Electric quadrupole moment and nucleus shap

[joebo]

The nuclear electric quadrupole moment is a parameter which describes the effective shape of the ellipsoid of nuclear charge distribution. A non-zero quadrupole moment Q indicates that the charge distribution is not spherically symmetric. By convention, the value of Q is taken to be positive if the ellipsoid is prolate and negative if it is oblate.

but i don't understand is that the real meaning of the shape, in my view, the prolate ellipsoid and oblate ellipsoid is the same, because if it represents the real shape of nucleus, we can change the coordinate ,then the oblate changes into oblate.or we can roate the nucleus and also can get the same result. so what's my fault?

 

[ansgar]

yes but the z-axis in referred to as the axis we quantize spin along

 

[Meir Achuz]

The assumption is made that the nuclear shape has axial symmetry, and the z-axis is chosen along the symmetry axis. You cannot rotate an (American) football into a (Swedish) pancake.

 

[bcrowell]

but i don't understand is that the real meaning of the shape, in my view, the prolate ellipsoid and oblate ellipsoid is the same, because if it represents the real shape of nucleus, we can change the coordinate ,then the oblate changes into oblate.or we can roate the nucleus and also can get the same result. so what's my fault?

Prolate means it has one long axis and two short axes. Oblate means it has one short and two long. You can also have shapes with all three axes unequal.

There is almost no empirical evidence for any stiff nuclear shape other than spherical or prolate. When theoretical calculations say that the minimum energy is achieved at an oblate deformation, what you generally see experimentally is that the nucleus has no well developed rotational bands, no significant ground-state quadrupole moment, and E2 transition strengths that are much smaller than would be expected for rotation. The interpretation is that the minimum is so soft that the fluctuations in deformation (basically due to the Heisenberg uncertainty principle) are bigger than the mean deformation.

 

[bcrowell]

yes but the z-axis in referred to as the axis we quantize spin along

This is incorrect. The difference between oblate and prolate is not just a change of spin axis. In the standard (\beta,\gamma) parametrization, there are six values of \gamma that represent axially symmetric shapes. Three of these are prolate shapes rotating about three different principal axes (two collective and one noncollective), and three are oblate shapes rotating about three different principal axes (again, two collective and one noncollective).

There are basically two possibilities that are important in practice. You can have noncollective rotation of a prolate ellipsoid about its symmetry axis, or collective end-over-end rotation of a prolate ellipsoid. The oblate cases are different, but we just don't observe them empirically.

 

[joebo]

thanks , but i still cannot understand it

 

[ansgar]

This is incorrect. The difference between oblate and prolate is not just a change of spin axis. In the standard (\beta,\gamma) parametrization, there are six values of \gamma that represent axially symmetric shapes. Three of these are prolate shapes rotating about three different principal axes (two collective and one noncollective), and three are oblate shapes rotating about three different principal axes (again, two collective and one noncollective).

There are basically two possibilities that are important in practice. You can have noncollective rotation of a prolate ellipsoid about its symmetry axis, or collective end-over-end rotation of a prolate ellipsoid. The oblate cases are different, but we just don't observe them empirically.

I have never said that it is a change of axis?...

 

[ansgar]

joebo you must take into account the spin-vector which breaks the apperant symmetry between oblate and prolate shape.

http://www.tulane.edu/~sanelson/images/uniaxialopticsign.gif

Imagine both of these shapes have their spin axis pointing upwards, then the difference is crystal clear.

 

[bcrowell]

joebo you must take into account the spin-vector which breaks the apperant symmetry between oblate and prolate shape.

http://www.tulane.edu/~sanelson/images/uniaxialopticsign.gif

Imagine both of these shapes have their spin axis pointing upwards, then the difference is crystal clear.

Maybe you could explain what the diagram is supposed to represent. Based on the content of the figure and the filename in the URL, I'm not sure it's even supposed to represent anything about nuclear physics. The filename in the URL seems to indicate that it has something to do with optics.

Sorry, but you really are incorrect about this idea of a symmetry between oblate and prolate. They're not congruent shapes, i.e., they can't be transformed into one another by rotation.

Here is a definition of the beta and gamma parameters I referred to above: http://www.pa.uky.edu/~jnorce/deformation/node3.html

The spin axis can break the symmetry between the different axes of a symmetric shape. There is no symmetry connecting a prolate shape to an oblate shape.