1. 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

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

3. 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

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

5. 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

6. 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

7. 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!?

8. 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.

9. 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

10. 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

11. 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!

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

13. 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

14. 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)]}

15. 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

16. 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)

17. 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)]

18. 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.

19. 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= ×10-15 [s-1], n = [GHz]

20. kn = c2/(8pn 2) j(n) n0A10 (g1/g0){1-exp[(-hn)/(kTex)]}
HI emission(1)
A10= ×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)

21. 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]

22. 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

23. 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

24. report Attach to your e-mail. Deadline : 15 Nov. Questions
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Questions
Show relations between Einstein coeff.
How long the mean time to transit a neutral hydrogen atom?