Some Quantum mechanicsuklein/teaching/ISM/Atomic... · 2009. 5. 7. · Klein/Kerp ISM lecture...

Some Quantum mechanics

Klein/Kerp ISM lecture Summer 2008

Bohr-Sommerfeld I• Bohr-Sommerfeld theory

(classical quantum mechanics• Assume a complete revolution

of an electron around the atomic nucleus

...3,2,1

h

=

=∫i

iii

n

ndqp


Bohr-Sommerfeld II

cm][105.0

e and e

h22

210

2

22

2

222

2

2

2

Zn

mZenr

mZn

mnr

nZ

nprrmpdqr

pr

mrZe

−⋅==

===

≡⋅==

==

∫

h

hh

h υυ

ππυ

υυ Columb force = centrifugal force

Bohr´s postulation

yields specific orbits


Bohr-Sommerfeld III

• The energy depends only on the separation between the nucleus and the electron

2

2

22

42

22

42

22

22

22

.

12

2

2

nZconst

nemZ

nmeZ

nmZeZe

mvr

ZeE

=

⋅−=

+⋅

−=

+−=

h

hh

222

4

ZeV 6.132

1 ⋅≅⋅= ZemRyh


The hydrogen atom

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=∆ 2

122

2 11.nn

ZconstE

Note: the photon has spin 1


Hydrogen atom (important transitions)

• Lyα 121.57 nm (UV)• Hα 656.27 nm (visible)• Hβ 486.13 nm (visible)• Hγ 434.05 nm (visible)• Paα 1.875 µm (ground base)• Brα 4.051 µm (ground base)• Brγ 2.166 µm (ground base)


Bohr-Sommerfeld IV• Separating the equation in a radial and a spherical

part yields

1...2,1,0mit 1

...3,2,1mit h

h

++=+=

=+≡

==

=

∫∫

lnnnnlln

nndp

ndrp

rr

rr

ϕ

ϕ

ϕϕϕ ϕ


Quantum mechanics I

∑ ∑∑≠

+−∇−=

Ψ=Ψ

i ji ijiii

e re

rZe

mH

Hdtdi

222

2 12h

h

Separating space and time yields the ansatz

t)independen (time HEyields )(),(

Ψ=ΨΨ=Ψ

−h

rr iEt

ertr


Quantum mechanics II• Central field

• Inserting this ansatz in the Schrödinger equation allows to separate the radial and the spherical term in two differential equations. This is only possible, when a constant is the result of each equation. The constant is called the separation constant.

For simplicity, the separation constant is chosen to l(l+1)

),()(),,( ϕϑϕϑ lmnl YrRr =Ψ


Quantum mechanics III

( ) ( )

( )1sin

1sinsin

11

121

2

2

2

2

22

+=⎭⎬⎫

⎩⎨⎧

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

−

+=−+⎟⎠⎞

⎜⎝⎛

ll

llh

ϕϑϑϑ

ϑϑYY

Y

VEmrdrdRr

drd

R

)()(),( ϕϑϕϑ Φ⋅Θ=YThe spherical coordinates can be separated in two


Quantum mechanics IV

polar term 0sin

)1(sinsin1

termazimuthal 1

1)1(sinsin1sin

2

2

2

2

22

22

=−++⎟⎠⎞

⎜⎝⎛ Θ

Θ

=Φ

Φ−

=Φ

Φ−=

⎭⎬⎫

⎩⎨⎧

++⎟⎠⎞

⎜⎝⎛ Θ

Θ

ϑϑϑ

ϑϑ

ϕ

ϕϑϑ

ϑϑϑ

mdd

dd

mdd

mdd

dd

dd

ll

ll


Quantum mechanics V• The energy levels are En = R·1/n2

• The quantum numbers are n, l and m• n = 1,2,3 … (principal quantum number)

• l = 0,1,2…n-1 (orbital quantum number)

• m =-l, -l+1 … +l (magnetic quantum number)

• ms = ±1/2 (Spin of the fermions)• The magnetic quantum number resembles a "hidden"

quantum number until the degeneracy is broken up


Visualisation of the wave function


Radial wave function


Hydrogen atom (example)

32S32P32D

3s3p0,3p±1

3d0,3d±1,3d±2

00,±10,±1,±2

012

3

22S22P

2s2p0, 2p±1

00,±1

01

2

12S1s001

n2S+1Lstatemln


Quantum mechanics VI• Nucleus with many electrons, we can use the

central field approximation

⎪⎩

⎪⎨

⎧

→

∞→+−

≈0for

rZ

for 1

)(r

rrNZ

rV


Quantum mechanics VII• Wave function for atoms with

many electrons

)mm,l,(n,awith)(...)2()1(

s=⋅⋅⋅=Ψ Nuuu kba


Spin-orbit coupling


Spin-orbit coupling• The magnetic moments of the electron and the orbital current

causes a small energetic shift leading to the fine structure splitting

• Spin-orbit coupling is responsible for one of the most important cooling processes in the ISM due to fine structure emission lines

( )

hydrogenfor eV 10 12

)(12

1

4-322

2

22

≈≅

⋅Φ

=⋅=Φ

rcmZe

lr

rdrcm

BlBls

rrrr σσµ


Spin-spin coupling• The magnetic moments of the nucleus and the electron spin

leads to a small shift of the energy levels and to overcome the degeneracy of the total angular momentum

• The hyper-fine structure line splitting is me/mH ~ 1/2000 smaller than the fine structure splitting! Most important for hydrogen in space, 21-cm line emission.

( ) ( ) ( )

eV 105

1112

6

0

−⋅≈

+−+−+−

=∆ ppp

npHFS ssjjFF

jsB

Eµµ


Spin-spin coupling


Quick summary• The atomic energy levels are determined primarily by Z and are

in the order of some eV• The spin-orbit coupling leads to small shifts of the energy levels

(α2 ~ 1/137 times electronic energy 10 < E < 100 eV ) yielding fine structure lines ∆E~10-3 eV to 10-1 eV. These fine structure emissions lines lie in the wavelength region between 10 < λ < 300µm

• The spin-spin coupling shifts the location of the energy level in the order of ∆E~10-6 eV yielding hyper-fine structure lines. Most important, easy to observe an THE diagnostic line of the Universe is the HI 21-cm line emission. The energetic separation between F=1 and F=0 is 5.9×10-6 eV which corresponds to a frequency of 1.421 GHz with an Einstein coefficient of 2.87×10-15 s-1


Quick summary II

• some eV denotes temperatures of a few thousand Kelvin

• Fine structure lines ∆E~10-4 eVare emitted by gas with a few hundred Kelvin

• Hyperfinestructure lines ∆E~10-6

eV are emitted by gas with a few Kelvins

⎥⎦⎤

⎢⎣⎡⋅≈

∝⇒==

KeV101.16k

kk

23h

4

ETTnE ν

Einstein Coefficients


Einstein coefficients



http://www.homepages.ucl.ac.uk/~ucapphj/EinsteinAandB.jpg



( )

mkkm

Tmk

km

T

Tmk

kmkmkkmmk

BA

eI

BA

eI

eNN

AIBNRR

32

3

k32

3

k

k

'

c

gives1

1c

)( BB using 1

1)(

yields

EquationBoltzmann )(

mequilibriu amicthermodynassume We

πν

πννν

ν

νν

ν

h

hhh

h

=

−=

−=

=

+==

−


Natural line width

( ) ( )

relation)y uncertaint time-(energy and 1gives

find enotation w mechanical quantum theUsing

givesequation aldifferenti homogenous thissolving

is level a from timeoffunction a as ns transitioofnumber The

20

2

0

hh

h

=Γ==Γ

Ψ=Ψ=

=

−=

Γ−

−

ττ

P

ettP

eNN

PNdt

dNkN

t

kkk

PtK

kk

k


Natural line width II

( )

( )ondistributi Lorentz

2

2)()(

shape line theand2

)(

isintensity lineThe

22

2

0

22

2

⎟⎠⎞

⎜⎝⎛ Γ+−

⎟⎠⎞

⎜⎝⎛ Γ

=

⎟⎠⎞

⎜⎝⎛ Γ+−

∝

k

k

EEII

EEI

νν

ν h


Transition propabilities• The Einstein coefficients define the transition

probabilities• The "lifetime" of an excited is inverse proportional to

the Einstein coefficient for spontaneous emission


Lorentz line shape


emission linesDipole approximation• ∆S=0• ∆L=0,±1• ∆l= ±1• ∆J=0,±1(not 0→0)• ∆MJ=0,±1(not 0→0 if ∆J=0)Line emission which do not follow these selection rules are denoted

as forbidden lines.Forbidden line emission does not imply that these emission lines are

not observable in space but characterized by a much lower transition probability in comparison to allowed transitions.


Doppler broadening


Doppler broadening• The gas temperature leads to

thermal motions of the emitting and absorbing ions and atoms, these motions can be described by the Maxwell-Boltzmann distribution

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= Tmv

r

r

eT

mNvn k2

2

k2)(

π

mT

vv

D

rr

k2c

and

1cc

0

000

νν

υυυυυ

=∆

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⇒=−

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

∆−

−

∆=

2

201)( DeI

D

ννν

πνν


Forbidden transitions• Some transitions are metastable (lifetime ~ 0.1 sec)

because they are "forbidden„ (∆J=±1, ∆L=0, ∆S=0), here 2-photon decay makes the work 22S -> 12S H(2S) -> H(1s)+hν1+hν2


Infrared and radio recombination lines• Important (forbidden) emission lines are in the radio

range of hydrogen-like ions, namely H+, He+, He++, C+

..., the wavelength of the emission line is

[ ]regime radio and infared in the are lines thesen´n

andn smallfor cm 110097.1

with´11

1-5

2221

´,

≈

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=

⎟⎠⎞

⎜⎝⎛ −=−

A

eA

Ann

mmR

nnZRλ


Infrared and radio recombination line• Transitions between fine structure levels are magnetic

dipol transitions (∆J=±1, ∆L=0, ∆S=0)• Their transition probability is about four orders of

magnitude smaller than of the allowed electric dipole transition

• Accordingly, the emission lines are usually optically thin and are excited by collisions of the atoms

• Because the electronic masses of the different isotopes are pretty much comparable, the different isotopes can not be differentiated by their corresponding fine structure line emission, example[13CII] and [12CII] 158µm emission line


Infrared and radio recombination line• Fine structure lines of

C+, N++, Ne+, Ar+, Si+, O+++ and S+++ have 1p or 5p electrons in their valence shell, this yields fine structure doublets and a single fine structure line is the ground state

C0, O0, Si0, N+, O++, S++ and Ne++ with 2 p and 4 p electrons show up with triplets and doublets in the ground state

Ne0, Ar0, He0, O+, S+ with 3p electrons have ground fine structure line emission


Infrared and radio recombination line

Radiation transfer


Radiation transfer• Emission

• For an isotropic emitter

dtdAddIdtdsdAdjdtdVdjdE Ω=Ω=Ω= ννν

[ ]cmHzsrscm erg 111-3 ⋅= −−−dsjdI νν

νν πPj

41

=


Radiative transfer• Absorption• Quantities: number of absorbing objects, cross section

of the absorbing objects with photons

• The absorbed amount of energy

absorbers ofnumber totaldsdAndVn ⋅⋅=⋅

absorbers of area totaldsdAn ⋅⋅⋅ νσ

dsIdsnIdIdtddAdsdIndtddAddI

ννννν

νν

κσνσν

−=−=Ω=Ω− or


Radiative transfer• The radiative transfer equation

• Only emission κν =0

νννν κ Ij

dsdI

−=

∫ ′′+=

=

s

sdsjII

jdsdI

0

)()0( ννν

νν


Radiative transfer• Only absorption jν = 0

• The optical depth is defined accordingly

• τ<1 optical thin• τ>1 optical thick

( )ν

ντ

κ

ν

ννν κ

−′′−

=∫

=⇒

−=

eIeIsI

IdsdI

s

sds

000)(

( ) ( ) snsdsnsdssss

⋅⋅=′⋅⋅′=′′= ∫∫ νννν σσκτ00

)(


Radiative transfer• Solution for mixed cases

( ) ( ) ( ) νν

ττττ

νν

νννν

ν

ν

ν

νννν

νν

ν

τττ

κτ

κτ

κ

ν

ννν ′′+=⇒

−=−=⇒

−==

∫ ′−−− dSeeII

ISIjddI

IjddI

dsdI

00

:


Radiative transfer• Thermodynamic equilibrium

• With

which is the Planck-function

ννν

ννν

BIS

ISdsdI

==⇒

=−= 0

1khexp

1ch2

2

3

−⎟⎠⎞

⎜⎝⎛

=

T

Bν

νν

Excitation and de-excitation


Einstein coefficients and collisions


Photoionization• (Z,z) + γ→(Z,z+1)+e–

• A photon with an energy in excess of 13.6 eV can ionize neutral hydrogenH+hν -> H++eEkin(e) = hν – 13.6 eV

• Cross section is roughly

( ) 23

18 cm 103.6−

− ⎟⎠⎞

⎜⎝⎛⋅≈νννσ ion


Collisional Ionization• (Z,z) + e– → (Z,z+1) + e– + e–

• The collision cross section is a function of the gas temperature ⇔ average velocity distribution and depends on the electron densityγcoll.(z,z+1)=neCz(z,Te) := <vσi>


Recombination• Inverse process of ionization • (Z,z+1) + e– → (Z,z) + γ + γ ...• Recombination rate depends on the electron and

nucleon volume densities n(H+)·n(e) and the average velocity distribution of the gas constituents

• The recombination cross section can be described by the Milne relation

( ) ( )νσνσ niz

nzrec mg

g 2

1

,

cvhv ⎟

⎠⎞

⎜⎝⎛⋅=

+


Dielectronic recombination• (Z,z+1) + e– → (Z,z+1)* → (Z,z) + γ + γ ...• The dielectric recombination describes a capture of an

electron by an ion which binds the electron in an excited state followed by a relaxation of the system by the emission of photons.

• The transition probabilities depend on the same physical parameters as for the recombination plus the ionization structure of the gas.


Recombination• The recombination rate can be approximated by the

empirical formula

• The recombination time is

[ ] [ ]1-343

413 scmK10104 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅≈ −

Tα

[ ] [ ]a)(

10n(e)

s103)(1 512

enen≈

⋅≈

⋅≈

ατ


Radiative transfer• The amount of energy emitted

spontaneously in a volume in a certain direction is

• The amount of energy absorbed by the same volume is

• The amount of energy emitted by stimulated emission is

• Using

( ) ( ) dtdddVANdE ee νπ

νϕνν4

h 2120Ω

=

( ) ( ) dtdddVIBNdE aa νπ

νϕπνν ν 4c4h 1210

Ω=

( ) ( ) dtdddVIBNdE es νπ

νϕπνν ν 4c4h 2120

Ω=

( ) ( ) ( ) dsddVae ⋅=== σνϕνϕνϕ and

( ) ( ) ( )

dsdtddIBNIBNAN

dsdtdddIdEdEdE ase

σϕπππν

σννν

ννν

ν

Ω⎥⎦⎤

⎢⎣⎡ −+=

Ω=−+

c4

c4

4h

121212212


Radiative transfer• This gives

• Using the relation between spontaneous emission and absorption, we find for the absorption

• And for the emission

• In case of LTE

( ) ( ) ( )

ννν

νν

εκ

νϕπννϕν

+−=

+−−=

I

ANIBNBNdsdI

2120

212121 4h

ch

( )νϕνκν ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

12

21121 1

ch

NgNgBN

( )νϕπνεν 212

0

ch AN=

⎭⎬⎫

⎩⎨⎧−=

Tgg

NN

khexp 0

1

2

1

2 ν

( )νϕννπ

κν ⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧−−=

TAN

gg

khexp11

8c 0

2111

2

0

2


Radiative transfer• The exponential correction factor for stimulated

emission becomes important in the ultraviolet wavelength regime only in very hot environments

• In the radio regime the correction term is important for mm-wavelength regime and extreme low temperature environments

Equilibrium considerations


ISM in equilibrium?• Thermal equilibrium denotes, that each excitation

process has the same likelihood as the corresponding de-excitation process. This is not realized in the ISM!

THE ISM IS NOT IN EQUILIBRIUM!


ISM in equilibrium?

• Maxwell velocity distribution– Collision thermalize the velocity distribution– Tkin = Te=Ti=Tn

• Boltzmann population of energy levels– Tex ≠Tkin

• Planck spectrum– ISM spectrum ≠ Black Body

Radiation transfer in non-equilibrium situations


Radiative transfer: rate equation• To quantify the emission and absorption coefficients it is

necessary to know both, the Einstein coefficients and the numberdensities N1 and N2. As claimed above, this is not the case under „normal“ astronomical conditions within the ISM.

• In the special case of Local Thermal Equilibrium (LTE) the population of the energy levels is according to the Boltzmanndistribution.

• In all other cases, one has to solve the rate equation

∑ ∑∑ ∑+−=k y

ykj

y kk

yjkj

j RNRNdt

dN


Rate equation I (absorption and emission)• Assuming the simple case

that only spontaneous emission and absorption changes the population of two different energy levels, the rate equation is

• With

• We find

• Accordingly, in this simple case the level population is determined by the Boltzmanndistribution. Note: the temperature is NOT the thermodynamic temperature of the considered system!

( )UBANUBN 10101010 +=

1010103

3

011

010 g and

ch8A ,

c4 BgBB

ggIU ===

νππ

⎟⎠⎞

⎜⎝⎛−=

+=

Tgg

I

IBB

NN

khexp

ch2 2

1

2

310

01

2

1 νν


Rate equation II (+ collisions)• Assuming that next to

emission and absorption processes also collisions are involved:

• The collision rate is

• Using• TK is the temperature which

determines the velocity distribution, the kinetic temperature

• It follows

( ) ( )212121212121 CUBANUBCN ++=+

( ) ( ) vvvv0

dfNNC ikikik ∫∞

== σγ

⎭⎬⎫

⎩⎨⎧−=

Kkikiki T

ggkhexp νγγ

⎭⎬⎫

⎩⎨⎧−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

K

r

K

r

Tm

Tmf

k2exp

k2)(

223

2 υυπ

υ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎭⎬⎫

⎩⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎭⎬⎫

⎩⎨⎧−=

⎭⎬⎫

⎩⎨⎧−=

b

bk

b

ex

TCA

TTCA

T

TgNgN

khexp1

1khexp

khexp

khexp

khexp

2121

2121

21

12

ν

ννν

ν

Line diagnostic


Diagnostics of the ISM• Ionization

– Photo ionization– Collisional ionization– Auger-Ionization

• Recombination– Radiative recombination– Dielectric recombination

• Continuum processes– Bremsstrahlung– Compton scattering


Line diagnostic (density)• Choose element with two energy levels about the same

excitation energy. Depending on the density the levels will be depopulated by radiation of collisional de-excitation

⎟⎠⎞

⎜⎝⎛−

+=

+=⇒

+=+=

TEC

gg

CnAnLTE

CnACn

nn

CnnAnCnnCnnAnCnn

e

e

e

e

ee

ee

kexp)( 12

211

2

21212121

12

1

2

313313131

212212121


Line diagnostic (density)• With

• We find

2122121 h4 νπ nAI =

331

221

31

21

312131331

21221

31

21 using hh

nAnA

II

nAnA

II

=

≈= νννν


Line diagnostic (density)






Line diagnostic (temperature)










Line diagnostic (density and temperature)

Practical example: soft X-ray radiation


• The improvement in thediagnostic in medicine was immediately recognized after thethe discovery of X-ray by Röntgen in 1895. The X-ray detector was thestandard photographic plate.

•The human bones consist manly of Calcium (Z=20) while the musclesconsists of Hydrogen (Z=1), Carbon(Z=6), Nitrogen (Z=7) and Oxigen(Z=8)

•Moseley-Gesetz:2)1( −∝ ZKαυ

Example: soft X-ray radiation




Example: Soft X-ray radiation

X-ray source interstellar cloud Observer

log I

log ν

log I

log ν


HI is THE tracer for the strength of thephotoelectric absorption, not the dominant absorber!

Example: Soft X-ray radiation


Example: soft X-ray absorption

‘τ’denotes the optical depth

This is the so-called effective photo electric absorption cross section

I = I0 exp( -σeff(E)·NH )

σeff =∑[-σZ(E) · nz/nH]



Kerp & Pietz (1998)

I =I0+ I1· exp(-σeff(E)·NH )


Diffuse X-ray plasma

HI column densitydistribution

Observed X-ray

Intensity distribution



HI 21-cm 0.25 keV ROSAT


Modelled ROSAT map based on theLeiden/Dwingeloo HI 21-cm line data.

(c) J. Kerp (c) J.Kerp

ROSAT 1/4 keV all-sky survey map

Kerp, Burton, Egger et al. 1999, A&A 342, 213



(c) J.Kerp(c) J.Kerp

Kerp, Burton, Egger et al. 1999, A&A 342, 213

ROSAT 1/4 keV all-sky survey map Modelled ROSAT map based on theLeiden/Dwingeloo HI 21-cm line data.


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