Quadrupole Hamiltonian

Here % specify the transformation from coordinate system j to system i. In Equation 3 only Dq q (Qdm) varies with the molecular motion. Since amphiphilic liquid crystalline systems generally are cylindrically symmetrical around the director Dq q (nDM) = 0 if qf 0. If it also is assumed that a nucleus stays within a domain of a given orientation of the director over a time that is long compared with the inverse of the quadrupole interaction, one now obtains for the static quadrupole hamiltonian... 

As an example of fermion hamiltonian we take the quadrupo-le-quadrupole hamiltonian... 

Figure 3. Comparison between the spectra generated by a quadrupole-quadrupole hamiltonian within the S-D subspace (N = 3, j = 23/2) and by the IBM hamiltonian obtained with this mapping procedure. Reproduced with permission from (SAM85a). Copyright 1986 North-Holland Physics Publishing Company.

The electrostatic quadrupole term in this sum is that for which ( = 2, so that the quadrupole Hamiltonian is given by... 

We now return to the spherical harmonic form of the quadrupole Hamiltonian, equation (4.29), which is... 

The theory of the magnetic hyperfine interactions in NCI is essentially the same as that already described for the PF radical in the previous section, except that the nuclear spins / are 1 for 14N and 3/2 for 35C1. The form of the effective Hamiltonian for the quadrupole interaction and its matrix elements for two different quadrupolar nuclei was described in some detail in chapter 8 when we discussed the electric resonance spectra of CsF and LiBr. We now use the same case (b) hyperfine-coupled basis set as was used for PF. The quadrupole Hamiltonian for the two nuclei can be written as the sum of two independent terms as follows ... 

When the Zeeman interaction is much larger than the quadrupole interaction, nonsecular terms may be discarded to give a relatively simple expression for the quadrupole Hamiltonian ... 

Henceforth the term t) will be taken to refer exclusively to the asymmetry parameter in the quadrupole interaction). The energy equation for the quadrupole interaction can be transformed into a form that makes it compatible with the other Hamiltonians above by substituting spatial operators with spin operators using the Wigner-Eckart theorem (Slichter 1990) which after some manipulation gives the quadrupole Hamiltonian in the PAS of this interaction... 

To obtain the quadrupole Hamiltonian of a spin in a magnetic field the Hamiltonian needs to be transformed from the PAS to the LAB frame, keeping only those terms that commute with L. This is called truncation of a Hamiltonian and is only valid when Hq << Hz (the high field approximation). To perform the transformation it is much more convenient if second-rank irreducible spherical tensors are used. The Cartesian and spherical tensor elements (T) can be related (see Schmidt-Rohr and Spiess 1994 and Eq. 8, in Man 2000), with two of the more common elements being... 

The electrostatic interaction between the nucleus and its environment is defined by the quadrupole Hamiltonian... 

In microwave spectroscopy, pure rotational transitions are studied. If one or more nuclei in the molecule has a nuclear quadrupole moment, the quadrupole Hamiltonian [Eq. (6)] has to be included in the quantum mechanical treatment because the field gradient q [Eq. (5)] is dependent on the rotational wave function. The nuclear quadrupole interaction, which causes the rotational transitions to split into hyperfine structure, can usually be treated as a perturbation to the rotational Hamiltonian. 

Pure NQR in the absence of a magnetic field is a simple theoretical problem.20 29 Only the quadrupole Hamiltonian [Eq. (6)] needs to be considered. For a spin 1= 1, the energy levels are... 

In accordance with the works we assume that for analysing the effect of RF pulses on the quadrupolar spin system, it is sufficient to consider only three t)q)es of interactions quadrupolar interaction, homonuclear dipole-dipole interaction and the nuclear-spin interaction with the magnetic component of the RF field. Before the initiation of the multi-pulse sequence the quadrupolar system is described by two Hamiltonians quadrupolar Hq and the part of the homonuclear dipole-dipole Hamiltonian Hj secular in relation to the quadrupole Hamiltonian, the sum of which can be regarded as the effective Hamiltonian of the spin system independent of the time factor. 

Let us consider a monocrystalline sample that contains chemically identical spin-1 nuclei. In our physical model it will be assumed that the quadrupole principal axes for each of the spins have the same orientation. The quadrupole Hamiltonian in the quadrupole principal axes frame is given as... 

The interpretation of a complex Mossbauer spectrum will obviously be simplified if the relative intensities of the various components are known. Once the energy levels of the Zeeman/quadrupole Hamiltonian have been calculated, and the spin quantum numbers for each state assigned (or appropriate linear combinations if the states are mixed), it is possible to calculate the intensities from the theory of the coupling of two angular momentum states [32, 33]. 

We defer writing the explicit form of Eq. (26) for specific cases until later. However, it is important to note the similarity of Eqs. (23)-(26) to Eqs. (11)-(13). The nuclear quadrupole Hamiltonian is identical in form to the... 

In an isotropic liquid the mean value of is zero and the quadrupole interaction contributes only to relaxation. In an anisotropic medium, on the other hand, the mean value of is no longer zero and a quadrupole splitting appears in the NMR spectrum. While the quadrupole hamiltonian in Eq. (7.3) may be evaluated in any coordinate system it is convenient to express the spin operators in a laboratory-fixed coordinate system and the electric field gradients in a principal axes coordinate system fixed at the nucleus. It is then suitable to rewrite the hamiltonian as... 

Under the quadrupole Hamiltonian, this state now processes to give... 

In the case of the quadrupole Hamiltonian, an axially symmetric electric field gradient tensor (77 0 in the solid state) is assumed, while 77 is iden-... 

Most experimental NMR work on phase biaxiality has employed a deuteron probe, whose time-averaged quadrupole Hamiltonian is... 

It is sometimes necessary to take into account the effects of centrifugal distortion [84Ah, 02Har]. The resulting formula for the diagonal element of the quadrupole Hamiltonian of a symmetric molecule is ... 

Here Q / are the elements of the nuclear electric quadrupole moment tensor. Equation [4] is of course a classical expression. A quantum-mechanical expression for the quadrupole Hamiltonian Hq is obtained from Equation [4] by replacing the tensor elements Q / by the operators... 

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