Vega Fitting

  • Set r as a fit parameter
  • Gaussian:
\begin{equation} f(y;\mu,\sigma) = \frac{1}{\sqrt{2\pi} \sigma^2} e^{\frac{-(y-\mu)^2}{2 \sigma^2}} \end{equation}
  • Lorentz:
\begin{equation} L(x;\gamma, x_{0}) = \frac{A}{\pi} \frac{\gamma/2}{(x-x_{0})^2 + (\gamma/2)^2} \end{equation}
  • Circle:
\begin{equation} f(x; A,r,\mu) = A\frac{2 \sqrt{r^2-(x - \mu)^{2}}}{\pi r^2}. \end{equation}
  • Background:
\begin{equation} f(x) = b_{0}+b_{1}x+b_{2}x^{2} \end{equation}


1.5.png
2.png

2.51.png
2.52.png

3.png
3.5.png

5.png
5.5.png

  • Table of parameters with r as a fit parameter
Parameters 1.5mm 2mm 2.5mm1 2.5mm2 3mm 3.5mm 5mm 5.5mm
$A$ -0.2884 ± 231.8705 -0.016 ± 0.2015 -0.0132 ± 1.3967 -0.0424 ± 0.6423 -0.0092 ± 480.4974 -0.0838 ± 0.2002 -0.0345 ± 2375.9143 -0.0005 ± 30.4944
$\mu$ 411.6233 ± 0.1001 435.6389 ± 0.4716 435.4525 ± 0.5694 487.2171 ± 0.1258 487.6716 ± 0.0762 485.9295 ± 0.0347 629.9951 ± 0.0227 657.3793 ± 0.0095
$\sigma$ 0.0182 ± 1.8745 -0.0466 ± 1.6901 0.0356 ± 5.2263 0.0354 ± 0.5943 0.0063 ± 324.846 0.0541 ± 0.9156 0.0113 ± 125.1331 -0.0002 ± 16.8113
$\gamma$ 2.3445 ± 0.2422 2.2945 ± 0.1918 2.2807 ± 0.2268 2.1437 ± 0.0788 2.2535 ± 0.1335 2.3628 ± 0.0648 0.5368 ± 0.0945 2.2522 ± 0.0698
$b_0$ 3.3135 ± 0.0057 3.2976 ± 0.0099 3.1694 ± 0.0085 4.4768 ± 0.0038 3.5605 ± 0.0044 5.5363 ± 0.0066 5.3204 ± 0.0038 4.432 ± 0.0061
$b_1$ 0.0228 ± 0.0002 0.0179 ± 0.0003 0.0178 ± 0.0003 0.0118 ± 0.0002 0.0083 ± 0.0002 0.0126 ± 0.0002 -0.0201 ± 0.0004 -0.0097 ± 0.0004
$b_2$ -0.0003 ± 0.0001 0.0003 ± 0.0 0.0002 ± 0.0 -0.0005 ± 0.0 -0.0003 ± 0.0001 -0.001 ± 0.0001 0.0021 ± 0.0001 -0.0 ± 0.0001
$r$ 0.814 ± 0.0561 0.2294 ± 0.6368 0.331 ± 0.8211 0.5778 ± 0.1526 0.226 ± 0.0971 0.2972 ± 0.0275 0.3611 ± 0.1473 0.1913 ± 4.7837
$\chi^2$ per dof 1.989 1.958 2.111 1.563 1.877 1.501 1.664 1.732
Hydrogen Lines 410.2 434.1 434.1 486.1 486.1 486.1   656.7

  • Convolution shifts the mean when not directly in the center
  • Conv.png:
    Conv.png

Errors

  • Use $p_i = \frac{n_i}{N}$, where $N$ is the total entries in histogram?
  • Use a multinomial
  • Variance of a multinomial is:
\begin{equation} V[p_i] = np_i(1-p_i)\end{equation}
  • where n is the number of trials
  • Not sure why we are using multinomial?
  • Not sure how the equation below was found?
\begin{equation}V[p_i] = \frac{V[n_i]}{N^2} = \frac{p_i(1-p_i)}{N} \end{equation}
  • This plot shows the rms of $p_i$ plotted agains the avearge, where
\begin{equation} p_{\mathrm{rms}} = \sqrt{\frac{\sum_{i}{(p_i - \hat{p_i})^2}}{n}} \end{equation}
  • pi_rms.png:
    pi_rms.png

Mapping

  • set D as a parameter in the fit
  • changed the degrees into radians
\begin{equation} \lambda = 2\frac{10^6}{kn} \sin{\left[ \tan^{-1}{\left( \frac{m_0 - m}{l} \right)} -\left(\alpha_0 \frac{\pi}{180}\right)+ \frac{1}{2} \left(\left(\rm{D}_{x = 0}\frac{\pi}{180}\right) + \tan^{-1}{\left(\frac{(p-382)(0.009)}{f_2}\right)} \right) \right]} \cos{\left[\frac{1}{2} \left( \left(\rm{D}_{x = 0} \frac{\pi}{180[^{\circ}]}\right)+ \tan^{-1}{\left(\frac{(p-382)(0.009)}{f_2}\right)}\right)\right]} \end{equation}
  • The quadratic equation is given by:
\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}

  • where the parameters are:

Parameter Guess Trig Fitted Trig Guess Polynomial Fitted Polynomial
$l$ 30mm 27.23±0.21 mm    
$\alpha_0$ 10 -174±4 degrees    
$m_0$ 10mm -0.428±0.426 mm    
$f_2$ 70mm -65.03±1.8 mm    
$D_{x = 0}$ 30 -7.57±7.7 degrees    
$\lambda_0$     278 264±3
$a_1$     0.1 0.12±0.01
$b_1$     50 62±1
$a_2$     0 (-1±0.1)e-05
$b_2$     0 (-0.68±0.01)
$c_2$     0 (-1±0.1)e-03
$\chi^2/\mathrm{DOF}$   9.61   9.68

  • Map.png:
    Map.png
-- JosephBayley - 19 Jan 2016
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Topic revision: r8 - 09 Feb 2016 - JosephBayley

 
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