Lesson5 Einstein coefficient & HI line Kagoshima Univ./ Ehime Univ. Galactic radio astronomy Lesson5 Einstein coefficient & HI line Toshihiro Handa Dept. of Phys. & Astron., Kagoshima University

Energy level Schrödinger equation of an electron Steady state = energy eigen value Parameter separation Schrödinger equation for steady state

Energy eigenvalue & bound states Solution depends on the boundary condition Continuous E is possible if E>0. Discreet values if E<0 (bound state) Continuous solution in quantum mechanics! In many cases, we consider bound states. Discreet eigenvalues and eigenfunctions

Electron in an atom/molecule Bound state → discreet energy levels Energy state

Energy transition and matter Electron transition between energy levels Emission & absorp of EM wave DE=hn Structure of energy levels=matter identify Wavelength of emission & absorption lines Matter identification with spectrum

Einstein coefficient(1) Emission & absorption: 2 level model Transition probability of emission A Independent of input intensity Transition probability of absorption B Proportional to input intensity dI = n2 A-n1 B I

Einstein coefficiant(2) Steady state dI=0 dI = n2 A-n1 B I → n2 A = n1 B I I =(n2/n1)(A/B) Thermal equilibrium b/w matter & radiation Therm. eq.→energy is in Boltzmann distribution n2/n1=exp(-DE/kT)=exp(-hn/kT) Therm. eq.→I must be blackbody radiation I= Bn (T)=(2hn 3/c2)(exp(hn/kT)-1)-1 Impossible to become a blackbody!?

Stimulated emission(1) Add another process Difficult to see the existence Introduction of “stimulated” emission spontaneous emission A21, absorption prob. B12 stimulated emission B21 Emission proportional to input intensity! dI = n2 A21-n1 B12 I+n2 B21 I = n2 A21-(n1B12-n2B21) I Seems to reduce the effective absorption coeff.

Stimulated emission(2) Steady state dI=0 n2 A21=(n1B12-n2B21) I I =A21/[(n1/n2) B12-B21] Therm. eq.→energy is in Boltzmann distribution n2/n1=exp(-DE/kT)=exp(-hn/kT) (Right hand)=(A21/B21)[exp(-hn/kT)(B12/B21)-1] Therm. eq.→I must be blackbody radiation I= Bn (T)=(2hn 3/c2)[exp(hn/kT)-1]-1 We got that B12/B21=1, A21/B21= 2hn 3/c2

Relation between Einstein coff. B12/B21=1, A21/B21= 2hn 3/c2 In the case with statistical weight g1, g2 Boltzmann dist.　n2/n1=(g2/g1)exp(-hn/kT) g1B12=g2B21, A21=(2hn 3/c2) B21 A21, B12, B21 are fixed for matter. Relation is valid if thermal non-equilibrium A21, B12, B21 : Einstein coefficients

Maser(1) If stimulated emission truly exist, then dI = n2 A21-(n1B12-n2B21) I Effective absorption increases. We cannot know it? What happens, if n2>n1(B12/B21)? n2/n1=exp(-DE/kT). Therefore, it means T<0. Negative temperature i.e. inverse population Stimulated emission can arise!

Maser(2) We can make T<0 for 3 level system pomping inverse population seed photon maser

Maser(3) MASER LASER Characteristics of stimulated photon First detected molecule with 3 atoms in space MASER Microwave Amplification by Stimulated Emission of Radiation Developed by Towns in 1954 He found ammonia in space, too. LASER Microwave →Light Characteristics of stimulated photon Same freq., phase, polarization as the seed photon

Excitation temperature In general, emission is I=(2hn 3/c2) [(n1/n2) -1] Thermal non-equilib.: n1/n2 is not Boltzmann But convenient expressed by “temperature” Excitation temperature Tex Define as (g2/g1)exp(-DE/kTex)= n2/n1 Tex=-DE/{k ln[(g1/g2)(n2/n1)]}

Emissivity Describe emissivity e by Einstein coeff. For isotropic radiation… Radiation energy dEn for dV dt dW dEn =hn j(n) n2 A21 dV dt dW/(4p) = (hn )/(4p) j(n) n2 A21 dS dx dt dW In the case of radiation only dIn　=dEn /(dt dS dW) =en dx It gives en = (hn )/(4p) j(n) n2 A21

Absorption coefficient(1) abs. coeff. k described with Einstein coeff. For isotropic absorption… Radiation energy dEn into dV dt dW dEn = -hn j(n) (n1 B12 -n2 B21) In dV dt dW/(4p) = -(hn )/(4p) j(n) ) (n1 B12 -n2 B21) In dS dx dt dW In the case that absorption only dIn　=-dEn /(dt dS dW) =-kn In dx It gives kn = (hn )/(4p) j(n) (n1 B12-n2 B21)

Absorption coefficient(2) (continued) kn = (hn )/(4p) j(n) (n1 B12-n2 B21) = (hn )/(4p) j(n) n1 B12[1-(g1n2)/(g2n1)] = (hn )/(4p) j(n) n1 B21(B12/B21-n2/n1) = c2/(8pn 2) j(n) n1A21 (g2/g1)[1-(g1n2)/(g2n1)] = c2/(8pn 2) j(n) n1A21 (g2/g1)[1-exp(-hn)/(kTex)]

Source function Source function Sn=en /kn Sn =(n2 A21 )/(n1B12-n2B21) =(2hn 3/c2)/[(g2n1)/(g1n2)-1] =(2hn 3/c2)/{exp[(hn)/(kTex)]-1} In therm. eq. Tex=T　(temperature in eq.) LTE: Local Thermal Equilibrium Tex’s are the same between all levels.

Neutral atomic hydrogen proton + electron Proton is a particle with spin 1/2  2 values electron is a particle with spin 1/2  2 values spin of a particle=should be related with mangetizm Interaction between two spins A10=2.86888×10-15 [s-1], n =1.420405751786[GHz]

kn = c2/(8pn 2) j(n) n0A10 (g1/g0){1-exp[(-hn)/(kTex)]} HI emission(1) A10=2.86888×10-15 [s-1] Enough slow transition to excite under ISM density Show maser if poping  hydrogen maser clock Absorption coefficient kn = c2/(8pn 2) j(n) n0A10 (g1/g0){1-exp[(-hn)/(kTex)]} g0=1←no degenerate, 　 g1=3←F=+1,0,-1 degen. nH=n0+n1= n0 {1+(g1/g0) exp(-hn/kTs)} For HI, Tex=Ts (spin temperature)

HI emission(2) Approximation as hn/kTs ≪ 1 First order approximation hn/k =0.07 [K]≪Ts~100 [K] First order approximation 1-exp[(-hn)/(kTs)]}= (hn)/(kTs) nH= n0 {1+(g1/g0) exp(-hn/kTs)}=4n0 kn = c2/(8pn 2) (3/4) nH A10 (hn)/(kTs) j(n) =2.6×10-15 (nH /Ts) j(n) [cgs]

NH[cm-2]=1.8224×1018 ∫ TB dv [K km s-1] HI emission(3) If Ts on the line-of-sight is const, NH=∫nH dx= 1/(2.6×10-15) ∫ Tstn dn [cgs] In optically thin case, TB= Ts(1-e-t)= Tstn NH= 1/(2.6×10-15) ∫ TB dn [cgs] Use Doppler velocity dn =(n /c) dv, NH[cm-2]=1.8224×1018 ∫ TB dv [K km s-1] Caution: this equation is valid only for Optically thin & constant Ts on the LOS

What is the natural width j(n)? Duration in quantum transition Dt Not 0　∵emitted EM wave is not d(t) time profile Not ∞　∵finish the transition in finite time span Gradually increase and gradually decrease Give a width after Fourier transformation wave particle wave packet

report Attach to your e-mail. Deadline : 15 Nov. Questions Submit to handa@sci.kagoshima-u.ac.jp Questions Show relations between Einstein coeff. How long the mean time to transit a neutral hydrogen atom?