Lecture 16' Emission and Absorbtion Spectra of Hydrogen

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Lecture 16. Emission and Absorbtion 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

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Emission and Absorption Spectra
Emission
sample of excited gas
Emission and Absorption spectra of Hydrogen gas
in the visible range
Absorption
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).
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Spectral Lines for H Atom
energy spectrum of H atom
All transitions that satisfy the selection rules
can emit dipole radiation (or be excited by
incident photons)
The emitted energy
The corresponding photon wavelength
- the Rydberg constant Rydberg discovered the
dependence 1/??(1/m2-1/n2) in 1888, this helped
Bohr to develop his model of atoms.
The wavelength of the transition between ni ?
and nf 1
- the ionization energy of H atom
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Spectral Series for H Atom (contd)
All spectral lines fall into the so-called
spectral series all lines within the spectral
series correspond to the same final state.
Lyman series
Balmer series
etc., etc.
Balmer
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
and all combinations with
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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). Lets take the transition
between the principal quantum numbers ni101 and
nf100. The radius of Bohr orbit for these states
The frequency which a classical electron should
emit while following this orbit
This is in line with the exact QM result
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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?
2. Which Lyman spectral lines will be emitted if
H atoms are excited by UV radiation with ?100nm?
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Complications spins, external fields, etc.
So far, weve 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 reality, the situation is more complicated.
The electrons magnetic moment is associated with
both orbital momentum and spin (an internal
degree of freedom). This magnetic moment
interacts with (a) external magnetic fields, (b)
the internal magnetic field (due to the
orbital motion of charges in H atom), and (c)
the protons magnetic moment (proton is also a
spin-1/2 particle). Thus, even in the absence of
external mag. field, the levels for spin-up and
spin-down electrons will be different.
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Angular Momentum and Magnetic Moment
Classically speaking, an orbiting electron in H
atom is equivalent to a circular current. The
magnetic moment of a current loop
Current charge which flows through the wire
cross-section in unit time
f the frequency of electron rotation around the
nucleus
the orbital magnetic moment of an electron
gyromagnetic ratio for electrons (minus L is
anti-parallel to ? because of -e)
In Q.M, orbits do not exist. Instead, the
magnetic moment is related to the operator
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
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Energy of Magnetic Dipole in Magnetic Field
Potential energy of a magnetic dipole in external
magnetic field
spin-up, magnetic moment down
? is anti-parallel to B
? is parallel to B
For an electron in H atom
Bohr magneton
Even in strong magnetic fields (10T), this
splitting is 10-3 eV (compare with the
inter-level distance 1 eV)
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H atom in a magnetic field B an example of a
non-central potential
10 Not-to-scale 20 Spin is ignored
- axial (but not central) symmetry
H atom
- magnetic moment due to the orbital motion of
an electron
B II z
- dependent on n and ml but not l
Degeneracy of the spectrum is partially lifted
(states with the same n and ml and different l
still have the same energy).
Coulomb B0
Coulomb B?0
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Problem (Midterm 2, 2008)
What is the maximum difference in potential
energy between H atoms in a 4d state when placed
in a magnetic field of 2T (ignore the spin
quantum number)?
4d state l2, ml2,1,0,-1,-2
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10 Not-to-scale 20 Spin is ignored
Selection rules
H atom
Which radiative transitions are allowed for n2
electron in H atom in magnetic field?
these dipole radiation transitions are also
allowed, but they correspond to the MW radiation
?
Coulomb B0
Coulomb B?0
H atom in magnetic field
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Normal Zeeman Effect
If H 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) see Figure. When we deal with
a single electron (H atom), the situation is
simplified by the fact that the electron spin
projection remains fixed in dipole transitions
(the electron interacts mostly with the electric
component of the field). Thus, the only splitting
we need to take into account is due to different
values of ml.
Discovery 1896 Nobel - 1902
Selection rules for the normal Zeeman effect
Spectral lines splitting
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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?
?? is due to the energy splitting between two
adjacent ml states
Two adjacent lines that correspond to ?ml1
correspond to the emission of photons with
energies
( list all the states that correspond to these
energy levels)
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Spin Goudsmit Uhlenbeck (1925), Dirac (1929)
Both electron and proton possess intrinsic
magnetic moments spins. They belong to
fermions particles with half-integer spin.
The spin angular momentum and its component along
the z axis are quantized
Electrons and Protons
ms1/2 spin up
ms-1/2 spin down
The electron spin magnetic moment
(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).
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, its 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)
Taking spin into account, we need four quantum
numbers (n, l, ml, ms) to completely describe
electron states in atoms.
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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
Current (charge which flows through the wire
cross-section in unit time)
(!)
Magnetic field of a current loop
Thus, the splitting of levels due to the
spin-orbital coupling is small (fine) and
this is why ? , which controls (among many other
things) the splitting, was called the fine
structure constant.
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The fine structure due to the interaction between
the magnetic moments associated with the electron
spin and orbital momentum.
the quantum electrodynamics corrections
The hyperfine structure due to the interactions
of the proton spin magnetic moment with both the
orbital and spin magnetic moments of an electron.
The hyperfine splitting is typically 100 times
smaller than even the fine splitting who
cares???
Astronomers do!
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21 cm (1.4 GHz) Hydrogen Line
The spins of the electron and proton can be
oriented in the same direction or in opposite
directions. Because of the interaction between
spins, an electron state that corresponds to
parallel spins has the energy slightly bigger
than the energy of anti-parallel configuration of
spins. This results in hyperfine splitting
100 times smaller than the fine splitting
??
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 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.
This line is of great importance for astronomy.
Because of its extremely small natural width (?E
h/?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)