From 023bc59c958e8ba28c16702badba4566668b7d24 Mon Sep 17 00:00:00 2001 From: Sam W <sw104@users.noreply.github.com> Date: Sat, 1 Dec 2018 17:47:42 +0000 Subject: [PATCH] Continued writing experiment 4B report. --- year2/4b/4b.pdf | Bin 129042 -> 140125 bytes year2/4b/4b.tex | 48 +++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 47 insertions(+), 1 deletion(-) diff --git a/year2/4b/4b.pdf b/year2/4b/4b.pdf index 343ceb4a56e537f141a5e7a612ddd1b0b146d7a4..b12f13c24487156a63a61c70288ba89f0a921e06 100644 GIT binary patch delta 74081 zcmV(`K-0gH@CV(-2(TF<1T!)<IF}Lk0Vsd%9LaJVxAo4i=-kMP2Eo3#N{2X+9jnS& zBsv$#hY>eT70IDzNG+Fqd)@;YXrKqZ3^}q?m0X%V4dCIuhqnVDZLS*H{CGs$zu%u8 z{pklG8b*nujy0#34cC-VYZ@aZF;q9F=gphrx0H#KUr%2`XVn<u4E23AF+w*BMznuu zJ8nWRrgD&m2nlm^y`G0&vgpMa;n>X)rd;T^#ePRGPLF;&Vt_AgSi_wr%(BL4322Ju z?B?jrue3Rb#+MBx5~ld*yWTWh5$Y_wUN^6g{#B*b8JH7*DGtCAt%98_7&|@-SSnLs zNr|xvfcn1wX_-<$RYY?Zd%UGw*QkF0o)~1Z0hHw^6&#kqlvI@L6;u;SMPr2ZGY@07 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c \hbar \mu}{\tilde{B_{\nu}}}} \alpha_B \end{equation} +\subsection{Determination of Vibrational Constants} + The values for the harmonic constant $\tilde{\nu_e}$ and the dimensionless anharmonicity constant $x_e$ in equation \ref{eq:vib-energy} (which gives the expected vibrational energy levels if the rotational spectrum is -ignored) can be determined by +ignored). \begin{equation}\label{eq:vib-energy} \tilde{E_{\nu}} = \tilde{\nu_e} \left( \nu + \frac{1}{2} \right) - \tilde{\nu_e} x_e \left( \nu + \frac{1}{2} \right)^2 \end{equation} +From equation \ref{eq:vib-energy} we can obtain the energy related to the pure vibrational transition +$\tilde{E}(\nu_f \leftarrow 0)$ (the $\nu = 0$ to $\nu = \nu_f$ transition) as equation \ref{eq:delta-e-coeff}. + +\begin{align} + \tilde{E}\left( \nu_f \leftarrow 0 \right) &= \tilde{E}_{\nu_f} - \tilde{E}_0 \nonumber \\ + &= \tilde{\nu_e}\left( \nu_f + \frac{1}{2} - \frac{1}{2} \right) - \tilde{\nu_e} x_e \left( + \frac{1}{4} - \left( \nu_f + \frac{1}{2} \right)^2 \right) \nonumber \\ + \label{eq:delta-e-coeff} + &= \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right) +\end{align} + +The energy related to the R branch transitions can be determined to yield +equation \ref{eq:r-trans-energy} where $J$ is the rotational state adopted in +the lower vibrational state ($\nu = 0$). + +\begin{equation}\label{eq:r-trans-energy} + R(J) = \Delta \tilde{E}(\nu_f \leftarrow 0) + (\tilde{B_{\nu_f}} + \tilde{B_0}) (J + 1) + (\tilde{B_{\nu_f}} - + \tilde{B_0}) ( J + 1)^2 +\end{equation} + +Setting $J = 0$ hence gives equation \ref{eq:r-trans-energy-j0}. + +\begin{equation}\label{eq:r-trans-energy-j0} + R(J) - 2 \tilde{B_{\nu_f}} = \Delta \tilde{E}(\nu_f \leftarrow 0) +\end{equation} + +Subsituting equation \ref{eq:delta-e-coeff} into equation \ref{eq:r-trans-energy-j0} yields equation +\ref{eq:r-trans-energy-sub}. + +\begin{equation}\label{eq:r-trans-energy-sub} + R(0) - 2 \tilde{B_{\nu_f}} = \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right) +\end{equation} + +Now from equation \ref{eq:r-trans-energy-sub} a system of linear equations can be +obtained by setting $\nu_f = 1$ and $\nu_f = 2$ (equations xx and xx) for which we can solve to give equations xx and xx. + + + + + + +for which setting $J = 0$ and $\nu_f = 1$ and subsequently $\nu_f = 2$ gives the +system of linear equations (equations xx and xx) for which we can solve to give equations xx and xx. + %TODO: Total energy equation, determination of \nu_e x_e, calculation of values, determination of k with error %propagation and table. -- 2.25.1