Vij D.R. Handbook of Applied Solid State Spectroscopy

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accomplished by a study of the angular dependence of the ENDOR lines,

which were observed in the 0.5–26 MHz range. The hyperfine couplings to

the nuclei in some shells have characteristically different angular

dependencies. When the angular dependence of the hyperfine coupling of

two shells is similar, the magnitudes of the hyperfine coupling are usually

very different and the line pairs can be assigned.

i. Signs of the hyperfine coupling constants: In general, only the magnitude

of the hyperfine coupling constants can be determined by EPR and ENDOR.

The sign, however, is especially important, because it is related to the sign of

the unpaired spin density and consequently to the electronic structure of the

radical. The general TRIPLE resonance method is very helpful in

determining the relative sign.

ii. Second-order shifts: Second-order effects in the hyperfine couplings

measured by ENDOR can generally be detected for A  30 MHz. Although

they do not contain any additional information, second order shifts must be

included in the analysis in order to avoid errors in the calculated values of the

elements of the hyperfine tensors.

4.3.1.2 Quadrupole Couplings

Nuclei having spin I  1 (e.g.,

Cu, see Tables 4.1 and 4.2) possess

an electric quadrupole moment in addition to a magnetic moment. If the

interaction of a quadrupole moment with the electric field gradient at the

nucleus is smaller than the hyperfine interaction, it will not affect the EPR

spectra, to first order. However, in ENDOR spectra this interaction will lead

to additional splittings. For an electron and a single nucleus with I  1, the

quadrupole term, H

, should be added to the spin-Hamiltonian.

H = H

HF Z

(4.29)

where Q is the quadrupole coupling tensor. It is often assumed that the

hyperfine coupling tensor Ã and the quadrupole coupling tensor Q have the

same principal axis system. This coincidence of these Ã and Q tensors may

be imposed by symmetry.

The energy level diagrams for an electron coupled to a nucleus with I = 1

are given in [1] page 185. The resonance condition for the four ENDOR

transitions is given by

QQ AQ

(4.30)

4. ENDOR Spectroscopy

+ H + hIQ I

166

Table 4.2 Electronic and nuclear properties of the transition metal ions.

Ion

Electron

configuration

Total

spin

Spin-

orbit

constant

(cm

–1

)

Nucleus

Abun-

dance

(%)

Nuclear

spin I

Quadruple

moment

(10

)

1/2 154

7.32 5/2 –0.3153 ...

5.46 7/2 –0.3154 ...

1 209

99.8 7/2 1.471 0.25

3/2 167

3/2 273

9.55 3/2 –0.3163 ...

2 230

2 352

Mn 100 5/2 1.387 0.3

5/2 347

5/2 ...

2.21 1/2 <0.10 ...

2 410

2 ...

100 7/2 1.328 0.5

3/2 533

1 649

1/2 829

69.1 3/2 1.484 –0.16

Cu 30.9 3/2 1.590 –0.15

The number of observed lines depends on which levels are common for a

given EPR transition. All four lines are expected for the m

= 0 EPR

transition. Two ENDOR lines are observed for the m

= 1 EPR (low field)

transition, while for the m

= –1 EPR (high field) transition another two

ENDOR lines are observed. This separation of the ENDOR frequencies into

two groups allows determination of the relative signs of the quadrupole and

hyperfine interactions.

For I > 1, there are 4 × I ENDOR transitions given by

Q =

A ± Q

± Q(2m

+ 1) (4.31)

Quadrupole couplings result from the electric field gradients at the nucleus

and therefore give information about charge distribution in nearby chemical

bonds.

4.3.2 Organic Free Radicals

Free radicals produced by ionizing radiation of organic single crystals, have

been extensively studied. An ENDOR study results in the radical’s

identification and its orientation in the crystal. The radical formation and

transformation mechanism can also be derived. The occurrence of

167

4.3 ENDOR in the Solid State

overlapping radical spectra is more common in single crystals than in liquids,

hence the ENDOR-induced EPR technique shown in [1], pages 97–98 is

especially useful. Most radicals in single crystals exhibit g values with little

anisotropy and range from 2.0023 to 2.0080.

4.3.2.1 Interpretation of the Hyperfine Tensors for Organic Free Radicals

The magnitudes of the hyperfine tensor components are determined from the

ENDOR spectra. The relative signs can be obtained with the general TRIPLE

method. The absolute sign of the principal values is often given on the basis

of theoretical considerations. The isotropic and anisotropic parts of the tensor

are usually separately listed. The orientation of the principal axes relative to

the crystal axis is defined in terms of the direction cosines.

Because protons are the most extensively studied nuclei, the proton

hyperfine tensors will be considered. Protons are labeled by Greek letters

depending on how many bonds away they are from the radical center; the

proton attached directly to the free radical center is labeled D. The D-proton

dipolar tensors exhibit large anisotropy amounting to about 50% of the

isotropic coupling for unit spin density at the radical center. The anisotropy

results from electron dipole-nuclear dipole magnetic interactions. The a

iso

D-protons is characterized by a negative sign.

The anisotropy of protons attached to a Ecarbon atom is typically about

10% of the a

iso

, for unit spin density. This small anisotropy is the result of the

greater (two bonds) distance of the Eproton from the atom on which the

unpaired spin density is located. E-proton coupling constants are characterized

by a positive sign of a

iso

. The isotropic coupling to E-protons varies as cos

equation (4.33) where Tis the dihedral angle between two planes, both of

which contain the D-carbon-E-carbon bond axis. The magnitude of a

iso

protons is related to the radical geometry. It is therefore possible to study the

internal rotation of the radical fragment by analyzing the E-proton coupling

constants. A typical example is the methyl group rotation. Methyl protons

appear equivalent at high temperatures, i.e., their hyperfine interaction can be

described by one tensor with axial symmetry around the C

axis of rotation.

On lowering the temperature, however, it is often observed that methyl group

rotation is stopped (on the EPR time scale). The hyperfine interactions for the

protons are then described by their individual hyperfine tensors. The

orientation of the “static” protons with respect to the rest of the radical can be

determined.

J-proton couplings are generally much smaller than those of Dprotons (by

about two orders of magnitude), so they usually cannot be directly measured

by EPR, but can be detected by ENDOR methods.

The special advantage of single-crystal ENDOR over EPR measurements

is the detection of small hyperfine coupling constants (typically from 100

4. ENDOR Spectroscopy

168

kHz). These small couplings are assigned to weakly coupled protons within

the molecule or lattice protons that are dipolar-coupled to the free radical.

From such couplings accurate dipole-dipole distances can be deduced and

from these, in turn, the precise structure and location of the radical can be

determined.

4.3.3 Transition Metal Ions

EPR spectra are routinely observed for paramagnetic transition metal ions in

crystals, chemical complexes, and biomolecules. For ions with large spin-

orbit coupling constants (see Table 4.2), the spin system is strongly coupled to

the lattice vibrations so that spin relaxation is very effective. Consequently,

the EPR lines are sometimes too broad to be observed at room temperature,

and EPR and ENDOR studies of such compounds are usually carried out at

much lower temperatures.

The central problem in EPR studies of transition metal ions is the effect of

the surrounding atoms, by means of symmetry, on their electronic levels. The

EPR spectra are characteristic of the transition metal ion itself rather than the

hidden in the inhomogeneously broadened EPR line. In contrast, ENDOR

spectra provide both strong and weak hyperfine as well as quadrupole

transition metal ions in crystals or in molecular environments are anisotropic.

the usual hyperfine term: H

= hS

I, where I is the spin of the nucleus (see

The isotropic coupling is due to mixing of the ground and excited state

configurations. In the absence of covalent bonding, the spin densities at the

ion are not very sensitive to the environment of the ion. The typical situation

for transition metal ion ENDOR is that H

> H

, so the ENDOR lines

appear asymmetrically around the value of A/2 (for S =½) because of the

second-order effects. On the other hand, for ligand-ENDOR often H

< H

so the ENDOR lines are centered about the free nuclear frequency Q

. The

identity of the ligand nuclei can be determined from the number of ENDOR

transitions and the value of Q

As an example, the

Cu ENDOR study for a Cu mixed-ligand complex

doped into the corresponding Ni complex single crystal is shown in Figure 4.5

[5]. The observed

Cu ENDOR spectrum Figure (4.5c) depends on which

EPR line is saturated, provided both EPR and NMR transitions have levels in

common Figure (4.5a). The

P ENDOR lines appear in a different frequency

range (5.5–7 and 17–18.5 MHz).] The spectra were analyzed in terms of the

Table 4.2). The isotropic and dipolar part of the Ã tensor can be separated.

surroundings. Weak hyperfine couplings to more distant nuclei are usually

mostly localized on the metal ion. The magnetic moments and the g factors of

The hyperfine interaction for the transition metal ion itself is described by

coupling tensors. The unpaired electron in transition metal complexes is

169

4.3 ENDOR in the Solid State

Hamiltonian expressed as the sum of the following interactions: electron

Zeeman, Cu hyperfine, Cu nuclear Zeeman, Cu quadrupole, P hyperfine, and

P nuclear Zeeman:

H = EB

S + S

–

+ I

+ S

– g

(4.32)

Figure 4.5 (a)

Cu energy levels and ENDOR transitions for each saturated EPR line. To the

right is a typical single-crystal EPR spectrum of the Cu complex I at 27.2 K. The Cu hyperfine

lines are split into doublets because of hyperfine interaction of the unpaired electron with

Proton spin flip transitions correspond to 'm

= 1, and 'm

= 1, the so-called forbidden

transitions (b) Complete

Cu ENDOR spectrum recorded for each EPR line saturated as

indicated in (a) (for B

i.e., in the plane of smallest g anisotropy). From [5].

The analysis gave very precise values of the Cu hyperfine and quadrupole

tensors as well as of the phosphorus hyperfine coupling tensor. The principal

axes Ã

tensor were found to coincide with those of g. Further, the relative

orientations of the principal axes of Q

and Ã

with respect to those of g

were determined so that the structure and symmetry of the Cu complex were

established [5].

4. ENDOR Spectroscopy

170

4.3.3.1 Triplet State ENDOR

An atom or a molecule with the total spin of the electrons S = 1 is said to be

in a triplet state. The multiplicity of such a state is (2S + 1) = 3. Triplet

systems occur in both excited and ground state molecules, in transition metal

ions with S = 1, in pairs of paramagnetic centers, and in some defects in

solids.

For a system with S = 1, there are three sublevels characterized by m

= ±

1 and m

= 0. In the absence of an external magnetic field, these sublevels, in

contrast to the systems with S = ½, may not be degenerate. The lifting of

degeneracy of the spin states in the absence of a magnetic field is called zero-

field splitting, and it is common for systems with S  1.

The zero-field splitting (or fine structure) term must be included in a spin-

Hamiltonian when S  1. It can be expressed as a tensor interaction of the

total effective spin with itself, i.e.,

= S

S. (4.33)

The spin-spin coupling (or zero-field splitting) tensor D is a symmetric and

traceless tensor. The fine structure term is often written in terms of zero-field

constants D and E in a principal axis system:





222

(1) ,

ÎÞ



Ïß

Ðà

SSS SS

HD E (4.34)

where D is the axial and E is the rhombic zero-field parameter. When

222

1, ( ) ( ) SSSS

HD E . With B parallel to either the x, y, or z

axis, a pair of EPR lines will be observed with couplings equal to ѿ D – E,

ѿD + E, and –2/3D. For axial symmetry, E = 0, and two of the states will

remain degenerate at zero magnetic field. The zero-field parameters D and E

can, in general, be determined from the EPR spectrum (for H

< H

ENDOR studies of triplet states are especially useful because the hyperfine

structure is rarely revealed in the EPR spectrum.

The total spin-Hamiltonian for the triplet system is given by:

H = H

+ H

(4.35)

with the H

term the second largest one because typically H

> H

and H

< H

. For triplet states of aromatic molecules, g is nearly isotropic

because of the small spin-orbit interaction. Thus, most of the observed

anisotropy is due to spin-spin coupling as a result of magnetic dipolar

interactions of the unpaired electrons in the triplet state.

In general, for metal ions with S  1 in crystals, g may be quite anisotropic.

The spin-orbit couplings are appreciable and account for the zero-field

splitting. The zero-field splittings for transition metal ions may be much

larger than for organic molecules, and may even exceed the Zeeman energy.

171

4.3 ENDOR in the Solid State

The energy level diagram for the triplet state corresponding to the zero-

field and electron Zeeman splittings is given in Figure 4.6, while the energy

levels with hyperfine and nuclear Zeeman interactions, to illustrate the

ENDOR spectrum, are shown in Figure 4.7. Saturation of the EPR lines will

produce an ENDOR line at the free nuclear frequency, and one other ENDOR

line. When the high-field EPR transitions are observed, the respective two

ENDOR lines will be detected, one of them at the free nuclear frequency.

Neglecting the ENDOR line at Q

, the general expression for the ENDOR

frequencies is

triplet ENDOR

= »A ± Q

», (4.36)

where either the plus or the minus sign is used when a single EPR triplet

transition is observed.

ENDOR studies of triplet systems must be preceded by an EPR study to

determine the D tensor. In general, it appears that ENDOR lines of triplets

are observable only when the magnetic field is almost perpendicular to a

principal axis of the hyperfine tensor, often being parallel to the principal axis

of the D tensor.

Figure 4.6 Energy of the triplet state with axial symmetry (E = 0, D > 0) for B»» z. The EPR

= ±1 transitions are indicated by LF (low-field) and HF (high- field). The electron Zeeman

and spin-spin interaction are taken into account, and H

< H

. At zero magnetic field the

triplet energy levels m

= 0 and m

= ±1 are split by D. Because the symmetry is assumed to

be axial, two of the states remain degenerate.

4. ENDOR Spectroscopy

172

Figure 4.7 Energy levels for the triplet state at a fixed magnetic field and microwave frequency

appropriate for the indicated EPR transitions. The EPR transitions correspond to the low-field

(LF) EPR line in Figure 4.6. The additional splittings of the triplet state energy levels come

from hyperfine (H

) and nuclear Zeeman (H

) interactions. The two upper NMR transitions

will be observed as an ENDOR spectrum.

In the ideal case, only one ENDOR line below or above the free nuclear

frequency is observed for each group of equivalent protons. So if the sign of

the dipolar tensor D is known, the sign of the hyperfine component can be

determined from the ENDOR spectra. From the hyperfine tensors both spin

density and geometrical information can be obtained.

4.3.4 Disordered Solids

ENDOR experiments in disordered solids are usually carried out in two

cases: the paramagnetic ion or molecule can be used to probe its environment

or the paramagnetic system is studied in frozen or solid solution because

dilute single crystals are not available.

There are, in general, two different types of ENDOR spectra in disordered

solids: (1) A powder-type spectrum is observed when all orientations

contribute equally to the spectrum; the ENDOR spectrum is the same no

matter which portion of the EPR line is saturated; a powder-type spectrum is

also obtained when all regions of the EPR spectrum are saturated at the same

time even if not all orientations contribute equally to the EPR powder pattern.

(2) A crystal-like spectrum can be observed when the various orientations of

paramagnetic molecules do not contribute equally to the EPR powder

173

4.3 ENDOR in the Solid State

spectrum; the ENDOR pattern depends on the portion of the EPR spectrum

being saturated; the EPR transitions can be saturated selectively.

A characteristic feature of ENDOR spectra of disordered solids is the

appearance of a strong line occurring at the free nuclear frequency, called a

matrix ENDOR line. It originates from purely dipolar coupling of the unpaired

electron with magnetic nuclei of the surrounding matrix at distances within

roughly 6 Å. In liquids this line is averaged to zero by the rapid tumbling of

the radical. The molecular motions in disordered solids, in general, affect the

intensity of the matrix ENDOR.

4.3.4.1 Powder-Type Spectra

When all orientations of the paramagnetic molecules with respect to the

direction of the magnetic field are present, the ENDOR lines are broadened

and may be very weak. An average of all orientations of the hyperfine

coupling tensors for each individual nucleus can be measured. Thus, the

ENDOR spectrum for a nucleus extends over the whole range of the hyperfine

values with some build-up of intensity at the principal values of the hyperfine

tensor. Narrow lines are observed if the anisotropy is small. For an D proton

in organic radicals the anisotropy typically amounts to 50% of the isotropic

value of a unit spin density, and hyperfine principal values are roughly a/2, a,

and 3a/2. Thus the D-proton powder lines are rarely observed. The anisotropy

of E protons is much smaller, typically about 10% of the isotropic value.

For a system of S = ½ and one proton with axially symmetric hyperfine

coupling to the electron spin, if the coupling is small, the ENDOR frequency

can be expressed by

(T) = Q

»»

cos

T + A

sin

T], (4.49)

where T is the angle between the magnetic field and the symmetry axis.

When all orientations are equally probable, the powder-type spectrum is

observed (Figure 4.8) giving the principal values of the hyperfine tensor.

This favorable situation typically occurs for the protons of a rotating

methyl group. When the protons are equivalent on the EPR time scale, the

hyperfine tensor is often axially symmetric with the largest component along

the C-C bond direction.

4. ENDOR Spectroscopy

174

Figure 4.8 ENDOR powder lines for a proton with axially symmetric hyperfine interaction;

(i) A

dip

˙˙

= 2A

dip

and (ii) A

dip

˙˙

= – 2A

dip

. From [6].

4.3.4.2 Crystal-like Spectra

Crystal-like ENDOR spectra in disordered systems can be observed when the

magnetic field is set at the so-called turning points in the EPR powder pattern

[6]. This is realized in the systems with considerable anisotropy of the

magnetic interaction, i.e., in the following favorable situations: (1) the

anisotropy of the g tensor dominates other magnetic interactions; this situation

is typical for biomolecules; (2) when the anisotropy of the g tensor is small,

but the hyperfine anisotropy is considerable (free radicals); (3) the anisotropy

of the electron spin-spin interaction (described by the D tensor) is dominant

(organic biradicals); and (4) the anisotropies of two magnetic interactions are

of the same order of magnitude, but the axes with the largest anisotropy

coincide (e.g., Cu

complexes).

Because of the anisotropy of the magnetic interaction, different orientations

of the paramagnetic molecules with respect to the magnetic field correspond

to different resonance conditions. Thus, a characteristic powder pattern,

sometimes well resolved, is given by EPR spectroscopy. The EPR spectrum

should be analyzed to locate the appropriate EPR field values where

orientational selection of the molecules can be observed. The magnetic field

set to these so-called turning points induces resonance in molecules having a

unique orientation. The ENDOR spectra taken at these field settings will

result in hyperfine tensor components relative to the tensor axes of the

respective dominating interactions. In contrast to powder-type spectra, the

crystal-like lines are characterized by linewidths close to those found in single

crystal spectra. At other settings of the magnetic field, the EPR spectrum

arises from molecules in many orientations and the respective ENDOR

linewidths are much broader.

Figure 4.9 gives an example of how the frequency and width of the

observed ENDOR lines depend on the portion of the EPR spectrum that is

saturated [7]. In this system [Cu(pic)

in Zn(pic)

O powder] the g tensor

and Cu hyperfine tensor are nearly coaxial. The symmetry axes of both

tensors are oriented perpendicular to the plane of the complex. Because of the

axial symmetry, there is only one turning point in the EPR spectrum. The

EPR resonances with the magnetic field parallel to the molecular plane

correspond to the high field end of the powder pattern, while the transitions

175

4.3 ENDOR in the Solid State