AN ELECTRON MICROSCOPE

In an electron microscope, the nonrelativistic electron beam is formed by a setup similar to the electron gun used in the Davisson–Germer experiment (see Fig. 39.2). The electrons have negligible kinetic energy before they are accelerated. What accelerating voltage is needed to produce electrons with wavelength 10 pm = 0.010 nm (roughly 50,000 times smaller than typical visible-light wavelengths)?

Question

AN ELECTRON MICROSCOPE

In an electron microscope, the nonrelativistic electron beam is formed by a setup similar to the electron gun used in the Davisson–Germer experiment (see Fig. 39.2). The electrons have negligible kinetic energy before they are accelerated. What accelerating voltage is needed to produce electrons with wavelength 10 pm = 0.010 nm (roughly 50,000 times smaller than typical visible-light wavelengths)?

Accepted Answer

IDENTIFY, SET UP, and EXECUTE:

We can use the same concepts we used to understand the Davisson–Germer experiment. The accelerating voltage is the quantity [latex]V_{ba}[/latex] in Eq. (39.3).

[latex]\lambda=\frac{h}{p}=\frac{h}{\sqrt{2 m e V_{b a}}}[/latex] (39.3)

Rewrite this equation to solve for [latex]V_{ba}[/latex]:

[latex]\begin{aligned}V_{b a} &=\frac{h^{2}}{2 m e \lambda^{2}} \&=\frac{\left(6.626 imes 10^{-34}\mathrm{~J} \cdot \mathrm{s} ight)^{2}}{2\left(9.109 imes 10^{-31} \mathrm{~kg} ight)\left(1.602 imes 10^{-19} \mathrm{C} ight)\left(10 imes 10^{-12} \mathrm{~m} ight)^{2}} \&=1.5 imes 10^{4} \mathrm{~V}=15,000 \mathrm{~V}\end{aligned}[/latex]

EVALUATE: It is easy to attain 15-kV accelerating voltages from 120-V or 240-V line voltage by using a step-up transformer (Section 31.6) and a rectifier (Section 31.1). The accelerated electrons have kinetic energy 15 keV; since the electron rest energy is 0.511 MeV = 511 keV, these electrons are indeed nonrelativistic.

Question 39.3: AN ELECTRON MICROSCOPE In an electron microscope, the nonrel...

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