Maser Radiation in an Astrophysical Context (Overview)
MMaser Radiation in an Astrophysical Context
Eric S. Gentry
MIT Department of Physics, 77 Massachusetts Ave., Cambridge, MA 02139-4307 (Dated: May 6, 2013)In this paper we will look at the phenomenon of Microwave Amplification by Stimulated Emission ofRadiation (a maser system). We begin by deriving amplification by stimulated emission using time-dependent perturbation theory, in which the perturbation provided by external radiation. Whenthis perturbation is applied to an ensemble of particles exhibiting a population inversion, the resultis stimulated microwave radiation. We will explore both unsaturated and saturated masers andcompare their properties. By understanding their gain, as well as the effect of line broadening ,astronomers are to identify astrophysical masers. By studying such masers, we gain new insightinto poorly understood physical environments, particularly those around young and old stars, andcompact stellar bodies.
I. INTRODUCTION
Typically when electromagnetic radiation passesthrough a medium, the intensity is attenuated as pho-tons are absorbed. Certain systems do the opposite how-ever; rather than attenuating a signal they amplify it atselect frequencies. Such a phenomenon is called a laser ,Light Amplification by Stimulated Emission of Radiation(lasers in the microwave band are a special case, called masers ). These systems are uncommon; they require en-ergy levels to be populated in ways not possible under theconditions of thermodynamic equilibrium (they require a population inversion of energy levels; see Figure 1 for apictorial explanation of population inversions and the re-lationship between amplification and absorption). Some-thing must be forcing these systems away from equilib-rium. In studying those forcing mechanisms, we benefitfrom the amplifying properties of lasers. While particlesin thermal equilibrium typically absorb light, systemswith population inversions can become shining beacons.For this paper we will look at their application to astro-physical systems — systems which we have no methodto directly probe, for which we must rely on observa-tion, inference, and theory. Astronomers typically study masers , microwave lasers, since microwave signals tendsto penetrate well through atmospheric gases on earth andinterstellar dust in space [1].In this paper, Section II will examine why masers areuncommon physical phenomenae, and why it might be in-teresting to study the physical conditions to which theyare sensitive. Section III looks at the theory of stimu-lated emission, and will identify those conditions to whichmasers are sensitive. That knowledge guides both oursearch for astrophysical masers, and our ability to inferphysical conditions of distant environments. To conclude,several topics of current research will be discussed in Sec-tion IV.
FIG. 1: Absorption occurs as one photon on Side A is ab-sorbed to raise a particle to the higher energy level. Masersuse population inversions and stimulating radiation to inducethe reverse process — an excited particle on Side B dropsto a lower energy level, producing a photon in the process.Diagram produced by Elitzur [2].
II. THERMODYNAMIC EQUILIBRIUM; WHYARE MASERS UNCOMMON?
Population inversions are necessary for amplificationby stimulated emission (discussed more fully in SectionIII 2).
Population inversions are when a higher energylevel is populated with more states than a lower energylevel. The reason why this does not typically occur, andwhy masers are so special, is because the canonical en-semble of thermodynamics predicts energy level popula-tion ratios: N N (cid:20) − E − E k B T (cid:21) (1)For a system with a ground state | (cid:105) of energy E and excited state | (cid:105) of energy E , we expect N < N .A population inverse is the exact opposite; it requires N > N which cannot be predicted by equilibrium ther-modynamics. To leave equilibrium, these systems require pumping , the addition of states to the higher energy levelfaster than quenching processes can restore equilibrium. a r X i v : . [ a s t r o - ph . GA ] D ec aser Radiation in an Astrophysical Context colli-sional quenching which increases with density and pres-sure [3]. III. PRODUCTION OF MASERS
The theory underlying stimulated emission is first-order time-dependent perturbation theory. Gases even assimple can have intrinsic electric dipoles, and if our gasparticle has an intrinsic electric dipole, then an externalelectromagnetic field will produce a time-dependent per-turbation to the system’s energy. Such a time-dependentperturbation is particularly useful, because it allows fortransitions between pure energy eigenstates. If this tran-sition is to a lower energy level, a photon will be emit-ted with a frequency determined by the energy differencebetween the initial and final states. In Section III 1 wewill use time-dependent perturbation theory to derive theexpected transition rate, and its dependence on externalfactors (such as applied field strength) and intrinsic prop-erties (such as permanent electric dipole moment of theparticle). In Section III 2 we look at how those transi-tions manifest themselves as amplification of an incidentsignal.
1. Transition Rates
For simplicity we will deal with 2 quantum states ofa gas (either atomic or molecular). We will label theseorthonormal states | (cid:105) and | (cid:105) with non-degenerate en-ergies E and E , such that ω ≡ | E − E | ¯ h (cid:54) = 0. Thoseenergies are determined by a Hamiltonian, H which caninclude rotational, vibrational, coulombic, and spin com-ponents of the system’s energy. The exact form of theHamiltonian is not necessary (and indeed can be incred-ibly complex); we only need an approximate spectrum ofenergy levels and states.In the case of stimulated emission, a traveling electro-magnetic wave produces a time-dependent perturbation, H (cid:48) , to the Hamiltonian. This time-dependent perturba-tion will typically be dominated by electric dipole interac-tions, rather than magnetic or higher multipole moments[4]. For a linearly polarized monochromatic plane wave,traveling in + (cid:98) e x , we expect fields of the form: E = E sin( ωt − kx ) (cid:98) e y (2) B = E c sin( ωt − kx ) (cid:98) e z (3)Which gives an electric dipole perturbation to theHamiltonian: H (cid:48) = qyE sin( ωt − kx ) (4)Ideally, we would like to factorize the perturbationinto a time-dependent term and a space-dependent term. This will allow us to deal with the spatial dependenceseparately from the periodic time dependence, which isa well-studied topic. While sin( ωt − kx ) is not directlyfactorizable, the gas particle can only feel spatial vari-ations in the electric field on the order of the particlediameter (picometers) whereas the electric field varies onmuch larger spatial scales (centimeters in the case of mi-crowaves). That difference in scales allows us to approx-imate the fields as being spatially independent (as far asthe particles can tell): E = E sin( ωt ) (cid:98) e y (5) B = E c sin( ωt ) (cid:98) e z (6)Which yields a perturbation which we can factor intoits spatial term, V ( r ), and its sinusoidal time depen-dence, sin( ωt ) H (cid:48) = qyE sin( ωt ) H (cid:48) = V ( r ) sin( ωt ) (7)With: V ( r ) = qyE = p ( r ) E (8)Where p encodes the intrinsic dipole moment.As desired, this time-dependent perturbation exhibitssinusoidal time-dependence. Transitions induced by asinusoidal time-dependent perturbation are well studied[5], but before fully solving the system, it is useful to qual-itatively understand the system. To do so, we can findthe transition probability between states to first-orderusing off-diagonal elements of the perturbation followingFermi’s Golden Rule: T → ∝ |(cid:104) | H (cid:48) | (cid:105)| ≡ |H (cid:48) | = |H (cid:48) | (9)By the hermeticity of H (cid:48) , we find that for a singleparticle, the | (cid:105) → | (cid:105) transition is just as likely as the | (cid:105) → | (cid:105) transition. The net transition of a system (a netabsorption or emission of photons) is only determined bythe initial population of each state, N and N . It is ulti-mately this fact that explains why population inversionsare so crucial to amplification by stimulated emission;in solving the system fully, we only learn how much thesignal is amplified, whereas energy level populations de-termine whether we observe net absorption or emission.But the strength of amplification is important, as it al-lows us to infer many details about the system, so wewill continue to fully solve for the complete transitionprobability.Our perturbing Hamiltonian has already been factoredinto a form: H (cid:48) = V ( r ) cos ( ωt ), and we are focusing on2 energy levels with an energy level gap of ¯ hω . If theperturbation is sufficiently small ( | V | (cid:28) ¯ h ( ω − ω ) ),then we can solve for the transition probability to first-order: aser Radiation in an Astrophysical Context P → ≈ | V | ¯ h sin (cid:2) ( ω − ω ) t (cid:3) ( ω − ω ) = | p E | ¯ h sin (cid:2) ( ω − ω ) t (cid:3) ( ω − ω ) (10)We can simplify the expression slightly by introductionthe notation sinc( x ) ≡ sin( x ) x . For monochromatic light,this leaves us with the transition probability: P → = | p E | ¯ h sinc (cid:20) ( ω − ω ) t (cid:21) t (cid:99) e n , averaged over allpossible directions: | p | = |(cid:104) | qy | (cid:105)| → (cid:104)|(cid:104) | q r | (cid:105) · (cid:99) e n | (cid:105) (cid:99) e n (12) | p | → | p | E is proportional tothe monochromatic field energy density, u = (cid:15) E , wecan rewrite Equation 11 as: P → = 2 | p | u (cid:15) ¯ h sinc (cid:20) ( ω − ω ) t (cid:21) t u → ρ ( ω ) dω , we now have a generalization for stimulatedemission by polychromatic light: P → = 2 | p | (cid:15) ¯ h (cid:90) ∞ ρ ( ω ) sinc (cid:20) ( ω − ω ) t (cid:21) t dω (15)Our work is not yet complete; we can do better thanjust leaving ourselves with an integral over an unknowndistribution ρ ( ω ). Noticing that the sinc gain profile willbe become more sharply peaked at ω as time passes, we can assume that after a long time has passed, sinc willbe sensitive to ρ ( ω ) only at ω = ω (the sinc term willeffectively have become proportional to a delta functionin frequency-space). This allows us to pull ρ ( ω ) out ofthe integral: P → = 2 | p | (cid:15) ¯ h ρ ( ω ) (cid:90) ∞ sinc (cid:20) ( ω − ω ) t (cid:21) t dω (16)We can now non-dimensionalize it, using µ = ( ω − ω ) t : P → = 2 | p | (cid:15) ¯ h ρ ( ω ) t (cid:90) ∞ sinc (cid:20) ( ω − ω ) t (cid:21) d (cid:18) ωt (cid:19) = | p | (cid:15) ¯ h ρ ( ω ) t (cid:90) ∞− ω t/ sinc [ µ ] dµ (17)But we are still left with a definite integral dependenton physical conditions (namely the energy gap, ¯ hω ), andwe want a closed-form algebraic expression for the genericcase. Note the limits of integration are far from the centerof the sinc profile, and after a long time has passed con-tributions far from the center will be negligibility small.This safely allows us to extend the limits of integrationto ω ∈ ( −∞ , ∞ ): P → = | p | (cid:15) ¯ h ρ ( ω ) t (cid:90) ∞−∞ sinc [ µ ] dµ = | p | (cid:15) ¯ h ρ ( ω ) tπ (18)One concern at this point is that we waited an arbitrar-ily long time in order to allow the sinc profile to becomesharply peaked around ω , but now we see that proba-bility grows linearly with time. This seems to suggestthat the probability is unbounded, allowing a transitionprobability greater than 1 if we only waited long enough.This is just an artifact of the first-order approximation.For a truly small perturbation, we require: | p | (cid:15) ¯ h ρ ( ω ) t (cid:28) | V | (cid:28) ¯ h ( ω − ω ) but the full derivationis inconsequential to astrophysical masers. In future sec-tions we will not be interested the evolution of a singlestate over a long period of time, but rather the averagelifetime of an ensemble of states)One benefit of the linear growth of P → is that itsuggest a probabilistic transition rate : R → ≡ ∂P ( t ) ∂t = π | p | (cid:15) ¯ h ρ ( ω ) (20)Two things should be noticed about Equation 20: aser Radiation in an Astrophysical Context R → = R → , as anticipated by Equation 9. Thismeans stimulated emission of a single excited par-ticle is equally as likely as absorption by a singleunexcited particle.2. This is a first-order approximation for small per-turbations (expansion criterion given by Equation19). While initially the intensity of light, ρ ( ω ),might be small, it will amplify the very frequencyto which it is most sensitive. In Section III 2, we’llsee exactly how this can lead to amplification, andhow amplification can lead to a regime of satura-tion, in which ρ ( ω ) is no longer “small.”3. R → appears to be sensitive to ρ ( ω ) only at res-onance, ω = ω . The derivation thus far predictsonly delta-function sensitivity in the maser and acorresponding delta-function emission profile. Sec-tion III A will discuss the non-zero widths in thesensitivity and absorption profiles.
2. Amplification
Our work does not stop with the single particle tran-sition rate (Eq. 20). We want to understand the behav-ior of an ensemble of states | (cid:105) and | (cid:105) , rather than theevolution of a single particle. Under constant irradiation,Equation 20 predicts a single particle will transition back-and-forth between energy levels. We want steady-stateamplification, which we only observe with population in-versions. For this section, we will adopt the traditionalEinstein notation for stimulated emission and follow thework of Goldreich and Kwan [3] in solving for how masersamplify signals over distances.First it is beneficial to define some notation. We willfirst switch to using variables and units more familiar toastronomers. For frequency we will use ν = ω π , and in-stead of energy density per unit frequency, ρ ( ω ), we usespecific intensity, I ν (specific intensity and energy den-sity are related by: ρ ( ν ) = c (cid:82) I ν ( ˆΩ) d Ω). In the partic-ular application of masers, the radiation will be a highlybeamed across a small solid angle — we will approximateit as a beam of constant intensity over a solid angle, ∆Ω.This leaves a mean intensity J ν = I ν ∆Ω4 π . (A more pre-cise treatment of beam shape is possible, but does notprovide any new insight.)Our ensemble of particles will be built entirely of statesin 2 energy levels, E and E such that E < E . Thesestates have corresponding number densities, n , n .There is a chance that an excited state, | (cid:105) , will spon-taneously decay (in the absence of external radiation) tothe lower energy state, | (cid:105) , releasing a photon in the pro-cess. This rate is proportional to the number density ofparticles in state | (cid:105) , with a proportionality constant of A . The rate of spontaneous decay is therefore: A n .There is also the possibility of stimulated transitions.These are the transitions derived in Section III 1, cul-minating with the rate equation (Eq. 20). Whereas R → contained both intrinsic information on the quan-tum states | (cid:105) and | (cid:105) as well as information about theexternal radiation, ρ ( ω ), we can factor out the intrin-sic properties ( B ) from the external conditions ( J ν ), sothat transitions occur at rates of B n J ν and B n J ν for absorption and stimulated emission respectively. Ex-plicitly, this factorization is given by: B = R → ρ ( ω ) = π | p | (cid:15) ¯ h (21)Just as R → = R → , we now find B = B .In the literature of quantum absorption and emission, A and B are commonly known as the Einstein coef-ficients [5]. From Equation 21 we know the form of B ;knowing that they are Einstein coefficients then implies[5]: A = 8 πhν c B (22)While B corresponds to transitions stimulated by ex-ternal radiation, A is independent of external radia-tion. A , the spontaneous emission coefficient, can beinterpreted as transitions stimulated by QED backgroundfluctuations [5], which have an effective intensity: πhν c .At low frequencies (such as microwave frequencies) thefactor of ν will suppress spontaneous emission, and wewill neglect it in our calculations.So far we have treated the system as a closed system(we have only considered transitions directly between | (cid:105) and | (cid:105) ). In order to provide steady-state populationinversions we need to generalize to an open system, onewhich allows particles to enter and leave the system. Wedefine λ i , the pumping rate , for the rate of addition ofparticles of state | i (cid:105) . Likewise, Γ n i , the quenching rate ,will describe the removal of particles of state | i (cid:105) .(For conditions common to astrophysical masers, Γ (cid:29) A [3]. Any excited state which does not undergo stim-ulated emission will most likely be quenched long beforeit undergoes spontaneous emission. This is another indi-cator that we can safely neglect A . Whenever an equa-tion is introduced in the following derivation, we will firstshow how to properly include A and then drop it fromthe subsequent algebra.)We now have our model for how the populations of ourtwo state system evolves: dn dt = λ − Γ n − B J ν ( n − n ) + A n (23) dn dt = λ − Γ n − B J ν ( n − n ) − A n (24)We are interested with the transitions, only insofar asthey give us information about amplification of the stim-ulating radiation. For that, we build an equation de-scribing the radiative transport for plane-wave radiationtraveling in (cid:98) e x : aser Radiation in an Astrophysical Context FIG. 2: Optical depth, τ , follows the convention τ = κx . Notethe exponential growth until τ ≈ I ν /I s . dI ν ( x ) dx = hc π ([ n − n ] B I ν ( x ) + A n ) (25)For a steady state amplification ( dn i dt = 0) we can com-bine Equations 23, 24, and 25 (dropping A ): dI ν ( x ) dx = hc π [ λ − λ ][Γ + 2 B J ν ( x )] B I ν ( x ) (26)Equation 26 contains all the information we need aboutthe differential amplification, but it is rather unwieldy inits current form. In order to simplify Equation 26, wedefine an apparent opacity, κ ν , and saturation intensity, I s [3, 6]: κ ν ≡ hc π [ λ − λ ] B Γ (27) I s ≡ π Γ∆Ω B (28)Definitions 27 and 28 significantly simplify Equation26 to the form: dI ν dx = κ ν I ν I ν /I s (29)Which can be integrated to find: I ν ( x ) = I ν (0) exp (cid:20) κ ν x − I ν ( x ) − I ν (0) I s (cid:21) (30)Equation 30 has two asymptotic behaviors (shown inFigure 2): 1. When the intensity is far below saturation of themedium ( I ν (cid:28) I s ) we observe exponential ampli-fication (as a function of distance) of the originalsignal. This regime will be the case more importantto us in the coming sections because small changesin path length or opacity, κ , will produce significantchanges in observed signals.2. Close to, and above saturation ( I ν > ∼ I s ), the am-plification slows down, growing linearly with dis-tance. At this point the intensity becomes so strongthat every particle which enters the system will un-dergo a stimulated transition (now both quenchingand spontaneous emission are negligible). Takingboth stimulated emission and absorption into ac-count, we find a net emission rate of λ − λ whichhas decoupled from the intensity. This shows thatthe saturation intensity, I s , is not the maximumintensity, but rather the point at which decouplingfrom intensity occurs.(If λ > λ , then κ ν <
0, indicating absorptive condi-tions. This also is equivalent to our statement that pop-ulation inversions are required for net absorption. In ourcurrent model of pumping and quenching, if λ > λ ,then we can never observe a population inversion. If λ < λ then we must observe a population inversion.For the net absorptive regime, Equation 30 reproducesthe Lambert-Beer Law: I ν ( x ) = I ν e −| κ ν | x , which de-scribes exponential absorption at thermodynamic equi-librium.)We have now found that under the conditions of a pop-ulation inversion, there can be exponential amplificationof faint signals. This exponential amplification is sen-sitive to the total path length, the pumping rates ( λ i ),the quenching rate (Γ), and the intrinsic electric dipolemoment matrix element ( p ; implicit in B , Eq. 21).Those sensitivities can allow us to trace small changesin amplifying systems, which we can use to probe thedetails of astrophysical systems. A. Line Narrowing
In Section III 1 we approximated the sensitivity andemission profiles of the transitions to be delta functions,centered at the energy of the transition between states.In Section III 2, we saw amplification precisely at I ν and nowhere else. In reality, we expect both the sen-sitivity and the emission to have some non-zero width.In this section we will take into account those non-zerowidths. We will see line narrowing at low saturations,and rebroadening of spectral lines at large saturations.Experimentally, line narrowing and subsequent rebroad-ening is an important feature — it can help us identifyamplifying systems due to unexpectedly narrow spectrallines. If not accounted for, one would infer unphysicallow surface temperatures of emitting bodies due to thenarrow line width. aser Radiation in an Astrophysical Context δv = (cid:113) k B Tm : v = v exp (cid:20) − v δv (cid:21) (31)(We assume there is still a pumping mechanism push-ing the energy level populations away from equilibrium,but that the pumping mechanism supplies particles witha thermal distribution of velocities.)If each particle was emitting at the resonant frequency, ν in its rest frame, we would expect the spectrum tobe Doppler-broadened when viewed in the center-of-massframe. This Doppler-broadening comes from redshifts atlow velocites: vc = ν − ν ν . Using that conversion we canfind the spectrum of photon frequencies seen by the gasparticles, and a characteristic Doppler-broadened width: I ν = I exp (cid:20) − ( ν − ν ) δν D (cid:21) (32) δν D = ν δvc (33)Since amplification by stimulated emission is sensitiveto precisely the same frequencies which are emitted, wecan infer that the opacities, κ ν , and optical depths τ ν = κ ν x follow the same profile as Equation 32: κ ν = κ exp (cid:20) − ( ν − ν ) δν D (cid:21) (34) τ ν = κ ν x = τ exp (cid:20) − ( ν − ν ) δν D (cid:21) (35)This τ ν gives the extent of exponential amplificationas a function of frequency, but we don’t yet have a spec-trum to amplify. A reasonable assumption for astrophys-ical masers is that any preexisting spectral profiles werecaused by Doppler-broadened spontaneous emission witha width of δν [7, 8]: I ν (0) = I exp (cid:20) − ( ν − ν ) δν (cid:21) (36)Which at low saturations is amplified to: I ν = I exp (cid:34) − ( ν − ν ) δν + τ exp (cid:34) − (cid:18) ν − ν δν D (cid:19) (cid:35)(cid:35) ≈ I exp [ τ ] exp (cid:34) − (cid:18) ν − ν δν (cid:19) (cid:18) δν δν D + τ (cid:19)(cid:35) (37) FIG. 3: Simulated line narrowing of a Gaussian Dopper-broadened spectrum at low saturation.
Which gives us an effective line width: δν = δν (cid:16) δν δν D + τ (cid:17) / (38)Even if the amplifying gas has the same tempera-ture (and thus the same velocity dispersion) as the ra-diation source gas (so δν δν D = 1) we would still ob-serve line narrowing! This line narrowing, of the formFWHM ∼ δν ∝ τ ) / , can be seen in Figure 3. Suchline narrowing has been clearly seen narrowing spectralfeatures by at least a factor of 2-3 [9]. Without a popu-lation inversion we cannot explain one gas of thermallydistributed velocities narrowing the radiation given off byanother cloud of gas with the exact same thermal velocitydispersion.As masers begin to saturate though, the intensity atresonant frequency begins to grow linearly with depth,while the wings continue to grow exponentially. At thispoint line narrowing ceases, and the wings begin to growrelative to the peak, resulting in rebroadening of the spec-tral line.It can be difficult to know whether a spectral line wassimply initially narrow or has been narrowed through am-plification by stimulated emission. Looking at the widthsof multiple lines can help identify lines which have beennarrowed while the rest have remained untouched. An-other approach is using the observed width to infer asurface temperature of the radiation, and compare thatto temperatures predicted by other measurements [9].A more complete treatment of line narrowing requiresmore complete models of opacity, source radiation spec-trum, velocity dispersion and geometry of the medium[9–11]. More complicate spectra begin to lose their Gaus-sian shape and even the symmetry in their tails. Sucheffects cannot be explained simply by differences in thesource emission. This further enables us to identify stim-ulated amplification; by comparing relative line widths, aser Radiation in an Astrophysical Context IV. RESEARCH APPLICATIONS ANDCONCLUSION
In Section III 2 we found that small variations of thelocal conditions along the path of light could cause ra-diation to be very path-dependent. This phenomenonis called anisotropic beaming . By itself, it is difficult todistinguish between anisotropic beaming of amplified ra-diation and radiation which was naturally collimated byanother process. Amplification by stimulated emissionalso has a special property of line narrowing : amplifiedspectral lines have a much narrower width than their ini-tial Doppler-broadened line shape.Anisotropic beaming and line narrowing allow us torecognize when amplification by stimulated emission istaking place. The extent of these effects gives informa-tion about the physical environment. This allows us avaluable look into poorly understood systems.One particular application is the magnetic field config-uration. These magnetic fields, though Zeeman, param- agnetic and other effects, can induce energy level splittingwhile also making the amplifying system sensitive to par-ticular polarizations of stimulating radiation [13]. Thestudy of magnetic field lines and polarization in masershas been applied to both early- and late-type stellar ob-jects, offering a valuable look into objects which are less-understood than more typical Main Sequence stars, suchas the Sun [14]. Objects being studied include regions ofrapid star formation, planetary nebulae, stellar envelopsof asymptotic giant branch stars, supernovae remnants,and millisecond pulsars [13, 15].By understanding and studying stimulated microwaveemissions, we gain new insight into physical environ-ments, especially around young and old stars and com-pact stellar bodies.
Acknowledgments
I would like to thank Ethan Stanley Dyer and SophieWeber for their guidance and feedback during the courseof writing this paper. I would also like to thank Pro-fessor Thaler for his clarifications of technical aspects ofunderstanding amplification by stimulated emission. [1] I. Shklovsky,
Cosmic Radio Waves (Harvard UniversityPress, 1960).[2] M. Elitzur, , 71 (2005), URL .[3] P. Goldreich and J. Kwan, Astrophys. J. , 27 (1974).[4] A. Siegman,
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Introduction to Quantum Mechanics (Prentice-Hall, 1995), ISBN 0-13-124405-1.[6] M. M. Litvak, Phys. Rev. A , 937 (1970).[7] M. M. Litvak, Astrophys. J. , 711 (1973).[8] P. Goldreich and D. A. Keeley, Astrophys. J. , 517(1972).[9] M. Elitzur, Astronomical Masers (Springer, 1992),ISBN 978-94-011-2394-5, URL http://books.google.com/books?id=kajIH3t_750C&lpg=PA111&ots=d9_vRIbEm6&dq=line%20narrowing%20masers&pg=PA106 .[10] A. M. Archibald,
Astrophysical masers , RadiativeProcesses in Astrophysics Term Paper (2006), McGillUniversity, URL .[11] E. Lekht, N. Silantev, J. Mendoza-Torres, and A. Tol-machev, Astronomy Letters , 89 (2002), ISSN 1063-7737, URL http://dx.doi.org/10.1134/1.1448845 .[12] J. M. Weisberg, S. Johnston, B. Koribalski, and S. Stan- imirovi, , 106 (2005), URL .[13] W. H. T. Vlemmings, in IAU Symposium , edited by J. M.Chapman and W. A. Baan (2007), vol. 242 of
IAU Sym-posium , pp. 37–46, 0705.0885.[14] M. Elitzur,
Masers, interstellar and circumstellar,theory , URL http://ned.ipac.caltech.edu/level5/ESSAYS/Elitzur/elitzur.html .[15] P´erez-S´anchez, A. F. and Vlemmings, W. H. T., A&A , A15 (2013), URL http://dx.doi.org/10.1051/0004-6361/201220735 .[16] This derivation followed the standard stimulated emis-sion conventions [5] in assuming an isotropic distributionof photon polarizations. This assumption is rarely truefor astrophysical masers, which typically have very pro-nounced anisotropies, with beams of radiation over verysmall solid angles; see Section III 2 below. It is possibleto make a better estimation, but we only find constantfactors of order unity, which do not provide any more in-sight. If you are interested in making these geometric cor-rections, the substitution is still given by Substitution 12,but for some anisotropic distribution of polarizations. Forinstance, if integrated over only 1 polar angle (i.e. a circu-larly polarized plane wave) we find | p | → sin θ | p | where θ is the angle between the dipole moment matrixelement, p21