7. Ch. 3, part 1

Lecture 9 - Electric Multipoles

In many cases we are interested in the “exterior” electric field of a charge distribution ρ(x′) that is contained in a sphere of radius R^{∗} centered at x = 0. If the charge is spherically symmetric, the potential Φ for r > R^{∗} is just q/r, where

q = ∫ ρ(x′) d^{3}x′

is the total charge. Quite generally it is

Φ(x) = ∫ \frac{ρ(x′)}{|x − x′|} d^{3}x′

Using the expansion

\frac{1}{|x − x′|} = ∑_{lm} \frac{4π}{2l + 1} r^{′ l}Y_{lm}^{∗}(θ′, ϕ′) \frac{Y_{lm}(θ, ϕ)}{r^{l+1}}

we obtain

Φ(x) = ∑_{lm} \frac{4π}{2l + 1} q_{lm} \frac{Y_{lm}(θ, ϕ)}{r^{l+1}}  (7.1)

where the spherical multipole moments q_{lm} are by definition

q_{lm} = ∫ ρ(x′) r^{′ l}Y_{lm}^{∗}(θ′, ϕ′) d^{3}x′

For r >> R^{∗} this multipolar expansion is very good and often only the first non-vanishing term is kept. It is also used for nuclei, atoms and molecules, where the charge tails off exponentially. In this case the multipolar expansion is only an asymptotic approximation, because it handles incorrectly the charge outside the sphere of radius r. However, it is often an excellent approximation, making a percentage error exp(−r/a), where a is of the order of 10^{−8} cm for atoms.

The multipolar expansion in spherical harmonics is convenient for formal manipulations, but the moments q_{lm} (as defined by Jackson) carry unwieldy normalization coefficients that are hard to remember. When only dipole and quadrupole moments are kept, it is more convenient to use the Taylor expansion

\frac{1}{|x − x′|} = \frac{1}{r} + ∑_{i} x′_{i} ∂_{i} \frac{1}{r} + \frac{1}{2} ∑_{ij} x′_{i}x′_{j} ∂_{i}∂_{j} \frac{1}{r} + ⋯

where ∂_{i} = ∂/∂x_{i} for short. Inserted into Coulomb’s integral, this gives

Φ(x) = \frac{q}{r} + ∑_{i} p_{i} ∂_{i} \frac{1}{r} + \frac{1}{2} ∑_{ij} \frac{1}{3}Q_{ij} ∂_{i}∂_{j} \frac{1}{r} + ⋯  (7.2)

where

p_{i} = ∫ ρ(x′) x′_{i} d^{3}x′

are the Cartesian components of the dipole moment p and

Q_{ij} = ∫ (3x′_{i}x′_{j} − r^{′ 2}δ_{ij}) ρ(x′) d^{3}x′

are the Cartesian components of the quadrupole moment tensor. We see that the quadrupole tensor is not the second moment tensor M_{ij} = ∫ x′_{i}x′_{j}ρ(x′) d^{3}x′, but only the traceless part of it, apart from the factor of 3 that is for convenience. (Any tensor T_{ij} has trace T = ∑_{i} T_{ii} and can be written as the sum of an isotropic tensor \frac{1}{3}T δ_{ij} and a traceless tensor T_{ij} − \frac{1}{3}T δ_{ij}). The simple reason why we can discard the isotropic part of M_{ij} without altering Φ(x) is that

∑_{ij} δ_{ij} ∂_{i}∂_{j} \frac{1}{r} = ∇^{2} \frac{1}{r} = 0 for r > R^{∗}

The deeper reason has to do with the symmetries of the multipole fields, as we now discuss.

7.0.1. Tensors and multipoles

We see that x′_{i} are the components of a vector, x′_{i}x′_{j} are components of a second-rank tensor, and so on. The connection between x′_{i} and the “interior dipole fields” r′Y_{1m}^{∗}(θ′, ϕ′) is obvious and was discussed in Lecture 7. The connection between x′_{i}x′_{j} and the “interior quadrupole fields” r^{′ 2}Y_{2m}^{∗}(θ′, ϕ′) was also given there, but is a little subtler, because there are 6 components of x′_{i}x′_{j} and only 5 harmonics Y_{2m}^{∗}. However, the combination ∑_{i} x′_{i}x′_{i} = r^{′ 2} does not appear in the Taylor expansion of 1/ |x − x′| and we can write

\frac{1}{|x − x′|} = \frac{1}{r} + ∑_{i} x′_{i} ∂_{i} \frac{1}{r} + \frac{1}{2} ∑_{ij} (x′_{i}x′_{j} − \frac{1}{3} r^{′ 2}δ_{ij}) ∂_{i}∂_{j} \frac{1}{r} + ⋯

In tensorial jargon, r^{′ 2} is the trace of the tensor x′_{i}x′_{j} and x′_{i}x′_{j} − \frac{1}{3} r^{′ 2}δ_{ij} is its traceless part.

Just as r^{′ 2}Y_{2m}^{∗}(θ′, ϕ′) corresponds to the traceless part of x′_{i}x′_{j}, so in general the (2l +1) quantities r^{′ l} Y_{lm}^{∗}(θ′, ϕ′) correspond to the traceless part of the rank-l tensor x′_{i_{1}}x′_{i_{2}} … x′_{i_{l}}. The general symmetric tensor of rank l has (l + 1)(l + 2)/2 components T_{i_{1}i_{2}...i_{l}}, but the zero-trace requirement imposes the (l − 1)l/2 conditions ∑_{i_{1}} T_{i_{1}i_{1}i_{3}...i_{l}} = 0. One can check that the numbers are right:

\frac{1}{2} [(l + 1)(l + 2) − (l − 1)l] = 2l + 1

7.0.2. Multipolar expansion of energy and forces

Not only the field external to a charge distribution, but also the forces acting on a charge distribution can be expressed in terms of its multipoles. Consider a distribution localized near x_{0} and write its density at the point x as ρ(x − x_{0}). Expand the external potential acting on it as

Φ(x_{0}+x′) = Φ(x_{0}) + ∑_{i} x′_{i} ∂_{i}Φ(x_{0})+\frac{1}{2} ∑_{ij} (x′_{i}x′_{j} − \frac{1}{3}r^{′ 2}δ_{ij}) ∂_{i} ∂_{j}Φ(x_{0}) + ⋯

where we have played the usual trick of keeping only the traceless part. Inserting in the expression for the energy

W (x_{0}) = ∫ Φ(x)ρ(x − x_{0}) d^{3}x = ∫ Φ(x_{0}+x′)ρ(x′) d^{3}x′

we obtain

W (x_{0}) = qΦ(x_{0}) + ∑_{i} p_{i} ∂_{i}Φ(x_{0})+\frac{1}{6} ∑_{ij} Q_{ij}∂_{i} ∂_{j}Φ(x_{0}) + ⋯

which can also be written

W (x_{0}) = qΦ(x_{0}) − p · E(x_{0})−\frac{1}{6} ∑_{ij} Q_{ij}∂_{i} E_{j}(x_{0}) + ⋯

In words, the total charge couples to the potential, the dipole moment to the field, the quadrupole moment to the field gradient, and so on.

Differentiating with respect to x_{0}, we find that the force acting on the distribution has components

F_{k}(x_{0}) = −q∂_{k}Φ(x_{0}) − ∑_{i} p_{i} ∂_{i}∂_{k}Φ(x_{0}) − \frac{1}{6} ∑_{ij} Q_{ij}∂_{i} ∂_{j}∂_{k}Φ(x_{0}) + ⋯

or, in vector notation,

F = qE + ∑_{i} p_{i} ∂_{i}E+\frac{1}{6} ∑_{ij} Q_{ij}∂_{i} ∂_{j}E + ⋯

7.0.3. Reduction of multipolar components

The dipole p, like any vector, has a magnitude and a direction. By a suitable rotation of coordinates, the direction can be chosen to be along the z axis, for instance. Thus, essentially, the magnitude of the dipole moment is all we need to know about the shape of the charge distribution: at this level, just one number.

The tensor Q_{ij}, like any symmetric tensor, has principal axes x, y, z , which can be found by the usual methods. The corresponding principal values are Q_{xx}, Q_{yy}, Q_{zz}. Because Q_{ij} is traceless, Q_{xx} + Q_{yy} + Q_{zz} = 0. Thus, for instance, Q_{zz} and \frac{1}{2} (Q_{xx} − Q_{yy}) are all we need to know about the shape of the charge distribution: at this level, just two numbers. Often the distribution is sufficiently symmetrical about the z axis to make Q_{xx} = Q_{yy}. Then all we need to know is Q_{zz}, which is simply called the quadrupole moment. The other principal values Q_{xx} and Q_{yy} are each equal to −\frac{1}{2}Q_{zz}.