Vij D.R. Handbook of Applied Solid State Spectroscopy
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an Orbach mechanism predicting 1/T
1
~ exp (–'E/kT). The energy gap to the
excited state involved in the relaxation process is found to be 'E = 126.1 r
0.5 K. Two-pulse ESE measurements of isotopically purified
28
Si:P showed
exceptionally long transverse relaxation times T
2
. Below ~ 10 K the spin
decoherence is controlled by instantaneous diffusion, and at higher tempera-
tures by an Orbach process. T
2
is found to be 14 ms at 7 K for small turning
angles in the samples investigated with the concentration ~10
16
P/cm
3
that
extrapolates to ~ 60 ms for an isolated spin. The observation of such an
extremely long T
2
has important implications for efforts to use bound electron
spins in Si:P as qubits for quantum information processing.
3.4.5 Interaction Between Electron Spins
Special attention in recent years has been given to the development of EPR-
based techniques for the measurements of long-range distances between
electron spins. Traditional CW-EPR techniques for the measurement of
distances between spins that do not produce observable splittings in the EPR
spectra (called fine structure) are based on measurements of relaxation times
using a saturation approach.
More sophisticated approaches for the measurement of distances between
electron spins are based on pulsed-EPR techniques. Those include pulsed
electron-electron double resonance (PELDOR) or double electron-electron
resonance (DEER), “2 + 1” sequence, and double-quantum EPR. All of them
are described in detail in a recent book [69].
Among these techniques, increasing attention has been given to PELDOR.
This technique uses pulses at two different microwave frequencies to measure
the distance in a pair of paramagnetic species by exploiting the dipole-dipole
spin interaction. Two different sequences were employed in the experiments.
In the PELDOR sequence originally proposed [70], two microwave pulses
induce the ESE signal at frequency Q
A
in a spin system. Between pulses 1 and
2 a pumping pulse is applied at frequency Q
B
at time T. The pumping pulse
induces transitions between the Zeeman levels of spins B and thus changes
local magnetic field at spins A. This results in additional dephasing of spins
A, and hence gives rise to a decrease of the PELDOR signal.
In the approximation of point dipoles, assuming that delocalization of the
electrons is small on a scale compared to the distance between them, the
interaction between spins is characterized by the frequency
2
AB dd AB
13cos
J
Q Q T (3.57)
where
2
ABe
3
dd
AB
gg
h
E
Q
r
145
3.4 Applications of EPR Spectroscopy
3. Electron Paramagnetic Resonance Spectroscopy
is distant dependent dipole-dipole coupling, T
AB
is the angle between the field
B
0
and r
AB
vectors, and J is the isotropic exchange coupling. In the powder
sample, splittings for different orientations form a typical powder pattern with
the two pairs of singularities corresponding to T
AB
= 0 and 90
o
with the
splittings Q
AB
(__) = _J – 2Q
dd
_ and Q
AB
(A) = _J + Q
dd
_, respectively. Fourier
transformation of the echo envelope should provide a powder pattern of Q
AB
splittings. For spin-spin distances above ~ 2 nm the exchange coupling J is
usually negligible and thus observation of even one singularity Q
AB
(A) with
larger intensity would provide a measurement of the r
AB
.
Milov et al. [71] performed model PELDOR experiments with the
biradicals of the rigid structure in frozen solution. They found the appearance
of echo oscillations with the frequency ~ 7.5 MHz. These oscillations were
sent in the sample with mononitroxide. Thus, assigned to Q
AB
(A) it gave the
distance between the nitroxide fragments equal to 1.9 r 0.16 nm. Schienman
et al. [72] in a PELDOR experiment with the RNA containing spin-labeled
uridines in frozen solution detected an oscillation of the echo envelope
with
1.2 r 0.2 MHz. This was also assigned to the Q
AB
(A), and gave
r
AB
= 3.5 r 0.2
In addition to the PELDOR sequence described above, a four-pulse
sequence has been proposed that avoids the problem of a large dead-time that
influences the three-pulse experiment [73].
The PELDOR techniques are capable of measuring distances in the range
from 1.5 up to 10 nm. Recent applications of PELDOR include measurements
of the distances between spins in biradicals [74], nitroxides in spin-labeled
polymers [75] and peptides [76], the redox-active components in the PSII
reaction center [77], and between Mo(V) and Fe(III) centers in chicken liver
sulfite oxidase [78].
ACKNOWLEDGEMENTS
Support for the authors’ research by grants from the National Institute of
Health (GM 62954 to S.A. Dikanov and GM 35438 to A.R. Crofts), by
Fogarty NIH grant (PHS 1RO3 TW 01495 to A.R. Crofts with Dr. Rimma
Samoilova, Institute of Chemical Kinetics and Combustion, Novosibirsk,
Russia) and by NIH/NCRR grant S10-RR15878 for instrumentation is
gratefully acknowledged.
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CHAPTER 4
ENDOR SPECTROSCOPY
Lowell D. Kispert and Lidia Piekara-Sady
The University of Alabama, Chemistry Department, Tuscaloosa, Alabama, USA
4.1 INTRODUCTION
To understand what electron nuclear double resonance (ENDOR) transitions
are, let us consider the energy levels of a system containing one unpaired
electron (S = ½) and one nuclear spin (I = ½), as in a hydrogen atom. Since
the electron spin and nuclear spin can each be oriented with or against the
external magnetic field, there are four possible energy levels as shown in
Figure 4.1. The two electron spin eigenfunctions are denoted by D
e
and E
e
,
whereas the two nuclear spin wavefunctions are denoted D
n
and E
n
. In the
presence of a large external applied field, the selection rule in electron
paramagnetic resonance (EPR) is that the electron flips (changes directions)
but the nuclear spin does not flip in an EPR transition. Thus, in Figure 4.1, an
EPR transition will occur between energy levels 1 and 4 and a second EPR
transitions will occur between energy levels 2 and 3. The intensity of the
transition between energy levels 1 and 4 will depend on the population
differences of these two energy levels. If one then applies a nuclear frequency
to the system, which corresponds to either the energy difference between
levels 4 and 3 or the difference between lines 2 and 1, one will induce nuclear
spin flip that will change the populations in energy levels 4 and 1,
respectively. This changes the intensity of the observed EPR signal. This
change in intensity is referred to as the ENDOR response.
Figure 4.1 Energy diagram for S = 1/2 and I = 1/2.
Analytically, the energy levels are obtained from the spin-Hamiltonian
appropriate for a system consisting of a single unpaired electron (S = ½) and a
single magnetic nucleus (I = ½).
H = H
EZ
+ H
NZ
+ H
HF
(4.1)
where H
EZ
is the electronic Zeeman term that describes the interaction of the
electron spin (S) with a magnetic field B according to
H
EZ
= EB
.
g
.
S. (4.2)
g is a dimensionless number called the g-factor and E is the Bohr magneton.
For a free electron, the g-factor is equal to 2.0023. However, the g value of
an electron in an atomic or molecular orbital is different and may also be
anisotropic, i.e., dependent on the orientation of the magnetic field B relative
to the system.
H
NZ
is the nuclear Zeeman Hamiltonian where
H
NZ
= –g
n
E
n
B
.
I (4.3)
In contrast to the g value in EPR measurements, g
n
is an inherent property
of the nucleus and any corrections to the nuclear Zeeman energy are referred
to as screening effects in NMR spectroscopy. The interaction of the unpaired
electron (S) with nearby magnetic nuclei (I) gives rise to a hyperfine structure
in the EPR spectrum. This electron-nuclear hyperfine interaction is described
by the H
HF
= hS
.
Ã
.
I Hamiltonian where h is the Planck constant so that the
elements of the hyperfine tensor à are expressed in frequency units. This
tensor can be separated into the sum of two terms giving
H
HF
= haS
.
I + hS
.
Ã
dip
.
I (4.4)
where the first term describes the isotropic hyperfine interaction (a is its
constant) and the second describes the anisotropic (dipolar) interaction
represented by the hyperfine dipolar tensor Ã
dip
.
The magnetic moments of the electron and nuclei are coupled via the
contact
interaction is the hydrogen atom, with a hyperfine constant of 1420 MHz.
In
more complex systems, i.e., in molecules, the general expression of the
isotropic hyperfine constant is
a = 8/3SgEg
n
E
n
U(r
n
) (4.5)
where U(r
n
) is the net unpaired electron-spin density at the nucleus. The
magnetic coupling also exists between the magnetic moments of an electron
and a nucleus. The magnitude of this interaction depends on the orientation of
the paramagnetic molecule relative to the external magnetic field B. In
isotropic fluid solutions, the dipolar interactions are averaged out and only the
orientation-independent isotropic couplings give rise to the observed
hyperfine coupling in a spectrum. The complete Hamiltonian for an electron
coupled to an oriented nucleus with spin I = ½ in the magnetic field is given
by
4. ENDOR Spectroscopy
(Fermi) interaction. The simplest system exhibiting a nuclear hyper-
fine
152
H = H
EZ
+ H
NZ
+ H
HF
= ES
.
g
.
B – g
n
E
n
B
.
I + hS
.
Ã
.
I (4.6)
A nucleus with I 1, exhibits an electric quadrupole moment Q, which
interacts with the gradient of the electric field due to all surrounding electrons.
This interaction is described by the quadrupole interaction term
H
Q
= hI
.
Q
.
I (4.7)
where Q is the quadrupole coupling tensor. In liquids, this interaction is
average out to zero. For the isotropic case of g and hyperfine coupling, as
occurs for radicals in solution, the Hamiltonian given in equation (4.1)
becomes:
H = gEBS – g
n
E
n
BI + haS
.
I (4.8)
In the high-field approximation, i.e., H
EZ
>>H
NZ
and H
HF
, the Hamiltonian
corresponds to a set of four energy levels:
E = m
s
gEB – m
I
g
n
E
n
B + ham
s
m
I
, (4.9)
where g
n
and a may be positive or negative, thus leading to a different
ordering of the levels. The EPR transitions are governed by the selection rules
'm
s
= ±1 and 'm
I
= 0. Adopting the notation
hQ
e
= gEB (4.10)
hQ
n
= g
n
E
n
B, (4.11)
two EPR transitions are obtained:
Q
EPR
= Q
e
±
2
a
(4.12)
and two NMR transitions:
Q
±
= »Q
n
± |
2
a
. (4.13)
The two NMR transitions defined by equation (4.13) are referred to as
ENDOR transitions. The term “ENDOR’’ is used to describe the indirect
detection of the NMR transitions where the ENDOR transitions obey the
selection rules 'm
s
= 0 and 'm
I
= ±1. The energy levels are separated into
two cases: »a»/2 > Q
n
and »a»/2 < Q
n
.
ENDOR spectra are shown schematically in Figure 4.2. For Q
n
>»a»/2, the
two ENDOR transitions are given by Q
n
± »a»/2. Two ENDOR lines, separated
by »a» and centered at Q
n
, are observed. The same two ENDOR transitions are
observed when either EPR line is monitored, and in both cases the ENDOR
lines are separated by exactly »a». However, since Q
n
depends on B, there will
be a shift in the center of the spectrum as different EPR lines are monitored.
For Q
n
< »a»/2, the two ENDOR transitions are given by »a» /2 ±Q
n
. Two lines
are observed separated by 2Q
n
and centered at »a» /2.
When there is more than one nucleus with I = ½ and a particular a value,
the interactions responsible for the various isotropic splittings are described
by the following Hamiltonian:
153
4.1 Introduction
H = gEB
.
S – 6g
n
E
n
B
.
I
i
+ h 6 a
i
S
.
I
i.
(4.14)
For a group of equivalent nuclei having the same nuclear Larmor frequency Q
n
and the same hyperfine constant a
i
= a, the EPR transitions are
Q
EPR
= Q
e
+ M
Ia
, (4.15)
where
II
1
m
n
i
i
Ç
M
and the summation is over n equivalent nuclei. The NMR transitions remain
unchanged, Q
±
= »Q
n
±
2
a
_ no matter how many equivalent nuclei are coupled
to the unpaired electron.
Figure 4.2 ENDOR line positions for (a) »a»/2 > Q
n
and (b) »a»/2 < Q
n
. The solid lines give the
line positions when the high field EPR line is monitored, the dashed lines when the low field
EPR is monitored.
In general [1] the EPR spectrum will consist of:
(1) (2I + 1)
N
hyperfine components, if there are N nuclei with nuclear
spin I.
(2) 2
N
different hyperfine components for N nuclei with I = ½, if all the
coupling constants differ and no degeneracy occurs.
(3) 3
i
(2I
i
+ 1)
N
i
hyperfine components, if there are several nuclei with
the individual values I
i
and N
i
; the symbol 3
i
denotes the formation
of a product; when some nuclei are equivalent, the number of lines is
decreased but the amplitude of the components in the hyperfine
pattern is increased.
4. ENDOR Spectroscopy
=
154
experiences the hyperfine field of only one unpaired electron, whereas an
electron can couple with all nuclei.
ENDOR measurements are always useful in cases where poor resolution
makes it impossible to determine the magnetic parameters from an EPR
spectrum. In addition, the ENDOR lines for nuclei with different magnetic
moments occur in different frequency ranges on account of their different
Larmor frequencies Q
n
= g
n
E
n
B/h (see Table 4.1). The dramatic gain in
resolution compared to EPR is presented in Figure 4.3.
From an ENDOR spectrum: (1) precise measurement of hyperfine
coupling constants »a» is possible when these are not obtainable from EPR
spectra, and (2) the nucleus responsible for the hyperfine coupling can be
identified from the relation Q
+
± Q
–
= 2Q
n
= 2g
n
E
n
B/h. Although only the
absolute value of g
n
(or J
n
) can be evaluated from the free nuclear frequency
(Q
n
), this value provides sufficient information to identify (see Table 4.1) the
nucleus. Conversely, at a given experimental field value B, the frequencies of
like nuclei can be calculated (Table 4.l) before the measurement is made, and
the ENDOR spectrum obtained can be interpreted by comparison. ENDOR
spectra also yield anisotropic hyperfine and quadrupole (for nuclei with I > 1)
coupling constants in solids.
Table 4.1 Properties of selected nuclei.
Nucleus I
Nat.
abund. (%)
J
n
(MHz/T)
a
Q
n
(MHz)
b
Q
(10
24
cm
2
)
1
H 1/2 99.99 267.51 14.00
2
H 1 0.015 41.06 2.15 0.00277
6
Li 1 7.42 39.37 2.06 0.007
7
Li 3/2 92.58 103.96 5.44 –0.03
10
B 3 19.78 28.75 1.50 00.74
11
B 3/2 80.22 85.83 4.49 0.036
13
C 1/2 1.11 67.26 3.52
14
N 1 99.63 19.32 1.01 0.016
15
N 1/2 0.37 –27.11 1.42
19
F 1/2 100 251.67 13.17
23
Na 3/2 100 70.76 3.70 ~0.14
25
Mg 5/2 10.13 –16.37 0.86 0.22
27
Al 5/2 100 69.71 3.65 0.15
29
Si 1/2 4.70 –53.14 2.78
31
P 1/2 100 108.29 5.67
39
K 3/2 93.10 12.48 0.65 0.11
85
Rb 5/2 72.15 25.83 1.35 0.27
87
Rb 3/2 27.85 87.53 4.58 0.13
117
Sn 1/2 7.61 –95.30 4.99
119
Sn 1/2 8.58 –99.71 5.22
133
Cs 7/2 100 35.09 1.84 –0.003
203
Tl 1/2 29.5 152.88 8.00
205
Tl 1/2 70.5 154.38 8.08
a
1 T (tesla) = 10
4
G.
b
For a magnetic field value of 3288 G.
group of equivalent nuclei no matter how many nuclei there are or what their
spin quantum number is. This results from the fact that each nucleus
In ENDOR there are always two lines separated by »a» or 2Q
n
for any
155
155
4.1 Introduction