Electron paramagnetic resonance (EPR) or
electron spin resonance (ESR)
spectroscopyis a technique for studying
chemical species that have one or
more unpaired
electrons, such as organic
and inorganic
free radicals or
inorganic complexes possessing a
transition metal ion.
The basic physical concepts of EPR are analogous to those of
nuclear magnetic
resonance (NMR), but it is electron spins that are excited
instead of
spin of
atomic nuclei. Because most stable molecules
have all their electrons paired, the EPR technique is less widely
used than NMR. However, this limitation to
paramagnetic species also means that the EPR
technique is one of great specificity, since ordinary chemical
solvents and matrices do not give rise to EPR spectra.
EPR was
first observed in Kazan State University by a Soviet physicist
Yevgeny Zavoisky in 1944, and was
developed independently at the same time by Brebis Bleaney at Oxford
University.
Theory
Origin of an EPR signal
Every electron has a
magnetic moment
and
spin quantum number s = 1/2, with magnetic
components m
_{s} = +1/2 and m
_{s} = -1/2. In the
presence of an external magnetic field with strength
B_{0}, the electron's magnetic moment aligns
itself either parallel (m
_{s} = -1/2) or antiparallel
(m
_{s} = +1/2) to the field, each alignment having a
specific energy (see the
Zeeman
effect). The parallel alignment corresponds to the lower energy
state, and the separation between it and the upper state is
\Delta
E = g_{e}μ_{B}B_{0}, where
g_{e} is the electron's so-called
g-factor (see also the
Landé g-factor) and
μ
_{B} is the
Bohr magneton.
This equation implies that the splitting of the energy levels is
directly proportional to the
magnetic
field's strength, as shown in the diagram below.
An unpaired electron can move between the two energy levels by
either absorbing or emitting electromagnetic radiation of energy
\epsilon =
h\nu such that the resonance condition,
\epsilon = \DeltaE, is obeyed. Substituting in \epsilon =
h\nu and \Delta
E =
g_{e}μ_{B}B_{0} leads to the
fundamental equation of EPR spectroscopy:
h\nu =
g_{e}μ_{B}B_{0}. Experimentally, this
equation permits a large combination of frequency and magnetic
field values, but the great majority of EPR measurements are made
with microwaves in the 9000 – 10000 MHz (9 – 10 GHz)
region, with fields corresponding to about 3500 G (0.35 T). See
below for other field-frequency combinations.
In principle, EPR spectra can be generated by either varying the
photon frequency incident on a sample while holding the magnetic
field constant, or doing the reverse. In practice, it is usually
the frequency which is kept fixed. A collection of
paramagnetic centers, such as free radicals, is
exposed to microwaves at a fixed frequency. By increasing an
external magnetic field, the gap between the
m_{s}
= +1/2 and
m_{s} = −1/2 energy states is widened
until it matches the energy of the microwaves, as represented by
the double-arrow in the diagram above. At this point the unpaired
electrons can move between their two spin states. Since there
typically are more electrons in the lower state, due to the
Maxwell-Boltzmann distribution (see below), there is a net
absorption of energy, and it is this absorption which is monitored
and converted into a spectrum.
As an example of how
h\nu =
g_{e}μ_{B}B_{0} can be used, consider
the case of a free electron, which has
g_{e} =
2.0023, and the simulated spectrum shown at the right in two
different forms. For the microwave frequency of 9388.2 MHz,
the predicted resonance position is a magnetic field of about
B_{0} =
h\nu /
g_{e}μ_{B} = 0.3350 tesla = 3350
gauss, as shown. Note that while two forms of the same spectrum are
presented in the figure, most EPR spectra are recorded and
published only as first derivatives.
Because of electron-nuclear mass differences, the
magnetic moment of an electron is
substantially larger than the corresponding quantity for any
nucleus, so that a much higher electromagnetic frequency is needed
to bring about a spin resonance with an electron than with a
nucleus, at identical magnetic field strengths. For example, for
the field of 3350 G shown at the right, spin resonance occurs near
9388.2 MHz for an electron compared to only about
14.3 MHz for
^{1}H nuclei. (For NMR spectroscopy, the
corresponding resonance equation is
h\nu =
g_{N}μ_{N}B_{0} where
g_{N} and
μ_{N} depend on the
nucleus under study.)
Maxwell-Boltzmann distribution
In practice, EPR samples consist of collections of many
paramagnetic species, and not single isolated paramagnetic centers.
If the population of radicals is in thermodynamic equilibrium, its
statistical distribution is described by the
Maxwell-Boltzmann
equation
- \frac{ n_\text{upper} }{ n_\text{lower} } = \exp{ \left(
-\frac{ E_\text{upper}-E_\text{lower} }{ kT } \right) } = \exp{
\left( -\frac{ \Delta E }{ kT } \right) } = \exp{ \left( -\frac{
\epsilon }{ kT } \right) } = \exp{ \left( -\frac{ h\nu }{ kT
}\right) }
where n_\text{upper} is the number of paramagnetic centers
occupying the upper energy state, k is the
Boltzmann constant, and T is the
temperature in
kelvins. At 298 K, X-band
microwave frequencies (\nu ≈ 9.75 GHz) give n_\text{upper} /
n_\text{lower} ≈ 0.998, meaning that the upper energy level has a
smaller population than the lower one. Therefore, transitions from
the lower to the higher level are more probable than the reverse,
which is why there is a net absorption of energy.
The sensitivity of the EPR method (i.e., the minimum number of
detectable spins N_\text{min}) depends on the photon frequency \nu
according to
- N_\text{min} = \frac{k_1V}{Q_0k_f \nu^2 P^{1/2}}
where k_1 is a constant, V is the sample's volume, Q_0 is the
unloaded
quality factor of the microwave
cavity (sample chamber), k_f is the cavity filling coefficient, and
P is the microwave power in the spectrometer cavity. With k_f and P
being constants, N_\text{min} ~ (Q_0\nu^2)^{-1}, i.e., N_\text{min}
~ \nu^{-\alpha}, where \alpha ≈ 1.5. In practice, \alpha can change
varying from 0.5 to 4.5 depending on spectrometer characteristics,
resonance conditions, and sample size. In other words, the higher
the spectrometer frequency the lower the detection limit
(N_\text{min}), meaning greater sensitivity.
Spectral parameters
In real systems, electrons are normally not solitary, but are
associated with one or more atoms. There are several important
consequences of this:
- An unpaired electron can gain or lose angular momentum, which
can change the value of its g-factor, causing it to differ
from g_{e}. This is especially significant for
chemical systems with transition-metal ions.
- If an atom with which an unpaired electron is associated has a
non-zero nuclear spin, then its magnetic moment will affect the
electron. This leads to the phenomenon of hyperfine coupling, analogous to J-coupling in NMR, splitting the EPR resonance
signal into doublets, triplets and so forth.
- Interactions of an unpaired electron with its environment
influence the shape of an EPR spectral line. Line shapes can yield
information about, for example, rates of chemical reactions.
- The g-factor and hyperfine coupling in an atom or
molecule may not be the same for all orientations of an unpaired
electron in an external magnetic field. This anisotropy depends upon the electronic structure
of the atom or molecule (e.g., free radical) in question, and so
can provide information about the atomic or molecular orbital
containing the unpaired electron.
The g factor
Knowledge of the
g-factor can give information
about a paramagnetic center's electronic structure. An unpaired
electron responds not only to a spectrometer's applied magnetic
field
B_{0}, but also to any local magnetic fields
of atoms or molecules. The effective field
B_{eff}
experienced by an electron is thus written
- B_{\mathrm{eff}} = B_0(1-\sigma) \,
where
\sigma includes the effects of local fields (\sigma
can be positive or negative). Therefore, the
h\nu =
g_{e}μ_{B}B_{eff} resonance condition
(above) is rewritten as follows:
- h\nu = g_\mathrm{e} \mu_B B_\mathrm{eff} = g_\mathrm{e} \mu_B
B_0 (1 - \sigma) \,
The quantity
g_{e}(1 - σ) is denoted
g
and called simply the
g-factor, so that the final
resonance equation becomes
- h\nu = g \mu_B B_0 \,
This last equation is used to determine
g in an EPR
experiment by measuring the field and the frequency at which
resonance occurs. If
g does not equal
g_{e} the implication is that the ratio of the
unpaired electron's spin magnetic moment to its angular momentum
differs from the free electron value. Since an electron's spin
magnetic moment is constant (approximately the Bohr magneton), then
the electron must have gained or lost angular momentum through
spin-orbit coupling. Because the mechanisms of spin-orbit coupling
are well understood, the magnitude of the change gives information
about the nature of the atomic or molecular orbital containing the
unpaired electron.
Hyperfine coupling
Since the source of an EPR spectrum is a change in an electron's
spin state, it might be thought that all EPR spectra would consist
of a single line. However, the interaction of an unpaired electron,
by way of its magnetic moment, with nearby nuclear spins, results
in additional allowed energy states and, in turn, multi-lined
spectra. In such cases, the spacing between the EPR spectral lines
indicates the degree of interaction between the unpaired electron
and the perturbing nuclei. The
hyperfine coupling constant of a nucleus
is directly related to the spectral line spacing and, in the
simplest cases, is essentially the spacing itself.
Two common mechanisms by which electrons and nuclei interact are
the
Fermi contact
interaction and by dipolar interaction. The former applies
largely to the case of isotropic interactions (independent of
sample orientation in a magnetic field) and the latter to the case
of anisotropic interactions (spectra dependent on sample
orientation in a magnetic field). Spin polarization is a third
mechanism for interactions between an unpaired electron and a
nuclear spin, being especially important for \pi-electron organic
radicals, such as the benzene radical anion. The symbols
"
a" or "
A" are used for isotropic hyperfine
coupling constants while "
B" is usually employed for
anisotropic hyperfine coupling constants.
In many cases, the isotropic hyperfine splitting pattern for a
radical freely tumbling in a solution (isotropic system) can be
predicted.
- For a radical having M equivalent nuclei, each with a
spin of I, the number of EPR lines expected is
2MI + 1. As an example, the methyl radical,
CH_{3}, has three ^{1}H nuclei each with I
= 1/2, and so the number of lines expected is 2MI + 1 =
2(3)(1/2) + 1 = 4, which is as observed.
- For a radical having M_{1} equivalent nuclei, each with
a spin of I_{1}, and a group of M_{2} equivalent
nuclei, each with a spin of I_{2}, the number of lines
expected is (2M_{1}I_{1} + 1)
(2M_{2}I_{2} + 1). As an example, the methoxymethyl
radical, H_{2}C(OCH_{3}), has two equivalent
^{1}H nuclei each with I = 1/2 and three equivalent
^{1}H nuclei each with I = 1/2, and so the number of lines
expected is (2M_{1}I_{1} + 1)
(2M_{2}I_{2} + 1) = [2(2)(1/2) + 1][2(3)(1/2) + 1]
= [3][4] = 12, again as observed.
- The above can be extended to predict the number of lines for
any number of nuclei.
While it is easy to predict the number of lines a radical's EPR
spectrum should show, the reverse problem, unraveling a complex
multi-line EPR spectrum and assigning the various spacings to
specific nuclei, is more difficult.
In the oft-encountered case of
I = 1/2 nuclei (e.g.,
^{1}H,
^{19}F,
^{31}P), the line
intensities produced by a population of radicals, each possessing
M equivalent nuclei, will follow
Pascal's triangle. For example, the
spectrum at the right shows that the three
^{1}H nuclei of
the CH
_{3} radical give rise to 2
MI + 1 =
2(3)(1/2) + 1 = 4 lines with a 1:3:3:1 ratio. The line spacing
gives a hyperfine coupling constant of
a_{H} = 23
G for each of the three
^{1}H nuclei. Note again
that the lines in this spectrum are
first derivatives of
absorptions.
As a second example, consider the methoxymethyl radical,
H
_{2}C(OCH
_{3}). The two equivalent methyl
hydrogens will give an overall 1:2:1 EPR pattern, each component of
which is further split by the three methoxy hydrogens into a
1:3:3:1 pattern to give a total of 3 x 4 = 12 lines, a triplet of
quartets. A simulation of the observed EPR spectrum is shown at the
right, and agrees with the 12-line prediction and the expected line
intensities. Note that the smaller coupling constant (smaller line
spacing) is due to the three methoxy hydrogens, while the larger
coupling constant (line spacing) is from the two hydrogens bonded
directly to the carbon atom bearing the unpaired electron. It is
often the case that coupling constants decrease in size with
distance from a radical's unpaired electron, but there are some
notable exceptions, such as the ethyl radical
(CH
_{2}CH
_{3}).
Resonance linewidth definition
Resonance linewidths are defined in terms of the magnetic induction
B, and its corresponding units, and are measured along the
x axis of an EPR spectrum, from a line's center to a
chosen reference point of the line. These defined widths are called
halfwidths and possess some advantages: for asymmetric lines values
of left and right halfwidth can be given. The halfwidth \Delta B_h
is the distance measured from the line's center to the point in
which
absorption value has
half of maximal absorption value in the center of
resonance line. First inclination width \Delta
B_{1/2} is a distance from center of the line to the point of
maximal absorption curve inclination. In practice, a full
definition of linewidth is used. For symmetric lines, halfwidth
\Delta B_{1/2} = 2\Delta B_h, and full inclination width \Delta
B_{max} = 2\Delta B_{1s}
Applications
EPR spectroscopy is used in various branches of science, such as
chemistry and
physics, for the detection and identification of
free radical and paramagnetic
centers. EPR is a sensitive, specific method for studying both
radicals formed in chemical reactions and the reactions themselves.
For example, when frozen water (solid H
_{2}O) is decomposed
by exposure to high-energy radiation, radicals such as H, OH, and
HO
_{2} are produced. Such radicals can be identified and
studied by EPR. Organic and inorganic radicals can be detected in
electrochemical systems and in materials exposed to
UV light. In many cases, the reactions to make the
radicals and the subsequent reactions of the radicals are of
interest, while in other cases EPR is used to provide information
on a radical's geometry and the orbital of the unpaired
electron.
Medical and
biological applications of EPR also exist. Although
radicals are very reactive, and so do not normally occur in high
concentrations in biology, special reagents have been developed to
spin-label molecules of interest. These reagents are particularly
useful in biological systems. Specially-designed nonreactive
radical molecules can attach to specific sites in a
biological cell, and EPR spectra can then
give information on the environment of these so-called
spin-label or
spin-probes.
A type of
dosimetry system has been
designed for reference standards and routine use in medicine, based
on EPR signals of radicals from irradiated polycrystalline
α-
alanine(the alanine deamination radical,
the hydrogen abstraction radical, and the
(CO
^{-}(OH))=C(CH
_{3})NH
_{2}^{+}
radical) . This method is suitable for measuring
gamma and
x-rays, electrons,
protons, and high-
linear energy
transfer (LET) radiation of
doses
in the 1
Gy to 100 kGy range.
EPR spectroscopy can only be applied to systems in which the
balance between radical decay and radical formation keeps the
free-radicals concentration above the detection limit of the
spectrometer used. This can be a particularly severe problem in
studying reactions in liquids. An alternative approach is to slow
down reactions by studying samples held at
cryogenic temperatures, such as 77 K (
liquid nitrogen) or 4.2 K (liquid helium).
An example of this work is the study of radical reactions in single
crystals of amino acids exposed to x-rays, work that sometimes
leads to
activation energies and
rate constants for radical reactions.
The study of radiation-induced free radicals in biological
substances (for cancer research) poses the additional problem that
tissue contains water, and water (due to its
electric dipole moment) has a strong
absorption band in the
microwave region
used in EPR spectrometers.
EPR also has been used by archaeologists for the dating of teeth.
Radiation damage over long periods of time creates free radicals in
tooth enamel, which can then be examined by EPR and, after proper
calibration, dated. Alternatively, material extracted from the
teeth of people during dental procedures can be used to quantify
their cumulative exposure to ionizing radiation.
People exposed to
radiation from the Chernobyl disaster have been examined by this method.
Radiation-sterilized foods have been examined with EPR
spectroscopy, the aim being to develop methods to determine if a
particular food sample has been irradiated and to what dose.
Because of its high sensitivity, EPR was used recently to measure
the quantity of energy used locally during a mechanochemical
milling process.
High-field high-frequency measurements
High-field-high-frequency EPR measurements are sometimes needed to
detect subtle spectroscopic details. However, for many years the
use of electromagnets to produce the needed fields above 1.5 T was
impossible, due principally to limitations of traditional magnet
materials. The first multifunctional millimeter EPR spectrometer
with a superconducting solenoid was described in the early 1970s by
Prof. Y. S.
Lebedev's group (Russian Institute of Chemical Physics,
Moscow) in collaboration with L. G.
Oranski's group
(Ukrainian Physics and Technics Institute, Donetsk) which began
working in the Institute of Problems
of Chemical Physics, Chernogolovka around 1975. Two decades later, a W-band EPR
spectrometer was produced as a small commercial line by the German
Bruker Company, initiating the expansion of
W-band EPR techniques into medium-sized academic laboratories.
Today
there still are only a few scientific centers in the world capable
of high-field-high-frequency EPR, among them are the Grenoble High
Magnetic Field Laboratory in Grenoble, France, the Physics
Department in Freie Universität Berlin, the National High Magnetic
Field Laboratory in Tallahassee, US, the National Center for Advanced ESR
Technology (ACERT) at Cornell University in Ithaca, US, the
Department of Physiology and Biophysics at Albert
Einstein College of Medicine, Bronx, NY, the
IFW in Dresden, Germany, the
Institute of Physics of Complex Matter in Lausanne in Switzerland, and the Institute of Physics of the Leiden University, Netherlands.
Waveband |
L |
S |
C |
X |
P |
K |
Q |
U |
V |
E |
W |
F |
D |
— |
J |
— |
\lambda/\text{mm} |
300 |
100 |
75 |
30 |
20 |
12.5 |
8.5 |
6 |
4.6 |
4 |
3.2 |
2.7 |
2.1 |
1.6 |
1.1 |
0.83 |
\nu / \text{GHz} |
1 |
3 |
4 |
10 |
15 |
24 |
35 |
50 |
65 |
75 |
95 |
111 |
140 |
190 |
285 |
360 |
B_0 / \text{T} |
0.03 |
0.11 |
0.14 |
0.33 |
0.54 |
0.86 |
1.25 |
1.8 |
2.3 |
2.7 |
3.5 |
3.9 |
4.9 |
6.8 |
10.2 |
12.8 |
The EPR waveband is stipulated by the frequency or wavelength of a
spectrometer's microwave source (see Table).
EPR experiments often are conducted at X and, less commonly, Q
bands, mainly due to the ready availability of the necessary
microwave components (which originally were developed for radar
applications). A second reason for widespread X and Q band
measurements is that electromagnets can reliably generate fields up
to about 1 tesla. However, the low spectral resolution over
g-factor at these wavebands limits the study of
paramagnetic centers with comparatively low anisotropic magnetic
parameters. Measurements at \nu > 40 GHz, in the millimeter
wavelength region, offer the following advantages:
- EPR spectra are simplified due to the reduction of second-order
effects at high fields.
- Increase in orientation selectivity and sensitivity in the
investigation of disordered systems.
- The informativity and precision of pulse methods, e.g.,
ENDOR also increase at high magnetic
fields.
- Accessibility of spin systems with larger zero-field splitting
due to the larger microwave quantum energy h\nu.
- The higher spectral resolution over g-factor, which
increases with irradiation frequency \nu and external magnetic
field B_{0}. This is used to investigate the
structure, polarity, and dynamics of radical microenvironments in
spin-modified organic and biological systems through the spin label and probe method. The figure shows how
spectral resolution improves with increasing frequency.
- Saturation of paramagnetic centers occurs at a comparatively
low microwave polarizing field B_{1}, due to the
exponential dependence of the number of excited spins on the
radiation frequency \nu. This effect can be successfully used to
study the relaxation and dynamics of paramagnetic centers as well
as of superslow motion in the systems under study.
- The cross-relaxation of paramagnetic centers decreases
dramatically at high magnetic fields, making it easier to obtain
more-precise and more-complete information about the system under
study.
See also
References
- Strictly speaking, "a" refers to the hyperfine
splitting constant, a line spacing measured in magnetic field
units, while A and B refer to hyperfine coupling
constants measured in frequency units. Splitting and coupling
constants are proportional, but not identical. The book by Wertz
and Bolton has more information (pp. 46 and 442).
- EPR of
low-dimensional systems
Further reading
Many good books and papers are available on the subject of EPR
spectroscopy, including those listed here. Essentially all details
in this article can be found in these.
- - Provides an overview of the role of free radicals in biology
and of the use of electron spin resonance in their detection.
- Protein structure elucidation by EPR:
External links