Difference: 20151129_MSciSpectrocopyLab (6 vs. 7)

Revision 701 Dec 2015 - JosephBayley

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META TOPICPARENT name="StewartBoogertSpectroscopy2015"

Fixing errors

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  • Finally a quadratic term was fitted for the background of the image, which has the equation:
\begin{equation} f(x) = b_{0} + b_{1} x + b_{2} x^2 \end{equation}
  • These are plotted below
  • The quadratic can be seen to improve the fit to the background of the data as shown below:
  • The three different fits produced parameter values of:
Parameters Gaussian + circle Skewed Gaussian + Circle Skewed Gaussian + circle + Quadratic background
Line: 76 to 76
 \begin{equation} L(x;\gamma, x_{0}) = \frac{A}{\pi} \frac{\gamma/2}{(x-x_{0})^2 + (\gamma/2)^2} \end{equation}
  • Both of these fits are shown with the data in the plot below:
Paramater Skewed Gaussian + Circle + Quadratic Background Voigt + Circle + Quadratic Background
Gauss Amp 1.077 -1.48(1.39e05)
Line: 98 to 98
$N_{\mathrm{dof}}$ 288 285
$\chi^2$ per dof 4.75 3.91

Mapping Wavelength

  • The three different equations that were fitted were:
 \begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m \end{equation} \begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m + a_{2} \mu^{2} + b_{2} m^{2} + c_{2} \mu m \end{equation} \begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m + a_{2} \mu^{2} + b_{2} m^{2} + c_{2} \mu m + a_{3}\mu^{3} + b_{3} m^{3} + c_{3} \mu^{2} m + d_{3} \mu m^{2}\end{equation}
  • These were then plotted for each micrometer setting with the data.
  • Ask about what errors to put in??
  • The parameters were found to be:

Parameter Linear Squared Cubed
$\lambda_{0}$ 2782 2652.8 2757
$a_{1}$ 0.1050.002 0.1220.008 0.0740.028
$b_{1}$ 55.640.26 62.071.08 56.754.45
$a_{2}$   (-1.190.89)e-05 (6.736.16)e-05
$b_{2}$   -0.6890.104 0.080.94
$c_{2}$   (-1.610.98)e-03 0.0110.007
$a_{3}$     (-4.624.81)e-08
$b_{3}$     (-3.356.44)e-02
$c_{3}$     (-7.596.11)e-06
$d_{3}$     (-7.635.07)e-04
$\chi^2$ 109524 63944 60107
$N_{dof}$ 69 66 62
$\chi^2$ per dof 1587 968 969

  • Some of the grating setting are shown below

Halogen Lamp

  • For the halogen lamp the above fit parameters were used to set the wavelength of each frame, these were then plotted as shown below:
  • The below plot shows the linear and squared plots and it can clearly be seen to improve at larger and smaller wavelengths, however not much change in the center.
  • When the cubed parameters were used, the plot did no seem to improve, as the image shows
  • This may be due to an incorrect point which the cubed plot fits better to however is further from the true value
 -- AshleaKemp - 27 Nov 2015

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