13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Electron paramagnetic resonance | 2/7 | https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance | reference | science, encyclopedia | 2026-05-05T10:04:27.171248+00:00 | kb-cron |
=== 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 Boltzmann distribution:
n
upper
n
lower
=
exp
(
−
E
upper
−
E
lower
k
T
)
=
exp
(
−
Δ
E
k
T
)
=
exp
(
−
ϵ
k
T
)
=
exp
(
−
h
ν
k
T
)
{\displaystyle {\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
upper
{\displaystyle n_{\text{upper}}}
is the number of paramagnetic centers occupying the upper energy state,
k
{\displaystyle k}
is the Boltzmann constant, and
T
{\displaystyle T}
is the thermodynamic temperature. At 298 K, X-band microwave frequencies (
ν
{\displaystyle \nu }
≈ 9.75 GHz) give
n
upper
/
n
lower
{\displaystyle n_{\text{upper}}/n_{\text{lower}}}
≈ 0.998, meaning that the upper energy level has a slightly 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 minimal number of detectable spins
N
min
{\displaystyle N_{\text{min}}}
) depends on the photon frequency
ν
{\displaystyle \nu }
according to
N
min
=
k
1
V
Q
0
k
f
ν
2
P
1
/
2
,
(Eq. 2)
{\displaystyle N_{\text{min}}={\frac {k_{1}V}{Q_{0}k_{f}\nu ^{2}P^{1/2}}},\qquad {\text{(Eq. 2)}}}
where
k
1
{\displaystyle k_{1}}
is a constant,
V
{\displaystyle V}
is the sample's volume,
Q
0
{\displaystyle Q_{0}}
is the unloaded quality factor of the microwave cavity (sample chamber),
k
f
{\displaystyle k_{f}}
is the cavity filling coefficient, and
P
{\displaystyle P}
is the microwave power in the spectrometer cavity. With
k
f
{\displaystyle k_{f}}
and
P
{\displaystyle P}
being constants,
N
min
{\displaystyle N_{\text{min}}}
~
(
Q
0
ν
2
)
−
1
{\displaystyle (Q_{0}\nu ^{2})^{-1}}
, i.e.,
N
min
{\displaystyle N_{\text{min}}}
~
ν
−
α
{\displaystyle \nu ^{-\alpha }}
, where
α
{\displaystyle \alpha }
≈ 1.5. In practice,
α
{\displaystyle \alpha }
can change varying from 0.5 to 4.5 depending on spectrometer characteristics, resonance conditions, and sample size. A great sensitivity is therefore obtained with a low detection limit
N
min
{\displaystyle N_{\text{min}}}
and a large number of spins. Therefore, the required parameters are:
A high spectrometer frequency to minimize the Eq. 2. Common frequencies are discussed below A low temperature to decrease the number of spin at the high level of energy as shown in Eq. 1. This condition explains why spectra are often recorded on sample at the boiling point of liquid nitrogen or liquid helium.
== 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
{\displaystyle g_{e}}
. This is especially significant for chemical systems with transition-metal ions. Systems with multiple unpaired electrons experience electron–electron interactions that give rise to "fine" structure. This is realized as zero field splitting and exchange coupling, and can be large in magnitude. The magnetic moment of a nucleus with a non-zero nuclear spin will affect any unpaired electrons associated with that atom. 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. Additional smaller splittings from nearby nuclei is sometimes termed "superhyperfine" coupling. 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. These effects (g-factor, hyperfine coupling, zero field splitting, exchange 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
{\displaystyle B_{0}}
but also to any local magnetic fields of atoms or molecules. The effective field
B
eff
{\displaystyle B_{\text{eff}}}
experienced by an electron is thus written
B
eff
=
B
0
(
1
−
σ
)
,
{\displaystyle B_{\text{eff}}=B_{0}(1-\sigma ),}
where
σ
{\displaystyle \sigma }
includes the effects of local fields (
σ
{\displaystyle \sigma }
can be positive or negative). Therefore, the
h
ν
=
g
e
μ
B
B
eff
{\displaystyle h\nu =g_{e}\mu _{\text{B}}B_{\text{eff}}}
resonance condition (above) is rewritten as follows:
h
ν
=
g
e
μ
B
B
eff
=
g
e
μ
B
B
0
(
1
−
σ
)
.
{\displaystyle h\nu =g_{e}\mu _{B}B_{\text{eff}}=g_{e}\mu _{\text{B}}B_{0}(1-\sigma ).}
The quantity
g
e
(
1
−
σ
)
{\displaystyle g_{e}(1-\sigma )}
is denoted
g
{\displaystyle g}
and called simply the g-factor, so that the final resonance equation becomes
h
ν
=
g
μ
B
B
0
.
{\displaystyle h\nu =g\mu _{\text{B}}B_{0}.}
This last equation is used to determine
g
{\displaystyle g}
in an EPR experiment by measuring the field and the frequency at which resonance occurs. If
g
{\displaystyle g}
does not equal
g
e
{\displaystyle 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.