Potential energy of quadrupole moments

[Gavroy]

hi

i am looking for an equation that gives me the potential energy of 2 quadrupole moments and their mutual distance is assumed to be large.

and they have no total charge or dipole moments.

does such an equation exist?

 

[Bill_K]

Are you sure you want to see this? It's rather complicated. Let's write down the dipole-dipole interaction first. An electric dipole moment is described by a vector p. The electrostatic potential energy between two dipoles p and p' a distance r apart is

W = (p.p' -3 (n.p)(n.p'))/r³

where n is the unit vector pointing from one dipole to the other.

Ok similarly, the quadrupole moment of an object is described by a symmetric traceless tensor Q. The electrostatic potential energy between two quadrupoles Q and Q' is given by

W = (Q:Q' -10 n.Q.Q'.n + 35/3 (n.Q.n)(n.Q'.n))/r⁵

Maybe you'd rather see this written out explicitly with all the indices.

Φ = (Q_ijQ'_ij - 10 n_iQ_ijQ'_jkn_k + 35/3 n_iQ_ijn_j n_kQ'_kln_l)/r⁵

 

[Gavroy]

Are you sure you want to see this? It's rather complicated. Let's write down the dipole-dipole interaction first. An electric dipole moment is described by a vector p. The electrostatic potential energy between two dipoles p and p' a distance r apart is

W = (p.p' -3 (n.p)(n.p'))/r³

where n is the unit vector pointing from one dipole to the other.

Ok similarly, the quadrupole moment of an object is described by a symmetric traceless tensor Q. The electrostatic potential energy between two quadrupoles Q and Q' is given by

W = (Q:Q' -10 n.Q.Q'.n + 35/3 (n.Q.n)(n.Q'.n))/r⁵

Maybe you'd rather see this written out explicitly with all the indices.

Φ = (Q_ijQ'_ij - 10 n_iQ_ijQ'_jkn_k + 35/3 n_iQ_ijn_j n_kQ'_kln_l)/r⁵

thank you this exactly what i needed!

 

[Gavroy]

oh,

could you explain to me, why there are 4 indices? i,j,k and l

i thought that this tensor would only use 3 cartesian coordinates?

then why are there 4 different indices?

 

[liuxunchen]

Hi, this is very helpful.
could anyone tell me the dipole-quadrupole energy formula?
I have the dipole moment and explicit quadrupole moment terms but aren't sure how to calculate their interaction energy!