Spheroidal quadrupole

Show that the quadrupole moment Q of a uniformly charged spheroid about the axis of symmetry is Z(b2 — a2), where a, b are the semiaxes, b being along the axis of symmetry. Show that the quadrupole moment about an axis making an angle (3 with the axis of symmetry is [ cos2 (3 — ]Q. 

It has been found that nuclei with spin 1 = 0 have no multipole mom t. According to theory, nuclei with I = 1/2 can have a dipole moment, but this has not yet been shown experimentally. Nuclei with 7=1 have quadrupole moments they are fairly common. The quadrupole moment can be calculated for spheroidal (i.e. deformation not too far from a sphere) nuclei in terms of the electron charge, e, by... 

In the case of spheroidal nuclei the a-particle formation may depend on the direction (in a body-fixed coordinate system). Furthermore, the noncentral electrostatic field of the deformed daughter nucleus may interact with the emitted a particle, causing an increase or decrease of its energy. These problems were discussed by several authors (Hyde et al. 1964). On the basis of these works, one can conclude that the a-particle wave function on the nuclear surface maybe quite different for different nuclei. Furthermore, the electric quadrupole field of the strongly deformed daughter nucleus plays an important role in the formation of the intensity of different a-particle groups. 

Another shape often found in literature for its interesting spectra is that of triangular nanoparticles or nanoprisms. For these nanoparticles, the quadrupole resonances seem to be much more intense than for spheroidal ones of equivalent dimensions, this being a surprising feature or peculiarity of such kind of targets [1]. 

It is apparent that the angular distribution of electric quadrupole radiation is generally a rather complicated function of 0,<(> but a simple example will serve to illustrate the main features. We consider an oscillating spheroidal charge distribution. In this case the off-diagonal elements of the electric quadrupole moment tensor vanish because of the symmetry of the system. If the z-axis is taken as the axis of symmetry we have = Q22 since the tensor is... 

Fig.2.7. Angular distribution of quadrupole radiation from an oscillating spheroidal charge distribution.

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