Zeeman interaction energy matrices

We use these relations to write out the energy matrix initially in zero field (i.e., ignoring the Zeeman interaction) ... 

To find out what the X-band spectrum of such a system will look like, let us now complete the energy matrix with the Zeeman interaction using all the spin-operations written out in Equations 7.48a to 7.48m ... 

These represent the nuclear spin Zeeman interaction, the rotational Zeeman interaction, the nuclear spin-rotation interaction, the nuclear spin-nuclear spin dipolar interaction, and the diamagnetic interactions. Using irreducible tensor methods we examine the matrix elements of each of these five terms in turn, working first in the decoupled basis set rj J, Mj /, Mi), where rj specifies all other electronic and vibrational quantum numbers this is the basis which is most appropriate for high magnetic field studies. In due course we will also calculate the matrix elements and energy levels in a ry, J, I, F, Mf) coupled basis which is appropriate for low field investigations. Most of the experimental studies involved ortho-H2 in its lowest rotational level, J = 1. If the proton nuclear spins are denoted I and /2, each with value 1 /2, ortho-H2 has total nuclear spin / equal to 1. Para-H2 has a total nuclear spin / equal to 0. 

We are now in a position to examine the details of the Zeeman effect in the para-H2, TV = 2 level, and thereby to understand Lichten s magnetic resonance studies. For each Mj component we may set up an energy matrix, using equations (8.180) and (8.181) which describe the Zeeman interactions, and equations (8.201), (8.206) and (8.214) which give the zero-field energies. Since Mj = 3 components exist only for J = 3, diagonalisation in this case is not required. For Mj = 2 the J = 2 and 3 states are involved. For Mj = 0 and I, however, the matrices involve all three fine-structure states and take the form shown below in table 8.7. Note that /. is equal to a0 + 3 63-2/4 and the spin-rotation terms have been omitted. The diagonal Zeeman matrix elements are... 

The resulting interaction matrix is given in Table 7.8. Since the Zeeman interaction leads to a splitting of the levels that conserves the barycentre, the secular equation does not contain odd powers in the energy ... 

An important addition to the literature of EPR spectrum simulation was presented as an Appendix in this work. A spin system consisting of one 5 = 3 and four / = I results in an energy matrix of order 2592. Even with the approximation of collinear matrices for electronic Zeeman and hyperfine interactions, matrix diagonalisation techniques are effectively unrealistic. The authors argue that starting from an uncoupled spin-Hamiltonian representation ... 

Similar to the situation in [Ru(bpy)3] (Sect. 3.3.1), 11), II), and III) represent zero-field spUt components, which result in their main contributions from the same orbital parentage or fi om one specific MLCT state. This is indicated (1) by the fact that the splitting pattern does not strongly depend on the matrix, though the absolute energies are shifted over a range of more than 250 cm, when the different matrices are compared (Table 8) (2) states 11) and II) exhibit a strong Zeeman interaction (Sect. 4.1.4 [92]) and (3) both states are -within Umits of experimental error of < 1 cm /kbar - equally shifted under... 

Such a form of quasi-equilibrium distribution takes place due to the fact of the availability of two invariants of motion. In Equation 25 parameters a and p linked to the operators Hz and Hss are thermodynamically conjugative parameters for the Zeeman energy and the energy of spin-spin interactions respectively. We can expand the exponent in Equation 25 in jxjwers of xT-Lz and f Hss and keep only the linear terms. As we shall see later such a linearization corresponds to the high temperature approximation. In the linear approximation in x Hz and Hss, the density matrix is reduced to... 

Here 1 is the unit matrix. Therefore these expressions describe the expectation value of the Zeeman energy and the expectation value of the energy of spin-spin interactions ... 

For further details concerning the using of chemical potential in spin thermodynamics the reader should refer to (Philippot, 1964). To calculate the expectation values of energies Hz) and Hss) at low temperatures one has to take into account the higher-order terms in the expansion of the density matrix in Equation 26. As a consequence the factorization condition 28 is violated and the Zeeman subsystem and the reservoir of spin-spin interactions carmot be considered as independent. So the advantage of the above-mentioned choice of thermodynamic coordinates is lost. Besides at low temperatures the entropy written in terms a and fi... 

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