Electromagnetic Interactions.

We have been discussing the admixture of $L~=~2$ configurations in the deuteron induced by the tensor component of the one-pion exchange (OPE) potential. Such a component is necessary to explain the electric quadrupole moment possessed by the deuteron. We wish to be more quantitative regarding these and the magnetic moments of multi-nucleon systems and hence we will spend some time setting up some nice relations that hold for given configurations.

Lets start by recalling the multipole expansion of electromagnetic interactions. We consider a charge distribution $\rho(x)$ that has support in a given region of space interacting with an electromagnetic potential that is slowly varying over the volume of the charge distribution. The interaction hamiltonian is given by

 

$$H_{\rm int} = \int\ d^3x\ j_\mu (x) A^\mu (x)$$

$$= \int\ d^3x\ \left[ \rho (x) \phi(x) - {\bf j} (x) \cdot {\rm A} (x) \right]$$

$$= \int\ d^3x\ \left[ \rho (x) \left( \phi (0)\ +\ \partial_i\phi (0) x^i \ +\ {1\over 2}\partial_i\partial_j\phi (0) x^i x^j\ +\ ...\right) \right.$$

$$\left. \ \ - {\bf j}(x) \left( {\bf A} (0)\ +\ \partial_i{\bf A} ... ...1\over 2}\partial_i\partial_j{\bf A} (0) x^i x^j\ +\ ...\right) \right] \ \ \ .$$ (15)

Recalling the ${\bf E}^j = -\partial^j\phi$ and ${\bf B} = \nabla\times {\bf A}$ we have that

 

$$H_{\rm int} = Q\phi(0) - {\bf P}\cdot {\bf E} - {1\over 6}Q^{ij}\partial^i {\bf E}^j - {\bf M}\cdot {\bf B} \ +\ ... \ \ \ ,$$ (16)

where the moments are defined by

 

$${\bf P} = \int\ d^3x\ \rho (x) {\bf x}$$

$${\bf M} = {1\over 2} \int\ d^3x\ {\bf x}\times {\bf j}(x)$$

$$Q^{ij} = \int\ d^3x\ \left( 3 x^i x^j - \delta^{ij}x^2 \right) \rho (x) \ \ \ ,$$ (17)

and so forth.