Error bars
  • A gaussian function fitted over the test lamp's data.
  • Using the amended $\chi^2$ equation, error on the lamp's data found. This is due to Poisson error being an underestimate.
  • Some examples of the $\sigma_0$ values, and their corresponding error on the peak location.

Lamp Micrometer setting (mm) $\sigma_0$ Error in peak location $\chi^2$ value $N_{DOF}$ Error on Wavelength
Cadmium 02.50 2.62E5 0.0961 55.17 55 0.1054
Cadmium 05.00 5.48E5 0.0840 55.12 55 0.0984
Cadmium 06.50 4.30E5 0.0765 55.10 55 0.0944
Helium 02.00 6.22E5 0.0878 55.31 55 0.1068
Helium 04.00 5.04E5 0.0791 55.08 55 0.1014
  • To find the $\sigma_0$, its value was changed for every spectral line for different micormeter setting. This didn't make a lot of physical sense as $\sigma_0$ did not depend on exposure time but it only depended on the intensity of individual peaks.
  • plot of a spectral line for Helium lamp.:
    Heliumdata.png

  • Another method used to find $\sigma_0$.
  • Fitted Gaussian for different spectral lines and found which spectral line had the least $\chi^2$ value with just the poisson error. Lets call this spectral line $Spec_{min}$.
  • Then $\sigma_0$ value was added to $\chi^2$ equation and changed until $\chi^2=N_{DOF}$ for $Spec_{min}$.
  • Then this $\sigma_0$ value was used to determine the error on every peak location for that exposure time.
  • This process was repeated for different exposure times.
  • This shows the spectral line which had the best fit with a Gaussian curve for 0.1 s exposure time.:
    Bestfitcadmium.png

  • some examples of $\sigma_0$ values and their corresponding error on peak location and $\chi^2$ value.
lamp micrometer setting (mm) Exposure time (s) xrange $\chi^2$ Error on peak location $\sigma_0$ Error on wavelength
Cadmium 03.00 0.1 61 55.95 0.0175 6.40E04 0.1043
Cadmium 02.50 0.1 61 384.301 0.0203 6.40E4 0.10568
Cadmium 05.00 0.1 61 1061.95 0.0965 6.40E4 0.09866
Cadmium 06.50 0.1 61 274.833 0.0119 6.40E4 0.09446
Helium 02.00 1.0 61 1329.78 0.01011 7.28E4 0.107122
Helium 05.50 1.0 61 2.29957 0.12450 7.28E4 0.097027
High Mercury 04.00 1.0 81 76.62 0.0656 7.28E4 0.09995
High Mercury 07.00 10.0 91 2.94541 0.42922 3.22E5 0.09749

  • Last method to find different $\sigma_0$.
  • Know the background of vega to be third order polynomial.
  • Used this to find $\sigma_0$, when $\chi^2=N_{DOF}$, for different intensity.
  • Made plot of $\sigma_0$ against intensity.
  • plot of intensity against $\sigma_0$:
    intvsig.png
  • Used the fit to find different $\sigma_0$ values for all the spectral lines which had different intensity.
  • Plot of cadmium at 03.00mm showing the error bar derived from intens vs $\sigma_0$ plot:
    Caderror.png
  • some examples of $\sigma_0$ values and their corresponding error on peak location and $\chi^2$ value.
lamp micrometer setting (mm) Exposure time (s) xrange $\chi^2$ Error on peak location $\sigma_0$ Intensity Error on wavelength
Cadmium 03.00 0.1 61 21.7893 0.0281 1.02E5 7.203E6 0.104312632822
Cadmium 02.50 0.1 61 381.8706 0.0231 6.42E4 5.76E6 0.1057
Cadmium 05.00 0.1 61 32.5048 0.0695 4.57E5 1.47E7 0.0985
Cadmium 06.50 0.1 61 27.2785 0.0355 2.03E5 9.97E6 0.0944
Helium 02.00 1.0 61 37.6274 0.0601 4.33E5 1.43E7 0.1069
Helium 05.50 1.0 61 474.5811 0.0086 5.01E3 1.06E6 0.0973
High Mercury 04.00 1.0 81 7016.7390 0.0110 7.00E3 1.682E6 0.1015
High Mercury 07.00 10.0 91 4149.4471 0.0133 9.90E3 2.18E6 0.0930

G*C convolution

  • G*C fit to middle Cd peak, 6mm
  • Fixed: amp = 1E12, mu = 300, $\sigma$ = 1
  • Varying starting radius:

Parameter r = 5 10 20 47 50 1000
$\sigma$ 3.68 3.01 2.98 2.11 2.81 2.98
r 1.50 1.50 1.49 3.95 2.50 1.50
amp 3.60E11 2.68E11 2.62E11 1.46E11 2.20E11 2.64E11
mu 2.92E2 2.92E2 2.92E2 2.92E2 2.92E2 2.92E2
bg 2.77E5 3.45E5 3.50E5 3.57E5 3.51E5 3.50E5
$\chi^2$ 1.45E7 2.22E6 2.08E6 1.43E6 2.02E6 2.07E6
ndf 195 195 195 195 195 195
$\chi^2$ / ndf 7.43E4 1.14E4 1.06E4 7.32E3 1.03E4 1.06E4

  • Fixed: amp = 1E12, mu = 300, r = 1
  • Varying starting $\sigma$:

Parameter $\sigma$ = 2 5 10 40 100
$\sigma$ 2.98 2.11 2.98 2.82 2.98
r 1.44 3.95 1.09 2.50 1.40
amp 2.56E11 1.46E11 2.04E11 2.20E11 2.50E11
mu 2.92E2 2.92E2 2.92E2 2.92E2 2.92E2
bg 3.50E5 3.57E5 3.50E5 3.51E5 3.50E5
$\chi^2$ 2.07E6 1.43E6 2.07E6 2.02E6 2.07E6
ndf 195 195 195 195 195
$\chi^2$ / ndf 1.06E4 7.32E3 1.06E4 1.03E4 1.06E4

  • Fixed: amp = 1E12, mu = 300, $\sigma$ = 5, r = 5
  • Varying starting fit range:

Parameter 50 100 150 200
$\sigma$ 2.98 2.82 -2.10 2.82
r 1.48 2.50 3.95 2.50
amp 6.55E11 1.10E11 1.09E11 2.21E11
mu 2.92E2 2.92E2 2.92E2 2.92E2
bg 3.49E5 3.58E5 3.61E5 3.50E5
$\chi^2$ 2.01E6 1.92E6 1.37E6 2.05E6
ndf 45 95 145 195
$\chi^2$ / ndf 4.48E4 2.02E4 9.47E3 1.05E4

Normalised line shape

Layered normalised profiles for all lines in all lamps Gaussian fit to the mean values of the normalised layered lines from all .fit files

LS_all.png

LS_all_mean_gauss.png
Layered normalised profiles for 'best' lines in all lamps Gaussian fit to the mean values of the normalised layered lines from 'best' .fit files

LS_best.png

LS_best_mean_gauss.png

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

  • Gaussian fit values:
Parameter All lamps Best lamps
amp 6.01 $\pm$ 6.58E-2 7.93 $\pm$ 3.83E-2
$x_0$ 6.85E-1 $\pm$ 4.01E-2 6.30E-1 $\pm$ 1.61E-2
$\sigma$ 3.54 $\pm$ 4.16E-2 3.16 $\pm$ 1.67E-2
b 3.09E1 $\pm$ 1.52E-3 2.47E-2 $\pm$ 9.34E-4
Normalised $\chi^2$ 5.61E-2 6.60E-2
$N_{\rm{DOF}}$ 96 96
Normalised $\chi^2$ per N 5.85E-4 6.87E-4

  • $\chi^2$ used:
 \begin {equation} \chi^2 = \sum_{i = 1}^{N} \frac{(y_i - f(x_i; \bar{\theta}))^2}{\sigma_i^2} \end{equation}
  • No $\sigma_0$ used
-- WillBurrows - 22 Feb 2017
Topic attachments
I Attachment Action Size Date Who Comment
PNGpng Bestfitcadmium.png manage 115.1 K 25 Feb 2017 - 12:45 JamesAngthopo This shows the spectral line which had the best fit with a Gaussian curve for 0.1 s exposure time.
PNGpng Caderror.png manage 108.5 K 28 Feb 2017 - 22:44 JamesAngthopo Plot of cadmium at 03.00mm showing the error bar derived from intens vs simga0 plot
PNGpng Heliumdata.png manage 78.0 K 23 Feb 2017 - 15:54 JamesAngthopo plot of a spectral line for Helium lamp.
PNGpng LS_all.png manage 77.2 K 28 Feb 2017 - 21:17 WillBurrows Layered normalised profiles for all lines in all lamps
PNGpng LS_all_mean_gauss.png manage 38.3 K 28 Feb 2017 - 21:20 WillBurrows Gaussian fit to the mean values of the normalised layered lines from all .fit files
PNGpng LS_best.png manage 36.7 K 28 Feb 2017 - 21:19 WillBurrows Layered normalised profiles for 'best' lines in all lamps
PNGpng LS_best_mean_gauss.png manage 35.5 K 28 Feb 2017 - 21:21 WillBurrows Gaussian fit to the mean values of the normalised layered lines from 'best' .fit files
PNGpng bgfitsigplot.png manage 221.5 K 01 Mar 2017 - 00:34 JamesAngthopo This plot shows the error obtained for when sigma0 was found by using vega background
PNGpng expsigplot.png manage 216.6 K 01 Mar 2017 - 00:33 JamesAngthopo This plot shows the error obtained for when sigma0 was found for different exposure time
PNGpng indsigplot.png manage 177.6 K 01 Mar 2017 - 00:33 JamesAngthopo This plot shows the error obtained for when sigma0 was found for individual spectral lines
PNGpng intvsig.png manage 71.0 K 01 Mar 2017 - 10:43 JamesAngthopo plot of intensity against sigma_0

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Topic revision: r9 - 01 Mar 2017 - JamesAngthopo

 
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