Dielectrics: An Atomic View — Question 8

Question 8

Random dipoles have zero mean polarization but a nonzero finite-sample fluctuation

N independent isotropically oriented permanent dipoles of magnitude p occupy volume $\mathcal V$. At zero field, $\langle p_z\rangle=0$ and $\langle p_z^2\rangle=p^2/3$. Let n=N/$\mathcal V$.

1. Find the mean and variance of the total z-directed dipole moment.

2. Find the root-mean-square polarization component and its ratio to the saturation magnitude np.

Question 8 - Solution

Random dipoles have zero mean polarization but a nonzero finite-sample fluctuation

(a) Method and calculation

$$M_z=\sum_{i=1}^Np_{z,i},\quad\langle M_z\rangle=0.$$$$\langle M_z^2\rangle=\sum_i\langle p_{z,i}^2\rangle+\sum_{i\ne j}\langle p_{z,i}\rangle\langle p_{z,j}\rangle=Np^2/3.$$

(b) Interpretation and further calculation

$$P_{z,\mathrm{rms}}=\frac{p\sqrt{N/3}}{\mathcal V}.$$$$\frac{P_{z,\mathrm{rms}}}{np}=\frac1{\sqrt{3N}}.$$

Physical check

Zero ensemble mean does not imply zero moment in every microscopic snapshot; relative fluctuations shrink as inverse square root of sample size.