Hamiltonian zero-field splitting

The advantage of this high-temperature expansion is that we need not determine the individual energy levels but can use the theorem of the invariance of the trace of a matrix. For example, to find the value of e2 we square the Hamiltonian and pick out only the diagonal terms. On summing these terms over all possible values, many of the sums cancel. For example, the zero-field splitting Hamiltonian of the form... 

With the help of the angular momentum matrices (Appendix 2) it is easy to construct the zero-field splitting Hamiltonian matrix for the given value of S (Table 8.21). 

In the case of non-cubic symmetry of the coordination polyhedron, but close to the cubic one, the zero-field splitting Hamiltonian adopts the form of [10]... 

As mentioned above, the electron spin system is strongly coupled to the classical degrees of freedom, in the first place the orientation of a molecule-fixed frame with respect to the lab frame, through anisotropic interactions. We concentrate here on the case of S > 1, where the main anisotropic interaction is denoted as the zero-field splitting, ZFS. In the language of spin Hamiltonians (8,80), the ZFS interaction is written ... 

Values of the spin Hamiltonian reported for d4 ions are given in Table XI. The difficulty in detecting the ESR is due most likely to short spin-lattice-relaxation times and large zero-field splittings. In both octahedral and tetrahedral fields the 5 D state of d4 gives degenerate orbital states which,... 

Values for the spin Hamiltonian are given in Table XIV. The 5D state of d6 has three orbital states for the ground state in octahedral symmetry. Since these three states are connected by the spin-orbit coupling, the spin-lattice-relaxation time is quite short and the zero-field splitting very large. In a distorted octahedral field the large zero-field distortion makes detection of ESR difficult. In the case of ZnF2 the forbidden AM = 4 transition was measured and fitted to Eq. (164). 

The EPR spectra have always been interpreted2994 using an effective S = 2 spin Hamiltonian including the Zeeman term, /iBB g-S, and the hyperfine term, ICa-A-S, which describes the interaction of the unpaired electrons with the copper nucleus (7Cu = I). The spectra are very sensitive to the ratio between the isotropic coupling constant J and the local zero field splitting of nickel(II), Z)Ni.2982 In the limit J DNi it can easily be shown that the following relations hold ... 

Such splitting is called zero field splitting and indicated as ZFS. It adds up to the Zeeman energy. In the spin-Hamiltonian formalism, i.e. when the effects of the orbital angular momentum are parameterized, it is indicated as... 

Calculations were also performed for electron spin Hamiltonian dominated by zero field splitting [14,15]. 

Abbreviations CF - crystal field SH - spin Hamiltonian ZFS - zero-field splitting MA - magnetic anisotropy TIP - temperature-independent paramagnetism MP - magnetic parameter averaged (gav, /tip), axial (gz> g > D /up) CSC - complete space calculation. 

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