Line Radiative Transfer

Transcription

1 ine Radiative Transfer Einstein Coeffiients We used armor's equation to estimate the spontaneous emission oeffiients A U for À reombination lines. A U is just the photon emission rate (s ) for an undisturbed atom or moleule going from an upper (U) to lower () energy state. To solve the spetral-line radiative transfer problem, we also need the stimulated emission oeffiient B U and the absorption oeffiient B U. Einstein showed that both the stimulated-emission and absorption oeffiients an be alulated from the spontaneous emission oeffiient. The Einstein oeffiients for a two-level system: A U for spontaneous emission, B U for absorption, and B U for stimulated emission. E E Consider any two energy levels U and of a quantum system suh as a single atom or moleule. The photon emitted or absorbed during a transition between the upper and lower states will have energy E E U À E and ontribute to a spetral line with frequeny 0 Eh small but finite widths, so the spetral line has some narrow line profile R 0 ( )d and onventionally normalized suh that. 0. The energy levels atually have entered on If this system is in its lower energy state, it may absorb a photon of frequeny Ù 0 and go to the upper state. The rate for this proess is proportional to the profile-weighted mean energy density Z U Ö Ñ ( )d of the radiation field, so the Einstein absorption oeffiient 0 U B U ( ) is defined to make the produt B UU Ö (7B) of 7 /08/ :26 PM

2 À equal the rate (s ) of photon absorption from the radiation field by a single atomi or moleular system in its lower energy state. Einstein's ritial insight was that there must be a third proess in addition to spo ntaneous emission and absorption. It is stimulated emission, in whih a photon of energy h 0 stimulates the system in the upper energy state to emit a seond photon with the same energy and diretion. The rate for this proess is also proportional to U Ö, so by analogy with Equation 7B the Einstein stimulated-emission oeffiient B U is defined to make the produt B UU Ö (7B2) À equal the rate (s ) of stimulated photon emission from a single quantum system in its upper energy state. Stimulated emission is sometimes alled negative absorption. It is not intuitively familiar beause negative absorption is muh weaker than ordinary absorption in room-temperature objets at visible wavelengths, but it ompetes effetively with ordinary absorption at radio wavelengths where. h ( ) Ü Suppose we have a marosopi physial system ontaining a large number of atoms or moleules in full thermodynami equilibrium (TE) with the surrounding radiation field. TE is a stationary state, so the average rates of photon reation and destrution must be equal. If the marosopi system ontains ( U; ) atoms or moleules per unit volume in the (upper, lower) energy states, then the balane of photon reation by spontaneous emission or stimulated emission and photon destrution by absorption implies UAU + B U Ö U U B U UÖ (7B3) In TE, the ratio of U to is fixed by the Boltzmann equation: U g U exp g Ô (E U À À E Õ ) g U exp À ; g where g U and g are the numbers of states with energies E U and E. The quantities g U and g are alled the statistial weights of those energy states. Examples of statistial weights inlude: () Hydrogen atoms have gn 2n 2, where n ; 2; 3; ::: is the eletroni energy level. The 2 2 number 2n is the produt of the 2 eletron spin states and n orbital angular momentum states in the n th energy level. (2) Rotating linear moleules (e.g., arbon monoxide, CO) have g 2 J +, where J 0 ; ; 2; ::: is the angular momentum quantum number. For eah J, there are 2J + possible values of the z -omponent of the angular momentum:. Jz À J; À ( J ); :::; ; 0; ; :::; (J ); J Õ 2 (3) Hydrogen atoms have two hyperfine energy levels whose differene yields the m ( MHz) HI line; and :406 : : : g U 3 g 2 of 7 /08/ :26 PM

3 The balane of photon reation and destrution in TE onnets the mean energy density of blakbody radiation to properties of the quantum system (atom or moleule): UÖ U A U BU À UBU A U ( )B U U À B U Full TE at temperature T implies both Ô UÖ g AU exp B U g U À B U Õ À and U Ö 4Ù Z B (T ) ( )d : 0 Inserting the Plank radiation law for B (T ) near 0 gives Ó Õ À 3 Ô Ò UÖ 4Ù Ù 2 exp : 2 ext we equate these two expressions for This equality holds for all temperatures implies both U Ö at the line enter frequeny: Ô À 3 Ô Õ À g AU exp BU exp : g U UÕ À B 4Ù 2 T, so Ô A U g B À U exp Õ B U g U B U 2 8Ù 3 3 Ô exp Õ À g B U g U B U (7B4) and 3 A U 8Ù B U 3 (7B5) These two equations are alled the equations of detailed balane. They relate A U, B U, and B U, so all three quantities an be omputed if only one (e.g., the spontaneous emission 3 of 7 /08/ :26 PM

4 oeffiient A U) is known. Equations 7B4 and 7B5 also prove that B U is not zero; that is, spontaneous emission must our. ote that these equations are valid for any mirosopi physial system beause they relate onstants harateristi of individual atoms or moleules for whih the marosopi statistial onepts of TE or TE are meaningless. Even though TE was used to motivate the derivation, the dependenes on temperature T and frequeny dropped out for a line at a single frequeny 0. Thus these equations are also valid for marosopi systems whether or not they are in TE or TE. [Reall the derivation of Kirhoff's law, whih also made use of full TE but whih yielded relating the emission and absorption oeffiients of any matter in TE, independent o f the atual radiation field.] Quantum Radiative Transfer Ï (T ) Ô (T ) B We an use the two equations (7B4 and 7B5) relating the three Einstein oeffiients to solve the spetral-line radiative transfer problem in terms of the spontaneous emission oeffiient A U alone. The radiative transfer equation (Eq. 2B4) is: where I is the speifi intensity, Ô is the net fration of photons absorbed (the differene between ordinary absorption and negative absorption) per unit length, and Ï is the volume emission oeffiient. (T ) di À Ô I + ; ds Three proesses must be onsidered: () absorption from the lower to upper level, (2) stimulated emission, whih we treat as negative absorption from the upper to lower level, and (3) spontaneous emission. They ontribute the three terms in the equation below: di À [ B I ( )] [À B I ( )] A ( ) Ô I ds U À U U + 4Ù U U À + 4 of 7 /08/ :26 PM

5 The net absorption oeffiient is Ô ( B B ) ( ) U À U U Equation 7B4 an be used to eliminate the stimulated emission oeffiient B ( ) U À U g g U B U to yield The emission oeffiient is A ( ) : 4Ù U U The ratio of these emission and (net) absorption oeffiients is Equation 7B5 an be used to eliminate U: Finally, Equation 7B4 an be used to eliminate both B U and B U: 3 2 g U À 2 g U In TE, Kirhoff's law independently implies U A U 4Ù B A U 3 2 U 0 4Ù B U Ò (8Ùh )B U Ó À U g À g U Ò 3 Ò 2 B U À g 2 B U U g U 2h 3 Ô Ò B (T ) exp h Ó 2 Ó À U g À g U Ó À Õ À so g U exp g U and we reover the Boltzmann distribution for TE (not just for full TE): 5 of 7 /08/ :26 PM

6 U g U exp À g (7B6) Using B ( ) U À U g g U and the assumption of TE, we an substitute BU g U BU g g U A g 3 8Ù U 3 and U g exp À g U to get the line opaity oeffiient Ô Õ 2 g U A À ( ) in TE) 2 U À exp ( (7B7) g 8Ù 0 in terms of the spontaneous emission rate A U only; the stimulated emission oeffiient B U and absorption oeffiient B U have been eliminated. The quantity Ô Õ À exp À in the line opaity equation above has two terms. The positive term () omes from absorption and the negative exponential term represents the negative opaity of stimulated emission. In the Rayleigh-Jeans limit, h 0 Ü Ô Õ À exp À Ù Ü Thus stimulated emission nearly anels absorption and signifiantly redues the net line opaity. Sine Ô / T À, B / T 0. The brightness of an optially thin ( Ü Ü ) radio emission line may be proportional to the olumn density of emitting gas but nearly independent of its temperature. Even if a marosopi system is not in TE, we an define its exitation temperature T x by 6 of 7 /08/ :26 PM

7 U g Ñ U exp À g x (7B8) If for some reason the upper level is overpopulated; that is U g > U g ; then T x is atually negative, Ô Õ À exp À x is negative, and Equation 7B7 gives a negative net opaity oeffiient. egative opaity implies brightness gain instead of loss. At radio wavelengths this is alled maser (mirowave amplifiation by stimulated emission of radiation) amplifiation. Astrophysial masers are ommon at radio frequenies beause h Ü. They an have brightness temperatures as 5 high as K, muh higher than the kineti temperature of the masing gas. 0 Ô 7 of 7 /08/ :26 PM