Dipole-Dipole Interactions of High-spin Paramagnetic Centers in Disordered Systems

Dipole-dipole interactions between distant paramagnetic centers (PCs) where at least one PC has spin S>1/2 are examined. The results provide a basis for the application of pulsed DEER or PELDOR methods to the measurement of distances between PC involving high-spin species. A projection operator technique based on spectral decomposition of the secular Hamiltonian is used to calculate EPR line splitting caused by the dipole coupling. This allows calculation of operators projecting arbitrary wavefunction onto high PC eigenstates when the eigenvectors of the Hamiltonian are not known. The effective spin vectors-that is, the expectation values for vector spin operators in the PC eigenstates-are calculated. The dependence of these effective spin vectors on the external magnetic field is calculated. There is a qualitative difference between pairs having at least one integer spin (non Karmers PC) and pairs of two half-integer (Kramers PC) spins. With the help of these effective spin vectors, the dipolar lineshape of EPR lines is calculated. Analytical relations are obtained for PCs with spin S=1/2 and 1. The dependence of Pake patterns on variations of zero field splitting, Zeeman energy, temperature and dipolar coupling are illustrated.