Bragg reflection and de Broglie wavelength

Problem Statement

An electron with mass $m$m and electric charge (absolute value of charge) $e$e is accelerated from a resting state and incident on the surface of a crystal with a lattice spacing of $d$d.

The accelerated electron possesses not only particle properties but also wave properties, and its de Broglie wavelength $\lambda$λ is expressed as $\lambda = \frac{h}{p}$λ=ph​ (where $p$p is the magnitude of the electron's momentum), using Planck's constant $h$h. The condition for this electron wave to be reflected by the crystal lattice planes and reinforce each other (Bragg's condition) is expressed as follows, where $\theta$θ is the angle of incidence (angle between the lattice plane and the direction of incidence) and $n$n is the order of reflection ($n=1, 2, 3, ...).

$$2d \sin \theta = n\lambda$$2dsinθ=nλ

Now, an electron beam accelerated at a constant potential difference $V$V is incident on a crystal surface, and the angle $\theta$θ is gradually increased from $0$0. The first peak in reflection intensity (constructive interference) for $n=1$n=1 is observed when $\sin \theta = \frac{1}{8}$sinθ=81​.

Next, while keeping the incident angle fixed at $\sin \theta = \frac{1}{2}$sinθ=21​, the potential difference accelerating the electrons is changed from $V$V to $V'$V′. The first peak in reflection intensity for $n=1$n=1 is observed again. Find the ratio of the potential differences $\frac{V'}{V}$VV′​ at this time. Assume that the kinetic energy of the electrons can be treated non-relativistically.

Constraints

• $m = 9.1 \times 10^{-31} \ \text{kg}$m=9.1×10−31 kg
• $e = 1.6 \times 10^{-19} \ \text{C}$e=1.6×10−19 C
• $h = 6.6 \times 10^{-34} \ \text{J}\cdot\text{s}$h=6.6×10−34 J⋅s
• $d = 3.3 \times 10^{-10} \ \text{m}$d=3.3×10−10 m

Input Format

When the calculated ratio of potential differences $\frac{V}{V}$VV​ is expressed as an irreducible fraction $\frac{p}{q}$qp​, input the value of the product of the numerator and denominator $p \times q$p×q.