Hyperfine principal values

Signs of the hyperfine principal values of a single nucleus. According to (3.3) and (3.4) the first order ENDOR frequencies of a single nucleus with spin I = 1/2 are given by... 

The physical interpretation of the anisotropic hyperfine principal values is given by the classical magnetic dipolar interaction between the electron and nuclear spin angular momenta. This interaction energy is given by... 

The ESR spectrum of C6H6 " trapped in CFCI3 at 15 K is shown in Figure la and agrees with that reported previously [18]. The principal values of the hyperfine coupling were obtained from previous ESR and ENDOR measurements [17, 18]. The best agreement with experiment was obtained with the axes oriented as in Table 4. In the latter study, the simulated ENDOR spectra were insensitive to the orientation of the tensor axes, however, and the assignment was made on the basis of molecular orbital calculations [9]. The tensor data are reproduced here for convenience (see Table 4). 

Radical IV can be considered as a unique phosphorus radical species. Reduction of the parent macrocycle with sodium naphtalenide in THF at room temperature gave a purple solution. The FPR spectrum displayed a signal in a 1 2 1 pattern, with flp(2P)=0.38 mT. DFT calculations on radical IV models indicated a P-P distance of 2.763 A (P - P is3.256 A in the crystal structure of the parent compound and the average value of a single P-P bond is 2.2 A). According to the authors, the small coupling constant arises from the facts that the principal values of the hyperfine tensor are of opposite sign and that the a P P one electron bond results from overlap of two 3p orbitals [88]. 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

Since the g-matrix has only three principal values and there are almost always many potentially interacting molecular orbitals, there is rarely sufficient information to interpret a g-matrix with complete confidence. When a well-resolved and reliably assigned optical spectrum is available, the energy differences, E0—Em, are known and can be used in eqn (4.11) to estimate the contribution of the corresponding MOs to the g-matrix. Extended Hiickel MO (EHMO) calculations can be useful (but do not trust EHMO energies ), but one is most commonly reduced to arguments designed to show that the observed g-matrix is consistent with the interpretation placed on the hyperfine matrix. 

Since the energy differences, AEx2 y2 and AEXZ are expected to be comparable, the parameter Q is probably not far from unity. For Q — 1, eqn (4.47) has a particularly simple form, tan 2/1 = I 2a/b so that, for small b/a, we expect /i a 0 and /i 45°, which is entirely consistent with experiment. The axial hyperfine matrix is in agreement with experiment, and the principal values of the g-matrix can also be rationalized with reasonable values of Q and b/a. A small rotation of dyz about the y-axis might reflect the small displacements of the phosphorus atoms from the idealized octahedral positions. 

The polyerystalline spectrum of N02 on MgO is somewhat complex, but it yields an unambiguous g and hyperfine tensor which can be checked by comparison with data for NO2 in single crystals. For N02 on MgO, principal values of the hyperfine tensor are m = 53.0, 21 = 49.0, and a31 = 67.0 G (29). It should be noted here that neither the signs of the coupling constants nor their directions relative to the molecular coordinates... 

Figure 2A shows a pulsed ENDOR spectrum of an oxo-Cr(V) complex where the unpaired electron is interacting with a 1H nucleus with principal hyperfine values Hax = Hay = — 2 MHz and 1 a- 5 MHz. In this case, the isotropic hyperfine value is Haiso = (%+ % + V)/3 = 0.33 MHz ( 0.11x10 4 cm ), a value that is not resolved in the CW-EPR spectrum. The hyperfine contribution of this proton is, however, clearly resolved in the ENDOR spectrum. The ENDOR spectrum is centered around the proton Larmor frequency (vH), identifying the contribution as stemming from an interaction with a H nucleus. The principal values can be read directly from the spectrum, as indicated in Fig. 2A. 

The authors also reconsidered radical 20, which, in the current work, was formed by x-irradiation and studied at 77 K. The experimentally determined hyperfine coupling tensors for the two methylene protons have the same principal values and agreed with the tensors found in the earlier work.16 These tensors are presumed to be the average of the two individual proton tensors. By assuming that C5 is sp2 hybridized the authors were able to calculate, from the observed average tensors, that Ami = 1.06 mT and Amai = 3.45 mT for the methylene protons. Using the individual tensors for the methylene protons and a 3.6 mT isotropic coupling for the C4(H) proton, a satisfactory simulation of the... 

The spectra of polycrystalline samples are broad and represent the envelope of all the anisotropic couplings together with the g-anisotropy. Although in favourable cases it is possible to extract the principal values of the g- and hyperfine coupling tensors from such... 

Figure 18-8. Proton hyperfine couplings for a planar > C — Ha fragment showing the principal values and directions of the proton anisotropic hyperfine coupling...

Here it is interesting to continue with the discussion of using the ENDOR data to discern radical geometry. Results for the cytosine reduction product observed in irradiated single crystals of l-MethylCytosine 5-FluoroUracil are shown in Table 18-2 [34]. First one notes the three principal values of the hyperfine coupling tensor. For an ordinary rr-electron radical with unit spin density on the central... 

It is important to note that the proportional relationship between Amax, Amid, and Amin for these couplings is the same for 100% spin density, and for the present case with approximately 50% spin density. When this is so it indicates that there is no rocking motion at the radical site. This is good evidence therefore that the radical site is essentially planar. The best evidence for radical planarity comes from the analysis of the direction cosines associated with each principal values of the hyperfine coupling tensor. The direction of Amin (Table 18-2) is known to be associated with the direction of the >C-H bond, while the direction associated with the Amid indicates the direction of the n-clcctron orbital. These directions are easily calculated from the crystal structure, and are included in Table 18-2. One sees that the direction associated with Amid deviates only 2.0° from the computed perpendicular to the ring plane, while the direction of Amin, deviates only 2.8° from the computed direction of the C6-H bond. The errors listed on these values are at the 95% confidence level. This is very clear evidence that the radical shown here is planar in the solid-state. Any torsional motion of the C6-H would lead to asymmetries of the hyperfine coupling tensor, and would not produce the observed agreement between the direction cosines and the known directions obtained from the crystal structure. 

We have prepared the acetyl radical by the reaction between sodium atoms and acetyl chloride and trapped it in a matrix of water, benzene, benzene-dg, or cyclohexane (Bennett et al., 1969a). The spectrum of the acetyl radical is basically the same in all of the matrices and that in benzene is shown in Fig. 9. The spectrum shows that the orbital of the unpaired electron has approximately axial symmetry, and the principal values of the gr-tensors and hyperfine tensors are ... 

ESR spectra of La Cs2/ Y Cs2, Ho Cs2, and Tm C82 taken from the solid soot extract were reported by Bartl et al. (1994, 1995a,b, 1996) and showed low resolved but split hyperfine structure, indicating that the metal atoms exist in ionic form in the fullerene cage also in the solid state. The research group also reported (Knorr et al., 1998) the principal values of the hyperfine tensor A and the relative orientation of g and A tensors of M C82 (M = Sc, Y, La) applying three- and four-pulse electron spin-echo envelope modulation techniques (ESEEM). 

While the effective g value is expressed in terms of three principal values directed along three axes or directions in a single crystal, only the principal values of g can be extracted from the powder spectrum rather than the principal directions of the tensor with respect to the molecular axes. (Therefore it is more correct to label the observed g values as gi, g2, g3 rather than g gyy, in a powder sample.) In the simplest case, an isotropic g tensor can be observed, such that all three principal axes of the paramagnetic center are identical (x = y = z and therefore gi= gi = g-i). In this case, only a single EPR line would be observed (in the absence of any hyperfine interaction). With the exception of certain point defects in oxides and the presence of signals from conduction electrons, such high symmetry cases are rarely encountered in studies of oxides and surfaces. 

The cobalt hyperfine interaction tensors, and Aj, have identical sets of principal values, tentatively 30.1, 8.5, and 1.1 oersteds. The magnitude is the best evidence of the cobdt contribution to the total orbital and the chemical equivalence of these two atoms. The principal axis system for each cobalt atom is determined by the local symmetry about that nucleus—i.e., the direction of the bonds. Thus determination of these two tensors can yield particularly valuable information about the nuclear geometry. Unfortunately, the principal axis directions are sensitive to small errors in the measured experimental components. However, our present data are consistent with a structure for (p-3) (analogous to structure Id) ... 

Table 1. Principal values (in MHz) of the hyperfine tensor for NV-center in diamond 11, A22, A33 — our results of DFT calculations using the 3-21G basis set, vfll[5],., 422[5], /<33[5] -...

Table 5.1. Parameters (principal values of g and 0 hyperfine structure tensors) of EPR signals assigned to 0 species and formed upon O2 adsorption on Ce02 previously submitted to outgassing treatments. Axis assignment is based on experiments using O-enriched oxygen (see ref. 16 for more details).

Eq. (30) gives an isotropic hyperfine tensor. It is a common technique to divide the hyperfine tensor into an isotropic tensor with principal value As and a traceless tensor with components 2Ad, -(1 - eMd, and —(1 + eMd such that the hyperfine components can be written... 

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