# Exploring Nuclear Quadrupole Moment and Deformation

• patric44
In summary: The quadrupole moment can be nonzero for a spherically symmetric charge distribution if the charge distribution has an off-center charge. In summary, the nuclear quadrupole moment describes the deviation of a nucleus from a spherical shape. This is calculated using the electric quadrupole tensor, which is analogous to the moment of inertia tensor. The quadrupole term in the multiple expansion of the potential describes this deformation and is dominant due to the small parameter representing the level of deformation. To understand this concept further, it is recommended to read Judah Eisenberg and Walter Greiner's books on Nuclear Models. The monopole and dipole moments cannot accurately assess this deformation, making the quadrupole moment essential.
patric44
Homework Statement
How the nuclear quadruple moment describe the deformation of the nucleus?
Relevant Equations
Q =1/e∫∫∫(3z^2-r^2)dV
hi guys
I have read the other day about how the nuclear quadruple moment descries the deformation of the nucleus, however i can't get my head around how is that!, I am familiar with the multiple expansion in which we can describe the potential of an arbitrary charge distribution by the following expansion
$$\Phi(r') = \frac{1}{4\pi\epsilon_{o}}\sum_{n=0}^{\inf}\frac{1}{r^{n+1}}\int_{v}(r')^{n}P_{n}(cos\theta)\rho(r')dV$$
and this potential is then expanded as multiple terms : monopole,dipole,quadrupole..., Now how the quadrupole term in this expansion describes a deformation in the shape of the nucleus? and : isn't quadrupole means "generaly speaking" 4 charges two of them are negative and the other is positive, arranged in a specific way, how could such term describes the only positively charged neucles?
I have many questions surrounding this point, can anyone clarify them or suggest a specific book or a set of lecture notes that would help.

ergospherical
Forgive me if this is wrong but from my VERY cursory understanding we refer to a "deformation of a nuclei as deviation from a spherical shape". From my understanding the Electric Quadrupole Tensor is analogous to a Moment of Inertia Tensor. Remember finding principle axes and moments of the inertia tensor and the different cases of spherical tops, prolate, and oblate? I.e. if all the principle moments have the same value then we have a spherical top but if they are unequal we do not (i.e. deformation or deviation from spherical).

ergospherical and patric44
I did a little bit of research and I found the following reference very informative!
https://photonics101.com/multipole-moments-electric/nuclei-model-deformed-sphere-multipole-moments

Summarising briefly; scattering experiments suggested that atomic nuclei can be modeled by a homogeneous charge distribution with a shape given by an axially symmetric function\begin{align*}
R(\theta) = R_0(1+ \xi P_2(\cos{\theta}))
\end{align*}where ##\xi \ll 1## is a parameter representing the level of deformation. If you calculate the moments of this distribution, as the reference does, the dipole moment is zero due to the presence of mirror-symmetry and the first non-zero moment is the quadrupole moment. The multipole moments go as ##b_{2n} \propto \xi^n##, so the quadrupole is dominant.

vanhees71 and PhDeezNutz
Try to get a hold on the three volumes of Judah Eisenberg and Walter Greiner's books on Nuclear Models.
I stopped reading somewhere after 10-60 pages of the first volume.

yucheng
patric44 said:
isn't quadrupole means "generaly speaking" 4 charges two of them are negative and the other is positive, arranged in a specific way, how could such term describes the only positively charged neucles?

A "[pure] quadrupole" would be an arrangement of charges with zero net-charge (zero monopole) and zero dipole-moment. But a general charge distribution can have nonzero multipole moments... where the monopole term handles the net-charge.

A point charge that is not at the origin has nonzero multipole moments.

Possibly useful: has calculations for point-charge arrangements
http://www.physics.usu.edu/Wheeler/EM3600/Notes12MultipoleMoments.pdf
http://orca.phys.uvic.ca/~tatum/elmag/em03.pdf

vanhees71
@robphy if I understand your post correctly

1) If a nucleus is deformed from a spherical arrangement the monopole stays the same so we can’t use the monopole moment to gauge deformation.

2) Regarding the dipole moment we can imagine a deforming a sphere to an ellipsoid in such a way that the dipole moment stays the same. Likewise the dipole cannot be used to assess deformation from spherical.

3) A quadrupole is spherical if all principle moments are the same. If they aren’t we’ve deformed from spherical. This is why we use the quadrupole to assess deviation from spherical.

Is that correct?

vanhees71
PhDeezNutz said:
@robphy if I understand your post correctly

1) If a nucleus is deformed from a spherical arrangement the monopole stays the same so we can’t use the monopole moment to gauge deformation.
Yes.

PhDeezNutz said:
2) Regarding the dipole moment we can imagine a deforming a sphere to an ellipsoid in such a way that the dipole moment stays the same. Likewise the dipole cannot be used to assess deformation from spherical.

I think the dipole moment of a sphere that is centered at the origin is zero.
Deforming the sphere into an ellipse will result in a nonzero dipole moment.
However, the dipole moment doesn't have enough information for what you seek.
PhDeezNutz said:
3) A quadrupole is spherical if all principle moments are the same. If they aren’t we’ve deformed from spherical. This is why we use the quadrupole to assess deviation from spherical.

Is that correct?

So, the quadrupole moment is what you need.

PhDeezNutz

## 1. What is a nuclear quadrupole moment?

A nuclear quadrupole moment is a measure of the distribution of electric charge within a nucleus. It is a property of a nucleus that is related to its shape and deformation.

## 2. How is nuclear quadrupole moment measured?

Nuclear quadrupole moment is typically measured through nuclear magnetic resonance (NMR) spectroscopy or nuclear quadrupole resonance (NQR) spectroscopy. These techniques use the interaction between the nuclear quadrupole moment and an external magnetic field to determine its value.

## 3. What factors affect the nuclear quadrupole moment?

The nuclear quadrupole moment is primarily affected by the shape and deformation of the nucleus, which can be influenced by the number of protons and neutrons in the nucleus, as well as the nuclear spin and energy levels.

## 4. Why is exploring nuclear quadrupole moment important?

Studying the nuclear quadrupole moment can provide valuable information about the structure and properties of atomic nuclei, which can have implications for fields such as nuclear physics, chemistry, and materials science. It can also help us better understand the fundamental forces and interactions within the nucleus.

## 5. What are some potential applications of understanding nuclear quadrupole moment?

Understanding the nuclear quadrupole moment can have practical applications in fields such as nuclear energy, nuclear medicine, and materials science. It can also aid in the development of new technologies, such as nuclear magnetic resonance imaging (MRI) and nuclear quadrupole resonance sensors.

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