
The spin Hamiltonian for a spin system consisting of a Cu^{2+} ion (S = 1/2, I = 3/2) with weakly coupled nuclei from ligands in the first and second coordination spheres (e. g. ^{13}C nuclei and protons) can be written as follows:
(26) 
The electron Zeeman interaction is much larger than the hyperfine coupling to the copper nucleus, expressed by A^{Cu}. The copper quadrupole interaction is neglected. The g and A^{Cu} matrices are assumed to be coaxial and axially symmetric and no distinction is made between the two copper isotopes. These simplifying assumptions are often justified in experimental work. The third term H_{N} describes the ^{13}C and proton interactions which are small compared to the first and second term. The CW EPR spectrum is then obtained by neglecting the last term and treating the copper hyperfine interaction e.g. by second order perturbation theory. The EPR spectrum is described by the principal values g_{}, g_{⊥}, A_{}, A_{⊥} of the g and A^{Cu} matrices.
A typical powder EPR spectrum for an axially elongated copper complex is shown in Fig. 10a. The parameters chosen for the simulation of the spectrum are those for a copperhexaquo complex with g_{} > g_{⊥} and A_{} > A_{⊥}. Due to the amplitude modulation of the B_{0} field, CW EPR spectra are usually recorded as first derivatives. In Fig. 10b, the corresponding absorption spectrum is shown. The spectrum is a superposition of four axial powder lines which can be assigned to the different m_{I} states. The 2D plot shows the contributions of the different m_{I} states as a function of the angle θ (defined in Fig. 3 as the angle between the zaxis and the external magnetic field vector B_{0}). In experimental spectra the splitting in the g_{⊥} region is often not resolved due to large linewidths whereas the g_{} and A_{} values can usually be determined without difficulties.
Fig. 10: (a) CW EPR spectrum for an axially elongated copper complex (first derivative); (b) same spectrum drawn as absorption spectrum with a 2D plot showing the different m_{I} transitions as a function of the angle θ. 
The interactions of the electron spin with the weakly coupled I = 1/2 nuclei, described by H_{N} are usually not resolved in the CW EPR spectra and can only be investigated by applying pulse EPR. Due to the limited bandwidth of the MW pulses only a small part of the EPR spectrum is excited. The dashed lines in Fig. 10b indicate the excitation bandwidth (~50 MHz) for a pulse EPR experiment performed at the B_{0} position indicated by the arrow. At a given B_{0} position spin packets from different m_{I} states with different θ values can thus contribute to the spectrum. The pulse EPR spectra of the ^{13}C nuclei and protons can be calculated by treating the spin Hamiltonian as for the S = 1/2, I = 1/2 system. More accurate results are obtained by including the g anisotropy in the computation. Diagonalization of the EZI while neglecting the second term in Eq. (26) as the copper hyperfine interaction has no direct influence on the nuclear frequencies leads to the two nuclear Hamiltonians
(27) 
where the hyperfine interaction matrix A_{i}' is now expressed in the PAS of the EZI. Diagonalization of these Hamiltonians yields the nuclear frequencies as functions of the magnitudes and relative orientations of the magnetic field and the g and A_{i}' matrices.
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