# Some Math polarizability

1. Jul 24, 2008

### iLIKEstuff

in using rayleigh scattering theory to calculate the polarizability of particles of arbitrary forms with sizes smaller than the wavelength, my text entitled "light scattering by small particles" by van de hulst states

"$${|\alpha|}^{2}$$ is defined by $${|\alpha|}^{2} = {l}^{2} {|\alpha_1|}^{2} + {m}^{2} {|\alpha_2|}^{2} + {n}^{2} {|\alpha_3|}^{2}$$ and l,m, n are the direction cosines of $$E_0$$ with respect to the three main axes of the polarizability tensor.

THe directions appearing in this problem must be clearly distinguished. The value of $$|\alpha|$$ is determined by the orientation of $$E_0$$ with respect to the particle; the direction of propagation of the incident light is irrelevant. "

my question: what are direction cosines? (if you know a better way of calculating polarizability, please let me know!)

thanks guys.

2. Jul 24, 2008

### HallsofIvy

The unit vector pointing in a given direction can always be written as
$$cos(\theta)\vec{i}+ cos(\phi)\vec{j}+ cos(\chi)\vec{k}$$
where $\theta$, $\phi$, and $\chi$ are the angles a line in that direction would make with the x, y, and z-axes respectively. Those cosines are the "direction" cosines. They are simply the x, y, and z components of a unit vector in a given direction.