Hamiltonian isotropic exchange

What is the coimection between the HDW exchange interaction of eqnation (27) and the complexities of an ion in a crystal field The isotropic exchange Hamiltonian is always correct since it is a direct conseqnence of the Panli Exclusion Principle. However, a modified form of the exchange interaction is frequently nsed to describe those cases in which the individnal moments are no longer isotropic. 

This result proves to be valid for all MV — d" pairs with n < 4, provided that the spin core is defined as an ion without extra electron. It is also valid for n > 4 but in such case the spin core must be defined as an ion without extra hole. The ferromagnetic effect of the double exchange is illustrated by Fig. 7. Along with the double exchange, the isotropic exchange interaction plays an important role in MV clusters. This interaction is described by the HDVV spin Hamiltonian ... 

An isotropic exchange Hamiltonian was employed appropriate to the rectangular disposition of the four Mn atoms, as in Eq. (16) and Fig. 24. ITO calculations were required. 

Some magnetic parameters enter the matrix elements of the Hamiltonian concerned these are magnetogyric factors (g-factors, components of the g-tensors) and the coupling constants (components of the spin-spin coupling tensors D, in the case of an isotropic exchange abbreviated as J). 

Restricting ourselves only to pair interactions, the spin Hamiltonian appropriate for the isotropic exchange interaction in polynuclear systems becomes... 

In the weak exchange limit the effect of isotropic exchange is small. This is well fulfilled when the magnetic centres are rather far from each other (say 800 pm apart). Thus the eigenvectors S, Ms) of S2 and Sz do not give a correct description of the spin states the proper molecular states are obtained by the diagonalisation of the full Hamiltonian matrix. In fact, the variation method is applied. 

The spin Hamiltonian of bilinear isotropic exchange for a trinuclear cluster adopts the form of... 

Using these expressions, the isotropic exchange Hamiltonian of a trinuclear system has the following diagonal elements... 

Coefficients of the non-zero matrix elements of the isotropic exchange Hamiltonian for general triads of S, = S2 = S3... 

In order to exemplify the construction of the zero-field exchange Hamiltonian matrix let us consider the simplest case of 5, = S2 = 53 = 1 /2. Then the structure of the isotropic exchange matrix is... 

It is obvious that the isotropic exchange Hamiltonian is diagonal (for the symmetric triad, regular triad, symmetric linear triad) unless J23 7 7,3 (a general triad). The number of combinations for 5, S2 53 becomes... 

The bilinear isotropic exchange Hamiltonian for a tetranuclear cluster is written in the form... 

As a framework for the discussion of the magnetization, magnetic anisotropy and magnetostriction we consider a hamiltonian which comprises the following isotropic exchange, effective single-ion anisotropy, elastic, magnetoelastic and Zeeman terms ... 

The magnetic anisotropy of RRh2Si2 (R = Nd, Tb, Ho, Er) which has an API type magnetic structure can be explained by using a R ion Hamiltonian which includes the crystal field term and the isotropic exchange term under the two-dimensional molecular field approximation, when the appropriate crystal field parameters are adopted. The crystal field parameters calculated are listed in table 20 (Takano et al. 1987a). 

Apart from the isotropic exchange which has been discussed up till now, the R-3d compounds are also characterized by anisotropic interactions. Considering the magnetocrystalline anisotropy to arise mainly from the interactions of the 4f shells with the crystal field, we can express the hamiltonian of a 4f ion by... 

The conventional interaction which is usually assumed to take place between RE-ions and conduction electrons is the isotropic exchange interaction of the form -IJnsSii. Here s, S t denote the conduction electron and RE-ion spin. In terms of electron creation and annihilation operators c, Cka the interaction hamiltonian is written as... 

Since the Eu chalcogenides are considered to be models for Heisenberg magnets, the isotropic exchange hamiltonian... 

In a previous study at P- and X-band, the temperature dependence of the magnetically coupled spectrum and its analysis based on a triplet-state spin-Hamiltonian were used to propose the range (0.8-100 cm ) for the parameter Jo of the isotropic exchange interaction JoSa Sb, but neither the magnitude of Jo nor its sign could be further specified. ... 

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