fundamentals of spectroscopy - atomic physics · forces in atoms and molecules • forces in the...
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Fundamentals of spectroscopy
1
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Spectral bands from the electromagnetic
spectrum
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Outline
• Interactions between electro-magnetic fields and
matter
– Visible, outer electrons
– X-rays, inner electrons
– Infrared, molecular vibrations
– Micro- and radiowaves, electron & nuclear spins
• Line widths
• Detection modes
3
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Interactions between electromagnetic
radiation and sample
• Electro-magnetic radiation transfer energy
• Sample is composed of atoms, molecules
• By examining the resulting electro-
magnetic radiation after it has intracted
with the sample - conclusions can be
drawn about the object under study
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Forces in atoms and molecules
• Forces in the universe
– Gravity and the electro-magnetic, weak and
strong forces
• In atoms (and for most processes in our
daily life) the forces have electric &
magnetic character
• We have direct attractive and repulsive
forces between the charged particles in the
atoms, but we also have magnetic
interactions
5
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Spectral bands from the electromagnetic
spectrum
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Energy levels in an atom
λ
Energy
Principal quantum number n determines
the distance from the nucleus
How would the atmosphere look at ~120 nm?
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What electron orbits are allowed?
An accelerating/deccelerating charge emits radiation
For example, a synchrotron sends out radiation in every bend
Why does an electron orbiting a nucleus then not send out
radiation, lose energy and finally collapse and hit the nucleus?
Fig 2.3
The de Broglie wavelength, l, is given by mv=h/l Eq. 2.2.
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What electron orbits are allowed?
Schrödinger Equation
Erwin
Schrödinger
1925
The electron cloud is described by a wave-function, Y
Niels Bohr
1913
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Hydrogen atom wave functions
http://sevencolors.org/post/hydrogen-atom-orbitals
Wavefunctions in QM:
probability distribution of the
electron, i.e. the electron cannot be
seen as a localized particle
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Generation of electro-magnetic fields
Figure from G Jönsson & E Nilsson
Våglära och Optik, Teach Support
11
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Hydrogen atom wave functions
http://sevencolors.org/post/hydrogen-atom-orbitals
Wavefunctions in QM:
probability distribution of the
electron, i.e. the electron cannot be
seen as a localized particle
Quantum numbers: (n, l, ml)
n - pricipal quantum number
n = 1, 2, ...
l - angular quantum number
l = 0,1,2,...,n-1; s, p, d, f
ml - magnetic quantum number
ml = -l, ..., l
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14
Energy level diagram for the sodium atom
s, p, d, f corresponds to
different orbital angular
momentum (0, ℏ, 2ℏ, 3ℏ) of
the electron
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Why can’t we go from one state
to any other state?
15
The transition probability from one stationary state to
another is proportional to
𝐹𝑖𝑛𝑎𝑙 𝑠𝑡𝑎𝑡𝑒 × 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑓𝑖𝑒𝑙𝑑 × 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑡𝑎𝑡𝑒
𝑆𝑝𝑎𝑐𝑒
Odd parity
Wave functions with even angular momentum quantum numbers have even parity
and
Wave functions with odd angular momentum quantum numbers have odd parity
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16
Energy level diagram for the sodium atom
The photon carries one unit of
angular momentum and can
therefore take a p-electron to
s or d but for example not an
s-electron to f
Strongest line
Even Odd Even Odd
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Energy levels in Lithium
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Spin
• Electrons have spin, s=1/2. In an atom with
two outer electrons, these can have opposite
or (if allowed by the Pauli principle) equal
spin directions
http://cwx.prenhall.com/bookbind/pubbooks/
hillchem3/medialib/media_portfolio/07.html 18
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Energy levels in Helium
Singlets
Total spin is zero
Triplets
Total spin is one
Fig 2.6
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Fig 2.7
Energy levels in Calcium
Which line for detection?
When to use
intercombination line?
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Spectral bands from the electromagnetic
spectrum
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Oscillations
• The oscillation frequency of a system
depends on the mass(es) involved and the
restoring force
22
https://en.wikipedia.
org/wiki/Oscillation
𝑓 =1
2𝜋
𝑘
𝑚 f = oscillation frequency
k = spring constant
m = mass
The spring exerts a force F = kx on a mass, m, where
x is the displacement from the equilibrium position
How could we change the electron ”spring constant” in an atom?
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How do we get from the visible
to the X-ray region?
• Increase the ”spring constant” that is the
restoring force on the electron
• In fact, the energy of the innermost electron
increases as Z2
23
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Spectral bands from the electromagnetic
spectrum
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X-ray production
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Page and/or figure references
In green: Sune Svanberg, Atomic and molecular spectroscopy,
Springer Verlag
In blue: Wolfgang Demtröder, Atoms, Molecules and Photons,
Springer
26
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27
Bremsstrahlung
X-rays can be produced by
accelerating/deccelerating
charges. The radiated power
is proportional to the
acceleration/deccelation
squared
Section 7.5.1
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Collisions can excite inner shell electrons to highly excited states. X-ray
radiation is emitted when these electrons decay back to the inner shells.
Characteristic lines , Section 7.5.2
Sune Svanberg, Atomic
and molecular
spectroscopy, Springer
Verlag, Fig 5.1
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29
Collisions can excite inner shell electons to highly excited states. X-ray
radiation is emitted when these electrons or other bound electrons decay
back to the inner shells. These characteristic lines are superposed on the
continuous brehmsstrahlung background
Characteristic lines , Section 7.5.2
Fig 2.8
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30
Absorption of X-ray radiation as a function
of X-ray wave length
Cu, Z = 29
Ag, Z = 47
page 276
Sune Svanberg, Atomic
and molecular
spectroscopy, Springer
Verlag, Fig 5.1
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31
The water window Fig 10.25, page 271
The short wavelength offers very good resolution. Operating in the water
window provides very good contrast between water and proteins in e.g.,
cells or tissue. Developing good microscopic techniques and sources in
this wavelength region is an active research field.
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Spectral bands from the electromagnetic
spectrum
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Molecular spectra
• For molecules we can in addition to
electronic transitions have vibrational and
rotational transitions
33
Fig 2.15
Page 25
How could we estimate the vibration frequency?
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Molecular spectra
• The proton/electron mass ratio is ~103
• The atomic nuclei in a molecule are ”glued”
together by the outer electrons, ”force
constant” should be similar as for outer
electrons where the electronic transitions
are a few electron volts
• Outer electron transitions in atoms are
typically a few eV, thus vibrational energies
are ~0.1 eV
34
𝑓 =1
2𝜋
𝑘
𝑚
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Energy separation in molecules
Fig 2.10, page 23
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Spectral bands from the electromagnetic
spectrum
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Molecular energies
37
Distance between nuclei is ~1Å
Some orbitals are bonding
and some are anti-bonding
Energy scale is in cm-1
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Energy conversions
Unit nm Joule eV
Hz cm-1
1 nm 1 1.99∙10-16 1.24∙103 3.00∙1017 1.00∙109
1 Joule
1.99∙10-16 1 6.24∙1018 1.51∙1033 5.03∙1022
1 eV 1.24∙103 1.60∙10-19 1 2.42∙1014 8.07∙103
1 Hz 3.00∙1017 6.63∙10-34 4.14∙10-15 1 3.34∙10-11
1 cm-1 1.00∙109 1.99∙10-23 1.24∙10-4 3.00∙1010 1
Wavelength Energy Frequency Wavenumber
𝐸 = ℎ𝑣 𝐸(𝑒𝑉) =ℎ𝑣
𝑒
Compare Eq 2.1, page 16 & Fig 2.2, page 17
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Vibration frequencies for
different molecular groups
39 Page 161
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Spectral bands from the electromagnetic
spectrum
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Forces in atoms and molecules
• Forces in the universe
– Gravity and the electro-magnetic, weak and
strong forces
• In atoms (and for most processes in our
daily life) the forces have electric &
magnetic character
• We have direct attractive and repulsive
forces between the charged particles in the
atoms, but we also have magnetic
interactions
41
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Magnetic moment
Figure adapted from page http://www.sr.bham.ac.uk/xmm/fmc2.html
A current, I, enclosing an area, A,
generates a magnetic moment m = IAân,
where ân is a unit vector normal to the
surface A.
m
42
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Spin
• Electrons have spin, s=1/2.
http://cwx.prenhall.com/bookbind/pubbooks/
hillchem3/medialib/media_portfolio/07.html
43
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Magnetic moments in atoms
• Orbital magnetic moment, 𝝁𝐿 = −𝜇𝐵L
• Spin magnetic moment, 𝝁𝑠 = −𝑔𝑠𝜇𝐵S
• 𝑔𝑠2,
• Nuclear magnetic moment, 𝝁𝐼 = 𝑔𝐼𝜇N I
• I is the nuclear spin
44
𝜇𝑁𝜇𝐵=𝑚𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑚𝑝𝑟𝑜𝑡𝑜𝑛
≈1
2000
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Interaction between a magnetic
moment and a magnetic field
Figure adapted from page http://www.sr.bham.ac.uk/xmm/fmc2.html
m
The energy, E, of a magnetic moment, m, in
a magnetic field B is given by the scalar
product E=-mB 45
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Electron & nuclear spins in a
magnetic field
46
https://wiki.metropolia.fi/display/Physics/Nuclear+magnetic+resonance
As E=-mB the energy difference between electron spin-up & spin-down for, e. g., B=1T is
consequently about 11.6*10-5 eV (~30 GHz, ~1 cm)
p
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Spectral bands from the electromagnetic
spectrum
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Outline
• Interactions between electro-magnetic fields and
matter
– Visible, outer electrons
– X-rays, inner electrons
– Infrared, molecular vibrations
– Micro- and radiowaves, electron & nuclear spins
• Line widths
• Detection modes
48
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Line widths of spectroscopic signals – Optical frequencies are close to 1015 Hz.
– The frequency width of an atomic/molecular transition in
gas at low pressure is ~1 GHz due to Doppler
broadening and 10-100 GHz due to collisions at
atmospheric pressure
– Below, part of solar spectrum. Many spectral lines can be
discerned within a narrow interval
49
nanometers Fig 6.87, page 178
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Line widths of spectroscopic signals – In liquids and solid state materials atoms/molecules are
much closer. Outer electrons interact from different
atoms/molecules interact strongly, lifetimes are short and
lines are much broader
50
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– However, electrons in deeper shells are shielded by the
outer electrons. Lines can then still be narrow also in
liquids and solids. E.g. in rare earth doped materials.
51
Line widths of spectroscopic signals
Fig 2.22
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Outline
• Interactions between electro-magnetic fields and
matter
– Visible, outer electrons
– X-rays, inner electrons
– Infrared, molecular vibrations
– Micro- and radiowaves, electron & nuclear spins
• Line widths
• Detection modes
52
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Detection modes
• Fluorescence
• Absorption
• Scattering
• Reflection
53
Fig 2.16
Page 25
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Fluorescence spectroscopy
0
1 2 3 4 5
S0
0
1 2 3 4 5
S1
0
1 2 3 4 5
S2
Ener
gy
Ab
so
rpti
on
Flu
ore
sc
en
ce
Vibrational
relaxation
Solids & liquids typically have significant
vibrational (and rotational) relaxation
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Absorption spectroscopy
Beer-Lambert law
Absorption coefficient: μa [cm-1]
”probability for absorption event
per unit length”
μa = s × N
s: cross section [cm2]
N: concentration [cm-3]
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Absorption measurement, example
600 700 800 900
(nm)
10 0
10 1
10 2
a (
c m
- 1 )
Absorption coefficients
Hb
HbO2
Muscle
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Scattering
• Elastic scattering (wavelength, l, unchanged in
the scattering process)
– Rayleigh scattering, scattering on objects (atoms,
molecules, particles . . . etc.) much smaller than the
wavelength, scattering cross section ~l-4
– Mie scattering, scattering on larger particles
• Inelastic scattering (the wavelength, l, is changed
in the scattering process)
– Raman scattering
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In Raman scattering molecules undergo transitions in which an
incident photon is absorbed and another scattered photon is emitted
at a different wavelength
Raman Scattering
1930
Fig 2.18
58
Chandrasekhara
Venkata Raman
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Vibration frequencies for
different molecular groups
59 Page 161
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Cross sections (s) (page 69)
• Resonant absorption s = 10-16 cm2
• Rayleigh scattering s = 10-26 cm2
• Raman scattering s = 10-29 cm2
• Mie scattering s = 10-26-10-8 cm2
• With 1015 photons/cm2 the probability for
resonant absorption equals 10% etc.
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Outline
• Interactions between electro-magnetic fields and
matter
– Visible, outer electrons
– X-rays, inner electrons
– Infrared, molecular vibrations
– Micro- and radiowaves, electron & nuclear spins
• Line widths
• Detection modes
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End
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