Nuclear quadrupole energy levels

In nuclear quadrupole resonance the molecules, and therefore the field-gradient axes, are fixed in space and, in the absence of any external fields, the solutions to the nuclear quadrupole Schrodinger equation are straightforward. If the asymmetry parameter is equal to zero the energy levels for any nuclear spin are given by  

Except when m = 0 for integral spins the energy-levels thus occur in doubly-degenerate pairs. Transitions can occur between these levels with the selection rule Am = 1 and the quadrupole resonance frequencies for a transition m m+1 are given by  

When the asymmetry parameter is not equal to zero, analytic solutions are possible only for I = 1 and I = 3/2. For integral spin the asymmetry ptu-ameter lifts the degeneracy of the m = 1 levels so that there are now two transitions for I = 1 and the two observed resonance frequencies yield the values of the coupling constant and of the asymmetry parameter. 

For half-integral spins the two-fold degeneracy of the quadrupole levels persists despite the non-zero asymmetry parameter and, for I = 3/2, there is still only one transition. 

The resonance frequency is a now function of two independent parameters, although the influence of the asymmetry parameter on the resonance frequency is small. To separate them, and, above all, to discover the value of r , it is necessary to apply a small magnetic field, thus lifting the degeneracy of the m = 1 levels. It is preferable to have a monocrystalline sample, for the orientation of the field-gradient tensor with respect to the crystalline axes is thereby revealed. Methods using polycrystalline samples are also available [3-4], but yield only the magnitude of the asymmetry parameter. 

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We saw in Chapter 7 that the resonance frequency of a quadrupolar nucleus is dependent on the orientation of the molecule in which it resides. Molecular tumbling now causes fluctuating electric fields, which induce transitions among the nuclear quadrupole energy levels. The resulting nuclear relaxation is observed in the NMR just as though the relaxation had occurred by a magnetic mechanism. 

Two difierent situations arise from the efiFect of the nuclear electric quadrupole. The examination of the resonance of a nucleus with a quadrupole moment shows a usually small value of which is determined by the relaxation process associated with the nuclear quadrupole energy levels in the electric field gradient at the nucleus. The larger this field gradient and the larger the quadrupole moment, the more efBcient is the relaxation process. Moniz and Gutowsky (1963) have measured, as an example, the Ti values for in typical oi anic compounds. The... 

Nuclear quadrupole energy levels and resonance frequencies... 

We shall first consider nuclei with an integer spin 1, 2 and 3. In all these cases the energies of the nuclear quadrupole energy levels can be calculated analytically. 

The energies E of the three nuclear quadrupole energy levels and the expansion coefficients of the corresponding eigenstates xp) = the repre-... 

The energies of the nuclear quadrupole energy levels and the NQR frequencies as functions of t] are shown in Figure 1. 

Table 3 Energies of the nuclear quadrupole energy levels in units of e qQIS and the expansion coefficients cfor / = 2...

The highest half-integer nuclear spin of a stable nucleus is / = 9/2. The energy of a nuclear quadrupole energy level is given as = e qQxjlA where x is a solution of the secular equation... 

G. W. Proctor and W. H. Tanttila, Saturation of nuclear electric quadrupole energy levels by ultrasonic excitation, Phys. Rev., 1955, 98, 1854. 

Figure 4 Schematic representation of the populations of the nuclear spin energy levels of a quadrupolar nucleus with spin 5/2 (such as Mg) under a strong magnetic field and a perturbative quadrupole coupling showing (A) populations at thermal equilibrium, (B) populations after complete saturation of the satellite transitions, and (C) populations after complete inversion of the satellite transitions, following the order first, inversion of STl and ST4 and then inversion of ST2 and ST3. The numbers at left of each level (named pj in the text) are proportional to the population of that level, with —hVl/ 2k T= 0. ...

Fig. 7.4 Top Nuclear energy levels of Fe as shifted by electrical monopole (left), or as split by electrical quadrupole (center) or by magnetic dipole interaction (right), schematized for hematite at room temperature (5 > 0 vs. a-Fe, EQ < 0, Bhf 0). Bottom Schematic Mossbauer spectra corresponding to the energy levels schematized on top (J. FriedI, unpubl.).

Static quadrupole effects in NMR are observed in solids (9) and also in anisotropic liquid crystals (10, 11, 12). For nuclei with spin quantum numbers, I, greater than V2, the distribution of positive charge over the nucleus can be nonspherical and the situation can be described in terms of a nuclear electric quadrupole moment. The interaction between the quadrupole moment, eQ and electric field gradients, eq, shifts the energy levels of the nuclear spin states. 

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