Bragg reflection and electron kinetic energy

Problem Statement

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

The accelerated electron exhibits wave properties, and its de Broglie wavelength $\lambda$λ is expressed as $\lambda = \frac{h}{p}$λ=ph​, where $h$h is Planck's constant and $p$p is the magnitude of the electron's momentum. The condition under which this electron wave is reflected by the crystal lattice planes and reinforces 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 by a 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 $\sin \theta = \frac{1}{9}$sinθ=91​.

Next, while keeping the incident angle fixed at $\sin \theta = \frac{1}{3}$sinθ=31​, 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 dimensionless ratio $\frac{V'}{V}$VV′​ of the potential difference after the change to the potential difference $V$V before the change. 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 plus $1000$1000.