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<p>Created page with &quot;&lt;!--\label{prob:reflectivity_coefficient}--&gt; =====Question 1===== Prove <a href="/wiki/Neutron_reflectivity#label-reflectivityeq:rq4" title="Neutron reflectivity">this equation</a> from the <a href="/wiki/Neutron_reflectivity" title="Neutron reflectivity">Neutron reflectivity</a> pa...&quot;</p> <p><b>New page</b></p><div>&lt;!--\label{prob:reflectivity_coefficient}--&gt;<br /> <br /> <br /> =====Question 1=====<br /> <br /> Prove [[Neutron reflectivity#label-reflectivityeq:rq4|this equation]] from the [[Neutron reflectivity]] page&lt;!--(\ref{reflectivityeq:rq4})--&gt;, \(r=\frac{k^2_{1z}-k^2_{2z}}{\left(k_{1z}+k_{2z}\right)^2}=\frac{q^2_c}{4\left(k_{1z}+k_{2z}\right)^2}\).<br /> <br /> <br /> {{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}<br /> <br /> Starting with [[Neutron reflectivity#label-reflectivityeq:Snell|this equation]] from the [[Neutron reflectivity]] page&lt;!--(\ref{reflectivityeq:rq4})--&gt;, \(n_1 \cos \theta_1 = n_2 \cos \theta_2\),<br /> \begin{equation}<br /> n^2_1\left(1-\sin^2\theta_1\right)=n^2_2\left(1-\sin^2\theta_2\right) \\<br /> \end{equation}<br /> Substitute [[Neutron reflectivity#label-reflectivityeq:|this equation]], \(n_m=\frac{\lambda_0}{\lambda_m}=\frac{k_m}{k_0}\):<br /> \begin{equation}<br /> k^2_{1\perp}-k^2_{2\perp}=k^2_0\left(n^2_1-n^2_2\right)<br /> \end{equation}<br /> Substitute [[Neutron reflectivity#label-reflectivityeq:n2rhob|this equation]], \(n^2=1-\frac{4\pi \rho\overline{b}}{k_0^2}=1-\frac{\lambda_0^2}{\pi}\rho\overline{b}\):<br /> \begin{equation}<br /> k^2_{1\perp}-k^2_{2\perp}=4\pi\left(\rho_2\overline{b}_2-\rho_1\overline{b}_1\right)<br /> \end{equation}<br /> Substitute [[Neutron reflectivity#label-reflectivityeq:Critqc|this equation]], \(q_c=\sqrt{16\pi\left(\rho_2\overline{b}_2-\rho_1\overline{b}_1\right)}\):<br /> \begin{equation}<br /> k^2_{1\perp}-k^2_{2\perp}=\frac{q^2_c}{4}<br /> \end{equation}<br /> Note that:<br /> \begin{equation}<br /> r=\left(\frac{k_{1z}-k_{2z}}{k_{1z}+k_{2z}}\right)\left(\frac{k_{1z}+k_{2z}}{k_{1z}+k_{2z}}\right)=\frac{k^2_{1z}-k^2_{2z}}{\left(k_{1z}+k_{2z}\right)^2}<br /> \end{equation}<br /> {{hidden end}}</div>

ucph>Tommy