Download Higher-Order Moments: Quadrupole and Octupole Charge Distributions and more Essays (high school) Physics in PDF only on Docsity!
Higher-Order Moments
Adrian Down
March 01, 2006
1 Quadrupole field, l = 2, V ∝ 1
r^3
1.1 Review: Definition of the moments
We examine some examples of a charge configurations for which the lowest order moment is the quadrupole moment. The following two examples are the most commonly observed distributions used to create quadrupole moments. Recall the definition of the moments in spherical coordinates,
qlm =
dτ ′^ ρ(r′)r′lY (^) lm∗(θ′, φ′)
where Y (^) lm∗ ∝ e−ımφ. Using Euler’s identity,
e−ımφ^ = cos mφ − ı sin mφ
Note. When the integral is taken over a charge distribution of discrete point charges, it becomes a summation of the integrand evaluated at the location of the point charges.
We also defined the moments in terms of the Cartesian components,
Qij =
dτ ′^ ρ(r′)
3 x′ ix′ j − r′^2 δij
In the case of l = 2, m can take the values 0, ± 1 , ±2. Hence there are five possible moments qlm. However, Qij is a symmetric 3 by 3 matrix, and thus contains six components. There must be some linear dependence in the Q
moments. This dependence can be found by taking the sum of the diagonal elements,
Q 11 + Q 22 + Q 33 =
dτ ′^ ρ(r′)(3x^21 − r^2 + 3x^22 − r^2 + 3x^23 − r^2 )
dτ ′^ ρ(r′)(3r^2 − 3 r^2 ) = 0
1.2 q 20 , where l = 2 and m = 0
Such a charge distribution consists of two equal charges of magnitude +q placed equidistant along an axis from a charge of magnitude − 2 q.
1.3 Pure Q 12
Such a charge distribution consists of four charges equidistant from the origin. There are two charges of magnitude +q on the line y = x and two charges of magnitude −q on the line y = −x.
Note. The plane of the charge distribution determines which Cartesian mo- ment this configuration corresponds to. For example, considering the same configuration in the yz plane gives a pure Q 23 moment.
Considering the spherical moments, rather than the Cartesian represen- tation,
q 22 ∝ Q 11 − 2 ıQ 12 − Q 22
2 Octupoles, l = 3, m ∈ { 0 , ± 1 , ± 2 , ± 3 }
It is possible to define the Cartesian moments,
Qijk =
dτ ′^
5 x′ ix′ j x′ k − r′^3
x′ iδkj + x′ j δki + x′ kδij
ρ(r′)
Qijk can be represented as a 3 by 3 by 3 symmetric matrix, with three diago- nal elements and 24 off diagonal elements. However, only six are independent, since cyclic permutations of the indices leave the Qijk unchanged,
Qijk = Qjki = Qkij = Qjik = Qikj = Qkji
