Lecture 1-2: Introduction to Atomic Spectroscopy
o Line spectra
Bulb
o Emission spectra
Sun
o Absorption spectra
Na
o Hydrogen spectrum
o Balmer Formula
Emission
spectra
H
Hg
Cs
o Bohr’s Model
Absorption
spectra
Chlorophyll
Diethylthiacarbocyaniodid
Diethylthiadicarbocyaniodid
PY3P05
Types of Spectra
o Continuous spectrum: Produced by solids, liquids & dense gases produce - no
“gaps” in wavelength of light produced:
o Emission spectrum: Produced by rarefied gases – emission only in narrow
wavelength regions:
o Absorption spectrum: Gas atoms absorb the same wavelengths as they
usually emit and results in an absorption line spectrum:
PY3P05
Emission and Absorption Spectroscopy
Gas cloud
3
1
2
PY3P05
Line Spectra
o
Electron transition between energy levels result in emission or absorption lines.
o
Different elements produce different spectra due to differing atomic structure.
H
He
C

PY3P05
Emission/Absorption of Radiation by Atoms
o
Emission/absorption lines are due to radiative transitions:
1. Radiative (or Stimulated) absorption:
Photon with energy (E = h = E2 - E1) excites electron from lower energy
level.
E =h
E2
E2
E1
E1
Can only occur if E = h = E2 - E1
2. Radiative recombination/emission:
Electron makes transition to lower energy level and emits photon with energy
h’ = E2 - E1.
PY3P05
Emission/Absorption of Radiation by Atoms
o
Radiative recombination can be either:
a) Spontaneous emission: Electron minimizes its total energy by emitting photon and
making transition from E2 to E1.
E2
E2
E1
E1
E’ =h’
Emitted photon has energy E’ = h’ = E2 - E1
b) Stimulated emission: If photon is strongly coupled with electron, cause electron to
decay to lower energy level, releasing a photon of the same energy.
E =h
E2
E2
E’ =h’
E1
E1
E =h
Can only occur if E = h = E2 - E1 Also, h’ = h
PY3P05
Simplest Atomic Spectrum: Hydrogen
o
In ~1850’s, optical spectrum of hydrogen was found to contain strong lines at 6563,
4861 and 4340 Å.
H
H
H
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
6563
 (Å)
4861
4340
o
Lines found to fall closer and closer as wavelength decreases.
o
Line separation converges at a particular wavelength, called the series limit.
o
Balmer found that the wavelength of lines could be written
 1 1 
1/   RH  2  2 
2 n 
where n is an integer >2, and RH is the Rydberg constant.

PY3P05
Simplest Atomic Spectrum: Hydrogen
 1 1 
1/   RH  2  2    6563 Å
2 3 
o
If n =3, =>
o
Called H - first line of Balmer series.
o
 in Balmer series:
Other lines
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Name
Transitions
Wavelength (Å)
H
3-2
6562.8
H
4-2
4861.3
H
5-2
4340.5
Highway 6563 in New
Mexico
o Balmer Series limit occurs when n .
 1 1 
1/   RH  2    ~ 3646 Å
2  
PY3P05
Simplest Atomic Spectrum: Hydrogen
o
o
o
Other series of hydrogen:
Lyman
UV
nf = 1, ni2
Balmer
Visible/UV
nf = 2, ni3
Paschen
IR
nf = 3, ni4
Brackett
IR
nf = 4, ni5
Pfund
IR
nf = 5, ni6
Rydberg showed that all series above could be
reproduced using
 1
1 
1/   RH 
n 2  n 2 

 f
i 
Series Limits
Series limit occurs when ni = ∞, nf = 1, 2, …

PY3P05
Simplest Atomic Spectrum: Hydrogen
o
Term or Grotrian diagram for
hydrogen.
o
Spectral lines can be considered as
transition between terms.
o
A consequence of atomic energy levels,
is that transitions can only occur
between certain terms. Called a
selection rule. Selection rule for
hydrogen: n = 1, 2, 3, …
PY3P05
Bohr Model for Hydrogen
o
Simplest atomic system, consisting of single electron-proton pair.
o
First model put forward by Bohr in 1913. He postulated that:
1. Electron moves in circular orbit about proton under Coulomb attraction.
2. Only possible for electron to orbits for which angular momentum is quantised,
ie., L  mvr  n
n = 1, 2, 3, …
3. Total energy (KE + V) of electron in orbit remains constant.

4. Quantized radiation is emitted/absorbed if an electron moves changes its
orbit.
PY3P05
Bohr Model for Hydrogen
o
Consider atom consisting of a nucleus of charge +Ze and mass M, and an electron
on charge -e and mass m. Assume M>>m so nucleus remains at fixed position in
space.
Ze 2
v2
m
4 0 r 2
r
1
o
As Coulomb force is a centripetal, can write
o
As angular momentum is quantised (2nd postulate):
o
o
mvr  n
n = 1, 2, 3, …
n2 2

Solving for v and substituting into Eqn. 1 => r  40
(2)
mZe2

n
1 Ze 2
 v 

mr 4 0 n
The total mechanical energy is:
E  1/2mv 2  V
 E n 

o
(1)
mZ 2e 4
40  2
2
2
1
n2
n = 1, 2, 3, …
(3)
Therefore, quantization of AM leads to quantisation of total energy.

PY3P05
Bohr Model for Hydrogen
o
Substituting in for constants, Eqn. 3 can be written
and Eqn. 2 can be written
o
n 2 a0
r
Z
13.6Z 2
eV
En 
n2
where a0 = 0.529 Å = “Bohr radius”.

Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1):

o
For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV
PY3P05
Bohr Model for Hydrogen
o
The wavelength of radiation emitted when an electron makes a transition, is (from
4th postulate):
4



2
1/  
 1
1 
1/   R Z 
or
n 2  n 2 


f
i 

 1 2 me4
where R  

3
40  4 c
2

o
Ei  E f
1
me
1
1
 
Z 2
 2 

3
2


hc
4 0  4 c n f n i 
(4)
Theoretical derivation of Rydberg formula.

o
Essential predictions of Bohr model
are contained in Eqns. 3 and 4.
PY3P05
Correction for Motion of the Nucleus
o
Spectroscopically measured RH does not agree exactly with theoretically derived
R∞ .
o
But, we assumed that M>>m => nucleus fixed. In reality, electron and proton
move about common centre of mass. Must use electron’s reduced mass ():
mM

m M
o
As m only appears in R∞, must replace by:

RM 
M

R  R
m M
m
o
It is found spectroscopically that RM = RH to within three parts in 100,000.
o

Therefore, different isotopes of same element have slightly different spectral lines.
PY3P05
Correction for Motion of the Nucleus
o
o
Consider 1H (hydrogen) and 2H (deuterium):
RH  R
1
 109677.584
1 m / M H
cm-1
RD  R
1
 109707.419
1 m / M D
cm-1
Using Eqn. 4, the wavelength difference is
therefore:

   H  D  H (1 D /  H )
  H (1 RH /RD )
o
Called an isotope shift.

o
H and D are separated by about 1Å.
o
Intensity of D line is proportional to fraction of
D in the sample.
Balmer line of H and D
PY3P05
Spectra of Hydrogen-like Atoms
o
Bohr model works well for H and H-like atoms (e.g., 4He+, 7Li2+, 7Be3+, etc).
o
Spectrum of 4He+ is almost identical to H, but just offset by a factor of four (Z2).
o
For He+, Fowler found the following
in stellar spectra:
Z=1
H
0
 1 1 
1/   4RHe  2  2 
3 n 
20
See Fig. 8.7 in Haken & Wolf.

n
2
1
Z=3
Li2+
n
3
2
n
4
3
2
40
Energy
(eV)
o
13.6 eV
Z=2
He+
60
54.4 eV
1
80
100
1
120
122.5 eV
PY3P05
Spectra of Hydrogen-like Atoms
o
Z=1
H
Hydrogenic or hydrogen-like ions:
0
o
o
o
o
He+ (Z=2)
Li2+ (Z=3)
Be3+ (Z=4)
…
Hydrogenic
isoelectronic
sequences
20
13.6 eV
Z=2
He+
n
2
1
Z=3
Li2+
n
3
2
n
4
3
2
40
Energy
(eV)
o From Bohr model, the ionization energy is:
60
54.4 eV
1
80
100
E1 = -13.59 Z2 eV
o
1
120
122.5 eV
Ionization potential therefore increases rapidly with Z.
PY3P05
Implications of Bohr Model
o
We also find that the orbital radius and velocity are quantised:
n2 m
rn 
a0
Z 
o
and
vn  
Z
c
n
Bohr radius (a0) and fine structure constant () are fundamental constants:

e2
40 2 
a0 
and   4  c
2
me
0
a0 
o
Constants are related by
o


With Rydberg constant, define gross atomic characteristics of the atom.
mc
 energy
Rydberg
RH
13.6 eV
Bohr radius
a0
5.26x10-11 m
Fine structure constant

1/137.04
PY3P05
Exotic Atoms
o
Positronium
o electron (e-) and positron (e+) enter a short-lived bound state, before they
annihilate each other with the emission of two -rays (discovered in 1949).
o Parapositronium (S=0) has a lifetime of ~1.25 x 10-10 s. Orthopositronium
(S=1) has lifetime of ~1.4 x 10-7 s.
o Energy levels proportional to reduced mass => energy levels half of hydrogen.
o
Muonium:
o Replace proton in H atom with a  meson (a “muon”).
o Bound state has a lifetime of ~2.2 x 10-6 s.
o According to Bohr’s theory (Eqn. 3), the binding energy is 13.5 eV.
o From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10.15 eV.
o
Antihydrogen:
o Consists of a positron bound to an antiproton - first observed in 1996 at CERN!
o Antimatter should behave like ordinary matter according to QM.
o Have not been investigated spectroscopically … yet.
PY3P05
Failures of Bohr Model
o
Bohr model was a major step toward understanding the quantum theory of the atom
- not in fact a correct description of the nature of electron orbits.
o
Some of the shortcomings of the model are:
1. Fails describe why certain spectral lines are brighter than others => no
mechanism for calculating transition probabilities.
2. Violates the uncertainty principal which dictates that position and momentum
cannot be simultaneously determined.
o
Bohr model gives a basic conceptual model of electrons orbits and energies. The
precise details can only be solved using the Schrödinger equation.
PY3P05
Failures of Bohr Model
o
From the Bohr model, the linear momentum of the electron is
 Ze 2  n
p  mv  m

40 n  r
o
However, know from Hiesenberg Uncertainty Principle, that

p ~
o
~
x r
Comparing the two Eqns. above => p ~ np
o

This shows that the magnitude
of p is undefined except when n is large.
o
Bohr model only valid when we approach the classical limit at large n.
o
Must therefore use full quantum mechanical treatment to model electron in H atom.
PY3P05
Hydrogen Spectrum
o
Transitions actually depend on more than a
single quantum number (i.e., more than n).
o Quantum mechanics leads to introduction on
four quntum numbers.
o Principal quantum number: n
o Azimuthal quantum number: l
o Magnetic quantum number: ml
o Spin quantum number: s
o
Selection rules must also be modified.
PY3P05
Atomic Energy Scales
Energy scale
Energy (eV)
Effects
Gross structure
1-10
electron-nuclear
attraction
Electron kinetic energy
Electron-electron
repulsion
Fine structure
0.001 - 0.01
Spin-orbit interaction
Relativistic corrections
Hyperfine structure
10-6 - 10-5
Nuclear interactions
PY3P05