Matrix Zeeman interaction

We now notice that we could write a Hamiltonian operator that would give the same matrix elements we have here, but as a first-order result. Including the electron Zeeman interaction term, we have the resulting spin Hamiltonian ... 

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 ... 

The orbital Zeeman interaction involves the matrix element, which is reduced as follows ... 

Hamiltonian matrix for the cubic ligand held, spin-orbit coupling and the electronic Zeeman interaction in the real cubic bases of a 2D term. The g-factor for the free electron has been set to two for clarity (Table A.l). 

A general treatment of spin coupled with quadrupolar spins was given using density matrix theory.27 This formulation enables one to calculate, on the same theoretical basis, the lineshapes of the systems with different ratios of the quadrupolar interaction to the Zeeman interaction. Additionally, it includes the spinning sidebands very naturally. 

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

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 q = 1 components of T (L) will be neglected because they involve the mixing of excited states with the ground vibronic state, giving rise to temperature-independent paramagnetism. With this simplification the matrix elements of the orbital Zeeman interaction operating within the ground vibronic state are... 

The final term in equation (9.55) is the rotational Zeeman interaction whose matrix elements are again obtained by remaining in the space-fixed axis system ... 

The matrix elements of the anisotropic spin Zeeman interaction are ... 

By replacing /V by ( J — S), we see that the rotational Zeeman interaction has straightforward diagonal matrix elements only ... 

In a Hund s case (b) coupling scheme, the Zeeman interaction is described by the following matrix element ... 

The first and the second terms on the right-hand side of Eq.(32) describe a light absorption and the dynamic Stark shift, the third term — the stimulated light emission, the fourth term — the relaxation processes in the ground state, the fifth term — the Zeeman interaction, the sixth term — the repopulation by spontaneous transitions at a rate m m i the seventh term — the relaxation of the density matrix of the ground state atoms interacting with the gas in a cell, not influenced bv the radiation. 

Fig. 20 Schematic diagram showing the effects of the Jahn-Teller, strain, and Zeeman interactions on the states of the Eg ground term. The diagram starts where Fig. 17 finishes. The states are specified as M , and the figure depicts the Jahn-Teller matrix elements between the...

In the presence of the magnetic field the Zeeman matrices are added to the zero-field Hamiltonian matrix. As the z-component of the Zeeman interaction enters the diagonal, factorisation to the 2 x 2 secular equations is possible and from the analytical roots the identification of the van Vleck coefficients is possible... 

Now, in the z-direction (the spherical component q = 0) we get the total interaction matrix, involving the isotropic exchange, asymmetric exchange and the Zeeman interaction, in the form... 

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