91Ó°ÊÓ

So, we'll use the relationship shown in Example $\mathbf{\text{4.3}}$directly, linking the particle's wavelength $\mathbf{\text{(}}\mathit{λ}\mathbf{\text{)}}$to the potential for rapid growth (V). can use Bragg's law to find the equivalent values of once have the wavelength, such that,

$\mathit{λ}\mathbf{\text{=}}\sqrt{\frac{{\mathbf{\text{h}}}^{\mathbf{\text{2}}}}{\mathbf{\text{2mqV}}}}$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦..(1)

Substitute the given data in equation (1), and we get,

$\begin{array}{c}\mathrm{λ}=\sqrt{\text{1.5}×{\text{10}}^{\text{-21}}{\text{m}}^{\text{2}}}\\ =\text{3.89}×{\text{10}}^{\text{-11}}\text{m}\end{array}$

$\begin{array}{c}\text{2»å²õ¾±²Ô(θ)}=\text{n}\mathrm{λ}\\ \text{²õ¾±²Ô(θ)}=\left(\frac{\mathrm{λ}}{\text{2d}}\right)\text{n}\\ \text{²õ¾±²Ô(θ)}=\left(\frac{\text{3.89}×{\text{10}}^{\text{-11}}\text{m}}{\text{2}×\text{0.1}×{\text{10}}^{\text{-9}}\text{m}}\right)×\text{n}\\ \text{θ}={\text{sin}}^{\text{-1}}\left(\text{0.194n}\right)\end{array}$

$\begin{array}{l}{\text{θ}}_{\text{1}}=\text{11.2}°\\ {\text{θ}}_{\text{2}}=\text{22.83}°\\ {\text{θ}}_{\text{3}}=\text{35.59}°\\ {\text{θ}}_{\text{4}}=\text{50.89}°\\ {\text{θ}}_{\text{5}}=\text{75.93}°\end{array}$

Keep in mind that the value of$\text{n=6}$ isn't allowed because arcsine's parameter exceeds$\text{1}$ , but used the integer n instead of m for the integer counter so it doesn't get mixed up with the unit of measurement ($\text{m}$ or meter).

(b)

To achieve the same maxima, both the electron and the photons must have the same wavelength value.

${\text{E}}_{\text{p}}\text{=}\frac{\text{hc}}{\mathrm{λ}}$

$\begin{array}{c}{\text{E}}_{\text{p}}=\left(\frac{4.14×{10}^{−23}\text{ }eVâ‹\dots s×\text{3}×{\text{10}}^{\text{8}}\text{″¾}/s}{\text{3.89}×{\text{10}}^{\text{-11}}\text{m}}\right)\text{ }\\ =\text{31.88keV}\end{array}$

Therefore, the energy of the photon is${\text{E}}_{\text{p}}=\text{31.88keV}$ .