KAIST Astrophysics (PH481) - Part 1

Week 3a Sep. 16 (Mon), 2019

Kwang-il Seon (선광일)Korea Astronomy & Space Science Institute (KASI)

University of Science and Technology (UST)

1

Sunyaev-Zeldovich effect• The Sunyaev-Zeldovich effect is Comptonization of the Cosmic Microwave

Background (CMB), the distortion of the blackbody spectrum (T = 2.73 K) of the CMB owing to the Inverse Compton (IC) scattering of the CMB photons by the energetic electrons in the galaxy clusters.

- Thermal SZ effects, where the CMB photons interact with thermal electrons that have high energies due to their “high” temperature.

- Kinematic SZ effects (Ostriker-Vishniac effect), a second-order effect where the CMB photons interact with electrons that have high energies due to their bulk motion (peculiar motion). The motions of galaxies and clusters of galaxies relative to the Hubble flow are called peculiar velocities. The plasma electrons in the cluster also have this velocity. The energies of the CMB photons that scattered by the electrons reflect this motion.

- The SZ effect should be taken into account in analyzing the CMB data.- Determinations of the peculiar velocities of clusters enable astronomers to map out the

growth of large-scale structure in the universe. This topic is fundamental importance, and the kinetic SZ effect is a promising method for approaching it.

2

Thermal SZ effect• The net effect of the IC scattering on the photon

spectrum is obtained by multiplying the photon number spectrum by the scattering kernel � and integrating over the spectrum.

- The net effect is that the BB spectrum is shifted to the right and distorted (see Figure (c)).

- Observations of the CMB are most easily carried out in the low-frequency Rayleigh-Jeans region of the spectrum � .

- Measurement of the CMB temperature as a function of position on the sky would thus exhibit brightness (antenna) temperature dips in the directions of clusters that contain hot plasmas (see Figure (c) and (d) .

- Note that the scattered spectrum is not a BB spectrum. The effective temperature increases. But, the total number of photons detected in a given time over the entire spectrum remains constant.

K(ν/ν0)

(I(ν) ∝ T for hν ≪ kTCMB)

3

The scattering kernel is the scattered spectrum for a monochromatic photon with frequency of � .

The result of such scatterings for an initial blackbody photon spectrum is shown in Figure (c) for the value:

ν0

kTemc2

τ = 0.5

Recall that at low frequency regime, the brightness temperature is

But, the effective temperature:

Tb =c2

2ν2kBIν

∫ Fnudν = σT4eff

Nscatt(ν) = ∫∞

0N(ν0)K(ν/ν0)dν0 where N(ν) = I(ν)/hν

P1: RPU/... P2: RPU

9780521846561c09.xml CUFX241-Bradt October 3, 2007 19:52

344 Compton scattering

!

!s

!

!s

e–

Plasma

(a)

(c) (d)

""cluster

Tr

# Tr

log !

log I

!obs

# Tr

!/!00.6 0.8 1.0 1.2 1.4 1.6

K(!/!0)

4

3

2

1

0

(b)

IBB

Iscatt

Fig. 9.7: Sunyaev–Zeldovich effect. (a) Scattering of CMB photons in a cloud of plasma. The clouddoes not affect the number of photons observed. In reality, only a small fraction of the observedCMB photons from the direction of a cluster are scattered by cluster electrons. (b) Distributionof scattered photon energies for initial monochromatic beam and kTe = 5.1 keV. (c) Sketch ofunscattered blackbody (dark line) and singly scattered (light line) spectra. The frequency shifts areto the right, and the shape is distorted. The shift is drawn for the unrealistically high value of theparameter y = kTet /mc2 = 0.15. (d) Antenna temperature as a function of position on the sky forradio measurements in the Rayleigh–Jeans (low-frequency) portion of the spectrum. [(b,c) afterR. Sunyaev and Y. Zel’dovich, in ARAA 18, 537 (1980), with permission]

A head-on collision between a low-energy CMB photon and an energetic electron willresult in the photon’s gaining energy. An overtaking collision (a photon catching an electron)will result in its losing energy. In this nonrelativistic case, these effects will cancel to firstorder in v/c. Examination of the Doppler shifts in these collisions, however, indicates thatthere is a net fractional photon frequency gain of order v2/c2; more energy is gained inhead-on collisions than is lost in overtaking collisions (Prob. 51). Because the energy ofa photon is hn , a fractional frequency increase equals the fractional energy increase. In aMaxwell–Boltzmann plasma of temperature Te, the average kinetic energy of the electronsis ⟨mv2/2⟩av = 3kTe/2, and so this gain is of order v2/c2 ≈ 3kTe/mc2.

A proper calculation must take into account the effect on monochromatic (n = n0) photonsof electron scatters averaged over all collision angles and electron speeds of a Maxwelliandistribution. The result, under the assumption that each photon is scattered just once inelectrons of 5.1 keV, is a distribution of photon frequency K(n /n0) that ranges up and down

scattering kernel

• Change of the BB temperature- In the Rayleigh-Jeans region,

- If the spectrum is shifted parallel to itself on a log-log plot, the fractional frequency change of a scattered photon is constant.

- Total photon number is conserved:

- The properly calculated result is .

4

I(⌫) =2⌫2

c2kBTCMB

AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR6sIyUwu6EUq70YVQoTfojCWTZtrQTGZIMkIZ5inc+CpuXCjiVtz5NqaXhbYeSPj4/3NIzu9FjEplWd9GZmV1bX0ju5nb2t7Z3TP3D1oyjAUmTRyyUHQ8JAmjnDQVVYx0IkFQ4DHS9ka1id9+IELSkDfUOCJugAac+hQjpaWeeXZTcHh8euX4AuGkpPm+lCZYX6Ne4ogAVtPGDGq31bRn5q2iNS24DPYc8mBe9Z755fRDHAeEK8yQlF3bipSbIKEoZiTNObEkEcIjNCBdjRwFRLrJdK0UnmilD/1Q6MMVnKq/JxIUSDkOPN0ZIDWUi95E/M/rxsq/dBPKo1gRjmcP+TGDKoSTjGCfCoIVG2tAWFD9V4iHSAekdJI5HYK9uPIytEpF+7xo35XzlfI8jiw4AsegAGxwASrgGtRBE2DwCJ7BK3gznowX4934mLVmjPnMIfhTxucP57ieiw==

" =�⌫

⌫=

⌫0 � ⌫⌫

= constant or ⌫0 = ⌫(1 + ")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

d⌫0 = d⌫(1 + ")AAACAnicbVDLSgMxFM3UV62vUVfiJljEilBmtKAboeDGZQX7gM5QMplMG5pJhiRTKENx46+4caGIW7/CnX9jpu1Cqwfu5XDOvST3BAmjSjvOl1VYWl5ZXSuulzY2t7Z37N29lhKpxKSJBROyEyBFGOWkqalmpJNIguKAkXYwvMn99ohIRQW/1+OE+DHqcxpRjLSRevZB6PH05DrvFffMGyFJEkWZ4Kc9u+xUnSngX+LOSRnM0ejZn14ocBoTrjFDSnVdJ9F+hqSmmJFJyUsVSRAeoj7pGspRTJSfTU+YwGOjhDAS0hTXcKr+3MhQrNQ4DsxkjPRALXq5+J/XTXV05WeUJ6kmHM8eilIGtYB5HjCkkmDNxoYgLKn5K8QDJBHWJrWSCcFdPPkvaZ1X3Yuqe1cr12vzOIrgEByBCnDBJaiDW9AATYDBA3gCL+DVerSerTfrfTZasOY7++AXrI9vHKuWkA==

N 0(⌫0)d⌫0 = N(⌫)d⌫ ! I0(⌫0)

h⌫0d⌫0 =

I(⌫)

h⌫d⌫

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

�TCMBTCMB

⇡ �2kBTCMBmc2

⌧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

ε =Δνν

=kBTCMB

mc2τ

∴ I′�(ν′�) = I(ν)

I′�(ν) = I ( ν1 + ε ) = 2ν2

c2 (1 + ε)2kBTCMB

ΔII

=I′�(ν) − I(ν)

I(ν)=

1(1 + ε)2

− 1 ≈ − 2ε = − 2Δνν

ΔTCMBTCMB

=ΔII

≈ − 2Δνν

We compare the spectrum at one frequency � in the cluster direction with that at the same frequency in an off-cluster direction.

� in the cluster direction

� in an off-cluster direction

ν

I′�(ν)I(ν)

5

P1: RPU/... P2: RPU

9780521846561c09.xml CUFX241-Bradt October 3, 2007 19:52

9.5 Sunyaev–Zeldovich effect 349

Fig. 9.9: Interferometric images at 30 GHz of six clusters of galaxies at redshifts ranging fromz = 0.17 to z = 0.89. The solid white contours indicate negative decrements to the CMB. The sixclusters have comparable x-ray luminosities, suggesting similar values of ne and R. The expecteddistance independence of the S-Z effect is apparent; the contours reach to similar depths despitethe factor of five range in redshift. [BIMA and OVRA arrays; J. Carlstrom et al., ARAA 40, 643(2002), with permission]

it to our assumption of uniform density and temperature by

Ix(nx, Te) =C14π

g(n, Te)

T 1/2eexp(−hnx/kTe)n2e2R.

(Thermal bremsstrahlung; W m−2 Hz−1 sr−1) (9.37)

A typical cluster havean average electron density of � ,a core radius of � ,and an electron temperature of � .

A typical optical depth is thus

The expected antenna temperature change is

This effect has been measured in dozens ofclusters.

∼ 2.5 × 10−3 cm−3

Rc ∼ 1024 cm ∼ 320 kpckBT ≈ 5 keV

�TCMBTCMB

⇡ �1⇥ 10�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

�TCMB ⇡ �0.3 mK for TCMB = 2.7 KAAACMHicbVDLSsNAFJ34rPVVdelmsAhuDIkKuhHECgoiKFgVmlIm0xsdnMmEmRuxhHySGz9FNwqKuPUrTGsWvs7qcM653HtPmEhh0fOenaHhkdGx8cpEdXJqema2Njd/ZnVqODS5ltpchMyCFDE0UaCEi8QAU6GE8/C60ffPb8BYoeNT7CXQVuwyFpHgDAupU9sP9kAio6edLDCKNo5284AlidG3dNVz1wOEW8yoOqSRNjT/lqLbdM3dLP3DvFOre643AP1L/JLUSYnjTu0h6GqeKoiRS2Zty/cSbGfMoOAS8mqQWkgYv2aX0CpozBTYdjZ4OKfLhdIdnBTpGOlA/T6RMWVtT4VFUjG8sr+9vvif10ox2mpnIk5ShJh/LYpSSVHTfnu0KwxwlL2CMG5EcSvlV8wwjkXH1aIE//fLf8nZmuuvu/7JRn1no6yjQhbJElkhPtkkO+SAHJMm4eSOPJIX8urcO0/Om/P+FR1yypkF8gPOxyeDr6dg

Interferometric images at 30 GHz of six clusters of galaxies. The solid white contours indicate negative decrements to the CMB. (Carlstrom et al. 2002, ARAA, 40, 643)

τ ≈ 3σTneRc ≈ 0.005 (σT = 6.652 × 10−25 cm2)

Hubble Constant• A value of the Hubble constant can be obtained for a given galaxy only if one has

independent measures of a recession speed � and a distance � of a galaxy.

- Recession speed is readily obtained from the spectral redshift.- Distance:‣ X-ray observations:

‣ S-Z CMB decrement:

‣ The radio and X-ray measurements yield absolute values of the electron density � and cluster radius � without a priori knowledge of the cluster distance.

‣ Imaging of the cluster in the radio or X-ray band yields the angular size of the cluster 𝜃. Then the distance � to the cluster is obtained by

- The SZ effect (at radio frequencies) in conjunction with X-ray measurements can give distances to clusters of galaxies. This can be used to derive the Hubble constant.

υ d

neR

d

6

H0 =v

dAAAB+nicbVDLSsNAFL2pr1pfqS7dDBbBVUm0oBuh4KbLCvYBbQiTyaQdOnkwM6mUmE9x40IRt36JO//GaZuFth64cDjnXu69x0s4k8qyvo3SxubW9k55t7K3f3B4ZFaPuzJOBaEdEvNY9D0sKWcR7SimOO0nguLQ47TnTe7mfm9KhWRx9KBmCXVCPIpYwAhWWnLNasu10C0aBgKTbJpnfu6aNatuLYDWiV2QGhRou+bX0I9JGtJIEY6lHNhWopwMC8UIp3llmEqaYDLBIzrQNMIhlU62OD1H51rxURALXZFCC/X3RIZDKWehpztDrMZy1ZuL/3mDVAU3TsaiJFU0IstFQcqRitE8B+QzQYniM00wEUzfisgY6xSUTquiQ7BXX14n3cu6fVW37xu1ZqOIowyncAYXYMM1NKEFbegAgUd4hld4M56MF+Pd+Fi2loxi5gT+wPj8AVeik2A=

I(⌫, Te) = Cg(⌫, Te)

T 1/2eexp(�h⌫/kTe)n2e(2R)

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

�TCMBTCMB

= �2 kTemc2

⌧ = �2 kTemc2

(�Tne2R)AAACWXicbVFNaxsxFNRuk9Rx08Zpjr2ImEJ6iNl1As0lEJoeeikkxU4CXnd5K7+1hSXtIr0tmGX/ZA+F0r/SQ+WPQ5pkQDCamYekUVYq6SiKfgfhi63tnZet3farvddv9jsHb29dUVmBQ1Gowt5n4FBJg0OSpPC+tAg6U3iXza+W/t0PtE4WZkCLEscapkbmUgB5Ke2USW5B1MlnVAR8kNaJ1fzq66emqR9uLk766+B8kGJTa/G93yQEFb/gzzkKczpOnJxqSAfcpMj73xIrpzP6kHa6US9agT8l8YZ02QbXaednMilEpdGQUODcKI5KGtdgSQqFTTupHJYg5jDFkacGNLpxvWqm4e+9MuF5Yf0yxFfqw4katHMLnfmkBpq5x95SfM4bVZSfj2tpyorQiPVBeaU4FXxZM59Ii4LUwhMQVvq7cjED3xP5z2j7EuLHT35Kbvu9+LQX35x1L882dbTYO3bEjlnMPrJL9oVdsyET7Bf7G2wHO8GfMAhbYXsdDYPNzCH7D+HhP44Qs6A=

d =R

✓AAAB+3icbVBNS8NAEN3Ur1q/Yj16WSyCp5JoQS9CwYvHKvYDmlA22027dLMJuxOxhPwVLx4U8eof8ea/cdvmoK0PBh7vzTAzL0gE1+A431ZpbX1jc6u8XdnZ3ds/sA+rHR2nirI2jUWsegHRTHDJ2sBBsF6iGIkCwbrB5Gbmdx+Z0jyWDzBNmB+RkeQhpwSMNLCrw2svVIRm93nmwZgByQd2zak7c+BV4hakhgq0BvaXN4xpGjEJVBCt+66TgJ8RBZwKlle8VLOE0AkZsb6hkkRM+9n89hyfGmWIw1iZkoDn6u+JjERaT6PAdEYExnrZm4n/ef0Uwis/4zJJgUm6WBSmAkOMZ0HgIVeMgpgaQqji5lZMx8REASauignBXX55lXTO6+5F3b1r1JqNIo4yOkYn6Ay56BI10S1qoTai6Ak9o1f0ZuXWi/VufSxaS1Yxc4T+wPr8AWZnlKE=

90 Radiation f i m Mmhg Charges orbiting in a circle with angular frequency oo, the function j,(r) actually contains frequencies not only at wo but also at all harmonics 200, 3w0.. . . In the dipole approximation only oo contributes, in the quadrupole ap- proximation only 20, contributes, and so on (see problem 3.7).

3.4 THOMSON SCATIERING (ELECTRON SCATTERING)

An important application of the dipole formula is to the process in which a free charge radiates in response to an incident electromagnetic wave. If the charge oscillates at nonrelativistic velocities, u

• We obtain the time-averaged power per solid angle � :

- Note that the time-averaged incident flux (Poynting flux) is � .

- The differential cross section for linearly polarized radiation is obtained by

where the quantity � gives a measure of the “size” of the point charge.(Note electrostatic potential energy � . The size is obtained by setting � ).

- For an electron, the classical electron radius has a value � .- The total cross section is found by integrating over solid angle.

- For an electron, the scattering process is then called Thomson scattering or electron scattering, and the Thomson cross section is

(⟨sin2 ω0t⟩t = 1/2)

⟨S⟩ =c

8πE20

r0eϕ = e2/r0 e2/r0 = mc2

r0 = 2.82 × 10−13 cm

8

⟨ dPdΩ ⟩t =e4E20

8πm2c3sin2 Θ, ⟨P⟩t =

e4E203m2c3

dσdΩ

= ⟨ dPdΩ ⟩t/⟨S⟩dσdΩ

=e2

m2c4sin2 Θ = r20 sin

2 Θ, r0 ≡e2

mc2

σ = ∫dσdΩ

dΩ = 2πr20 ∫1

−1(1 − μ2) dμ = 8π3 r

20

σT =8π3

r20 = 6.652 × 10−25 cm2

92 Radiation from Moving Charges

For an electron u = uT = Thomson cross section =0.665 X cm’. The above scattering process is then called Thomson scattering or electron scattering.

Note that the total and differential cross sections above are frequency independent, so that the scattering is equally effective at all frequencies. However, this is really only valid for sufficiently low frequencies, so that a classical description is valid. At high frequencies, where the energy of emitted photons hv becomes comparable to or larger than m2, then the quantum mechanical cross sections must be used; this occurs for X-rays of energies hv20.511 MeV for electron scattering (see Chapter 7). Also, for sufficiently intense radiation fields the electron moves relativistically; then the dipole approximation ceases to be valid.

We note that the scattered radiation is linearly polarized in the plane of the incident polarization vector E and the direction of scattering n.

It is easy to get the differential cross section for scattering of unpolarized radiation by recognizing that an unpolarized beam can be regarded as the independent superposition of two linear-polarized beams with perpendicu- lar axes. Let us choose one such beam along E , , which is in the plane of the incident and scattered directions, and the second along c2, perpendicular to this plane. (See Fig. 3.7.) Let 0 be the angle between E , and n. Note that the angle between c2 and n is 7 / 2 . We also have introduced the angle B = n / 2 - 0 , which is the angle between the scattered wave and incident wave. Now the differential cross section for unpolarized radiation is the average of the cross sections for scattering of linear-polarized radiation

Figure 3.7 Geotnety for scattering impohtized mdiatim

• Note:- The total and differential cross sections are frequency independent.- The scattered radiation is linearly polarized in the plane of the incident polarization vector �

and the direction of scattering � .

- � : electron scattering is larger than ions by a factor of � � � .

- We have implicitly assumed that electron recoil is negligible. This is only valid for nonrelativistic energies. For higher energies, the (quantum-mechanical) Klein-Nishina cross section has to be used.

• What is the cross section for scattering of unpolarized radiation?- An unpolarized beam can be regarded as the independent superposition of two linear-

polarized beams with perpendicular axes.- Let us assume that � = direction of scattered radiation

� = direction of incident radiation- Choose

(1) the first electric field along � , which is in the plane defined by � and � . The plane is called “scattering plane”. (2) the second one along � orthogonal to the scattering plane and to � .

̂ϵn

σ ∝ 1/m2 (mp /me)2 = (1836)2 ≈ 3.4 × 106

nk

̂ϵ1n k

̂ϵ2n

9

• Let � = angle between � and � . ( � is the angle between the scattered direction and the acceleration direction)- Note that angle between � and � is � , and � is the angle between the

scattered wave and incident wave.- Then, the differential cross section for unpolarized radiation is the average of the cross

sections for scattering of two electric fields.

- This depends only on the angle between the incident and scattered directions.- Total cross section:

Θ ̂ϵ1 n Θ̂ϵ2 n π/2 θ = π/2 − Θ

10

92 Radiation from Moving Charges

For an electron u = uT = Thomson cross section =0.665 X cm’. The above scattering process is then called Thomson scattering or electron scattering.

Note that the total and differential cross sections above are frequency independent, so that the scattering is equally effective at all frequencies. However, this is really only valid for sufficiently low frequencies, so that a classical description is valid. At high frequencies, where the energy of emitted photons hv becomes comparable to or larger than m2, then the quantum mechanical cross sections must be used; this occurs for X-rays of energies hv20.511 MeV for electron scattering (see Chapter 7). Also, for sufficiently intense radiation fields the electron moves relativistically; then the dipole approximation ceases to be valid.

We note that the scattered radiation is linearly polarized in the plane of the incident polarization vector E and the direction of scattering n.

It is easy to get the differential cross section for scattering of unpolarized radiation by recognizing that an unpolarized beam can be regarded as the independent superposition of two linear-polarized beams with perpendicu- lar axes. Let us choose one such beam along E , , which is in the plane of the incident and scattered directions, and the second along c2, perpendicular to this plane. (See Fig. 3.7.) Let 0 be the angle between E , and n. Note that the angle between c2 and n is 7 / 2 . We also have introduced the angle B = n / 2 - 0 , which is the angle between the scattered wave and incident wave. Now the differential cross section for unpolarized radiation is the average of the cross sections for scattering of linear-polarized radiation

Figure 3.7 Geotnety for scattering impohtized mdiatim

Properties of Thomson Scattering• Forward-backward symmetry:

- The differential cross section is symmetric under � .• Total cross section of unpolarized incident radiation = total cross

section for polarized incident radiation. This is because the electron at rest has no preferred direction defined.

• Scattering creates polarization- The scattered intensity is proportional to � , of which � arises

from the incident electric field along � and � from the incident electric field along � .

- The portion of “� ” of the polarization along � will be cancelled out by the independent polarization along � .

- Therefore, the degree of polarization of the scattered wave:

• Electron scattering of a completely unpolarized incident wave produces a scattered wave with some degree of polarization.- No net polarization along the incident direction � , since, by

symmetry, all directions are equivalent.

- 100% polarization perpendicular to the incident direction � , since the electron’s motion is confined to a plane normal to the incident direction.

θ → − θ

1 + cos2 θ 1̂ϵ2 cos2 θ

̂ϵ1cos2 θ ̂ϵ2

̂ϵ2 × n

(θ = 0)

(θ = π /2)

11

Π =1 − cos2 θ1 + cos2 θ

scattereddirection

incidentdirection

Astrophysical Applications of Polarization by Scattering• Detection of a concentric pattern of polarization vectors in an extended region indicates that the

light comes via scattering from a central point source.

• Left map shows the IR intensity map at 3.8 𝜇m of the Becklin-Neugebauer/Kleinmann-Low region of Orion. It is not easy to identify which bright spots correspond to locations of possible protostars.- However, the polarization map singles out only two positions of intrinsic luminosity: IRc2 (now known to be an

intense protostellar wind) and BN (suspected to be a relatively high-mass star)- All the other bright spots (IRc3 through 7) correspond to IR reflection nebulae.

• The above example is not for Thompson scattering. But, dust scattering also produces polarization from unpolarized light.

12

Werner et al. (1983, ApJL, 265, L13)

1983ApJ...265L..13W

1983ApJ...265L..13W

Atomic Structure, Radiative Transitions

13

History: Fraunhofer Lines• In 1814, Joseph von Fraunhofer (1789-1826) used

one of the high-quality prisms he had manufactured to diffract a beam of sunlight onto a whitewashed wall.- Besides the characteristic colors of the rainbow, he

saw many dark lines.- He catalogued the exact wavelength of each dark

line and labelled the strongest of them with letters. These are still known today as Fraunhofer lines. Many of these labels, such as the sodium D lines (5896Å, 5890Å; Na I D1, D2) are still used today.

- He did not know what caused the dark lines he observed.

- However, he performed a similar experiment using light from the nearby star Betelgeuse and found that the pattern of dark lines changed significantly. He concluded correctly that most of those features were somehow related to the composition of the object.

14

The solar spectrum as recorded by Fraunhofer (color overlaid).

- The first real step in understanding Fraunhofer’s observations came in the middle of the 19th century with the experiments of Gustav Kirchhoff (1824-1887) and Robert Bunsen (1811-18911). They studied the color of the light emitted when metals were burnt in flames. In certain cases, the wavelength of the emitted light gave an exact match with the Fraunhofer lines. These experiments demonstrated that the Fraunhofer lines were a direct consequence of the atomic composition of the Sun.

- In fact, some of the lines were due to the Earth’s atmosphere, the so-called telluric lines.

History: Nebulium?• In 1918, extensive studies of the emission spectra of

nebulae found a series of lines which had not been observed in the laboratory.- Particularly strong were features at 4959Å and 5007Å. For

a long time, this pair could not be identified and these lines were attributed to a new element, ‘nebulium’.

- In 1927, Ira Bowen (1898-1973) discovered that the lines were not really due to a new chemical element but instead forbidden lines from doubly ionized oxygen [O III].

- He realized that in the diffuse conditions found in nebulae, atoms and ions could survive a long time without undergoing collisions. Indeed, under typical nebula conditions the mean time between collisions is in the range 10-10,000 secs. This means that there is sufficient time for excited, metastable states to decay via weak, forbidden line emissions.

- The forbidden lines could not be observed in the laboratory where it was not possible to produce collision-free conditions over this long timeframe.

- Other ‘nebulium’ lines turned out to be forbidden lines originating from singly ionized oxygen [O II] and nitrogen [N II].

15

112 Astronomical Spectroscopy {Third Edition)

101

4000

=: = 0. u

=

n.

5000 >. (A)

. :c s -= . ., .,-

1.D.

. :i::

Fig. 7.1. Optical spectra of NGC 6153 from 3540 to 7400 A, obtained for a deep (IO-minute) exposure using the ESO 1.52 m telescope in Chile. The two spectra plotted are (a) obtained by uniformly scanning the long-slit across the entire nebula, and (b) taken with a fixed slit centred on the central star. [Reproduced from X.-W. Liu et al., Mon. Not. R. Astron. Soc. 312 , 585 (2000).]

7.1. Nebulium

In 1918, extensive studies of the emission spectra of nebulae found a series of lines which had not been observed in the laboratory. Particularly strong were features at 4959 A and 5007 A. For a long time, this pair could not be identified and, inspired by Lockyer's success with helium, these lines were attributed to a new element, 'nebulium'.

Ten years after their original observation, Ira Bowen (1898- 1973) found the true explanation. Bowen realised that in the diffuse conditions found in nebulae, atoms and ions could survive a long time without undergoing collisions. Indeed , under typical nebula conditions the mean time between collisions is in the range 10- 10000 s. This means that there is sufficient time for excited, metastable states to decay via weak , forbidden line emissions. These lines could not be observed in the laboratory where it was not possible to produce collision-free conditions over this long timeframe.

Optical spectra of NGC 6153, Liu et al. (2000, MNRAS)

forbidden transition

allowed transition

[O III], [O II], [N II], etc:We use a pair of square brackets for a forbidden line.

metastable state

Atomic Hydrogen• Spectra of atomic hydrogen, H, are of paramount astronomical importance.

- This is because approximately 90% of atomic matter by number is hydrogen.

• The spectrum of atomic hydrogen also plays an important role in the theory of quantum mechanics.- Hydrogen is the only atom for which exact quantum mechanical solutions can be found for its

energy levels and wavefunctions.

• Schrodinger equation- momentum operator:- Hamiltonian operator:- Schrodinger equation:

- Solution:

16

� ~2

2mr2 � Ze

2

r = E

AAACIXicbVDLSsNAFJ34rPVVdekmKIIgliQV7EYoiOBSwbZi05ab6aQdOpmEmYlQQn7AD3Dpxl9x40KR7sS1/+Gk7UJbD9zL4Zx7mbnHixiVyrI+jbn5hcWl5dxKfnVtfWOzsLVdk2EsMKnikIXi1gNJGOWkqqhi5DYSBAKPkbrXP8/8+j0Rkob8Rg0i0gygy6lPMSgttQvlY9cXgBO354FoOWniBKnLwWPQctxI0ol9RzJPpJl0dpH1dmHfKlojmLPEnpD9ytFz7fH7oXTVLgzdTojjgHCFGUjZsK1INRMQimJG0rwbSxIB7kOXNDTlEBDZTEYXpuaBVjqmHwpdXJkj9fdGAoGUg8DTkwGonpz2MvE/rxErv9xMKI9iRTgeP+THzFShmcVldqggWLGBJoAF1X81cQ90JEqHmtch2NMnz5KaU7RLRedap3GCxsihXbSHDpGNTlEFXaIrVEUYPaEX9IbejWfj1fgwhuPROWOys4P+wPj6AU7fp+A=

n,l,m(r, ✓,�) = Rnl(r)Ylm(✓,�)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

p =~ir

AAACCXicbVC7SgNBFJ31GeNr1dJmMAiCEHYTQRshYGMZwTwgG8LdyWwyZHZ2mZkVwrKthfb+gK2NhSK2/oGdH+B/OHkUmnjgwuGce7n3Hj/mTGnH+bIWFpeWV1Zza/n1jc2tbXtnt66iRBJaIxGPZNMHRTkTtKaZ5rQZSwqhz2nDH1yM/MYNlYpF4loPY9oOoSdYwAhoI3Vs7IWg+36Qxtm5F0ggqdf3QWYpyzwBPoeOXXCKzhh4nrhTUqgcP9Yfvu/K1Y796XUjkoRUaMJBqZbrxLqdgtSMcJrlvUTRGMgAerRlqICQqnY6/iTDh0bp4iCSpoTGY/X3RAqhUsPQN52ju9WsNxL/81qJDs7aKRNxoqkgk0VBwrGO8CgW3GWSEs2HhgCRzNyKSR9MHtqElzchuLMvz5N6qeiWi6Urk8YJmiCH9tEBOkIuOkUVdImqqIYIukVP6AW9WvfWs/VmvU9aF6zpzB76A+vjB53QnpA=

H =p2

2m+ V = � ~

2

2mr2 � Ze

2

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

Rnl(r) = �"✓

2Z

na0

◆3 (n� l � 1)!2n{(n+ l)!}3

#1/2e�⇢/2⇢lL2l+1n+l (⇢)

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

Ylm(✓,�) = (�1)(m+|m|)/2(l � |m|)!(l + |m|)!

2l + 1

4⇡

�1/2P |m|l (cos ✓)e

im�

AAACanicbVFNb9NAFFy7fJRAIS0SCPViCEiOQoMdIsEFqRIXjkEibVHsWOvNc7zqrm3tPiNFWx/4i9z4BVz4EazjHKDlSSuNZuZpdmfTSnCNQfDTcfdu3b5zd/9e7/6Dg4eP+odHZ7qsFYM5K0WpLlKqQfAC5shRwEWlgMpUwHl6+bHVz7+B0rwsvuCmgljSdcEzzihaKul//5oYIRs/whyQvo6qnA8/+CfhcGl8ObqSV8M3kyYSkOEiyhRlxhcnLfu8sWjUoU6YiFHYmGlU8SZSfJ1jvDShXfZmiVga67QhrNRd0BCWhss2rUn6g2AcbMe7CcIdGJDdzJL+j2hVslpCgUxQrRdhUGFsqELOBDS9qNZQUXZJ17CwsKASdGy2VTXeK8usvKxU9hTobdm/NwyVWm9kap2SYq6vay35P21RY/Y+NryoaoSCdUFZLTwsvbZ3b8UVMBQbCyhT3N7VYzm1xaH9nZ4tIbz+5JvgbDIO344nn6eD0+mujn1yTF4Qn4TkHTkln8iMzAkjv5wD54nz1PntHrnP3OPO6jq7ncfkn3Ff/gG5qbly

spherical harmonics:

radial function: ⇢ = 2Zna0

r, a0 ⌘~2

mec2= 0.529Å (Bohr radius),

L2l+1n+l = associated Laguerre 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

P |m|l = associated Legendre 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

• Each bound state of the hydrogen atom is characterized by a set of four quantum numbers ( � )- � : principal quantum number- � : orbital angular momentum quantum number

‣ By convention, the values of � are usually designated by letters.

- � : magnetic quantum number.‣ It determines the behavior of the energy levels in the presence of a magnetic field.

‣ This is the projection of the electron orbital angular momentum along the � -axis of the system.

• Spin

- The electron possesses an intrinsic angular momentum with the magnitude of � .

- There are two states, � , for the spin.

• Degeneracy for a given n:

n, l, m, msn = 1, 2, 3,⋯l = 0, 1, 2,⋯, n − 1

l

m = − l, − l + 1,⋯,0,⋯, l − 1, l

z

|s | =12

ms = ±12

17

26 Astronomical Spectroscopy (Third Edition)

is the electron orbital angular momentum quantum number. The actual angular momentum is given by n[l(l + 1)]½. l can take the values 0, 1, 2, ... , n - 1. By convention, the values of l are usually designated by letters (see Table 3.1).

m is the magnetic quantum number , so called because it determines the behaviour of the energy levels in the presence of a magnetic field. mn is the projection of the electron orbital angular momentum , given by l , along the z-axis of the system. It can take (2l + 1) values -l, -l + 1, ... , 0, ... , l - 1, l.

2 s 1=0 p 1=1 p 1=1

m=O

B m=O m=1

0 0 m N -2

d 1=2 d 1=2 d 1=2 m=O

~ m= + 0

-

-2 -2 0 -2 0 -2 0 2

X

Fig. 3.2. Wavefunctions for the angular motions of the hydrogen atom which are given in terms of spherical harmonics , Yi, m(0, ). Plots are for the x - z plane except for the l = 2, m = 2 plot for which the x--y plane is shown. The plots give the absolute value of the spherical harmonic as a distance from the origin for each angle; the signs indicate the sign of the wavefunction in each region. (S.A. Morgan, private communication.)

Table 3.1. Letter designations for orbital angular momentum quantum number l.

0 s

1 p

2 d

3 f

4 g

5 h

6 7 k

8 1

2 ×n−1

∑l=0

(2l + 1) = 2n2

• Wavefunctions for a hydrogen-like atom

1828 The hydrogen atom

Table 2.2 Radial hydrogenic wavefunctions Rn,l in terms of the variable ρ =Zr/(na0), which gives a scaling that varies with n. The Bohr radius a0 is defined ineqn 1.40.

R1,0 =

(Za0

)3/22 e−ρ

R2,0 =

(Z

2a0

)3/22 (1− ρ) e−ρ

R2,1 =

(Z

2a0

)3/2 2√3

ρ e−ρ

R3,0 =

(Z

3a0

)3/22

(1− 2ρ + 2

3ρ2)

e−ρ

R3,1 =

(Z

3a0

)3/2 4√

23

ρ

(1− 1

2ρ

)e−ρ

R3,2 =

(Z

3a0

)3/2 2√

2

3√

5ρ2 e−ρ

Normalisation:

∫ ∞

0

R2n,l r2 dr = 1

These show a general a feature of hydrogenic wavefunctions, namelythat the radial functions for l = 0 have a finite value at the origin, i.e.the power series in ρ starts at the zeroth power. Thus electrons withl = 0 (called s-electrons) have a finite probability of being found at theposition of the nucleus and this has important consequences in atomicphysics.

Inserting |E| from eqn 2.20 into eqn 2.17 gives the scaled coordinate

ρ =Z

n

r

a0, (2.21)

where the atomic number has been incorporated by the replacemente2/4πϵ0 → Ze2/4πϵ0 (as in Chapter 1). There are some important prop-erties of the radial wavefunctions that require a general form of thesolution and for future reference we state these results. The probabilitydensity of electrons with l = 0 at the origin is

|ψn,l=0 (0)|2 =1π

(Z

na0

)3. (2.22)

For electrons with l ̸= 0 the expectation value of 1/r3 is〈

1r3

〉=∫ ∞

0

1r3

R2n,l (r) r2 dr =

1l(l + 12

)(l + 1)

(Z

na0

)3. (2.23)

These results have been written in a form that is easy to remember;they must both depend on 1/a30 in order to have the correct dimensionsand the dependence on Z follows from the scaling of the Schrödinger

Radial functions

May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03

20 Astronomical Spectroscopy

3.5 Wavefunctions for HydrogenThe Schrödinger equation (3.4) can be solved analytically by working inspherical polar coordinates, i.e. r = (r, θ, φ). In these coordinates the wave-function is separable into radial and angular solutions

ψ(r, θ, φ) = Rnl(r)Ylm(θ, φ) . (3.7)

The radial solutions, Rnl, can be expressed analytically in terms ofLaguerre polynomials [see Rae (2002) in further reading]. Figure 3.1 showsboth the wavefunctions and the probability distribution of the lowest few

0 10 20

0.5

1

1.5

0 10 20 30 400

0.5

1

1.5

00.20.40.60.8

0 10 20 30 400

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

0 10 20 30 400

0.05

0.1

0.15

0.2

0 10 20 30 40 50

0

0.1

0.2

0 10 20 30 400

0.05

0.1

0.15

0.2

(rRnl(r))2Rnl(r)

1s

r (a0)

2s

1s

2s

2p 2p

3s 3s

0 30 40

0 10 20 30 40

0 10 20 30 40

Fig. 3.1. Wavefunctions (left) and probability distribution (right) for the radialcoordinate of the hydrogen atom. (T.S. Monteiro, private communication.)

May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03

20 Astronomical Spectroscopy

3.5 Wavefunctions for HydrogenThe Schrödinger equation (3.4) can be solved analytically by working inspherical polar coordinates, i.e. r = (r, θ, φ). In these coordinates the wave-function is separable into radial and angular solutions

ψ(r, θ, φ) = Rnl(r)Ylm(θ, φ) . (3.7)

The radial solutions, Rnl, can be expressed analytically in terms ofLaguerre polynomials [see Rae (2002) in further reading]. Figure 3.1 showsboth the wavefunctions and the probability distribution of the lowest few

0 10 20

0.5

1

1.5

0 10 20 30 400

0.5

1

1.5

00.20.40.60.8

0 10 20 30 400

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

0 10 20 30 400

0.05

0.1

0.15

0.2

0 10 20 30 40 50

0

0.1

0.2

0 10 20 30 400

0.05

0.1

0.15

0.2

(rRnl(r))2Rnl(r)

1s

r (a0)

2s

1s

2s

2p 2p

3s 3s

0 30 40

0 10 20 30 40

0 10 20 30 40

Fig. 3.1. Wavefunctions (left) and probability distribution (right) for the radialcoordinate of the hydrogen atom. (T.S. Monteiro, private communication.)

26 Astronomical Spectroscopy (Third Edition)

is the electron orbital angular momentum quantum number. The actual angular momentum is given by n[l(l + 1)]½. l can take the values 0, 1, 2, ... , n - 1. By convention, the values of l are usually designated by letters (see Table 3.1).

m is the magnetic quantum number , so called because it determines the behaviour of the energy levels in the presence of a magnetic field. mn is the projection of the electron orbital angular momentum , given by l , along the z-axis of the system. It can take (2l + 1) values -l, -l + 1, ... , 0, ... , l - 1, l.

2 s 1=0 p 1=1 p 1=1

m=O

B m=O m=1

0 0 m N -2

d 1=2 d 1=2 d 1=2 m=O

~ m= + 0

-

-2 -2 0 -2 0 -2 0 2

X

Fig. 3.2. Wavefunctions for the angular motions of the hydrogen atom which are given in terms of spherical harmonics , Yi, m(0, ). Plots are for the x - z plane except for the l = 2, m = 2 plot for which the x--y plane is shown. The plots give the absolute value of the spherical harmonic as a distance from the origin for each angle; the signs indicate the sign of the wavefunction in each region. (S.A. Morgan, private communication.)

Table 3.1. Letter designations for orbital angular momentum quantum number l.

0 s

1 p

2 d

3 f

4 g

5 h

6 7 k

8 1

Spherical harmonics

• The spherical harmonics are eigenfunctions of the orbital angular momentum operator .

• The sizes of the angular moment and � -component are• Wavefunction (with � ) at �

• Probability:

• Properties of the spherical harmonics:


• This implies that closed shells are spherically symmetric and have very little interaction with external electrons.

zl = 0 r = 0

19

L = r⇥ pAAACDnicbVDLSsNAFJ3UV62vqEs3g6XgqiS1oBuh4MaFiwr2AW0ok+mkHTp5MHMjlJAvcOOvuHGhiFvX7vwbJ20K2npg4Mw593LvPW4kuALL+jYKa+sbm1vF7dLO7t7+gXl41FZhLClr0VCEsusSxQQPWAs4CNaNJCO+K1jHnVxnfueBScXD4B6mEXN8Mgq4xykBLQ3MSt8nMHa95Da9WlCZ9oH7TC3+UTowy1bVmgGvEjsnZZSjOTC/+sOQxj4LgAqiVM+2InASIoFTwdJSP1YsInRCRqynaUD0OCeZnZPiilaG2AulfgHgmfq7IyG+UlPf1ZXZhmrZy8T/vF4M3qWT8CCKgQV0PsiLBYYQZ9ngIZeMgphqQqjkeldMx0QSCjrBkg7BXj55lbRrVfu8Wrurlxv1PI4iOkGn6AzZ6AI10A1qohai6BE9o1f0ZjwZL8a78TEvLRh5zzH6A+PzBwDznU4=

|L| =pl(l + 1)~, |Lz| = m~

AAACF3icbVBLS8NAGNz4rPUV9eglWISKUpJa0Euh4MVDDxXsA5oQNttNu3TzcHcj1KT9FV78K148KOJVb/4bt2kO2jqwMDvzfezOOCElXOj6t7K0vLK6tp7byG9ube/sqnv7LR5EDOEmCmjAOg7kmBIfNwURFHdChqHnUNx2hldTv32PGSeBfytGIbY82PeJSxAUUrLVUmJ6UAwcN66Pk6rJ75iIaZGeGidjc+BAdjaZJHX7Ial66dVWC3pJT6EtEiMjBZChYatfZi9AkYd9gSjkvGvoobBiyARBFI/zZsRxCNEQ9nFXUh96mFtxmmusHUulp7kBk8cXWqr+3oihx/nIc+TkNASf96bif143Eu6lFRM/jAT20ewhN6KaCLRpSVqPMIwEHUkCESPyrxoaQAaRkFXmZQnGfORF0iqXjPNS+aZSqFWyOnLgEByBIjDABaiBa9AATYDAI3gGr+BNeVJelHflYza6pGQ7B+APlM8fus2gOA==

| (r, ✓,�)|2d3x = R2nl(r)|Ylm(✓,�)|2r2 sin ✓drd✓d�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

Z ⇡

0

Z 2⇡

0|Ylm(✓,�)|2 sin ✓d✓d� = 1

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

lX

m=�l|Ylm(✓,�)|2 =

2l + 1

4⇡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

R10(r) =2

(a0/Z)3/2e�r/(a0/Z)

R20(r) =1

(2a0/Z)3/2

✓2� r

a0/Z

◆e�r/(2a0/Z)

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

Ylm(✓,�) =

r1

4⇡AAACEnicbVDLSsNAFJ3UV62vqEs3wSK0ICWpBd0IBTcuK9iHNCFMppN26OThzI1QQr7Bjb/ixoUibl2582+ctllo64ELh3Pu5d57vJgzCab5rRVWVtfWN4qbpa3tnd09ff+gI6NEENomEY9Ez8OSchbSNjDgtBcLigOP0643vpr63QcqJIvCW5jE1AnwMGQ+IxiU5OrVOzflQVaxYUQBn9rxiFUvbXkvILV9gUlqZWnDjlmWuXrZrJkzGMvEykkZ5Wi5+pc9iEgS0BAIx1L2LTMGJ8UCGOE0K9mJpDEmYzykfUVDHFDppLOXMuNEKQPDj4SqEIyZ+nsixYGUk8BTnQGGkVz0puJ/Xj8B/8JJWRgnQEMyX+Qn3IDImOZjDJigBPhEEUwEU7caZIRVEqBSLKkQrMWXl0mnXrPOavWbRrnZyOMooiN0jCrIQueoia5RC7URQY/oGb2iN+1Je9HetY95a0HLZw7RH2ifP9hHni0=

This indicates that the electron can interact with the nucleus � hyperfine structure!→

| n,l=0(0)|2 =1

⇡

✓Z

na0

◆36= 0

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

L2Ylm = l(l + 1)~2Ylm, LzYlm = m~YlmAAACMXicbVBLSwMxGMzWV62v+rh5CRahopTdKuhFKHjpoYcK9iFtXbJptg1NdpckK9Sl/Ule/CfipQdFvPonTLfFR+tAYDLzfSQzTsCoVKY5MhILi0vLK8nV1Nr6xuZWenunKv1QYFLBPvNF3UGSMOqRiqKKkXogCOIOIzWndzX2a/dESOp7N6ofkBZHHY+6FCOlJTtdbHKkuo4blQZ3eXhrR4wP4CVkWXZsHTW7DhLf8slwOCzZDz9DPPandzudMXNmDDhPrCnJFPbcGGU7/dxs+zjkxFOYISkblhmoVoSEopiRQaoZShIg3EMd0tDUQ5zIVhQnHsBDrbSh6wt9PAVj9fdGhLiUfe7oyXE+OeuNxf+8Rqjci1ZEvSBUxMOTh9yQQeXDcX2wTQXBivU1QVhQ/VeIu0ggrHTJKV2CNRt5nlTzOes0l7/WbZyBCZJgHxyALLDAOSiAIiiDCsDgEbyAV/BmPBkj4934mIwmjOnOLvgD4/MLxv+rpA==

• The spectrum of the hydrogen atom comes from electrons jumping between different levels in the atom.- If very small contributions from the electron and nuclear spins are neglected, the energy levels

depend only on the principal quantum number � .

- The electronic spectrum of the H atom comes from changes in � .

- The wavelength, � , for the transition � is given by the Rydberg formula:

• Spectral series of the H atom- The spectrum of H is divided into a number of series linking different upper levels � with a single lower

level � value. Each series is denoted according to its � value and is named after its discoverer.- Within a given series, individual transitions are labelled by Greek letters.

nn

λ n2 → n1

n2n1 n1

H-atom Spectra 20

1λ

= RH ( 1n21 −1n22 ), where n2 > n1 and RH =

4/31215.68Å

= 1.09677 × 107 m−1

Atomic Hydrogen 29

Table3.2. Spectral series of the H atom. Each series comprises the transitions n2 - n1, where n1 < n2 < oo.

Range (cm- 1 )

n1 Name Symbol Spectral region n2 = n1 + 1 n2 = oo 1 Lyman Ly ultraviolet 82257 109677 2 Balmer H visible 15237 27427 3 Paschen p infrared 5532 12186 4 Brackett Br infrared 2468 6855 5 Pfund Pf infrared 1340 4387 6 Humphreys Hu infrared 808 3047

according to its n 1 value and is named after its discoverer. Table 3.2 sum-marises the six main H-atom series.

The range of each H-atom series is given by the lowest frequency transi-tion, between the levels n1 and n2 = n1 + 1, n2 = n1 + 2, n2 = n1 + 3 and so on up to the series limit, which is given by the transition between n 1 and n 2 = oo. As discussed in Sec. 3.8.1, the series limit is not always observable. Table 3.2 gives the spectral region in which each series is observed. As the Balmer series lies in the visible region, it is particularly easy to observe from Earth. As a result, Balmer lines have been particularly important in the study of H-atom spectra.

Within a given series, individual transitions are labelled by Greek letters. These letters denote the change in n or D.n. In this notation:

D.n = 1 is a, D.n = 2 is (3, D.n = 3 is 'Y, D.n = 4 is 8, D.n = 5 is 1:.

Thus Lya is the transition between n 1 = 1 and n 2 = 2, and H"( is that between n 1 = 2 and n 2 = 5. Greek letters are usually only used for the most important transitions with low D.n. Transitions with high D.n are commonly labelled by the number n 2 . Thus, H15 is the Balmer series tran-sition between n1 = 2 and n2 = 15.

Atomic Hydrogen 29

Table3.2. Spectral series of the H atom. Each series comprises the transitions n2 - n1, where n1 < n2 < oo.

Range (cm- 1 )

n1 Name Symbol Spectral region n2 = n1 + 1 n2 = oo 1 Lyman Ly ultraviolet 82257 109677 2 Balmer H visible 15237 27427 3 Paschen p infrared 5532 12186 4 Brackett Br infrared 2468 6855 5 Pfund Pf infrared 1340 4387 6 Humphreys Hu infrared 808 3047

according to its n 1 value and is named after its discoverer. Table 3.2 sum-marises the six main H-atom series.

The range of each H-atom series is given by the lowest frequency transi-tion, between the levels n1 and n2 = n1 + 1, n2 = n1 + 2, n2 = n1 + 3 and so on up to the series limit, which is given by the transition between n 1 and n 2 = oo. As discussed in Sec. 3.8.1, the series limit is not always observable. Table 3.2 gives the spectral region in which each series is observed. As the Balmer series lies in the visible region, it is particularly easy to observe from Earth. As a result, Balmer lines have been particularly important in the study of H-atom spectra.

Within a given series, individual transitions are labelled by Greek letters. These letters denote the change in n or D.n. In this notation:

D.n = 1 is a, D.n = 2 is (3, D.n = 3 is 'Y, D.n = 4 is 8, D.n = 5 is 1:.

Thus Lya is the transition between n 1 = 1 and n 2 = 2, and H"( is that between n 1 = 2 and n 2 = 5. Greek letters are usually only used for the most important transitions with low D.n. Transitions with high D.n are commonly labelled by the number n 2 . Thus, H15 is the Balmer series tran-sition between n1 = 2 and n2 = 15.

�n ⌘ n2 � n1AAACAnicbVC7SgNBFJ31GeNrjZXYDAmCjWE3CloGTGEZwTwgCcvs5CYZMju7zswGwhJs/BUtLBSx1T+wsvNvnDwKTTxwL4dz7mXmHj/iTGnH+baWlldW19ZTG+nNre2dXXsvU1VhLClUaMhDWfeJAs4EVDTTHOqRBBL4HGp+/3Ls1wYgFQvFjR5G0ApIV7AOo0QbybMPmiXgmmCBm3AbswEWXgGfmO56ds7JOxPgReLOSK6Y+vzIlB6zZc/+arZDGgcgNOVEqYbrRLqVEKkZ5TBKN2MFEaF90oWGoYIEoFrJ5IQRPjJKG3dCaUpoPFF/byQkUGoY+GYyILqn5r2x+J/XiHXnopUwEcUaBJ0+1Ik51iEe54HbTALVfGgIoZKZv2LaI5JQbVJLmxDc+ZMXSbWQd0/zhWuTxhmaIoUOURYdIxedoyK6QmVUQRTdoQf0jF6se+vJerXepqNL1mxnH/2B9f4DnRyYig==

Lyman series : �Balmer series : �Paschen series: �Brackett series : �

Transitions with high � are labelled by the � . Thus, � is the Balmer series transition between � and � .

Lyα, Lyβ, Lyγ, ⋯Hα, Hβ, Hγ, ⋯Pα, Pβ, Pγ, ⋯Brα, Brβ, Brγ, ⋯

Δnn2 H15

n1 = 2 n2 = 15

21

Atomic Hydrogen 27

s is the electron spin quantum number. The electron spin angular momen-tum is given by n[s(s + 1)]½ which, for a one-electron system, equals 4"n, since an electron always has spin one-half.

Sz gives the projection of the electron spin angular momentum, given by s, along the z-axis of the system. This projection is actually nsz. In general, Sz can take (2s + 1) values given by -s , -s + 1, .. . , s - 1, s. For a one-electron system, this means Sz can take one of two values: -½

1 or +2 .

The simplest notations for the various states of H is to denote each state by its nl quantum numbers. Thus the ground state is denoted ls; the first excited states are 2s and 2p; the n = 3 states are 3s, 3p and 3d. (See Fig. 3.3.) These notations leave them and Sz quantum numbers unspecified since these quantum numbers are really only significant for H in the presence

!

I

6 4

100000-

7 5

3

2 80000-

eo000-

40000-

20000-

I I

I

I I I I I I I I

l !Bj S>Of? i I I "' ~;!ll l!l I O'Q. ,.."°.., .~ a,,_,

o-,-~~~~-

I 11 I I - - :§ i~ :! ~~a,! ~gs &1 1 --- ... u:.i .., .., Brackett

Paschen

H Hydrogen Z • I

...... "' ..... "' "'"' ... "'

22

Model spectrum of an A5-type star showing both the Balmer and Paschen discontinuities (or jumps, breaks).

The Balmer discontinuity is prominent in A-type stars.

30 Astronomical Spectroscopy (Third Edition)

60000 r--.----,--.--,---.---r---r---,--r-----.---,.--.-----,--.---,---,--,--,---,---,

01 B Ori

50000

Ul J 40000

30000

Hl2 Hll HIO HE Ho H9 Hr

20000 H8

3600 3800 4000 4200 4400 A (A)

Fig. 3.4. Spectrum of B-type star 0 1 B Ori showing Balm er series absorptions record ed at the Anglo-Australian Telescop e . (X .-W. Liu , private communication.)

The wavelength of each individual transition can be predicted using the Rydberg formula. For example, Lya lies at

1 ( 1) 3 1 ~=RH 1 - 4 = 4RH = 82258.2 cm- , (3.11) meaning that

A = 1.21568 x 10- 5 cm = 1215.68 A = 121.568 nm.

All hydrogen series transitions between bound states are described as bound - bound transitions. Figures 3.4- 3. 7 give sample H-atom spec-tra recorded in very different astronomical environments. Figure 3.4 shows an optical spectrum of the B-type star 8 1 B Ori . Absorption by the Balmer series up to H14 (n 2 = 14) is clearly visible. Figure 3.5 shows Lyman series absorption in a shell of gas expanding about a hot, Wolf- Rayet star. Figure 3.6 shows higher members of the Paschen series recorded in absorption in a spectrum from a soft X-ray transient binary

Spectrum of B-type star � B Ori showing Balmer series absorption lines

Θ1

Atomic Hydrogen 37

10 8 ,----.....-----,,-----,------.---,-----.---,----,----,

10 6 '-----'---''-----'----'----'----'----'-----:-'":---:-'"'::-----::-:'. 100 200 300 400 500 600 700 800 900 1000

Wavelength (nm)

Fig. 3.11 . Model spectrum of an A5-type star showing both the Balmer and Paschen discontinuities. (R.J. Sylvester, private communication.)

the complete series is never observed in stars and the spectrum shows a discontinuity known as the Balmer jump (see Fig. 3.11). The jump occurs at the point where continuous (bound - free) absorption switches on. The position of the Balmer discontinuity shifts to longer wavelengths at higher pressure.

To understand why, it is necessary to consider the physical size of the H atom. The approximate radius of the electron orbit in the Bohr atom, rn, is given by the formula:

(3.16)

where for H(ls), r 1 = 1 a0 and a0 = 0.529 x 10- 10 m. At high pressure, the mean atom - atom distance will be smaller than some rn, and higher orbits will not be found since they will be destroyed by atom - atom collisions.

Worked Example: The Sun has a number density, N, of about 1017 cm- 3 . What is the highest H-atom n level that one would expect to find?

Balmer break ~ 3646Å

Paschen break ~ 8204Å

H �α

H �βH �γ

• Atoms contain several sources of angular momentum.- electron orbital angular momentum �

- electron spin angular momentum �

- nuclear spin angular momentum �- The nuclear spin arises from the spins of nucleons. Protons and neutrons both have

an intrinsic spin of a half.

• As in classical mechanics, only the total angular momentum is a conserved quantity.- It is therefore necessary to combine angular momenta together.

• Addition of two angular momenta:- The orbital and spin angular momenta are added vectorially as � . This gives

the total electron angular momentum.- One then combines the total electron and nuclear spin angular momenta to give the

final angular momentum � .

LSI

J = L + S

F = J + I

Angular Momentum Coupling 23

- In classical mechanics, adding vector � and vector � gives a vector � , whose length must lie in the range

- In quantum mechanics, a similar rule applies except that the results are quantized. The allowed values of the quantized angular momentum, � , span the range from the sum to the difference of � and � in steps of one:

- For example, add the two angular momenta � and � together to give � . The result is

a b c

ca b

L1 = 2 L2 = 3L = L1 + L2

24

c = |a − b | , |a − b | + 1, ⋯ , a + b − 1, a + b

|a − b | ≤ c ≤ a + b Here, � are the lengths of their respective vectors.a, b, c

L = 1, 2, 3, 4, 5.

c = |a−b | c = a+b

• So far the discussion on H-atom levels has assumed that all states with the same principal quantum number, � , have the same energy.- However, this is not correct: inclusion of relativistic (or magnetic) effects split these levels

according to the total angular momentum quantum number � . The splitting is called fine structure.

• For hydrogen,

• Spectroscopic notation:

- The above table shows the fine structure levels of the H atom.- Note that the states with principal quantum number � give rise to three fine-structure levels. In

spectroscopic notation, these levels are � , � and � .

n

J

n = 222S1/2 22Po1/2 2

2Po3/2

The Fine Structure of Hydrogen 25

S =12

→ J = L ±12

(2S+1)LJconfiguration L S J term level

ns 0 1/2 1/2 2S 2S1/2

np 1 1/2 1/2, 3/2 2Po 2P

o1/2,

2Po3/2

nd 2 1/2 3/2, 5/2 2D 2D3/2,2D5/2

nf 3 1/2 5/2, 7/2 2Fo 2F

o5/2,

2Do7/2

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

Note that the levels are called to be

singlet if 2S+1 = 1

doublet if 2S+1 = 2

triplet if 2S+1 = 3

(when L > 0)

S = 0, J = LS = 1/2, J = L ± 1/2

S = 1, J = L − 1, L, L + 1

• Fine structure of the hydrogen atom

• Splitting in the n = 2 levels of atomic hydrogen. The larger splitting is the fine structure and the smaller one the Lamb shift.- According to the Dirac equation, the 2S1/2 and 2P1/2 levels

should have the same energies.- However, the interaction between the electron and the vacuum

(which is not accounted for by the Dirac equation) causes a tiny energy shift on 2S1/2. (Quantum electrodynamics effect)

26

May 17, 2005 14:40 WSPC/SPI-B267: Astronomical Spectroscopy ch03

Atomic Hydrogen 45

Table 3.5. Fine structure effects in the hydrogen atom: splitting ofthe nl orbitals due to fine structure effect for l = 0,1, 2, 3. The result-ing levels are labelled using H atom, and the more general spectro-scopic notation of terms and levels (see Sec. 4.8).

Configuration l s j H atom Term Level

ns 0 1212 ns 12 n

2 S n 2 S 12

np 1 1212 ,

32 np 12 , np 32 n

2 Po n 2 Po12, n 2 Po3

2

nd 2 1232 ,

52 nd 32 , nd 52 n

2 D n 2 D 32, n 2 D 5

2

nf 3 1252 ,

72 nf 52 , nf 72 n

2 Fo n 2 Fo52, n 2 Fo7

2

For hydrogen, s = 12 so that, except for the l = 0 case, j = l ±12 (see

Table 3.5). This table labels the resulting levels with the common H-atomnotation nl j, where l is given by its letter designations, s, p, d, etc., andby spectroscopic notation for which labels of the (2 S +1)LJ are used. A fulldiscussion of spectroscopic notation can be found in Sec. 4.8.

Table 3.5 shows the fine structure levels of the H atom. This tableshows that the states with principal quantum number n = 2 give rise tothree fine-structure levels. In spectroscopic notation, these levels are 2 2 S 1

2,

2 2 Po12

and 2 2 Po32.

So far the discussion on H-atom levels has assumed that all those withthe same principal quantum number, n, have the same energy. In otherwords, the energy does not depend on l or j. This is not correct: inclusionof relativistic (or magnetic) effects split these levels according to the totalangular momentum quantum number j. This splitting, called ‘fine struc-ture’, has been well-studied in the laboratory. An even more subtle effectcalled the Lamb shift, which is due to quantum electrodynamics, can also beobserved. Values of these splittings for the n = 2 levels are given in Fig. 3.21.

0.365 cm–1

0.035 cm–12 2P1/2

2 2S1/2

2 2P3/2

Fig. 3.21. Splitting in the n = 2 levels of atomic hydrogen. The larger splitting isthe fine structure and the smaller one the Lamb shift.

configuration L S J term level

ns 0 1/2 1/2 2S 2S1/2

np 1 1/2 1/2, 3/2 2Po 2P

o1/2,

2Po3/2

nd 2 1/2 3/2, 5/2 2D 2D3/2,2D5/2

nf 3 1/2 5/2, 7/2 2Fo 2F

o5/2,

2Do7/2

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

12S1/2

22S1/2 = 22P1/2

22P3/2

Relativistic QM (Dirac’s eq)

Quantum Electrodynamics

What is the origin of the Spin-Orbit interaction (fine structure splitting)?

• Fine-structure splitting: Relativistic effects couple electron orbital angular momentum and electron spin to give the so-called fine structure in the energy levels. Inclusion of relativistic effects splits the terms into levels according to their J value.

• When the electron will move around the nucleus with a non relativistic velocity v, the electric field exerting on the electron will be . (Note that the nucleus has a positive charge � .)

- In the electron rest frame, this electric field will be perceived as a magnetic field

• This magnetic field will interacts with the electron’s magnetic moment, which is

Ze

27

Here, the magnetic field is perpendicular to the electron’s orbital plane.

(where � is the orbital angular momentum of electron)ℓ ≡ r × p = mer × v

μ = −e

mecs

B′� = B⊥ = γ (B⊥ − β × E)= −

vc

E = −Zec

v × rr3

=Ze

mecr3ℓ

E = Zerr3

• Then, the interaction energy is

• For the sum of the interactions of all electrons will be

- The individual spin and orbital angular momenta may be averaged over in such a way that an equivalent interaction is simply

- From the relation , we obtain

28

U = �µ ·B = Ze2

m2ec2r3

s · `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

Hso =X

i

⇠i (si · `i)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

Hso = ⇠ (S ·L) where S =X

i

si, L =X

i

`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

J2 = |L+ S|2 = L2 + S2 + 2S · 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

Hso =1

2⇠�J2 � L2 � S2

�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

U = �µ ·BAAACC3icbVDLSsNAFJ3UV42PRl26CS2CLixJFXQjFkVwWcG0haaUyWTSDp1kwsxEKCF7N/6KGxeKuvUH3PkJ/oWTtgttvTDM4Zx7ueceL6ZESMv60goLi0vLK8VVfW19Y7NkbG03BUs4wg5ilPG2BwWmJMKOJJLidswxDD2KW97wMtdbd5gLwqJbOYpxN4T9iAQEQamonlF2zg5dj1FfjEL1pW6YZC7ymXRDKAdekF5kPaNiVa1xmfPAnoJKvfR9fqC/XTV6xqfrM5SEOJKIQiE6thXLbgq5JIjiTHcTgWOIhrCPOwpGMMSim45vycw9xfhmwLh6kTTH7O+JFIYi96o6c4diVsvJ/7ROIoPTbkqiOJE4QpNFQUJNycw8GNMnHCNJRwpAxInyaqIB5BBJFZ+uQrBnT54HzVrVPqrWblQax2BSRbALymAf2OAE1ME1aAAHIHAPHsEzeNEetCftVXuftBa06cwO+FPaxw9zoZ4x

µ = IAAAAB7XicbVDJSgNBEO2JWxyXRD16aQyCXsJMFPQiRkTQWwSzQDKEnk5P0qaXobtHCEP+wYsHRbz6Ef6FNz/Bv7CzHDTxQcHjvSqq6oUxo9p43peTWVhcWl7Jrrpr6xubufzWdk3LRGFSxZJJ1QiRJowKUjXUMNKIFUE8ZKQe9i9Hfv2BKE2luDODmAQcdQWNKEbGSrUWT85uLtr5glf0xoDzxJ+SQjn3fX7oflxV2vnPVkfihBNhMENaN30vNkGKlKGYkaHbSjSJEe6jLmlaKhAnOkjH1w7hvlU6MJLKljBwrP6eSBHXesBD28mR6elZbyT+5zUTE50GKRVxYojAk0VRwqCRcPQ67FBFsGEDSxBW1N4KcQ8phI0NyLUh+LMvz5NaqegfFUu3No1jMEEW7II9cAB8cALK4BpUQBVgcA8ewTN4caTz5Lw6b5PWjDOd2QF/4Lz/AI+3kVc=