Exploring the Spherical Shape of Nuclei and Resources for Further Research

[talanum1]

Is the notion of spherical shape of nuclei well proven by experiment?

Could someone please direct me to a relevant source?

 

[Bob_for_short]

Is the notion of spherical shape of nuclei well proven by experiment? Could someone please direct me to a relevant source?

Nuclei come in different shapes - spherical, ellipsoidal, etc. Get textbooks on nuclear physics.

 

[talanum1]

I think that proof of nuclei being spherical (ellipsoid etc.) would need to be tested not by experiments that can be atributed to the field around the nucleus (this only proves that the field has spherical symmetry).

We would need experiments atributed to the degenerate pressure (exclusion principle) to be able to say that. Even then we would have a problem if using particles that can penetrate into the empty space between nucleons in a nucleus, and at the other extreme particles of the same size (wavelength) that would disturb the nucleus. Any comments?

The notion is not even well defined in terms of explaining energy lelvel spacing, parity and total angular momentum (clearly not supporting arbitrary configurations).

 

[humanino]

Any comments?

We measure cross-sections which in the elastic case are parameterized by form factors. In non-relativistic quantum mechanics, we can interpret form factors as Fourier transform of charge distributions : so if you know one you know the other, which you will notice sort of makes sense, if you know the distribution of charge, you should know how a probe will scatter off it. Of course, the window of validity for a non-relativistic approximation in high energy collision is quite narrow. However, we still know how to rigorously define theoretically and measure experimentally the various distributions you may be interested in (like mass, electric charge, or even the distribution of forces between constituents).

As Bob_for_short pointed out, there are quite a few textbooks out there, and form factors are usually addressed at the very beginning. For instance, page 58 in "Chapter 5 : geometric shapes of nuclei", in "Particles and nuclei" by Povh, Rith, Scholz and Zetsche (Springer-Verlag).

 

[talanum1]

I'm interested in what aspects of nuclei may be explained by assuming spherical/ellipsoid shape. Also is the charge distribution defined well enough to say: "one configuration is stable (possible) while another is not"? What about the neutrons (not having charge they do not count in the charge distribution)?

I have my model and it predicts exactly (actually some distances are left to be determined) the charge distribution (and mass distribution). In which case I can compare them to the form factors. Would such data be available in the textbooks?

 

[talanum1]

If those shapes do not explain anything then the concept is not much better than a starting guess.

 

[bcrowell]

The gold standard for measuring the shape of a nucleus experimentally is a measurement of the static electric quadrupole moment. I believe this can be done with electron scattering, or possibly by measuring hyperfine splittings. The next best thing is to measure a transition electric quadrupole moment. One often sees, e.g., a nucleus in a 6+ state emitting an E2 gamma-ray and ending up in a 4+ state, with the half-life being much too short to be explained except as a collective motion. (It is also possible to get collective E2 transitions from vibration, though.) Often the simplest and most straightforward way to see that a nucleus is deformed is that it has a ground-state rotational band with spins that go 0+, 2+, 4+,... (assuming even-even) and whose energies follow a J(J+1) pattern. So anyway, this is all extremely well established, and has been since probably the 1960's.

 

[talanum1]

I think a quadropole moment is compatible with a cylindrically symmetrical charge distribution (if you allow a type of imaging effect of the distribution), or two cylinders at right angles.

 

[Raghnar]

There are several experimental proofs of the nuclear shape.
Form factors, cross sections, gamma emission and the relative strenght BE2.
Most of all is the rotational spectrum and the relative moment of inertia underline a symmetry breaking of the spherical simmetric potential.
Some nuclei have a very pronunced rotational spectrum with pratically none single particle vibrational spetrcum, so you can deduce that they are very deformed and from the inertia (also watching not the whole spectrum but only the total energy loss per total angular momentum with a calorimeter) you can deduce if is an oblate (disc) or prolate (cigar) or a multi-axial deformation. In fact only a couple of nuclei are spherical and all of them have a lot of surface vibrations that deform the shape dynamically.
Some start with a slight deformation and then have a phase transition to an highly deformed shape (sometimes even of another type) with a very regular rotational spectrum, that are the superdeformed nuclei (like 152Dy, if I'm not wrong with the memory).
Obviously there are some issues also related by the intrinsic finite body system with a superflud state that rise some issues about the nuclear density in highly excited states, but in general the shape is a well understood property (one of the few) of nuclei.

 

[bcrowell]

In fact only a couple of nuclei are spherical and all of them have a lot of surface vibrations that deform the shape dynamically.

Raghnar, you're basically right in everything you say. Just a minor quibble: It's not just a couple of nuclei that are spherical. A significant percentage of all nuclei are spherical in their ground states. There are hundreds and hundreds of them. Basically any time either the proton number or the neutron number is close to a spherical magic number, the nucleus is going to be spherical, in the sense that the potential energy is minimized at a spherical shape, and the E2 transitions have non-collective strengths. You're correct that there are zero-point vibrations of the surface (this is basically the Heisenberg uncertainty principle).

 

[Raghnar]

Uhm... Maybe I remember wrong... But I think that there is a figure in Rowe that can asset this "minor" quibble! ;)

 

[bcrowell]

Uhm... Maybe I remember wrong... But I think that there is a figure in Rowe that can asset this "minor" quibble! ;)

You may be thinking of doubly magic nuclei. It's not just doubly magic nuclei that are spherical.

 

[talanum1]

Those surface vibrations can't be well defined in case of few nucleons.

My geometric model does not predict spherical nuclei but it is going to have to be taken seriously because it predicts correctly (to four decimal places, I only had four places accuracy on g^(s)) the nuclear magnetic moment of Na ( Z=11, N = 12), something that the Shell Model can't because it only consideres the "last" nucleon. I have not computed more yet, but the model is worked out for all nuclei from H to past Tc (Z = 43).

The superposition is: -1|A> + |B> where A and B are the two formulas derived from the formula for mu for s = 1/2, j = L + 1/2 and j = L - 1/2