Quivering Mirror

a) A very small mirror is suspended from a quartz strand whose elastic constant is D. (Hooke’s law for the torsional twist of the strand is τ = -Dθ, where θ is the angle of the twist.) In a real-life experiment the mirror reflects a beam of light in such a way that the angular fluctuations caused by the impact of surrounding molecules (Brownian motion) can be read on a suitable scale. The position of the equilibrium is ⟨θ⟩ = 0. One observes the average value ⟨θ²⟩, and the goal is to find Avogadro’s number (or, what is the same thing, determine the Boltzmann constant). The following are the data: At T = 278 K, for a strand with D = 9.43 × 10^­-9 dyn.cm, it was found that ⟨θ²⟩ = 4.20 × 10^-6. You may also use the universal gas constant R = 8.31 × 10⁷ erg/K mole. Calculate Avogadro’s number.

b) Can the amplitude of these fluctuations be reduced by reducing gas density? Explain your answer.

SOLUTION

a) When the mirror is in thermal equilibrium with gas in the chamber, one may again invoke the equipartition theorem and state that there is (1/2)τ of energy in the rotational degree of freedom of the torsional pendulum, where the torque is given by τ = -Dθ. The mean square fluctuation in the angle would then be given by

 (1)

So,

(2)

Now, Avogadro’s number N_A = R/k_B, and we obtain

(3)

b) Even if the gas density were reduced in the chamber, the mean square fluctuation ⟨θ²⟩ would not change. However, in order to determine whether individual fluctuations might have larger amplitudes, we cannot rely on the equipartition theorem. We instead will examine the fluctuations in the frequency domain. ⟨θ²⟩ may be written

(4)

where ω(f ) is the power spectral density of θ. At high gas density, ω(f ) is broader and smaller in amplitude, while the integral remains constant. This corresponds to more frequent collisions and smaller amplitudes, whereas, at low density, ω(f ) is more peaked around the natural frequency of the torsional pendulum  where I is its moment of inertia, still keeping the integral constant. It then appears that by reducing the density of the gas we actually increase the amplitude of fluctuations!