Lecture 16. Emission and Absorption Spectra of Hydrogen

Outline:

Emission and Absorbtion Spectra

Orbital Magnetic Momentum, Magnetic Dipole in External Magnetic Field

Normal Zeeman Effect

Spin, Spin-Orbit Coupling, and Fine Splitting of Spectral Lines

Hyperfine Splitting

Emission

Emission and Absorption Spectra

sample of excited

gas

source of broad-band

radiation

sample of gas

(e.g., the black-body radiation source)

When the frequency of incident radiation matches one of the transition frequencies, gas atoms absorb photons and re-emit them in arbitrary directions (also non-radiative relaxation is possible). As a result, the intensity of transmitted light sharply drops (black lines against bright background).

Absorption

Emission and Absorption spectra of Hydrogen gas

in the visible range

Spectral Lines for H Atom

13.6HhcR eV

- the ionization energy of H atom

The wavelength of the transition betweenni = and nf =1:

2 21

1 1 1

1H HR R

2

1n HE hcR

n

47 1

2 30

1.1 108H

meR m

h c

41

2 2 2 20

1

8e

n

m e EE

h n n

energy spectrum of H atom:

All transitions that satisfy the selection rules can emit dipole radiation (or be

excited by incident photons)

1

0, 1l

l

m

4

2 2 2 20

1 1

8i f

en n

f i

m ehcE E

h n n

The emitted energy:

The corresponding photon wavelength:

4

2 3 2 2 2 20

1 1 1 1 1

8i f

Hn n f i f i

meR

h c n n n n

- the Rydberg constant Rydberg discovered the dependence 1/(1/m2-1/n2) in 1888, this helped Bohr

to develop his model of atoms.

Spectral Series for H Atom (cont’d)All spectral lines fall into the so-called spectral series: all lines within the spectral series correspond to the same final state.

Lyman series:2 2

1

1 1 1

1Hn

Rn

Balmer series:2 2

2

1 1 1

2Hn

Rn

BrackettPfund

etc., etc.

Note that the spectral lines are due to all transitions between ni and nf allowed by selection rules: e.g., the Balmer line with max corresponds to

Balmer:

0, 1lm

1: 1 0, 2 1, 0 1i fl l l

and all combinations with

Correspondence Principle

The predictions of quantum mechanics approach those of classical physics in the limit of large quantum numbers.

In particular, this implies that the QM description of radiative transitions between levels with large ni and nf should give the same frequency of emitted radiation as the classical description (see §4.6). Let’s take the transition between the principal quantum numbers ni=101 and nf=100. The radius of Bohr orbit for these states

2 40 10 0.05 0.5nr n a nm m

The frequency which a “classical” electron should emit while following this orbit:

92 /2 /1

7 102 2 2

eeE mK mv

f K E HzT r r r

This is in line with the exact QM result:

9101 100 2 2

1 13.6 1 16.6 10

100 101

eVf E E Hz

h h

42 10

Problems

Beiser 4.21 A beam of electrons bombards a sample of hydrogen. Through what potential difference must the electrons have been accelerated if the first line of the Balmer series is to be excited?

3 2 4 2E E E E E

4 2 2 2

1 1 32.55

2 4 16H HE E hcR hcR eV 1.89 2.55V V V

2. Which Lyman spectral lines will be emitted if H atoms are excited by UV radiation with =100nm?

7 1 7 111 10 0.9 1.1 10H Hm R R m

2 2

1 10.9 1 0.1 2,3H HhcR hcR n

n n

1nE E E

112

1

1 1 412 3H HR R

1

222

1 1 913 8H HR R

3 2 2 2

1 1 5 513.6 1.89

2 3 36 36H HE E hcR hcR eV eV

H Atom in External Magnetic Field

So far, we’ve considered the simplest case: a spin-less electron in a Coulomb potential, no external magnetic field degenerate spectrum, only the principle quantum number, n, matters.

In the external magnetic field, the degeneracy of the spectrum is lifted and the spectral line splitting is observed (the Zeeman effect). We’ll consider the simplest case, the so-called “normal” Zeeman effect. Note that the spectral line splitting is small, and we need a high resolution of our measuring setup (e.g., a diffraction grating) to resolve it.

Angular Momentum and Magnetic MomentClassically speaking, an orbiting electron in H atom is equivalent to a circular current. The magnetic moment of a current loop: I area

Current: charge which flows through the wire cross-section in unit time:

1sI q qf

T f – the frequency of electron

rotation around the nucleus

2 22 2e

e

vr eef r rf v e L m vr L

m

2ee

eL

m

2 e

e

m

– gyromagnetic ratio for electrons (minus: L is anti-parallel to because of -e)

e

In Q.M, orbits do not exist. Instead, the magnetic moment is related to the operator

ˆ ˆ ˆ2

qr p

m

This operator coincides (within a constant q/2m) with the operator of angular moment, their properties are identical: e.g., the values of projections of along z axis are quantized: 2 2z z l

e e

e eL m

m m

– the orbital magnetic moment of an electron

Energy of Magnetic Dipole in Magnetic Field

0B B

orbital moment “up”, magnetic moment

“down”

Potential energy of a magnetic dipole in external magnetic field: cosU B B

2 B

is parallel to B

is anti-parallel to B

24 59.3 10 / 5.8 10 /2B

e

eJ T eV T

m

Bohr magneton

2 2z z le e

e eL m

m m

For an electron in H atom:

This gives the scale of the effect: even in a strong magnetic field (10T), this splitting is only ~ 10-3 eV (compare with the inter-level distance ~ 1 eV)

B

H atom in a magnetic field B (spin being ignored): an example of a non-central potential

2

04coulomb magnetic

eU U U B

r

Degeneracy of the spectrum is partially lifted (states with the same n and ml and different l still have the same energy).

2 e

eL

m

- magnetic moment due to the orbital motion of an electron

B II z:2 2magnetic z l

e e

e eU BL Bm

m m

22

, 2 20

1

2 4 2l

en m l

e

me eE m B

n m

- dependent on n and ml but not l

5, 2

13.65.8 10 /

ln m l

eVE eV T m B

n

CoulombB=0

nE E

1, 0n l

2, 0,1n l

H atom

CoulombB0

, ln mE E

1lm 0lm 1lm

Not-to-scale !

2lm

0lm 1lm

1lm

2lm

3, 0,1, 2n l

- axial (but not central) symmetry

CoulombB=0

nE E

1, 0n l

2, 0,1n l

H atom

CoulombB0

, ln mE E

22

, 2 20

1

2 4 2l

en m l

e

me eE m B

n m

1lm 0lm 1lm

10 Not-to-scale20 Spin is ignored

2lm

0lm 1lm

1lm

2lm

3, 0,1, 2n l

1n

2n

1, 1ll m 0, 0ll m 1, 0ll m 1, 1ll m

Which radiative transitions are allowed for n=2 electron in H atom in magnetic field?

1

0, 1l

l

m

0, 0ll m 1

1

1l

n

l

m

1

1

0l

n

l

m

1

0

0l

n

l

m

1

1

1l

n

l

m

Selection rules:

0

1

1l

n

l

m

0sm

H atom in magnetic field

these dipole radiation transitions are also allowed,

but they correspond tothe MW radiation

Problem (Midterm 2, 2009)A hydrogen atom is placed in a magnetic field. Ignore the spin.

(5) Draw the energy level diagram (not to scale) for all states with principal quantum number n=1 and n=2.

(5) Show by arrows on this diagram all the allowed transitions with emission of a photon. Explain how you have determined which are allowed transitions.

(10) Calculate the wavelengths corresponding to these allowed transitions if the field is B=5T.

n=2

n=1

l=1, ml=1l=1, ml=0 l=0, ml=0l=1, ml=-1

B=0 B0

1 2 3

45

Selection rules:

l=1, ml=0, 1

, ,2 2

1 1 1

4 l i l ff i

eBR m m

n n mc

#2: n=2, l=1, ml=0 n=1, l=0, ml=0 7

22

1 31.2 10

4R m

#4: n=2, l=1, ml=1 n=2, l=0, ml=0 2 1 3

4,54,5

12.3 10 4.3 10

4

eBm m

mc

#5: n=2, l=0, ml=0 n=2, l=1, ml=-1

l=0, ml=0

- microwave range

#1: n=2, l=1, ml=-1 n=1, l=0, ml=0 #3: n=2, l=1, ml=1 n=1, l=0, ml=0

1,3

1 1 1 31

1 4 4 4 4

eB eBR R

mc mc

2 22

1 1

4

eB

mc

2 121,3 3.3 10

4

eBm

mc

Normal Zeeman Effect

If an atom is placed in magnetic field, the degeneracy of its levels that correspond to the same n, l but different ml and ms will be removed (lifted).

Discovery – 1896Nobel - 1902

1

0, 1l

l

m

Selection rules for the normal Zeeman effect:

1 0

2 0

3 0

/

/

B

B

f f B h

f f

f f B h

Spectral lines splitting:

Problem

The red Balmer series line in hydrogen (=656.5nm) is observed to split into three different spectral lines with =0.04nm between two adjacent lines when placed in a magnetic field B. What is the value of B?

3 2 1, 1, 0l

hcE E n l m

is due to the energy splitting between two adjacent ml states:

Two adjacent lines that correspond to ml=1 correspond to the emission of photons with energies:

3 2*1, 1, 1B l

hcE E B n l m

* 2 2B BB

hc hc hcB hc B B

34 8 9

22 24 9

6.62 10 3 10 / 0.04 102

9.3 10 / 656.5 10B

Js m s mhcB T

J T m

( list all the states that correspond to these energy levels)

Spin: Goudsmit & Uhlenbeck (1925), Dirac (1929) Both electron and proton possess intrinsic magnetic moments – spins. They belong to fermions: particles with half-integer spin.

1S s s

The electron spin magnetic moment :

, 1,..., 1,z s sS m m s s s s

Electrons andProtons:

3

2S

1

2s

2 1s 1

2z sS m

se

eS

m

The gyromagnetic ratio for spin (-e/me) differs from that for angular momentum (-e/2me). This illustrates the fact that spin cannot be reduced to spinning of a charged sphere. In fact, if one uses an upper experimental limit on the electron size (most likely, it’s a point particle), the linear speed of the sphere equator should exceed c by four orders of magnitude to produce the spin magnetic moment! (Ehrenfest, see Example 7.1)

2sz Be

e

m

The spin angular momentum and its component along the z axis are quantized:

Taking spin into account, we need four quantum numbers (n, l, ml, ms) to completely describe electron states in atoms.

ms=1/2 – “spin up”

ms=-1/2 – “spin down”

(The direction of B is chosen up, so the energy of “spin-up” electron with sz=- B will be lower by 2 B than that for a similar “spin-down” electron).

Complication I: Spin-Orbit Coupling

Classical consideration: In the rest frame of an electron, it feels the magnetic field created by the orbital motion of a proton. This magnetic field affects the electron states in H atom (spin-orbital coupling). As a result, even if there is no external magnetic field, all electron levels with l 0 are split.

We can estimate this splitting by calculating the magnetic field at the position of the electron due to the proton motion:

+ -

v

Thus, the splitting of levels due to the spin-orbital coupling is small (“fine”) with respect to E1 – and this is

why , which controls (among many other things) the splitting, was called “the fine structure constant”.

ep

Current (charge which flows through the wire cross-section in unit time):

1

2evs

I q eT r

19 6

700 2 210

1.6 10 3 104 10 20

2 2 1 10

eI evB T

r r

Magnetic field of a current loop:

5 313 10 / 20 1 10

2e BE B B eV T T eV

(!)

B

2 5

1

5.3 10BB

E

In the absence of external magnetic field, there is always an “internal” magnetic field (due to the “orbital motion” of charges in H atom), which results in the energy shift of all levels due to spin-orbit coupling; this shift, being dependent on the “orbital” motion, is different for the levels with different l .

The fine structure is due to the interaction between the magnetic moments associated with the electron spin and orbital momentum. When the spectral lines of the hydrogen spectrum at B = 0 are examined at high resolution, they are found to have a complicated structure.

The H-α line has a wavelength of about 656 nm, the spin-orbit coupling results in splitting+shift of only 0.015 nm.

H-α

Combined Effect of S-O Coupling and Mag. Field

0B B

Complication II: Hyperfine Structure

The proton is also a spin-1/2 particle. The hyperfine structure is due to the interactions of the proton spin magnetic moment with both the orbital and spin magnetic moments of an electron.

61.4 0.07 7 10GHz K eV

the

qu

an

tum

e

lect

rod

yna

mic

s co

rre

ctio

ns

Astronomers do!

The hyperfine splitting is typically ~100 times smaller than even the

fine splitting – who cares???

the

fin

e s

tru

ctu

re

the

hyp

erf

ine

str

uct

ure

This line is of great importance for astronomy. Because of its extremely small natural width (E~ ħ/t), this line can be used for precision mapping of velocities of hydrogen gas in the Universe (Doppler shift). Fortunately, the Earth's atmosphere is transparent for this frequency range.

The Pioneer plaques: Pioneer 10 (1972) and Pioneer 11 (1973)

21 cm (1.4 GHz) Hydrogen Line

1 23

Ur

1s

The transition between these states is forbidden by selection rules for dipole radiation. The transitions are due to (very weak) interaction between electron spin and the magnetic component of radiation. As the result, the probability of this transition is tiny: the lifetime of the electron in the upper state is ~11 million years (!) (compare with ~10−8 s for excited states that correspond to allowed optical transitions). However, as the total number of H atoms in the interstellar medium is huge, this emission line is easily observed by radio telescopes.

Cold hydrogen gas, all atoms are in their ground state. Because of the hyperfine splitting, the ground state (n=1, l=0, ml

=0, ms=±1/2) is a doublet.