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Questions tagged [multipole-expansion]

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124 views

What is the minimal number of point charges required to model an electric octupole?

In an answer to this question, I showed that a general electric quadrupole can be modeled by a combination of seven point charges (or possibly less if one or two of the principal quadrupole moments is ...
3
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1answer
704 views

How does a electric quadrupole oscillate?

I know that in static a electric quadrupole is made of two positive charges and two negative charges, distributed as in the following figure: What I don't understand is when it oscillates and ...
3
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78 views

Can gravitational waves observed far from a black hole tell us anything about the multipole moments of a dynamical horizon?

In a paper by Ashtekar et al in 2013 on the approach to the final state to a stationary black hole they study the evolution of the multipole moments of dynamical horizons, which relax away (except for ...
3
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127 views

Higher order multipolar second harmonic generation in centrosymmetric materials

As is pointed in this question, second harmonic generation is forbidden in the bulk of the materials possessing centrosymmetry. In some papers it is said that in the dipolar approximation the SHG ...
3
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0answers
102 views

Physical significance of toroidal moments

The electric field of a toroidal dipole moment $\vec t$ is the same as the electric field of a electric dipole moment $\vec p$, except that it is scaled with the factor $ik$. The $i$ makes the ...
3
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628 views

Electrodynamic multipole expansion

I am reading Jackson, Classical Electrodynamics, and I have a question regarding the electrodynamic multipole expansion (with page numbers I refer to the 3rd edition). So on page 409, he gives in ...
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0answers
66 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 ...
2
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1answer
60 views

Can I replace a multipole expansion by a combination of separate dipoles?

If I want to be able to model a magnetic field flux density $\mathbf{B}$ from a magnetic source located at the origin at a position $\mathbf{r}$, it is my understanding that I can represent $\mathbf{B}...
2
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0answers
81 views

Multipole expansions: test on a function $\zeta=\zeta(t)$

Considering the potential $\psi(r)$ of a sphere of mass $M$ with density $\rho(\mathbf r')$, connected by a small volume positioned in the $P'$ point as shown in the figure: The $\psi(r)$ is: $$\...
2
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1answer
135 views

Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?

In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
2
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0answers
169 views

Quadrupole and Multipoles in Physics

I am a little confused over the notion of quadrupole and higher moments in physics in general. The first time I saw it was in electromagnetism, when we did multipole expansion to analyze higher ...
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0answers
258 views

Positive/negative octupole moment of nuclei?

Does octupole moment of nuclear charge distribution show any positive/negative character, like the quadrupole moment does? Quadrupole moment has prolate and oblate types, but what about ...
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45 views

Approximating electrostatic grids as a multipole expansion

Is there a known good summary, or a succinct algorithm to compute the far-field approximations of an arbitrary set of electrostatic surfaces set at different potentials? I'm looking to model a ...
2
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0answers
599 views

The anapole moment, derivation from Dirac current density

Basically I am looking for a way to expand the electromagnetic interaction energy $W = A_{\mu}j^{\mu}$ (both $A$ and $j$ obtained from the Dirac equation) similar to the classical expansion in ...
2
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0answers
88 views

Proton as superposition of hadrons: $\vert p\rangle = c_0\vert p_0\rangle+c_1\vert h\rangle+\cdots$

I have a question regarding hadron fluctuations. For instance on page 85 in Feynman's "Photon-Hadron Interactions" equation 15.2 reads: $$\tag1\vert \omega\rangle = \vert \omega_0\rangle+\frac{\...
2
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0answers
135 views

quadripolar moment in curved space

So, i'm going over the Thorne's derivation of the quadrupolar radiation term, and they write the core term as: $$ \frac{3 r_i r_j - 2 r^2 \delta_{ij}}{4 r^5} $$ But if i try to obtain this term by ...
2
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2answers
73 views

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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0answers
87 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 ...
1
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1answer
102 views

The greatest quadruple moment

Let us consider the following problem: it is necessary to find the shape of the body with fixed mass and density which at large distances compared with its characteristic dimensions would give the ...
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0answers
325 views

Why is the electric field of an axial quadrupole not the same as the electric field of two axial dipoles, at far distance?

An axial electric quadrupole, made of four inline charges $(+q, -q, -q, +q)$ with opposite charges a distance $a$ apart, and the two $-q$ charges adjacent, has an electric field at a remote point $P$ ...
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0answers
106 views

Tight coupling of photons and baryons before recombination

I am trying to understand the equations of CMB anisotropies. In the book I am studying with, the author wants to show that, when solving for the photon perturbations in the tightly coupled limit (...
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0answers
156 views

Does it make sense an inertial mass dipole moment for massless particles?

Electric and magnetic dipoles exist in nature even with zero net electric and magnetic charge Electrons have a unit of charge, about $10^{-31}$ kg of inertial mass, zero electric dipole moment, $\...
1
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1answer
83 views

Electrostatic polarization of an axially symmetric conductor

A single point charge at the origin induces charge onto a grounded Z-axis symmetric conductor. The induced charge is given in cylindrical coordinates as $\sigma(r, z)$ since the Z symmetry means there ...
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0answers
54 views

multipole moments of dipole with finite spacing

Can a dipole with finite spacing between poles be represented by pure multipoles centered at the origin? Say for example that I have a dipole with finite spacing $2\epsilon$ between the poles. I ...
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0answers
69 views

External and internal multipole expansion for axisymmetric potential - the region of convergence

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
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0answers
76 views

Multipole expansion of Woods-Saxon potential?

When can a distribution be expanded in multipoles? What is the basic requirement? Can it be done for a potential like the Woods-Saxon form?
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0answers
152 views

How does the Wigner-Eckart theorem rule Multipole Expansion?

I am wondering why a spin-S particle have only the term up to $k=2S$ in his multipole expansion ? It seems that the Wigner-Eckart theorem shows the relation between spin and multipole expansion but I ...
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0answers
220 views

Question about “quadrupole radiation” vector potential formula derivation

I tried to get an expression for $\mathbf A (\mathbf x )$ in quadrupole approximation. After some transformations of Liénard–Wiechert vector potential I got, as in many books, $$ \mathbf A \approx \...
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0answers
1k views

What is the magnetic quadrupole moment of a nucleus in cylindrical coordinates?

What is the magnetic quadruple moment of a nuclei in cylindrical coordinates? The quadrupole moment of a nucleus is zero in spherical coordinates but in the cylindrical coordinates it can't be ...
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0answers
14 views

Understanding the multipole Expansion for Quadrupoles

Potential of a Quadrupole is given as, $$V(r) = \frac{1}{4\pi e_0}(\frac{1}{R}q+\frac{1}{R^2}\sum_{i=x,y,z}\hat{R_i}\vec{p_i} + \frac{1}{R^3}\sum_{i,j=x,y,z}\hat{R_i}\hat{R_j}Q_{ij}$$ where $$Q_{ij} ...
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20 views

Multipole expansion

consider a charge distribution say q, 2q, 3q, kept on the vertices of an equilateral triangle. There exist monopole, dipole ,qudrupole etc terms in the potential expression. these 3 charges are ...
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21 views

Question about the Multipole Expansion second term (quadrupole)

I was wondering if the $n=2$ term in the potential of a multipole: $V(r)= \dfrac{1}{4 \pi \epsilon_0} \sum \dfrac{1}{r^{(n+1)}} \int (r')^n P_n(cos \alpha) \rho(r')dv'$ was the same as this ...
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12 views

What is the nature of the multipoles used to shape/re-direct an e-beam?

Imagine having an electron beam, just like in an electron microscope. Now imagine that we have some multipoles prepared in the beams path to influence the shape/path of the beam. My question is: ...
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1answer
38 views

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 ...
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0answers
42 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, ...
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22 views

Diffraction by induced quadrupoles

This is a very interesting problem that I've been struggling to solve for a week, so I decided to ask for some orientation, as I think it could also be of interest for the community. Let's consider a ...
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0answers
30 views

The magnetic and electric dipole moment 'popping out' from multipole expansions

I interpret electric dipole moments and magnetic dipole moments as intrinsic properties of certain materials, but in Griffiths it's literally picked out of expressing the potential and magnetic vector ...
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0answers
23 views

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....
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1answer
31 views

Relevance of electromagnetic multipole transitions

In what kind of systems higher electromagnetic multipole transitions (like electric quadrupole transitions) become important or at least measurable? Is it for antennas in radiofrequency? Is it in the ...
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0answers
18 views

Express polarization moments in terms of temperature quadrupoles

I am trying to compute the polarisation moments in the tight coupling limit which is an exercise from Dodelson's Modern cosmology where the evolution equation is given by $$\dot{\Theta}_p+ik\mu\...
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0answers
17 views

Relative strengths of multipole contributions in atom photo-excitations

I have heard and read many times that dipole contribution to the photo-absorption matrix element is orders of magnitude stronger than quadrupole. To be exact, they are related as: $\frac{\langle ...
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1answer
30 views

Expressing interaction between two classical charge distributions in terms of multipole moments

I am interested in expressing the interaction energy between two classical charge distributions in terms of the multipole moments of each of the charge distributions. For example, let's assume each of ...
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0answers
45 views

What is a force multipole?

In a recent talk about physics and mechanics inside the cell, I heard such terms as 'force monopole' and 'force dipole'. What do such terms mean? Are they talking about the angular distributions, ...
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68 views

Are there transitions of mixed multipolarity in charged ions?

In nuclear physics transitions with mixed multipolarity, ex. (M1+E2), play an important role due to d-wave admixture. I was wondering if analogous situation has ever been considered in atomic ...
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33 views

Are there atoms with compatible electric and magnetic multipole transitions

I know that usually E1-transition is dominant over all the higher multipoles in atomic spectroscopy. My question is: are there any natural/fabricated atomic systems where magnetic transitions are ...
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729 views

Quadrupole moment tensor

I am currently working on gravitational waves, and as in every lecture on general relativity I derived the symmetric trace-free perturbation of a Minkowski metric with the Lorenz gauge condition, ...
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0answers
207 views

Choice of Origin in Multipole Expansion in Electrostatics

In a multipole expansion in electrostatics, how do we select the origin? For instance, if there is a linear continuous distribution of charge $+Q$ (say from $x = -a$ to $x = +a$). Is there any rule ...
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44 views

strength of each dipole charge

If I have a two dimensional dipole whose line charges are located on the y axis, I know that the electric flux through a gaussian pill box containing both the charges will be zero, that is \begin{...
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0answers
68 views

Series expansion of potential due to a dipole displaced from the origin

I learn that we can expand the electric potential in an infinite series of $\rho$ and $\cos(n \phi)$ when solving the Laplace equation in polar coordinates. The problem I want to consider is the ...
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60 views

find cylindrical multipole coefficients

How does one find the coefficients to the cylindrical multipole expansion? I have the harmonic function $\omega = \dfrac{f(\theta)}{\sqrt{r}}$ where $f(\theta)$ is function with a period of $2\pi$ ...