Editing Quadrupole (section)

=== Electric quadrupole ===
A simple example of an electric quadrupole consists of alternating positive and negative charges, arranged on the corners of a square.  The monopole moment—the total charge—of this arrangement is zero.  Similarly, the [[Electric dipole moment|dipole moment]] is zero, regardless of the coordinate origin that has been chosen. A consequence of this is that a quadrupole in a uniform field experiences neither a net force nor a net torque, although it can experience a net force or torque in a non-uniform field depending on the field gradients at the different charge sites.<ref>{{Cite web |date=2016-12-20 |title=3.8: Quadrupole Moment |url=https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electricity_and_Magnetism_(Tatum)/03%3A_Dipole_and_Quadrupole_Moments/3.08%3A_Quadrupole_Moment |access-date=2024-11-16 |website=Physics LibreTexts |language=en}}</ref> As opposed to the monopole and dipole moments, the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, regardless of where we place the coordinate origin.  The [[electric potential]] of an electric charge quadrupole is given by<ref>{{Cite book| author= Jackson, John David| date= 1975| title= Classical Electrodynamics| publisher= [[John Wiley & Sons]]| isbn= 0-471-43132-X| url-access= registration| url= https://archive.org/details/classicalelectro00jack_0}}</ref>
<math display="block">V_\text{q}(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{|\mathbf{R}|^3} \sum_{i,j} \frac{1}{2} Q_{ij}\, \hat{R}_i \hat{R}_j\ ,</math>

where <math>\varepsilon_0</math> is the [[electric permittivity]], and <math>Q_{ij}</math> follows the definition above.

Alternatively, other sources<ref>{{Cite book| author= Griffiths, David J.| date= 2013| title= Introduction to Electrodynamics, 4th ed.| publisher= Pearson| page= 153,165}}</ref> include the factor of one half in the <math>Q_{ij}</math> tensor itself, such that:

<math display="block">Q_{ij} = \int\, \rho(\mathbf{r})\left(\frac{3}{2}r_i r_j - \frac{1}{2}\left\|\mathbf{r}\right\|^2\delta_{ij}\right)\, d^3\mathbf{r},</math>
and
<math display="block">V_\text{q}(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, \hat{R}_i \hat{R}_j\ ,</math>

which makes more explicit the connection to [[Legendre polynomials]] which result from the multipole expansion, namely here <math display="inline">P_2(x) = \frac{3}{2}x^2 - \frac{1}{2}.</math>

In atomic nuclei the electric quadrupole moment is used as a measure of the nucleus' obliquity, with the quadrupole moment in the nucleus given by
<math display="block">Q \equiv \frac{1}{e} \int r^2 (\cos^2(\theta) - 1) \rho(\mathbf{r})\,d^3 \mathbf{r}</math>
where <math> \mathbf{r}</math> is the position within the nucleus and <math>\rho</math> gives the charge density at <math> \mathbf{r}</math>.<ref>{{Cite book |last=Amsler |first=Claude |title=Nuclear and particle physics |date=2015 |publisher=IOP Publishing |isbn=978-0-7503-1140-3 |edition=Version: 20150501 |series=IOP expanding physics |location=Bristol, UK}}</ref>

An electric field constructed using four metal rods with an applied voltage forms the basis for the [[quadrupole mass analyzer]], in which the electric field separates ions based on their mass-to-charge ratio (m/z).<ref>{{Cite book |title=Mass spectrometry: an applied approach |date=2019 |publisher=Wiley |isbn=978-1-119-37733-7 |editor-last=Smoluch |editor-first=Marek |edition=Second |series=Wiley series on mass spectrometry |location=Hoboken, NJ |editor-last2=Grasso |editor-first2=Giuseppe |editor-last3=Suder |editor-first3=Piotr |editor-last4=Silberring |editor-first4=Jerzy}}</ref>