Error propagation
  • Assumed the photoelectrons distribution is a p.d.f therefore assumed error on the number of photoelectrons to be
 \begin{equation} \sigma=\sqrt{N_{PE}} \end{equation}
  • Plot showing the data and the gaussian fit for micrometer setting of 4.00mm and 6.00mm:
    plotgaussfit.png
  • Plot showing the error for Helium lamp at 6.0mm:
    enchancedgaussplot.png

  • The table shows the $\chi ^2$ value, $N_{DOF}$ and the error on the location of the peak.
micrometer setting (mm) $\chi ^2$ $ N_{DOF} $ width on guassian
$4.00 \pm 1e-4$ 1310296.532 764 2.44
$6.00 \pm 1e-4$ 847762.900 764 2.79
  • Equation used for $\chi ^2$:
 \begin{equation}  \chi ^2=\sum_{n=1}^{N}\frac{(y_i-f(y_i;a,b,c))^2}{\sigma_i^2}  \end{equation}
  • $y_i$ is the observed value, $f(y_i,a,b,c)$ is the expected values, $\sigma$ is the error.
  • Since the poisson distribution is assumed $\sigma ^2 = N_{PE} = \mu$ where $\mu$ is the gaussian mean.
  • Therefore if it is a poisson distribution, the equation below is true.
 \begin{equation} \sigma=\sqrt{\mu} \end{equation}

  • Compared $\sigma$ and $ \sqrt{\mu}$ of Helium lamp for 3 micrometer setting:
Micrometer setting (mm) $\sigma$ $ \sqrt{\mu}$
02.00 500.314 703.506
06.00 398.801 430.771
08.00 502.246 658.612

  • Conclusion - not poisson distribution.
Calibration

  • The equation for the propagation of the error is :
 \begin{equation}  \sigma ^2=\sum_{i,j=1}^N\left[\frac{dy}{dx_i}\frac{dy}{dx_j} \right] V_{ij} \end{equation}

  • Error on the lambda is given by
 \begin{equation} \Delta \lambda(p,m)^2=(-D_1-E_1(p-p_0)+2D_2(m-m_0))^2(\Delta m)^2+(-C_1-E_1(m-m_0)+2C_2(p-p_0))^2(\Delta p)^2 \end{equation}

  • On the left shows the fit and plots of data point with cadmium data, on the right it shows the same thing but the cadmium data has been excluded:

quadonly.png

quaddata0600.png

quad0700.png

quaddata0700.png
  • The values of the parameters:
Parameters Values(Without Cd data) Values (with Cd data)
$C_1$ 0.102 9.98e-02
$C_2$ -4.95e-6 1.05e-05
$D_1$ 52.8 53.3
$D_2$ -0.869 -8.69e-01
$E_1$ -2.80e-3 -2.65e-03
$p_0 $ 354 528
$m_0 $ 4.5 $\pm$ 1e-4 4.13 $\pm$ 1e-4
$\chi ^2$ at 6.00mm 0.000411 0.3102
$N_{DOF}$ at 6.00mm 2 4
$\chi ^2$ at 7.00mm 0.00321 0.0372
$N_{DOF}$ at 7.00mm 5 5

Fitting

  • Lorentzian + background equation used:
 \begin{equation}  L(x;a, \mu, \gamma, \bar{c}) = \frac{a}{\pi \gamma} \bigg[ \frac{\gamma ^2}{(x - \mu)^2 + \gamma ^2} \bigg] + \sum_{i = 0}^{N} c_i x^i  \end{equation}

  • Lorentzian + (3rd order polynomial) background fits to the peaks in Vega, centered at 2.90mm and 6.00mm:

Vega_0290_LB.png

Vega_0290_LB_zoomed.png

Vega_0600_LB.png

Vega_0600_LB_zoomed.png

  • Chi squared values for the Lorentzian + background fits to the centered Vega peaks, using both a second and third order polynomial:

Micrometer Position (mm) Chi Squared 2nd Order Chi Squared 3rd Order $\gamma$ (pixels)
0.80 1097009 1082766 37.0
1.60 1075022 1037859 125.1
2.30 2874083 2877388 32.1
2.90 1136163 926898 11.6
6.00 1152200 275787 10.0
7.00 1786081 1562246 11.3
  • Needs errors to properly test how good of a fit
Convoluting

  • Using a discrete convolution:
 \begin{equation}  (C * L)(x) = \sum_{t}^{N} C(t; r)L(x - t; a, \mu, \gamma, \bar{c})  \end{equation}

  • Where the functions L and C are the Lorentzian + polynomial and circle functions respectively.
  • Circle function used:
 \begin{equation}  C(x; r) = \frac{1}{\pi r^2} \sqrt{(r^2 - x^2)}  \end{equation}

  • Current convolution fit to the Vega peak centered at 6.00mm:
    Convolution_Vega0600_01.png
-- JamesAngthopo - 06 Feb 2017:
Topic attachments
I Attachment Action Size Date Who Comment
PNGpng Convolution_Vega0600_01.png manage 55.9 K 08 Feb 2017 - 10:15 WillBurrows Current convolution fit to the Vega peak centered at 6.00mm
PNGpng Vega_0290_LB.png manage 66.2 K 07 Feb 2017 - 21:55 WillBurrows Lorentzian + background fit to the peak in Vega, centered at 2.90mm
PNGpng Vega_0290_LB_zoomed.png manage 63.1 K 07 Feb 2017 - 21:56 WillBurrows Lorentzian + background fit to the peak in Vega, centered at 2.90mm
PNGpng Vega_0600_LB.png manage 67.4 K 07 Feb 2017 - 19:39 WillBurrows Lorentzian + background fit to the peak in Vega, centered at 6.00mm
PNGpng Vega_0600_LB_zoomed.png manage 62.7 K 07 Feb 2017 - 19:39 WillBurrows Lorentzian + background fit to the peak in Vega, centered at 6.00mm
PNGpng correctedplot.png manage 82.6 K 07 Feb 2017 - 23:44 JamesAngthopo This shows how the data point varies when cadmium data is excluded
PNGpng enchancedgaussplot.png manage 81.0 K 07 Feb 2017 - 23:02 JamesAngthopo Plot showing the error for Helium lamp at 6.0mm
PNGpng plotgaussfit.png manage 66.0 K 07 Feb 2017 - 22:39 JamesAngthopo Plot showing the data and the gaussian fit for micrometer setting of 4.00mm and 6.00mm
PNGpng quad0700.png manage 63.9 K 08 Feb 2017 - 10:53 JamesAngthopo plot for 7.00mm with cadmium data
PNGpng quadat0700.png manage 64.0 K 08 Feb 2017 - 00:24 JamesAngthopo Cadmium data included
PNGpng quaddata0600.png manage 80.9 K 08 Feb 2017 - 10:45 JamesAngthopo  
PNGpng quaddata0700.png manage 80.2 K 08 Feb 2017 - 10:48 JamesAngthopo plot for 7.00mm without cadmium data
PNGpng quadonly.png manage 81.6 K 08 Feb 2017 - 10:35 JamesAngthopo  
PNGpng quadonlydata.png manage 74.1 K 08 Feb 2017 - 00:15 JamesAngthopo This shows how the data point varies when cadmium data is excluded
PNGpng uncorrectedplot.png manage 99.2 K 07 Feb 2017 - 23:44 JamesAngthopo This shows how the data point varies when cadmium data is included

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Topic revision: r12 - 09 Feb 2017 - TomCrane

 
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