Electromagnetic multi-pole moment


Electromagnetic multipole comes from the charge and current distribution of the nucleons.

Magnetic multipole in nucleus has 2 origins, one is the spin of the nucleons, another is the relative orbital motion of the nucleons.  the magnetic charge or monopoles either not exist or very small. the next one is the magnetic dipole, which cause by the current loop of protons.

Electric multipole is solely by the proton charge.

From electromagnetism, we knew that the multipole has  different radial properties, from the potential of the fields:

\Psi(r) = \frac{1}{4\pi\epsilon_0} \int\frac{\rho(r')}{|r-r'|}d^3r'

A(r) = \frac{\mu_0}{4\pi}\int\frac{J(r')}{|r-r'|}d^3r'

and expand them into spherical harmonic by using:

\frac{1}{|r-r'|} = 4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l} \frac{1}{2l+1}\frac{r_{<}^l}{r_>^{l+1}} Y_{lm}^*(\theta',\phi')Y_{lm}(\theta,\phi)

we have

\Psi(r) = \frac{1}{\epsilon_0} \sum_{l,m}\frac{1}{2l+1}\int Y_{lm}^*(\theta',\phi') r'^l\rho(r')d^3r' \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}

A(r)=\mu_0 \sum_{l,m}\frac{1}{2l+1}\int Y_{lm}^*(\theta',\phi') r'^l J(r') d^3r' \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}

we can see the integral give us the required multipole moment. the magnetic and electric are just different by the charge density and the current density. we summarize in this way :

q_{lm} = \int Y^*_{lm}(\theta',\phi') r'^l \O(r') d^3 r'

where O can be either charge or current density. The l determine the order of multipole. and the potential will be simplified :

M(r)=\sum_{l,m}\frac{1}{2l+1} q_{lm} \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}

were M can be either electric or magnetic potential, and i dropped the constant. since the field is given by 1st derivative, thus we have:

  1. monopole has 1/r^2 dependence
  2. dipole has 1/r^3
  3. quadrapole has 1/r^4
  4. and so on

The above radial dependences are same for electric or magnetic. for easy name of the multipole, we use L-pole, which L can be 0 for monopole, 1 for dipole, 2 for quadrapole, etc.. and we use E0 for electric monopole, M0 for magnetic monopole.

Since the nucleus must preserver parity, and the parity for electric and magnetic moment are diffident.the different come from the charge density and current density has different parity. The parity for charge density is even, but for the current density is odd. and 1/r^2 has even parity, 1/r^3 has odd parity. therefore

  • electric L-pole — (-1)^{L}
  • magnetic L-pole — (-1)^{L+1}

for easy compare:

  • E0, E2, E4… and M1,M3, M5 … are even
  • E1,E3,E5…. and M0, M2, M4…. are odd

The expectation value for L-pole, we have to calculate :

\int \psi^* Q_{lm} \psi dx

where Q_{lm} is multipole operator ( which is NOT q_{lm}), and its parity is follow the same rule. the parity of the wave function will be canceled out due to the square of itself. thus, only even parity are non-Zero. those are:

  • E0, E2, E4…
  • M1,M3, M5 …

that make sense, think about a proton orbits in a circular loop, which is the case for E1, in time-average, the dipole momentum should be zero.

Informations we can extract

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in scattering experiment,  the raw informations we can know or observe are only few things:

  1. the number of particles counted at particular solid angle. ( when you have a unit sphere, the area on the surface is called solid angle)
  2. The polarization (spin)
  3. charge
  4. energy
  5. momentum (time of flight)

Since the number of particles counted is related to the intensity of the incident beam, the density of the target, the interaction and the differential cross section.

on the other hand, the number of particles counted should be related to intensity of incident beam, density of the target and interaction potential. Thus, the differential cross section is related to the interaction potential.

The polarization can be measured by 2nd scattering of known polarization target. or directly from a polarized primary target. or by a polarized beam.

For a nucleus there are 12 properties, and we can group them into 2, 1 is static properties, another is dynamic properties.

Static properties


  1. mass
  2. radius
  3. spin
  4. parity


  1. relative abundance
  2. decay half-live
  3. magnetic dipole
  4. electric quadrapole

Dynamic properties

  1. decay modes
  2. reaction mode
  3. cross section
  4. excited state

Of course, the final goal of nuclear physics is find the Hamiltonian for governing the motion of nuclear matter. Thus, we can base on the intrinsic static properties, to deduced the extrinsic and dynamic properties.

think about in atomic physics, we know the potential, the spin, then we can give out every things, like the radius, parity, decay half-live, cross section, etc….

so, the nuclear Hamiltonian is the KEY to open the door of understanding of nuclear matter.

Objects of Interest

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Nuclear Physics is a study on nuclear matter which is fundamental building block of the world.

electron, proton , neutron, deuteron, tritium, etc… those are objects in nuclear, we call them “particle”. the most simple particle in here is electron, proton and neutron.

The different between nuclear and atom is:

Nuclear core (sit in the center) + Electrons (moving around) = Atom

the mass of atom is almost contributed by nuclear. This is because the mass of proton is about 1830 times bigger than electron, and neutron’s mass is only heavier a bit then proton.

There are many properties contained in each particle. there are electric charge, mass, spin, kinetic energy, etc… and the objective of nuclear physics is understand all these properties and how these properties affect the inter-reaction among them. for example, how a proton and neutron form a nuclear core in deuteron? how they attract each other?

these properties, some may say, are ASSIGNED to the particles. Basically, we can only measure the effect or the result from each interaction. we think, there is a FORCE to make particles able to INTER-ACT with each others. simple to say, when an electron meets another electron, they affect each other by ELECTROMAGNETIC force. but when consider an electron meet a neutron, they don’t interact by electromagnetic(EM) force. in order to distinguish these. we assign an electric CHARGE to electron, and no charge for neutron.

so, basically, Nuclear Physics is study the PROPERTIES of particles and the INTERACTION among them.

There are 3 major forces/interactions, Weak force, EM force and Strong force. Until this moment, we only know the weak and EM force and not fully understand the strong. We neglect the gravity in here, because it is very weak and do no observable effect.






Strong 10,000 10-15m
EM 1000 long
Weak 1 10-18m