value will be used, again for the reasons of checking the procedures of computation. The values of absorption coefficients and source intensities, which are quadratic in the dipole moment, would be altered at most 10 per cent by the choice. 4. H H c -axi o 0 -o —- b-axis x HI ^H H a-axis FIG. 2-1: THE AXES OF SYMMETRY FOR THE H 0 MOLECULE II I I! i~ 1 1 i i 5' I I, I. I I _. I I~~~~~Z

THE UNIVERSITY OF MICHIGAN 7455-1-T III ROTATIONAL ENERGY LEVELS AND THE FREQUENCY OF THE RADIATION Since Ray's parameter of asymmetry is roughly K = 0.436 for the water molecule, this rotor is closer to a prolate symmetry limit than to an oblate symmetry limit. To demonstrate such a condition together with other parameters which affect the energy levels, Fig. 3-1 and Fig. 3-2 are given. Figure 3-1 is the diagram shown in usual textbooks (for example, Townes and Schawlow, 1955, p. 86) with added numerical values for our specific problem. The notations are the same as those in the latter reference. Figure 3-1 shows where our transition takes place with an arrow. By calculating the energy associated with the levels 313 and 220 for our value of K we will obtain the frequency of this radiation. There is no general analytical energy expressio which can be used for any value of (J, K1, K1, K). Hence we have to rely on available tabulations. The simplest of the possible approaches is to use a formula (Townes and Schawlow, 1955, p. 89) W 1 1 W= 2 (A + C) J(J+ 1) +- (A - C) E (K) (3.1) h 2 2,2J 7T where W is the energy of the level in energy units, h is Planck's constant T = K - K -1 1 (3.2) ______L__________________________i 6 _______[__.__ii

THE UNIVERSITY OF MICHIGAN - 7455-1-T The quantities on the right hand side of the Eq. (3. 1) must be expressed in units of frequency. If the result for W is wanted in terms of wave numbers the following formula may be used where v is the velocity of light: c/ W 1 1 v = - (A + C')J (J+ 1) + - (At - C')E (K) (3.3) hv 2 2 J, T c where = A B C A =- B' = - C? = v v v c c c The values of EJ (K)usedinEq. (3.1)or (3.3) aretabulated in various references J,T (King, et al, (1943, p. 39); Townes and Schawlow, (1955, p. 527); Allen and Cross, (1963, p. 235)) as a function of J, r and K. For our transition, we find the following values of EJ (K), on employing interpolation and symmetric character. For the 313 level: T = -2 EJ (K) = E (-0.43643)= -E (0.43643) 3, -2 3,2 tabulated values are E (0.43) = 8. 6535614 3, 2 E (0.44) = 8.6757937 3, 2 After the interpolation E 2(-0 43643) = -8. 66786 For the 220 level: T = 2 20 E J(K) = E (-0.43643)= -E (0.43643) J, T, 2,, -2 tabulations are E (0.43) = -2. 7092576 2, -2 E (0. 44) = -2. 6941292 2, -2 L —------------------------ 7 —- -- ----- ~ —

THE UNIVERSITY OF MICHIGAN 7544-1 -T after the interpolation E (-0.43643) = 2. 69953 2, 0 These final values of ET (K) are in good agreement with King, et al, (1947, p. 440 J,T With these values, energy values are computed by Eq. (3. 3) W -1 (V-) = 142.2175cm (3.4) hv c 3,1,3 W ( - ) = 136.2604 (3.5) h v c 2,2,0 W__ 1 -1 A( ) = 1 5,9571cm -1 (3.6) hv X C v = 178.59 GHz (3.7) The energy values are close to those of King, et al, (1947, p. 438), which are 142.30 cm and 136.15 cm while Randall, et al, (1937, p. 164) gave 142.17 cm and 136.10 cm. Although the final frequency value makes a 3 per cent difference from the 184. 5 GHz quoted by King, et al, (1947), there is no other possible combination of levels that results in a frequency close to this value; therefore we can say that we are looking at the right transition, The discrepancy in v is due to even smaller discrepancies in two energy values which alter the energy difference by the 3 per cent discrepancy found. Another way of calculating energy levels will be mentioned briefly, to see how two methods compare. According to Townes and Schawlow (1955, p. 86) they can also be calculated by using the following relations: -----------------------— 8 —-----------

TTHE UNIVERSITY OF MICHIGAN 7455-1-T W B'+ C' B'+ C''J (J + 1) + (A' - 2 )w (3.8) hv 2 2 c where 2 2 3 w = K + b +b + C b 3+.. (3.9) p 2 p 3 p and cn are tabulated in their Appendix III. After substitution of numerical values (bp, A', B' and C'), the following results are obtained: WWt, = -0.02869 (hv) = 142.2174cm c 3,1,3 W WO0 = 4.07906 ( W) = 136.2593cml 2 2,0 hv c 2,2,0 1 -1 c = 5.9581 cm - = 178. 62 GHz X X We notice they are quite close to those obtained by the first method (Eq. 3.4 through 3. 7) in spite of the fact that Eq. (3.8) is supposed to be good only for an asymmetric rotor slightly deviated from a prolate limit. This means that for our K = -0.43643, J = 2 and J = 3,Eq. (3.8)was essentially in agreement with Eq. (3.3) which on the other hand is said to be valid for any range of K. For higher values of J, however, Eqs. (3.8) and(3.3) are expected to yield different results for the same K, since our K = -0.43643 is far from a slightly asymmetric top. 9

THE UNIVERSITY OF MICHIGAN 7455-1-T W(J, K 1' K) K H20 K-1 1 2,330 J 3 1 Q 5~~~00 I rB= C+ 0.2818(A-C) I 00= Prolate Limit K = -0.43643 IO L it r 21 10 b = - 0.16400 b = - 0.56032 K_1 K are pseudo quantum numbers for an asymmetric top. The arrow indicates the transition we are interested in. FIG. 3-1: ENERGY LEVELS AND THE TRANSITION IN OUR PROBLEM ------------- 10" -------------

THE UNIVERSITY OF MICHIGAN 7455-1 -T K. IC~~~ HO0 1 <2"z A-C B = C+ 0.2818(A -C) 10B BBC^^ CA-C 0=+- B=A -0.43643 2__ 2 -1 I i Prolate Limit Oblate Limit b - 0 -0.16400 -0.33333 -1.0000 — b p p b -- -1.0000 -0.56032 -0.33333 0 deb o FIG. 3-2: THE RELATION BETWEEN ROTATIONAL CONSTANTS AND VARIOUS OTHER PARAMETERS FOR THE WATER MOLECULE 11

THE UNIVERSITY OF MICHIGAN 7455-1-T IV THE VALUE OF THE DIPOLE MOMENT MATRIX ELEMENT In the later chapters where we calculate the absorption coefficient and the sourc intensity we will need a value of the matrix element for our specific transition. The value of one of the factors in the matrix element is tabulated in many references and it is by interpreting the values from such tabulations that we will perform calculations later. Here in this section the theory behind such tabulations of matrix elements is briefly reviewed, inorder to facilitate the explanation of the calculating procedure later. No attempt will be made to give a full quantum-mechanical treatment which would reproduce and justify tabulated numerical values: the review is meant just to clarify notations which appear in formulae. For a full treatment, Allen and Cross (1963), King et al (1943), Cross et al, (1944) and Strandberg (1954) must be referred to. What we will need in the later calculations is the following: er,2E leXa2 I+ |eY |2 + eZ |2. (4.1) Here X is the operator corresponding to the X component of the position vector r in space-fixed co-ordinates; Y and Z are defined in similar manners. X, | Y 0 and Z O are corresponding matrix elements calculated between wave functions ip and i. Here a and 3 stand for the intitial and final states specified by sets of quantum numbers respectively. What we need is the right hand side of Eq. (4.1), and we designate this summation by the left hand side. In other words Eq. (4.1) is the definition of er3 2 (Schiff, 1955, p. 253). To write down the set of quantum indices that specify a and /,.......___________ 12 _______________

THE UNIVERSITY OF MICHIGAN 7455-1-T a = (R, V, E) (4.2) = (R', V', E') (4.3) where primes indicate the final state. R and R' are rotational quantum states, namely, R = (J, K 1 K1' M) (4.4) R' = (J' K', K, M) (4.5) Similarly, V, V': vibrational quantum states E, E': electronic quantum states Next we introduce a molecule-fixed co-ordinate system, the three axes being the principal axes a, b and c of the moment of inertia (Fig. (4-1)). In Fig. (4-1), the dipole moment vector p is drawn not to be parallel to the principal axes, in order to indicate a general case. i, j and k are unit vectors along space-fixed axes X, Y and Z. If we denote the direction cosines of i with respect to a, b and c axes by cos (aX), cos (bX) and cos (cX), eX = e a cos (aX) + e b cos (bX) + e c cos(cX), and similarly, for eY and eZ. 13..........

THE UNIVERSITY OF MICHIGAN 7455-1-T To write them down in a compact form, eF = " egcos(gF) (4.6) g where g = a, b and c F = X, Y or Z The wave function ia can be approximated satisfactorily by a product of wave functions for rotational, vibrational and electronic states. - R( ( )' V,E(q) (4. 7) I!E I ). 1(q) (4.8) 0I = OR'(() EV' ) (4.8) Here ) represents three Eulerian angles that describe relative positions of the (a, b, c) axes with respect to (X, Y, Z) axes. q represents sets of co-ordinates which describe the electronic and vibrational states of the molecule. Then by Eq. (4.6), eF, - < i|eF| a > = < 3 |egcos(gF)| a> 14: -------------------- 4 —-----------

THE UNIVERSITY OF MICHIGAN 7455-1-T = Vf DR' cos (gF)q Rdv j'V'E' egVE dvq (4.9) (dv and f dv mean the integrations over all the co-ordinates represented by () and q respectively. In our problem, the transition is purely rotational, in the sense that no electronic or vibrational transition is involved. Therefore V' = V, E' = E and the second factor in Eq. (4. 9) becomes Veg Edv <eg> ((4.10) V, E V, E %q V,E g V,E 410 Here p means the component of the electric dipole moment along the g-axis of the g molecule. As mentioned in Chapter II, for a water molecule p is parallel to the b-axis. Then <p > = 0 for g=a, or c g V,E >E = <b> = for g= b g V,E b =V, E Consequently, Eq. (4. 9) reduces to eFa < >VE iR cos (bF) Rdv ~ =< <> * <R'[cos(bF) R>. (4.11) V, E

THE UNIVERSITY OF MICHIGAN 7455-1-T In our case E is absolutely the ground electronic level E and V means predominantly the ground vibrational level: later in this investigation it is pointed out that the first and the second vibrationallevels shoul d be considered at high temperature. Since a different vibrational state means a different distribution of charges within the molecule, <P> and < > > may be all different. But V E V, E V2 E O O 1 o 2 o in this calculation we assume <> V <,> V < E> - E (4.12) oo V1 Eo V2 Eo This is not a bad approximation, as indicated by the close agreement of A, B, and C in the ground and V states. For the purpose of obtaining the numerical results required under this contract, the contribution of V2 is negligible due to its small occupation probability at our temperatures, and it is therefore not important that this approximation is made. With such assumptions, and from Eqs. (4.1) and (4.11), we have er I 2 2 " i<RI|cos(bF) R>. (4.13) Following the notation of Cross, et al, (1944) we put ~Fb = cos (bF) (4.14) Fb and write Eq. (4. 13) as er 2 = 2 <(FbB)R.R2 * (4.15) 16.......... 16 -...........

THE UNIVERSITY OF MICHIGAN -. 7455-1 -T Here A indicates that the calculation of the matrix element of Fb is carried out Fb with wave functions of an asymmetric rotor. Writing R' and R in full expression er 2 2 Fb) J'K, K A K K' M (4.16) F' -1, 1' -1, 1, This is the quantity we have to calculate, namely, the direction cosine matrices for the asymmetric rotor. In the case of symmetric rotors, corresponding matrix s ments, (Fg )R have been calculated and tabulated by Cross et al, (1944 Fg RIRs p. 212). Here s stands for the calculation performed with symmetric rotor wave functions, and R', R are rotational states of symmetric rotors, s being a s s suffix that stands for symmetric rotor. Just as before, F is either X, Y, or Z, g is either a, b, or c. Suppose RS S'' K' M' then the expression for (FO ) RT R is given in Cross, et al, (1944) as a funcg tion of J, K and M, which are in turn related to J', K' and M', by selecs A tion rules. If we could relate (F R to ( Fb) R then our problem Fg RI;, R F R', R' S S would be solved: we would be able to calculate the latter as a linear combination of the former, for example. This is done by the transformation of the matrix (- ), (P g) but not directly. The reason comes from the different symmetry property of the symmetric rotor wave function si (J,K,M ) and the asymmetric rotor wave A ss s function / (J,K_,K,M) (Cross, et al, (1944,p. 212); Allen and Cross, (1963, p. 105); King, et al, (1943, p. 32) ). We have to first form a linear combination of j to construct the "Wang function" S(J, K, M, y), to l — -.........i --- "..17.......

THE UNIVERSITY OF MICHIGAN 7455-1-T A which V (J, K, K, M) converges at a prolate or an oblate limit. In terms of such function S, the direction cosine matrices of asymmetric rotors can be correlated properly with those of the symmetric rotor as limiting cases. We denote a matrix which represents such a transformation of iS by U (Wang transformation). i/ = U S. (4.17) Furthermore we denote by T a transformation which diagonalizes the energy matrix of the asymmetric rotor after U is applied. Then the quantity of interest here, A, is written (Schiff, 1955, p. 129) Fg S Ag T(K-)U UT.(K) (118) qFg Tf f F g 1 1 ( F18) Here the asterisk designates the Hermitian adjoint of matrices, T. the transformation matrix T for the initial state with initial asymmetry Ki, and Tf for the final state. In our problem no change of asymmetry parameter takes place, so Tf = T. However, apart from our problem, if a transition occurs with a great change of energy, it may accompany a change in moment of inertia and hence a change in K.' "-.....-'-....-'-'-'" 18

THE UNIVERSITY OF MICHIGAN 7455-1-T Now by Eqs. (4.16) and (4.18) the value we need is written le la2 =M2 Z T (iK)U F b UT(K) (4.19) where F = X, Y and Z 3 = J, K' K' MT' -1' 1'I a = J,K K1' K M Cross, et al, (1944, p. 211) called this quantity 2,n,, n', 2 The value tabulated in references is related to the quantity in Eq.(4. 19), but not identical to it. The first modification is the summation of this quantity over the initial and the final quantum numbers M and M'. This is relevant because in the case of free rotation the energy levels involved are degenerate with respect to M and M'. Then by measurement we are observing the result of Eq.(4.19) summed over every possible combination of M and M'. Furthermore, this modification has another advantage, that the same tabulation could be used both for absorption and emission, as will become clear soon. The second modification is dividing Eq. 2 (4. 19)by p such that the tabulation could be used for any molecule. Hence the tabulated value is 1 e' 2 -s T U U 2 ml - {Tm FbUT / M - Oa (4.20) —.._...,.,,-. —--—. 19,..........

THE UNIVERSITY OF MICHIGAN 7455-1-T King, et al, (1947, p. 434) denoted this quantity by5 j2. In Eq. (4.20), M and M' are related by selection rules. After the summation over possible M' is carried out, L er s 2 (which is called /M2J l in Townes and Schawlow (1955, p. 97)) proves to be independent of M. Then Eq. (4.20) can be written L =[ 2= _ 2 (2J +1) er_ 2 (4.21) I-e M' 1 2 = (2J + 1) U J _I, (4.22) in which J' denotes K' and K, as J represents K and K1, and a summation -1 1' 1 over M' is understood To maintain thermal equilibrium it is necessary that (2J + 1) l er 2 = (2J' + 1) er 2 (4. 23) Ml' M Therefore the value d 2 tabulated can be used directly for both absorption and emission. A more general form of Eq. (4.20) is I — 20

THE UNIVERSITY OF MICHIGAN 7455-1-T g (a, b, or c). These values for possible transitions with various K, J, K 1' K1, Jt' K' 1 and K' are given in Townes and Schawlow ( 1955, Appendix V) or in' -1' 1 Cross, et al, (1944, p. 222). The quantity necessary for our calculation can be looked up as follows. If we are interested in the absorption, the transition is 220 -> 31. Hence AJ = + 1, which is classified as R branch. Also AK =-1 and AK = 3. Then our transition is written R1 3, the reverse process (absorption) of which is P1-3. This transition is possible only when ptfb, so in the listing it is tabulated under the entry R (Townes and Schawlow, 1955, p. 588). The tabulation is widely -1, 3 spaced in K, and the closest value of K to our K = -0.43643 is K = -0. 5, for which the value is found 2 L = (2J + 1) 0.1097 (4. 25) 2' In principle, in order to get an exact value of L for our K, we have to use Eq. (420) after calculating the elements of T (K). However, for our specific transition, i.e., for this branch with such a low value of J, the interpolation turns out to be accurate enough. This is in contrast to the 5 3 > 61 6 transition which is also b 2,3 1,6 designated by R but for which the interpolation gives a rather poor result. -1, 3 The statement has been verified by King et al, (1947, pp. 436-437). The present authors obtained by interpolation the value L (K = -0.436) = 0.1016, (Fig 4-2) whereas the exact value given by King et al, (1947) is 0. 1015. In the following calculation the latter will be used, since we feel we have confirmed it. 21

THE UNIVERSITY OF MICHIGAN z I c J molecule X Ad., \ b i FIG. 4-1: TWO CO-ORDINATE SYSTEMS USED TO DESCRIBE THE WAVE FUNCTION OF A MOLECULE |________________________ 22 ___

THE UNIVERSITY OF MICHIGAN 7455-1-T 1 2 0.40 =2 (2J+ 1) er 2 2 M' 0.35 \ 0.30 - 6 - 2 3 0.25 _ 0. 20 - I\(KH O I \,I~2 0.10 0.1016 0.05 3 2) 0\.- -13 20 -1 -0.5 0 0.5 1.0: FIG. 4-2: THE LINE STRENGTH FOR ROTATIONAL TRANSITIONS AS A FUNCTION OF K. 23 2 3

THE UNIVERSITY OF MICHIGAN 7455-1-T V CALCULATION OF EFFECTIVE ABSORPTION COEFFICIENT 5.1 Derivation of Formula If we take into account the broadening of the spectrum due to the collision of the emitting molecule (H20) with other molecules (H20, N2 and 02 in the air) the absorption coefficient at the peak of the broadened spectrum is given by King et. al., (1947) WR a(TP ) 8N(T) 2 2 kT ~12 P ( w(T 3kTQ(T)2 ()2 ge w (5.1) Av' (cm ) Primes in v and A v indicate that they are in units of cm. ForA v and v -1 expressed in sec WR 8ir N(T) 2 2 k1 j2 - (5.2) (T, Pw) 3kTv Q(T) M gT (5. 2) C _-1 (cm') Here, P: partial pressure of water molecules in units of atmosphere w N(T): number of water molecules per cm at T k, one atmosphere N(T)P: number of water molecules per cm at T k and at P w w atmosphere of water vapor. | 1f|2: Defined in (4.22). g: the factor which takescare of spin degeneracy of the initial state. For H20, (5 g= t2 - (-1)1 (5.3) Av: half width at half maximum of the absorption curve, at temper24

THE UNIVERSITY OF MICHIGAN 7455-1-T ature T and the total pressure Pt of the gas mixture. v: the frequency at the resonance, determined by W. -W. v = h L' (5.4) in our case, 184 GHz. Q(T): partition function, that is, the sum of all Boltzmann factors, including degeneracy due to quantum number M and spin. It is Q(T) = " ~ (2-(-1 ) ) ( 2J+1) exp -W(J,T)/ kT (5.5) J=0 T- = -J WR: the rotational energy of the initial state. W(J, T): energy of the rotational level with J and T To derive Eq. (5.1) and (5.2), we have to consider first of all the various sources for broadening of spectral lines. Among them are (Townes and Schawlow, 1955, p. 336). (1) natural line breadth (2) Doppler effect (3) pressure broadening (broadening due to collision) but in our case the first two are negligible (VanVleck, 1947 b,p. 416); (Townes and Schawlow, 1955, pp. 336-338) and only the third is considered, on the basis of which equation (5. 1) is derived (VanVleck and Weisskoff, 1945). After prodedures described by VanVleck and Weisskoff (1945), one obtains c _] i; eo ---- I -, (e25

THE UNIVERSITY OF MICHIGAN 7455-1 -T Where f(v..,v) = -f(v.., v) _ 1 v A Av VV-+ AV (5-7) 7r vij (v.. v)2 + AV2 (V. + v)2 +A^V2 i the final state j: the initial state N the number of atoms per cm3 W. -W. 1ij I. h v the frequency of the incident radiation: 2ir (5.8) T: the mean interval between the collisions For the simplicity of argument, let us assume that i2 is merely the matrix element squared, ignoring the degeneracy of the ith and jth level. The Eq. (5.6) gives the absorption coefficient for any incident frequency v. To calculate this value of a(v), all pairs i and j must be taken. However, when we are interested only in the peak value of a at a certain resonant frequency (vvij.), only two terms in Eq. (5.6) are important, due to the nature of the function f(vi.,v). In our case these are the terms for i =313 j = 220 and i' = 20 313 The former is absorption and the latter, induced emission. What we usually observe as absorption is the net effect of those two terms. Therefore, taking ------------------------- 26

THE UNIVERSITY OF MICHIGAN 7455-1-T the sum of two terms will give what we need. This is also in agreement with the purpose of the calculation of parameters such as "effective" absorption coefficient, which appear in radiative transfer considerations. In other words, we really want a result which corrects a for induced emission. Then we put v = vv. in Eq. (5.6) and take only the two domimant terms mentioned in the summation. We employ the property v.. = - v.. f(.., v) = -f(v.., v) 1j j1 1j l1 i j2 2 and the following terms appear: W. W. W. hv -J _1 J LJ kT kT kT kT e -e =e (1-e ) W. _2 hv.. kT where we have assumed the condition hvij/kT <<1 (5.10) By such an approximation in (5.6) one obtains an expression quite close to Eq. (5.2). The Eq. (5.2) contains additional considerations: the degeneracy of the initial and the final state with respect to M and M', and also that due to the spin of the initial state. The expression g LI( 12 takes care of all of them Moreover, the pressure dependence of N is explicity given by N(T)P For the range of validity of Eq. (5.6), one is advised to refer to VanVleck (1947 b); a brief discussion is given in chapter IX of this report. 27 _____________

THE UNIVERSITY OF MICHIGAN 7455-1-T 5.2 Calculations 5.2.1 Check of Numerical Calculations for t = 14.5 C Now we can calculate o(T). In this portion of the report we will use the same data as King et al. (1947) used and check their calculations, thus verifying our interpretations and procedures. If they do not provide the necessary data we will find it in other references and state the origin. Quantities used in Eq. (5.2) are: t= 14.5~C T = 287.7~K TN = 273.20K N NA TN N(T) = 22.4 x 103 T (5.11) 19 3 = 2.554 x 1019cm atm 23 NAV = Avogadro's number = 6.025 x 10 molecules/ ol. AV mol. -18 f = 1.84 x 10 (esu.cm) (VanVleck 1947a p. 427) 9 -1 v = 184.5 x 10 sec1 6.15 cm-1 v T for our initial state = K -K = 2 -0 = 2. As given in Eq. (4.22), et seq. 28

THE UNIVERSITY OF MICHIGAN 7455-1-T 2 = 2 (2J+1l) J 2j,_ = 0.1015 As for Q, we would like to avoid calculating Eq. (5.5) exactly, since that would be quite tedious. If we define Q'(T) which represents a similar function to Q except for the spin factor, namely Q'- " (2J+ 1) exp kMi (5.12) J= T= -J then an approximation gives ABC = A v (5. 13) (Townes and Schawlow, 1955, p. 101). This approximation is said to be good when kT T>>C A (5.14) In our case kT/hv = 200. O0 for t = 14.5 C i.e., T = 287. 7K. c a - 27.8 1 V Therefore Eq. (5. (5.14) is moderately well satisfied. Assuming that we use Eq. (5. 13), we still have to relate Q' to Q. For this purpose we consider a factor F and relate them as Q = FQ' 29

THE UNIVERSITY OF MICHIGAN 7455-1-T In other words F is an average weighting factor due to spin of the two hydrogen atoms within a water molecule. In a full expression W(J.) W(J, r) kT kT Q =T = (2J1+ 1) e (2J+) e (5.15) J, T TJ, T For H20, g = 3 when T is odd T g = 1 when T is even. As an approximation we here simply put F = 2. This is justified by the discussion in Gordon(1934, p. 72), and furthermore considered appropriate by the following reason: with the expression Eq. (5.13) and F = 2, we obtain from the previously given data Q' (t= 14.5~C) = 81.8 (5.16) Q (t = 14.5~ C) = 163.6 ( This result, which will be used in the following calculations, compares fairly well with Q (t = 20~ C) = 170 calculated numerically by VanVleck (1947a), considering that Q is an increasing function of temperature. In fact if we use the approximation Eq. (5.13) for t = 20 C, we realize the difference with the above Q = 170 is 1 percent. Av: This is given by Av = Pt'A v(T) (5.17) 30

- THE UNIVERSITY OF MICHIGAN. 7455-1-T where P is the total pressure of the gas mixture, and /Av is the value of 1t Av =2 at P 1 atmosphere at a certain temperature. Av must be 2ITT t o chosen carefully depending on the physical phase of the absorbing material(King et al., 1947, p. 435; VanVleck, 1947a, p. 428), but has been determined so far by experiments. For H20 in air Avo(T) = (0.1 + 0.02) cm per atmosphere (5.18) V c is fairly well established. King et al. (1947) used this value for 14.50C whereas VanVleck (1947a) used it for 20 C. Since its temperature dependence is King, et al., 1947, p. 435.) Avo(T) = d T1/2 (5.19) where d is a constant, this choice makes a 4 percent difference. The reason why they did not consider this variation with T seems due to the roughness in the value of Eq. (5.18). In our calculation we follow King et al., (1947). Av (T) -1 = 0.1 cm per atmosphere for 14.5 C v c Other numerical values for a checking calculation are Pt = PW. (This means that gas is pure water vapor), and W2 220 220 hv cm-h 20_ _ c c 136. 15cm-1 =kT kT = U A ~= 0.6808 kT kTcm-1 200.0-1 ~ hv c 31 _________________________

THE UNIVERSITY OF MICHIGAN 7455-1-T The energy value of King et al., (1947) mentioned in Chapter III was used. W2 220 kT e = 0.5061 kT = 3.962 x 10 erg After the substitution of all these numbers into Eq. (5.2) the value of ( 14.5 C, Pt PW ) = 6799 X 106 cm(5.20) was obtained, which differs by less than 1 percent from e( 14.5 C, Pt PW) = 6838 x 10 6 cm 1 (5.21) of King et al., (1947). A similar calculation for 52 —> 6 transition also gives a good agreement with King et al., (1947): the writer obtained 37.42 x 10 whereas their value is 37.83 x 10 cm 5.2.2. A general Expression' W' t) and a Test Calculation for T = 1000 K. p The dependence of ar on pressure comes through the factor - Av(Pt. T) in Eq. (5.2). This is written as P P W _ = f (5. 22) Av (T) * Pt v0(T) ( 22) where Pf is the fractional pressure of water vapor in the gas mixture. On the other hand the dependence on temperature is due to the quantitiesN, Q, v, and 1/kT in Eq. (5. 2). They result in the following functional form: ------------------------ - 32

THE UNIVERSITY OF MICHIGAN 7455-1 -T Where T is any reference temperature and T is the temperature where a(T) is sought as a whole factor, /TN 3 N(T) N(T) _ _N 0 (5. 23 kTQ(T)A vo(T) \ kT Q(To)AVo(T o) (T) 0 0 00 WR kT In addition, the Boltzmann factor e is temperature dependent. Another important factor about temperature dependence is that at high temperature, appreciable fraction of molecules could be in higher vibrational state. Herzberg (1945, p. 281) gives frequencies for three modes of vibration |of water molecules The energy levels can be computed in terms of the frequlencies (v1, v2, v3 ), the lowest three of which are stated to be c 2 = 155 cm-1 kT 0. at 1000 K - = 3151, e =0. 0107 at T = 1000 K v c hvl 1 3650cm-1 kT 0 - = 3650, e = 0.0052 at T = 1000 K. c

THE UNIVERSITY OF MICHIGAN 7455-1-T The Boltzmann factors calculated on the preceeding page for T = 1000 K indicated that we can not ignore the fration (11.7 percent) of molecules excited to these levels, at this temperature. In order to include such a situation in the calculation of a(T) in Eq. (5.2), we make the following assumption, although it may be crude: there is no interaction between rotational and vibrational levels, therefore the 220 313 transition at the ground vibrational level(nl=0, n2=0, n3=0) and 220 313 transition with (n l 0, n2 f 0, n3 0) give the same transitional frequency (Fig. 5-1). Here (n1, n2, n3) are the vibrational quantum numbers. This means at least that we assume rotational constants A, B, and C and the dipole moment u do not change even if the molecule is excited to higher vibrational levels. The tabulation of Herzberg (1945, p. 488) given in Chapter II of this report on rotational constants suggests A, B, and C do change due to vibration. The dependence of pI on vibrational state was discusse in Chapter IV. These contradict our above assumption, therefore our result should be used at high temperature with such a limitation of accuracy in mind. By virtue of such assumptions, the only modification that should be made in Eq. (5. 1) and (5.2) due to vibrational excitation is the form of the Baltzmann factor, but this makes no difference in our final result for the following reason. In Eq. (5.1) or (5.2) the factor WR k T g (2J+ 1) e Q(T)= f (5. 24) Q(T) is the fraction of molecules which are in the 220 rotational level. We wrote 0 2 in Eq. (5.24) the factor (2J+ 1) explicitly, which is included in/ If of the Eq. (5.2). This was correct when we completely ignored the vibrational 34.

THE UNIVERSITY OF MICHIGAN 7455-1-T excitation. As discribed in Townes and Schawlow (1955, p. 101) the fraction f must be written in general as T rot vib (5.25) where WR WR kT kT rot (2J + 1) e kT (2J + 1) e (5.26)'rot -WR, "rot a, R Q(T) i. g'(2J' + 1) e kT J' W hih d V n fvib= e kT T( e kT (5.27) and where W is the vibrational energy of the initial level, d is the degeneracy of a vibrational mode n. In H20, n = 1,2,3 and d d2 = d = 1. At low 2 1 2 3 temperature as in the previous calculation ( t = 14. 5 C) f = 1 and only the vib factor gf rotappeared in Eq. (5.2). At high temperature, if we are interested only in a particular vibrational mode f has to be calculated by Eq. (5.27) and vib substituted into Eq. (5. 25). But here we do not care which vibrational state the molecules are in, and by virtue of the assumptions previously stated, we can modify Eq. (5.25) in the following manner to include all possible vibrational states. f(T) 7g f 7 f ( T) -gfrot vib v =g frot fvib = g~rot (5.28)...... - 35

THE UNIVERSITY OF MICHIGAN 7455-1-T The notationL means the summation over various vibrational states and r ~ v. fvib must add up to unity. Then, Eq. (5.28) is exactly the same as the v vib previous one, Eq. (5.24). After such discussions we conclude that the temperature dependence of a comes about through Eq. (5. 23) and the Boltzmann factor for the rotational level. W Then 220 P P T kT P W W o3 e. W (T ) = (T - ) ( -) (5. 29) t t 2T t 20 kTo Taking the value we already calculated as a reference T = 287.7 K W = 136.15 c1 o 220 W -6 cm-1 a(T, P 1 ) = 6838 x 10 Pt W21 220 kT - 0.6808 e = e =0.5064 (8 e- 136.15 a(T, - 6838 x (5.30 ~~~~~~~P ~0.5061 P(5.30) t t (cm- ), where -1 k- = 0.6950 cm (5.31) hvc deg.............~.- ~~36

THE UNIVERSITY OF MICHIGAN 7455-1 -T 0 136.15 For T = 1000 K, = 0.1959 kT hv C - 136.15 exp 0kT ) 0.8221. hv Assuming P pW Pt P a (10000K, = 1) = 6838 x 10 x 3.865 x 102 t -6 -1 = 264.3 x 10 (cm ) (5.32) AE — --— 13 2 2-20 (nl 0, n2 1, n = 0) XI AE1 - 313 20 (n =0, n2 n = 0) FIG. 5-1: THE ASSUMPTION AE1 = A E. 37

THE UNIVERSITY OF MICHIGAN 7455-1-T VI VOLUMETRIC POWER PRODUCTION OF SPONTANEOUS EMISSION The power intensity, S, sought here is the amount of energy emitted per sec. per unit volume of water vapor pressure P. In the present calculation w we will not take into account any broadening (as will be discussed in section VII, the quantity actually required for calculation of emergent intensity is the spectral density of S. This will be obtained by simply multiplying the value obtained here for S by a factor. The result derived in this section is thus a necessary intermediate step, with line-broadening considerations to be incorperated in section VII). The transition probability for a molecule per unit time is given by quantum mechanics to be (Schiff, 1955, p. 400) 3 2 4w er 0 =.-e - * (6.1) 3 tv 3 c Then the source intensity S(T, PW) is given by S(T,PW) = 0 h G(TP) (6.2) where G(T, PW) is the number of water molecules in the initial 31 level, summed over the quantum number M, per unit volume at the temperature T and the water vapor pressure PW. It is given by T G(T P) NN T PW f (6 3) where NN = Loschmedt's number = 2.687 x 1019 molecules/cm3 at ST. P 38

THE UNIVERSITY OF MICHIGAN 7455-1-T T = 273.2~K N PW: expressed in the unit of atmospheres f: fraction of molecules in the upper state, i.e., 313state. f= _>gg fr f > 9T rot fvib ~T rot W3 313 kT g (2J + 1) e -~~! = ^ —O;!~T -- ^(6.4) Q(T) After substitution of Eq. (6.1), (6.3) and (6.4) into (6.2), we can use the relation (2 + 1) er 2 (6.5) 2 M er as before, and Eq. (6.2) becomes S(TPW) 3 N PW (66) W W 3 N 6 T W Q(T) (6.6) 3v c As an example we will calculate this quantity with the following data: T = 287. 7~K (t = 14. 5 C) P = 1 atmosphere..........L —------------- 39

THE UNIVERSITY OF MICHIGAN 7455-1-T Other values are the same as before: v' = 6.15 cm (the line frequency in wave-number units) -18,I = 1.84 x 10 1(esu cm) Q(T) = 163.6 13 142.30 kT 199. 95 e e 5 0.4908 Lh uIl2 = 0.1015 v = 2.998 x 101 cm/sec c The result is SS (T = 287. 7K, PW = 1 atm) = 5.870 x 10-4 erg/sec-cm3 (6. 7) The temperature-dependent expression will be derived next. The temperature dependence of Eq. (6. 6) comes from _ W kT e, and from Q(T) which is proportional to T/2, and T in the denominator of Eq. (6.6). Then W313 5T /2 - 13 S(T,P) =S(T,P =1) ePw (6.8) 313 kT e --------------------— 40 ------------

THE UNIVERSITY OF MICHIGAN 7455-1-T where T is any reference temperature and T is the temperatureatwhich S is sought. If we take T as 287. 70K, then with the aid of Eqs. (6. 7) and (6.8) 0 142. 30 kT /2 hvc -4 287. 7 e S(T, P) - 5.870 x 10 5/ p (6.9) 19W ~ ~T 0. 4908 (erg/ sec! cm3) ___________________________________ 41 _________________________

THE UNIVERSITY OF MICHIGAN 7455-1-T VII LINE-SHAPE AND SPECTRAL DENSITY FOR VOLUMETRIC POWER PRODUCTION The quantities calculated in this report are obtained for use in evaluating a solution to the radiative transfer equation. However, it is not really the volumetric power production which appears in that equation and its solution, but rather the spectral density of this quantity, evaluated near the line center. Therefore, we will state in this section the relation between the spectral density and volumetric power production. This relation, of course, is based on considerations of line-broadening or line-shape. It is necessary first, then, to establish what cause of line-broadening predominates, in order to deduce the line-shape associated with the volumetric power production by spontaneous emission. For this purpose, we shall assume a pressure of 1 atmosphere and temperature of 1000 K. The line-width due to pressure broadening as discussed in Eq. (5.17) through Eq. (5.19), will be roughly 1.6 GHz. On the other, the contribution from Doppler broadening is given (see Aller, 1963, p. 111) by - x 7.16 x 101 1 (7.1) 16 in which Pu is the molecular weight, or pu = 18 for H'O. The computed 2 Doppler half-width for a line at 183 GHz, at 10000K temperature, is thus Av - 1 MHz. and we conclude that pressure broadening alone need be considered for the conditions chosen and this particular line of water. The desired spectral density may then be obtained by dividing the volumetric power production due ---- ~ 42

THE UNIVERSITY OF MICHIGAN 7455-1-T to spontaneous emission Eq. (6.9) by 7TA v, as may easily be demonstrated.* Clearly, this factor is of correct dimensionality to convert from power per unit volume to power per unit frequency interval and per unit volume, and the corresponding solution for radiative intensity, which has the dimensions of power per unit volume per unit frequency interval divided by absorption coef ficient, is of dimensions power per unit area per unit frequency interval, or radiative intensity. The conversion discussed in this section is thus at least dimensionally consistent. * This proof has been given in the quarterly report ECOM-01378-6 on this contract. 43

THE UNIVERSITY OF MICHIGAN 7455-1-T vmI POPULATION OF STATES -ADDITIONAL COMMENTS As given in Eq.(6.3), the population of state can be calculated immediately if the fractional number of molecules, f, is calculated. Suppose we look for the fraction of molecules at (J, K_1,K1) rotational level and (nl, n2, n3) vibrational level. Then f is given by f gT fvib frot (8.1) W(J,K_, K1) (8.2) (2J + 1) e kT rot W WR, g' (2J' + 1) e kT W (n1 n2 n3) hvc f =e kT T (1-e kT) (8.3) vib n In the case of water molecules there is no degeneracy of vibrational mode. Vl, v2 and v3 are given by Herzberg (1945, p. 281). They are V1 cm-1 v2 cm -1 - 3650, = 1595 v v c c v3 cm-1 v- = 3756 v c From these data and Eq. (8. 3), fvib can be calculated for various temperatures. A convenient set of numbers to remember is.....L —-------- 44

THE UNIVERSITY OF MICHIGAN 7455-1-T kT = 199 95 cm-1 2 0 cm-1 at 14.5 C..99 95 ~ 200.0 at t = 14.5 C. C k c kT = 695.0 cm-1 at T = 1000~K. C As for the factor f t' the partition function in the denominator of Eq. (8.2) is given by W(J', K!, K' ); -1; 1 kT Q(T)- g7 (2J' + 1) e J'rT',,/7T fkT 3 / 7r kT 3 ^52 | 7ABCAT> = 2 LAC' (ik )3 (8.4) The energy W(J, K_ 1 K1) in the numerator of Eq. (8.2) is calculated by Eq. (3.1) or (3.8). g in Eq. (8. 1) has already been given in Eq. (5.3). Hence f in Eq. (8.1) can be completely determined. In our previous calculations of ca, the factor (2J + 1) in Eq. (8.2) was put together with 1 2 ~2 2^ e r x CNM' -fi' and appeared as 2 2 2 = -2 (2J + 1) f'er I (8.5) in Eq. (5.1) and (5.2). I||2 in Eq. (8.5) is the value looked up in a table. 45

THE UNIVERSITY OF MICHIGAN 7455-1-T IX APPROXIMA TIONS MADE AND POSSIBLE IMPROVEMENT ON THE PRESENT CALCULATIONS As pointed out already, the vibrational excitation is non-negligible at high temperatures and this is the point where the improvement should be made. The assumption that rotational constants A, B, C and the dipole moment J l stay the same in spite of vibrational excitation is apparently slightly wrong. In addition, it appears to the writer that a certain interaction between vibrational and rotational motion should exist for asymmetric rotors, just as that described in Townes and Schawlow (1955, p. 80) for symmetric rotors. Then the spectrum would be something like Fig. (9.1). Naturally, the line for the ground vibrational state would be the strongest. If we would like to obtain a for this strongest line, it can be calculated in similar manner to the previous calculations in Chapter V, with the factor fi of Eq. (8. 3) in addition. In that case all the constants such as A, B, C, and p should be all known for the ground vibrational level specifically. Another point of possible improvement is the derivation of Eq. (5.1). kT This depends on whether v..i<< is satisfied or not. since this inequality ij h was assumed in expanding the exponential. The approximation on Q(T) made by Eq. (5.13) and by putting F = 2 inEq. (515)seemsreasonableinourtemperaturerange, i.e., T greater than 100 K (Gordon, 1933; Townes and Schawlow, 1955). The last thing that should be mentioned is the centrifugal stretching of the rotor. However this should be of rather small effect since our transition takes place between small J(2 and 3), and at the same time the change in J is small. As a conclusion, unless one wishes to obtain the accuracy within several percent, the approximations mentioned above seem to have no consequence in our result, except at extremely high temperature, say T >>1000 K. One should 46

THE UNIVERSITY OF MICHIGAN 7455-1-T keep in mind that constants A, B, C and p should be examined on their accuracy to see if those refinements are worth doing. In addition the present ambignity in Av alone could nullify the meaning of the refinements, since it is of the order of 10 — 20 percent. _( (R R', V1) (R R', V2) Frequency dependence of a R,R': the initial and the final rotational levels V1 V2: the vibrational levels FIG. 9-1: ----------- ~ 47 --

THE UNIVERSITY OF MICHIGAN 7455-1-T X SUMMARY OF NUMERICAL RESULTS The quantities needed for construction of a numerical solution to the radiative transfer equation in cylindrical geometry are, as discussed in a previous report on this contract (Barasch, Chu, and LaRue, 1966), and in section VII of the present report, the spectral density S of volumetric power production originating in spontaneous emission and the effective absorption coefficient a. With these, evaluated near the line center, the specific intensity radiated near the peak frequency which would be encountered outside a gas body containing water molecules can be computed as a function of the gas temperature, partial pressure of water, and total pressure of the gas. The relations from which S and v may be evaluated have been developed earlier in this report; we collect them here for improved convenience in using them. 10.1 Expressions for Sv Combination of equation (6. 9) and the discussion of section VII determines this quantity. That is, in cgs units of erg cm sec / v d-142. 30 1 -4 (2877\5/2 exp kT/hv S (T PW) 1 5. 87 ~ 10-4 287) e c PW (10 1) S(T, Pw) = - 5.87' 1T 0. 4908 wherein T is the absolute temperature and PW the partial pressure of water vapor, in units of atmospheres. The line-width parameter Av has been defined in equations (5.17) through (5.19). The result of combining them is 288 1/2 Aiv 3(28 Pp GHz (10.2) ----------------- 48 l

THE UNIVERSITY OF MICHIGAN 7455-1-T in which P is the total gas pressure, measured in atmospheres, and T as always the absolute temperature. 10.2 The Effective Absorption Coefficient, a This has been given in equation(5. 30), which we reproduce here /__ xp kT/hv P -3 -1 287 7 3 Lx' /^ a(T, PW/P) 6.838 x 10 cm1 87 " L c Pw (10.3) 0. 5061 P ACKNOWLE DGMENTS The authors would like to express their thanks to Dr. R.K. Osborn and Dr. A. Z. Akcasu of The University of Michigan Nuclear Engineering Department for their clarifying discussions on many points in this work. 49

.. THE UNIVERSITY OF MICHIGAN 7455-1-T REFERENCES Allen, H. C. Jr., and P. C. Cross (1963), Molecular Vib-Rotors, John Wiley and Sons, Inc. New York. Aller, L. H. (1963) "Astrophysics - The atmospheres of The Sun and Stars", Ronald Press Co. New York. Barasch, M. L., C-M Chu, and J. J. LaRue, (1966) "Missile Plume Radiation Characteristics" (U), Technical Report ECOM-01378-2, January, 1966, SECRET. (Brief designation U. of M. Radiation Laboratory Report 7455-2-Q). Cross, P. C., R. M. Hainer and G. W. King (1944), "The Asymmetric Rotor, II, Calculation of Dipole Intensities and Line Classification," Journ. of Chemical Physics 12, 210-243. Gordon, A. R. (1934), "The Calculation of Thermodynamic Quantities from Spectroscopic Data for Polyatomic Molecules; the Free Energy, Entropy and Heat Capacity of Steam," Journ. of Chemical Physics 2 65-72. Herzberg, G. (1945), Infrared and Raman Spectra, D. Van Nostrand Company Inc., New York. King, G. W., R. M. Hainer and P. C. Cross (1943), "The Asymmetric Rotor, I, Calculation and Symmetry Classification of Energy Levels," Journ. of Chemical Physics 11, 27-42, King, G. W., R. M. Hainer and P. C. Cross (1947), "Expected Microwave Absorption Coefficients of Water and Related Molecules," Phys. Rev. 71 433-443. King, W. C. and W. Gordy (1954), " One-to-Two Millimeter Spectroscopy," Phys. Rev. 93, 407-412. Penner, S. S. (1959), Quantitative Molecular Spectroscopy and Gas Emissivities, Addison - Wesley Publishing Company, Inc., Reading Massachussetts. Randall, H. M., D. M. Dennison, N. Ginsburg and L. R. Weber (1937), "The Far Infrared Spectrum of Water Vapor", Phys. Rev. 52, 160-174. 50

THE UNIVERSITY OF MICHIGAN 7455-1-T Schiff, L. I. (1955), Quantum Mechanics 2nd Ed., McGraw-Hill Book Company, Inc., New York. Strandberg, M. W. P. (1954), Microwave Spectroscopy, Methuen and Co. Ltd., London. Townes, C. H. and A. L Schawlow (1955), Microwave Spectroscopy, McGrawHill Book Company, Inc., New York. VanVleck, J. H. (1947a), "The Absorption of Microwaves by Uncondesed Water Vapor," Phys. Rev. 71, 425-433. VanVleck, J. H. (1947b), " The Absorption of Microwaves by Oxygen," Phys. Rev. 71, 413-424. VanVleck, T. H. and V. R. Weisskoff (1945), "On the Shape of CollisionBroadened Lines", Revs. Mod. Phys. 17, 227-236. 51

UNIVERSITY OF MICHIGAN 3 9011111111115 03483 124111111 3 9015 03483 1241