Download Lectures 1-2: Introduction to Atomic Spectroscopy Types of Spectra

Types of Spectra
Lectures 1-2: Introduction to Atomic Spectroscopy
o Continuous spectrum: Produced by solids, liquids & dense gases produce - no
“gaps” in wavelength of light produced:
o Line spectra
o Emission spectra
o Absorption spectra
o Hydrogen spectrum
o Balmer formula
o Bohr’s model
Bulb
Sun
Na
Atomic emission
spectra
o Emission spectrum: Produced by rarefied gases – emission only in narrow
wavelength regions:
H
Hg
Cs
o Absorption spectrum: Gas atoms absorb the same wavelengths as they
usually emit and results in an absorption line spectrum:
Chlorophyll
Molecular absorption
spectra
Diethylthiacarbocyaniodid
Diethylthiadicarbocyaniodid
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Emission and Absorption Spectroscopy
Line Spectra
Gas cloud
o
Electron transitions between energy levels result in emission or absorption lines.
o
Different elements produce different spectra due to differing atomic structure
(discovered by Kirchhoff and Bunsen).
H
3
1
2
He
C
!"
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Emission/Absorption of Radiation by Atoms
Emission/Absorption of Radiation by Atoms
o
o
Radiative recombination can be either:
Emission/absorption lines are due to radiative transitions:
a) Spontaneous emission: Electron minimizes its total energy by emitting photon and
making transition from E2 to E1.
1. Radiative (or Stimulated) absorption:
Photon with energy (E# = h$ = E2 - E1) excites electron from lower energy
level.
E# =h$
E2
E2
E1
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.
Can only occur if E# = h$ = E2 - E1
2. Radiative recombination/emission:
Electron emits photon with energy (h$’ = E2 - E1) and makes transition to
lower energy level.
E# =h$
E2
E2
E#’ =h$’
E1
E1
E# =h$
Can only occur if E# = h$ = E2 - E1 Also, h$’ = h$
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Simplest Atomic Spectrum: Hydrogen
o
Simplest Atomic Spectrum: Hydrogen
In 1850s, the visible spectrum of hydrogen was found to contain strong lines at
6563, 4861 and 4340 Å.
H%
H&
6563
4861
!" (Å)
H#
o
If n =3, =>
o
Called H% - first line of Balmer series.
o
Other lines in Balmer series:
4340
o
Lines found to fall more closely as wavelength decreases.
o
Line separation converges at a particular wavelength, called the series limit.
o
In 1885, Balmer found that the wavelength of lines could be written
o
Name
Transitions
Wavelength (Å)
H%
3-2
6562.8
H&
4-2
4861.3
H#
5-2
4340.5
Road to National Solar
Observatory in New
Mexico: “Highway 6563”
Balmer series limit occurs when n '(.
where n is an integer >2, and RH is the Rydberg constant.
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Simplest Atomic Spectrum: Hydrogen
o
Simplest Atomic Spectrum: Hydrogen
Rydberg showed that all series above could be
reproduced using
o
Ritz Combination Principle: Transitions
occur between terms:
where n are principal quantum numbers.
o
Series limit occurs when ni = !, nf = 1, 2, …
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
where Tf = RH/nf2 etc.
o
Transitions can occur between certain terms
=> selection rule. Selection rule for
hydrogen: *n = 1, 2, 3, …
o
Transitions between ground state and first
excited state produce a resonance line.
o
Grotrian diagram shows terms and the
transitions.
Series limits
Energy level or term diagram for hydrogen
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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:
Bohr Model for Hydrogen
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 = nh where n = 1, 2, 3, …
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.
o
As Coulomb force is a centripetal:
o
And assuming angular momentum is quantised:
o
Solving for v and substituting into Eqn. 1 =>
(1)
(2)
3. Total energy (K + V) of electron in orbit remains constant.
and
!
4. Quantized radiation is emitted/absorbed if an electron changes orbit. The
frequency emitted is $ = (Ei - Ef ) / h.
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o
Quantized AM has restricted the possible circular orbits of the electron.
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Bohr Model for Hydrogen
o
The total mechanical energy is:
o
Using Eqn. 1,
o
The potential energy (V) can be calculated from
o
Total mechanical energy is therefore
o
Using Eqn. 2 for r,
o
Therefore, quantization of AM leads to quantisation of total energy.
Bohr Model for Hydrogen
o
Substituting in for constants, Eqn. 3 can be written
and Eqn. 2 can be written
where a0 = 0.529 Å = “Bohr radius”.
o
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
(3)
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Bohr Model for Hydrogen
o
Correction for Motion of the Nucleus
The wavelength of radiation emitted when an electron makes a transition, is (from
4th postulate):
or
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 (µ):
o
As m only appears in R!, must replace by:
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.
(4)
where
o
Theoretical derivation of Rydberg formula.
o
Essential predictions of Bohr model
are contained in Eqns. 3 and 4.
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Correction for Motion of the Nucleus
o
Spectra of Hydrogen-like Atoms
Consider 1H (hydrogen) and 2H (deuterium):
cm-1
o
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:
Using Eqn. 4, the wavelength difference is
therefore:
$1
1'
1/ " = 4R He & 2 # 2 )
%3 n (
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.
o
See Fig. 8.7 in Haken & Wolf.
Z=1
H
0
13.6 eV
20
Z=2
He+
n
2
1
n
3
2
Z=3
Li2+
2
Energy
(eV)
40
60
54.4 eV
1
80
!
100
Balmer line of H and D
1
120
122.5 eV
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Spectra of Hydrogen-like Atoms
o
o
Z=1
H
Hydrogenic or hydrogen-like ions:
o
o
o
o
He+ (Z=2)
Li2+ (Z=3)
Be3+ (Z=4)
…
0
Hydrogenic
isoelectronic
sequences
20
13.6 eV
n
2
Implications of Bohr Model
Z=2
He+
1
n
3
2
Z=3
Li2+
o
4
3
Energy
(eV)
o
eV
and
o
40
60
54.4 eV
Bohr radius (a0) and fine structure constant (%) are fundamental constants:
1
and
80
100
E1 = -13.59
We also find that the orbital radius and velocity are quantised:
n
2
From Bohr model, the ionization energy is:
Z2
n
4
3
120
1
o
Constants are related by
o
With Rydberg constant, define gross atomic characteristics of the atom.
122.5 eV
Ionization potential therefore increases rapidly with Z.
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Rydberg energy
RH
13.6 eV
Bohr radius
a0
5.26x10-11 m
Fine structure constant
!
1/137.04
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Exotic Atoms
o
o
o
Failures of Bohr Model
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
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.
Muonium:
o Replace proton in H atom with a µ 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.
2. Violates the uncertainty principal which dictates that position and momentum
cannot be simultaneously determined.
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.
o
Bohr model gives a basic conceptual model of electrons orbits and energies. The
precise details can only be solved using the Schrödinger equation.
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Failures of Bohr Model
o
o
Hydrogen Spectrum
From the Bohr model, the linear momentum of the electron is
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.
However, know from Hiesenberg Uncertainty Principle, that
o Principal quantum number: n
o
Comparing the two Eqns. above => p ~ n*p
o Azimuthal quantum number: l
o
This shows that the magnitude of p is undefined except when n is large.
o Magnetic quantum number: ml
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.
o Spin quantum number: s
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o
Selection rules must also be modified.
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Atomic Energy Scales
Energy scale
Energy (eV)
Effects
Gross structure
1-10
electron-nuclear
attraction
Electron-electron
repulsion
Electron kinetic energy
Fine structure
0.001 - 0.01
Spin-orbit interaction
Relativistic corrections
Hyperfine structure
10-6 - 10-5
Nuclear interactions
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