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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2answers
191 views

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 ...
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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 ...
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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 ...
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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 ...
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1answer
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 ...
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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(...
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1answer
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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54 views

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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1answer
34 views

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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1answer
71 views

Quadrupole moment

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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1answer
33 views

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 ...
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1answer
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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46 views

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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1answer
26 views

Potential due to a charge distribution and multipole expansion

I'm studying on the Griffiths' book Introduction to Electrodynamics and a doubt came to me reading about the multipole expansion. In chapter 2.3.4 this formula is shown $$ V(\vec{r}) = \frac{1}{4\pi\...
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1answer
33 views

Can a quadrupole form in a purely organic crystalline material?

I'm a chemist so bear with me here. This is a question about small molecules, such as biphenyls. In the solid-state biaromatic systems without steric hinderance forms planar conjugated structures. By ...
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Transforming from the non-traceless to the traceless quadrupole moment

Today I was working over the multipole expansion for electric potentials, and on my own, I got to the non-traceless definition of the quadrupole moment $ Q_{\alpha \beta} = \int{\rho(\mathbf{r'})\...
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1answer
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Why is $A \propto 1/r^3, B \propto 1/r^4$, far away from circular loops

Two equal circular current loops are placed coaxially with each other. The loops have equal but opposite currents $I$. $$ A \propto r^{-l} $$ $$ B \propto r^{-k} $$ , where $A$ is the vector ...
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1answer
113 views

Components of Electric Quadrupole Oscillator Strength

Fermi's Golden Rule states that the rate of a transition of an electron going from the ground state $0$ into some state $n$, is directly proportional to the square of the first order perturbation $\...
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1answer
79 views

How can a potential be non-central?

I'm studying nucleon-nucleon interactions and I'm reading that the potential for said interaction has a non-central (or tensor) component. Now, I understand that, when describing a 2-bodies problem, ...
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1answer
44 views

Electric octupole in cartesian coordinates

I am new to this: Does anyone know to define different electric octupole moments in cartesian coordinates? I am looking for expressions that look like this (for electric dipoles).
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Understanding the multipole Expansion for Quadrupoles

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

How to calculate the Quadrupole moment via integration?

I have following problem: calculate the quadrupole moment of following arrangement, where $e$ is the charge and $a$ is the distance between the charges. Hint: The quadrupole moment is defined ...
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25 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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1answer
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Clarification on Taylor expansion in linear quadrupole multipole expansion in cartesian coordinates

In my textbook, given the electric linear quadrupole on the z axis, as in the figure, the author considers the functions $$ \frac{1}{r_i} = \frac{1}{\sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}}, $$ with $r = ...
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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
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Reconstructing Charge Distribution from Multipole Expansion

Let $\rho$ be a static, discrete or continuous charge distribution, and $\phi(\mathbf{r})$ the corresponding electric potential. We may expand $\phi$ in a multipole series, $$ \phi(\mathbf{r}) = \frac{...
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1answer
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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 ...
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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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1answer
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 ...
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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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1answer
112 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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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
167 views

Quadrupole moment of Kerr spacetime

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 ...
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270 views

Total charge for Dipole and Quadrupole moment

I have been struggling on what to put for $Q(total)$ in the equations for the dipole and quadrupole moment for the potential: $$ V_{quad}(\mathbf r) \approx \frac{2Qd^2}{4\pi\epsilon_0} \left( \frac{...
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1answer
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Question about multipole expansion of electrostatic potential?

If I take a certain dipole and by dipole I mean that two charges of opposite sign differentiated by very small distance. And if I take the formula of multipole expansion of potential then I see no ...
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1answer
143 views

The fields of Liénard and Wiechert and Poynting vector

EDIT: I know that the electric and magnetic fields depend not only on speed but also on acceleration and can both be expressed as the sum of two contributions: \begin{equation} \overline{E} (\bar{r},...
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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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Applications of multipole expansion in gravitational problems [closed]

I want to know what exactly are practical applications of multipole expansion in some problems concerning gravitation. I have also read in Subtle is the Lord by Abraham Pais that Einstein had ...
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1answer
59 views

Looking for an understanding of Toroidal Moments [duplicate]

The wikipedia page is not so enlightening Apparantly the Neutrino, if it is a Dirac particle, has a Toroidal Moment. Does this mean the Dirac neutrino would interact electromagnetically as well as ...
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3answers
535 views

Electric quadrupole moment and electric octupole moment

From a Chinese electrodynamics textbook, first we take a $1/r$ Taylor expansion into the electric potential equation and simplify it, then we get the final equation, and we call each term different ...
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1answer
65 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}...
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3answers
263 views

Systematic expansion of $e^{i\vec{k}\cdot\vec{r}}$ in atomic physics in terms of Legendre polynomials and identifying different $l$ terms

In the context of light-matter interaction one often makes the approximation $e^{i\vec{k}\cdot\vec{r}}\approx 1$. Keeping higher order terms in $e^{i\vec{k}\cdot\vec{r}}$ give magnetic dipole, ...
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1answer
61 views

Electric quadrupole - tensor identity

In classical electrodynamics, we introduce the electric quadrupole moment $$D^{ij}\equiv\int y^i y^j \rho \mathrm{d}^3y$$ and its reduced (trace-less) version $$\mathcal{D}^{ij}\equiv D^{ij} - \frac{1}...