# Questions tagged [multipole-expansion]

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### Electric potential for half-grounded sphere [closed]

I have a sphere where the upper half surface has a potential of $V_0$ and the lower half is grounded and I have to find the potential everywhere (using the Laplace solution for spherical coordinates). ...
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### Electric Quadrupole Moment of the Deuteron. 1/4 factor

Sometimes in books I use to see this expression $$\widehat{Q}_{0}=\left(3 z^{2}-r^{2}\right)=\sqrt{\frac{16 \pi}{5}} r^{2} Y_{20}(\theta, \phi)$$ for quadrupole electric moment of the deuteron but ...
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### Partition function of quadrupole? [closed]

I would like to compute the partition function Z for the quadrupole moment, but it is not as easy as the one for the dipole. Could anyone help please?
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### Modelling forces between/on toroidal dipoles/anapoles

I've been trying to find something more concrete about these moments and how they might interact within fields, but haven't found anything clear. Supposedly these moments are "left over" ...
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### How to understand observed electric quadrupole moment of deuteron?

Deuteron is p-n, so naively should have zero electric quadrupole moment. However, experimentally it turns out quite large: $0.2859\ e\cdot fm^2$ from https://en.wikipedia.org/wiki/Deuterium#...
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### Multi-pole expansion for non-Green solution

In electrostatic, we use the Green function to describe the multipole expansion coefficient as: $\int \rho G = monopole + dipole + quadrupole + ...$ This is possible, because we can find a green ...
38 views

### What is the rank of the quadrupole moment tensor in QM and nuclear physics?

For an assignment on Quantum Mechanics, we have been given an expression for the quadrupole moment of the deuteron $$\langle Q\rangle = \langle(2z^2-x^2-y^2)\rangle$$ It is known that the quadrupole ...
28 views

### Can the ground state of an even-even nucleus have a quadrupole moment?

The question I'm stuck with is the following: The quadrupole moment in a way describes the shape of the electric charge distribution of a nucleus right? Oblate if negative, prolate if positive and 0 ...
72 views

### If all the multipole expansions apart from the monopole is zero, then is the charge density spherically symmetric?

Say I have a charge density $\rho(\vec{x})$ in some finite volume $V$, such that all of the multipole expansions apart from the monopole moment (meaning dipole, quadrupole and so forth) are zero. Does ...
50 views

### Multipole expansion of $V$ in powers of $\frac{1}{r}$

If the total charge is $0$, why should the dominant term be the dipole? Also, for a dipole consisting of only two opposite charges separated by a distance $d$, i dont understand how can there be ...
44 views

### Relating two different expressions for quadrupole moments [closed]

How do we get from the expression $$\frac{1}{2} (3(\vec{r} \cdot \vec{r}')^2-r'^2 r^2)$$ to $$\sum_{i,j=1}^3 r_ir_j \frac{1}{2} (3r'_ir'_j-r'^2\delta_{ij})$$ If someone could go step by step and ...
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### Multipole Translation Theorem in FMM [closed]

Consider the standard multipole expansion for the coulomb potential often used in the fast multipole method algorithm (Theorem 5.2). A translation theorem is used in the algorithm (Theorem 5.3). Is O(...
51 views

### Multipole expansion sphere

I saw the solution on griffiths, I didnt understand why in the monopole term they integrate from 0 to r, and then in the dipole they integrate on theta from 0 to pi and in the quadrupole the ...
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### Is linear response only the first term in a series expansion?

So in the theory of linear response, the goal is to look at how certain dynamical variables (or operators in QFT) respond to an external source. To be more concrete, suppose that $x$ obeys (in index ...
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### What would the multi-pole expansion of a spherical shell of magnets look like?

A singular magnet can be expanded in terms of the dipole term, while a set of four magnets with alternating sides facing the inside is described in terms of the quadrupole term. What would happen if ...
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What information does the quadrupole provide? I've seen many definitions on the internet, but I don't understand the relation between them. What is the relation between knowing that the quadrupole ...
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### Calculating multipole expansion outside uniformly charged sphere

When calculating the potential outside a uniformly charged sphere with charge density $\rho$ and radius R using the multipole expansion, it makes sense that only the monopole term survives (since it ...
67 views

### What do moments of inertia do in the potential terms of Lagrangians?

I am struggling to understand the Lagrangian computed in this paper. In particular, a binary spacecraft-debris system is assumed as below. The analysis goes as follows. 1- I am in trouble to ...
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### Multipole moment of an octupole

I have a cube with each side of length a centered on the origin. Each of the corners carries a charge of magnitude q, but the sign of the charge alternates such that no edge connects corners with the ...
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### Does a uniformly moving charge induce electric quadrupole moment densities in surrounding space?

The electric field of a uniformly moving charge is cylindrically symmetric around an axis parallel to its velocity vector. It varies inversely to the square of the distance. The electric field of a ...
44 views

### Is there a way to know the non-zero coefficient of $\cot(\theta)$ expansion in spherical harmonics?

I'm currently trying to find an analytical solution to the Poisson equation for a given distribution using a multipole expansion. During this task, I found the radial expansion and everything else, ...
462 views

### Difference between monopole moment and charge or mass itself?

I'm trying to understand the difference between monopole moment and charge or mass itself. What I found is related to magnetic monopole that is irrelevant. I want to know in multiple expansion of ...
114 views

### Why gravitational waves can only be generated by a time-varying quadrupole moment of the mass distribution?

The (rather old) source I dispose of "Sexl, Urbantke : Gravitation and Cosmology" describes the radiation of gravitational waves only rather sketchy. So why gravitational waves are only generated by ...
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### Electric quadrupole and octupole moments for nuclei

I am getting slightly confused as to which nuclei cab exhibit quadrupole and octupole excitations. In This link it says closed shell nuclei cannot exhibit quadrupole oscillations because if their ...
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### How to extract quadrupole moment and its error from $\chi^2 < \chi^2 + 1$ surface?

I have following info: plot of $\chi^2$ minimization of 208-Rn Coulomb excitation data, Surface corresponds to regions $\chi^2 < \chi^2 + 1$ with error bars of 1$\sigma$, mean lifetime is 8 $\pm$ 0....
In this paper this paper, the Kerr black hole is described as having quadrupole moment of $q=J^2/M$ (which means $q=a^2M$ using $J=aM$) whereas in this paper it says in the abstract that the limiting ...