$\sigma_0$ Error
  • $\chi^2$ equation used
\begin{equation} \chi^2=\sum_{i=1}^{N}\frac{y_i-f(x_i;\vec{\theta})}{\sigma_i^2+\sigma_0^2} \end{equation}
  • $\sigma_0$ chosen to be a constant at 5E3.
  • Returned plot of gaussian fit with error bars:

Calib_Cd_0300.png
Calib_He_0400.png
  • Cadmim at micrometer setting of 03.00mm had $\chi^2$ value of 8541.6015 abd Helium at micrometer setting of 4.00 had a $\chi^2$ value of 88793.4233.
Convolutions

  • Gaussian equation:
\begin{equation} G(x; a, x_0, \sigma, C_0, C_1) = \frac{1}{\sqrt{2\pi\sigma^2}} \rm{exp} \bigg[ {\frac{-(x - x_0)^2}{2\sigma^2}} \bigg] + C_1x + C_0 \end{equation}

  • Circle equation:
\begin{equation} C(x;r) = \frac{2}{\pi r^2}\sqrt{(r^2 - x^2)} \end{equation}

  • Convolution:
\begin{equation} (G * C)(x) = \sum_{i} G(x - i;a, x_0, \sigma, C_0, C_1)C(i;r) \end{equation}

  • G*C and Gaussian fits to the He line at 4.00mm:
    Conv_Gauss_He0400.png

  • Fitted parameter values:
Parameter Gaussian G*C
a 2.41E08 $\pm$ 3.48E06 1.10E07 $\pm$ 2.35E04
$x_0$ 2.68E02 $\pm$ 2.39E-02 2.68E02 $\pm$ 5.85E-03
$\sigma$ 8.27 $\pm$ 1.39E-01 2.08 $\pm$ 6.45E-02
r   4.51 $\pm$ 3.80E-03
$C_1$ 6.49E01 $\pm$ 4.04E02 8.19E01 $\pm$ 1.03E02
$C_0$ 1.15E05 $\pm$ 1.09E05 1.31E05 $\pm$ 2.78E04
$\chi^2$ 1.35E06 4.53E05
$N_{\rm{DOF}}$ 115 114
$\chi^2 /$ $N_{\rm{DOF}}$ 1.18E04 3.98E03

RMS Error

  • Fit-data difference as a function of intensity for G*C fits to lamp lines:
    Inensity_diff.png

  • RMS equation used:
\begin{equation} x_{\rm{rms}} = \sqrt{ \bigg( \frac{1}{N} \sum_{i = i}^{N} x_i^2 \bigg) } \end{equation}

  • RMS values of data in 20 bins of width 1003796 PE:
    RMS_diff.png

  • Form of the fit:
\begin{equation} f(x; C_1, C_0) = C_1x + C_0 \end{equation}

  • Fitted parameter values:
Parameter Value
$C_1$ 6.27E-03 $\pm$ 2.14E-03
$C_0$ 1.72E05 $\pm$ 2.48E04

-- WillBurrows - 07 Mar 2017

Attachments Attachments
Topic attachments
I Attachment History Action Size Date Who Comment
PNGpng Calib_Cd_0300.png r1 manage 112.1 K 08 Mar 2017 - 07:13 JamesAngthopo Cadmium data at micrometer setting of 3mm
PNGpng Calib_He_0400.png r1 manage 86.3 K 08 Mar 2017 - 07:13 JamesAngthopo Helium data mircometer setting of 4.00mm
PNGpng Conv_Gauss_He0400.png r1 manage 50.1 K 07 Mar 2017 - 23:59 WilliamBurrows G*C and Gaussian fits to the He line at 4.00mm
PNGpng Inensity_diff.png r1 manage 39.6 K 07 Mar 2017 - 23:53 WilliamBurrows Fit-data difference as a function of intensity for G*C fits to lamp lines
PDFpdf Part_spec.pdf r1 manage 279.4 K 08 Mar 2017 - 07:49 JamesAngthopo sigam0 for 60000 instead of 5000
PDFpdf Postage_y_f.pdf r2 r1 manage 52.8 K 08 Mar 2017 - 10:30 WilliamBurrows 'Postage stamp' plot of all y-fit differences
PNGpng RMS_diff.png r1 manage 44.8 K 07 Mar 2017 - 23:56 WilliamBurrows RMS values of data in 20 bins of width 1003796 PE
Texttxt allparams.txt r1 manage 5.1 K 08 Mar 2017 - 10:46 JamesAngthopo Parameters for all spectral lines used to calibrate
PDFpdf chivals.pdf r1 manage 30.0 K 08 Mar 2017 - 07:46 JamesAngthopo shows values for all cadmium and helium data
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Topic revision: r5 - 08 Mar 2017 - JamesAngthopo

 
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