Download 1 Spectroscopy and Quantum Mechanics Spectroscopy measures

Spectroscopy and Quantum Mechanics
Spectroscopy measures differences between stationary quantum states, or solutions of the
time-independent Schrödinger equation (those we’ve discussed so far).
Use electromagnetic energy (photons) to induce transitions between quantum energy
levels, which are characterized by eigenvalues and eigenfunctions of the Schrödinger
equation. Since the energy of a photon is equal to Planck’s constant times the oscillation
frequency of the electromagnetic radiation, or E=hν, then the energy for a spectroscopic
transition is:
∆E fi = E f − Ei = h(ν f − ν i ) = h∆ν
Range of electromagnetic wavelengths and frequencies used in molecular spectroscopy:
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How do we characterize induced spectroscopic transitions?
Absorption
Emission
Scattering
Absorption of Light
Intensity of light absorption by atoms and molecules depends on transition probability,
number density of molecules (in solution, concentration), and the path length of light
through the sample.
Beer’s law: Absorbance A = εcl.
Absorbance is –log10 of transmittance, which is T = I/I0, or A = -log10(I/I0).
The integral of the absorption lineshape is proportional to the number of photons
absorbed over that area.
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How do we connect the experimental observations, (Beer-Lambert empirical law for
absorption strength) with quantum mechanics?
A. Transition Dipole Moments
The absorbance A is proportional to the rate of transitions between stationary states. This
rate is described by the Einstein ‘B’ coefficient, given by
Bij =
µ ij
2
6ε 0 h 2
where µij is the transition dipole moment (or matrix element).
µ ij = j µˆ i = ∫ Ψ *j µˆ Ψi dτ
Here µ is simply the dipole moment operator,
µ̂ = ∑ qi ⋅ rˆi
i
For a transition ever to occur between stationary states |i> and |j> (or Ψi and Ψj ), the rate
must be non-zero. Otherwise, we wait for the age of the universe for the transition to
occur, but it never does. When µij = 0, this situation is referred to as a forbidden
transition. An explanation of how these results are derived from time-dependent
quantum mechanics is given later.
In a later section we will connect transition dipole moments (or matrix elements) with
spectroscopic selection rules that determine when the probability for transitions between
quantum states is non-zero.
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Emission of Light
The Einstein B coefficients for a vertical energy transition are the same in both
directions. I.e., the rate of transition from state |i> to state |j> will be the same as the
reverse transition from |j> to |i>. More mathematically, Bij = Bji.
Experimentally, emission of light can provide the same spectroscopic information about
the stationary (or time-independent) quantum states. This emission is called
fluorescence.
Fluorescence is a highly sensitive technique, especially when coupled with laser
excitation of the atomic or molecular quantum state.
Fluorescence can be resolved at the single photon level, providing extremely
sensitive detection.
Fluorescence is used to detect single molecules under a confocal microscope.
Time-resolved fluorescence is a means of characterizing the lifetime of the excited
state, which is essential for understanding excited-state photophysics, reactivity,
and photochemistry (such as photosynthesis). Measuring the change in
polarization of the emitted photons with time permits detection of the change in
orientation of the emitting transition dipole, a.k.a. the molecular reorientation
time correlation function.
Two types of light emission: stimulated and spontaneous
Stimulated emission is the kind that parallels absorption transitions. Photons from the
radiation field can stimulate an upwards or downwards vertical energy transition. The
upwards transition is (stimulated) absorption, and the downwards transition is stimulated
emission. Stimulated Emission are two of the words making up the acronym LASER:
Light Amplification by Stimulated Emission of Radiation.
Spontaneous emission only occurs from a pre-existing excited state. For example, light
from a laser photon can cause an upward stimulated absorbtion transition. After a
characteristic radiative lifetime of the excited state, the atom/molecule can undergo a
downward transition by spontaneous emission.
The rate of spontaneous emission is given by the Einstein ‘A’ coefficient,
A ji =
8πhν 3
Bij
c3
Note that the ratio of Einstein coefficients Aji/Bji = 8πhν3/c3. This tells us that the rate of
transitions caused by spontaneous emission depends on the frequency of light, or the
difference between energy levels. For transitions of higher energy (such as in the
ultraviolet), the radiative lifetime (or excited state lifetime) will be shorter than in the
visible region.
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Spectroscopy by Scattering
Energy differences between quantum states in atoms/molecules can also be detected
experimentally by observing changes in energy of scattered particles. Commonly, the
following particles are used in spectroscopy:
photons
x-rays
neutrons
electrons
Most commonly, photons, or in classical terms, electromagnetic radiation, are used in
spectroscopy, especially since the development of masers and lasers. Spectroscopy with
light is often called optical spectroscopy.
The Atkins text describes several of the common optical spectroscopy methods, including
dispersive methods as well as Fourier-transform InfraRed (FT-IR) spectroscopy,
absorption spectroscopy.
Dispersive spectroscopies rely on use of a grating or prism to spatially disperse
wavelengths linearly along a particular direction. Use of a slit permits recording the
intensity of a narrow bandwidth; use of an area detector (such as a linear photodiode
array or CCD detector as used in a portable video camera) permits recording of a broad
piece of the spectrum at once.
FT methods detect all wavelengths simultaneously, but by recording the interferogram
between a reference beam and once transmitted through the sample, the spectrum is
obtained by numerical Fourier-transform.
Optical detectors include photomultipliers, image intensifiers, charge-coupled devices
(CCDs), which are sophisticated grids of photodiode detectors, germanium diodes (for
microwaves), and even photographic films.
Ultrashort laser pulses have highly coherent beams with a broad spectral range that can
easily cover half of the visible spectrum. Use of these lasers with a coherent detection
method permits high resolution spectroscopy by Fourier-transform methods.
Some classic texts on optical spectroscopy include the series by Gerhard Herzberg, and
Molecular Vibrations by Wilson, Decius, and Cross. Historically, these works
concentrated on general solutions to small-molecule problems, up to molecules the size
of benzene. Now researchers commonly work on macromolecules including proteins,
DNA, and polymers, as well as advanced composites and nanostructures.
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X-ray Scattering Spectroscopy
X-rays are well known as probes of structure on atomic sizes, because common x-ray
wavelengths are about 1 Å wavelength. Synchrotron radiation at national user facilities
provides intense, fully tunable x-ray radiation. Structural information is obtained by
Fourier-inversion of the x-ray diffraction pattern, where the diffracted spots can be
predicted by Bragg’s scattering law. Diffraction is one extreme of x-ray scattering, in the
elastic scattering limit. In this limit, the energy of the x-ray is not changed, but the
direction of the x-ray photon is changed.
X-ray Spectroscopy is an inelastic scattering method, where the difference in energy
levels for different quantum states is probed by the change in energy experienced by an
x-ray photon. Since x-rays have energies similar to the binding energies of core electrons
of atoms, x-ray spectroscopy is an excellent means for determining the atomic
composition of a sample. Synchrotron radiation has made x-ray spectroscopy much more
accessible to chemists, physicists, biologists, and materials scientists. In the U.S.,
synchrotrons at Brookhaven, Argonne, Berkeley, Stanford, Cornell, and Wisconsin
provide tunable x-rays for scientists from around the world, who work on a 24-7/330
schedule.
X-ray scattering occurs mostly from the core electrons, rather than the shielded nuclei.
Since each element has a different pattern of filled-shell core electrons, x-ray scattering
spectra are quite sensitive elemental composition.
Several common x-ray spectroscopy methods include:
X-ray Fluorescence
EXAFS (Extended X-Ray Absorption Fine Structure)
XPS (X-Ray Photoelectron Spectroscopy)
Auger spectroscopy
ESCA (electron spectroscopy for chemical analysis- induced by x-rays)
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Neutron Scattering Spectroscopy
Like x-ray diffraction, neutron-diffraction is the elastic limit of neutron scattering, where
the scattered neutrons are observed in a new direction predicted by Bragg’s Law. In the
inelastic scattering limit, the energies of scattered neutrons change by the difference in
energy levels of the scattering quantum states of the atoms/molecules.
Neutrons have a larger scattering cross-section from nuclei than from electrons. Thus,
neutrons are sensitive to elemental composition. Thus, x-ray scattering (mostly from
core atomic electrons) and neutron scattering (mostly from atomic nuclei) are a perfect
complement to each other.
Neutron-scattering methods include
ineleastic neutron scattering
quasi-elastic neutron scattering (QENS)
Neutrons are very sensitive to hydrogen-bonds (actually, deuterium bonds), since
deuterium scattering is about 35 times greater than for hydrogen. As x-ray scattering
from the single electron in a hydrogen atom is tiny, neutron scattering permits resolution
of protons/deuterons in diffraction/scattering experiments. Since H-bonds are allimportant in aqueous solutions, and biological samples, neutron scattering is a most
powerful technique.
Neutron facilities are also found mostly at national laboratory facilities, including NIST
(Gaithersburg, MD), Los Alamos, NM, Argonne, and Oak Ridge, where the latest
generation Spallation Neutron Source is being built. Brookhaven had been a former
world-leader in neutron experiments until political considerations forced the U.S. Dept.
of Energy to shut down the Brookhaven High-Flux Neutron Reactor in 2000.
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