Marsch equation

Hence, in the present chapter the Hill and Marsch equations fractal analogs are obtained, which has shown, that cross-linked epoxy polymers microhardness is defined by their structure only, characterized by its fractal dimension. Tabor s criterion is only fulfilled for Euclidean (or close to them) solids. The cross-linking degree enhancement results to loosely packed matrix loosening and corresponding reduction of cross-linked epoxy polymers microhardness. The similar results obtained for linear polyethylene and nanocomposites on its basis, filled with organoclay [16]. 

Let us consider now the physical nature of the deviation of ratio HJc from the constant c 3 m Equation 9.35. The structure fractal dimension d can be calculated according to Equation 5.4. Combination of Equations 5.4,5.5, 9.36 and 9.37 allows fractal variants of the Hill and Marsch equations to be received, respectively [60] ... 

Thus, the fractal analogues of the Hill and Marsch equations obtained above have shown that the microhardness of crosslinked epoxy polymers is defined only by their structure, characterised by its fractal dimension. Tabor s criterion is correct for Euclidean (or close to Euclidean) solids only. The degree of increase in crosslinking results in loosening of the loosely packed matrix and to a corresponding reduction of the microhardness of crosslinked epoxy polymers [60]. 

Note that the derivation of the Dirac equation in chapter 5 holds for any freely moving spin-1 /2 fermion and hence also for the proton we come back to such general two-particle Hamiltonians at the beginning of chapter 8 (for an explicit solution of the corresponding two-fermion eigenvalue equation we refer the reader to the work by Marsch [112-115]). In the field-free Dirac Hamiltonian only the rest mass determines which fermion is considered. Accordingly, the total wave function of the hydrogen atom reads... 

E. Marsch. An effective Dirac equation for a binary of two fermions. /. Phys. A Math. Theor., 41 (2008) 185301. 

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