If electron is to be diffracted significantly by a crystal, its wavelength must be about equal to the spacing, \(d,\) of crystalline planes. Assuming \(d=0.250 \mathrm{nm}\) estimate the potential difference through which an electron must be accelerated from rest if it is to be diffracted by these planes.

To find the wavelength of the electron that will result in significant diffraction, we will use the de Broglie wavelength formula: \[\lambda = \frac{h}{p}\] where \(\lambda\) is the wavelength, \(h\) is the Planck's constant, and \(p\) is the momentum of the electron. Since we know that the wavelength must be equal to the spacing of the crystalline planes (\(d = 0.250 \text{ nm}\)), we can write: \[0.250 \mathrm{nm} = \frac{h}{p}\]

To find the momentum of the electron, we will use its relation to the wavelength: \[p = \frac{h}{\lambda}\] where \(\lambda = 0.250 \mathrm{nm}\). Substitute the values and convert the wavelength to meters: \[p = \frac{6.626 \times 10^{-34} \mathrm{J \cdot s}}{0.250 \times 10^{-9} \mathrm{m}}\] Now, calculate the momentum of the electron: \[p \approx 2.65 \times 10^{-24} \mathrm{kg \cdot m/s}\]

Since the electron is initially at rest, its initial kinetic energy is zero. We can use the momentum to find the final kinetic energy of the electron: \[K_f = \frac{p^2}{2m}\] where \(m\) is the mass of the electron. Substitute the values and calculate the final kinetic energy: \[K_f = \frac{(2.65 \times 10^{-24} \mathrm{kg \cdot m/s})^2}{2(9.11 \times 10^{-31} \mathrm{kg})}\] \[K_f \approx 9.62 \times 10^{-19} \mathrm{J}\]

Now that we have the kinetic energy, we can use the energy conservation principle to find the potential difference. The electron gains potential energy as it travels through the potential difference, and this potential energy is converted into kinetic energy. Thus, we have: \[eV = K_f\] where \(e\) is the electron charge and \(V\) is the potential difference. Solve for \(V\): \[V = \frac{K_f}{e}\] Substitute the values: \[V = \frac{9.62 \times 10^{-19} \mathrm{J}}{1.602 \times 10^{-19} \mathrm{C}}\] Now, calculate the potential difference: \[V \approx 60.06 \text{ V}\] So, the electron must be accelerated from rest through a potential difference of approximately \(60.06 \text{ V}\) to be diffracted by the crystalline planes.