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# From Tema 1 Optical Spectroscopy emission file
**Problem 1:** Using a mercury continuous calibration lamp (with well-defined lines in the literature), transform the X-axis of the spectrum into physical units of wavelength [nm].
[To solve this problem, use the data in the Excel document titled: “Lampa_Calibrare_Mercur.xlsx”]
**Problem 2:** Once the X-axis of the spectrum is calibrated, it will be removed from the optical system and a discharge in Argon will be positioned in its place. Identify at least 7 lines in the spectrum of the discharge in Argon using the NIST database.
[To solve this problem, use the data in the Excel document titled: “Spectru Descarcare Argon.xlsx”].
**Problem 3:** In order to extract quantitative data from an emission spectrum, it is necessary to calibrate the intensity axis in physical units. To achieve this, we will place a tungsten plate lamp in front of the optical system. Using a pyrometer, the plate temperature was obtained as T=1800 K.
Requirements:
3.1. Determine the theoretical spectrum of the tungsten plate lamp
3.2. Determine the measured spectrum of the tungsten plate lamp
3.3. Determine the curve that characterizes the response of the device for each wavelength.
3.4. Use the response curve of the device to calibrate the emission spectrum of a discharge in oxygen in radiance.
3.5. Measure the discharge temperature in oxygen using the Boltzman method for the emission lines at 777 nm and 844 nm. [Please comment on the temperature obtained!]
3.6. Determine the density of atoms in the excited state for the line at 844 nm. NOTE: in order to calculate the excited state density it is very important to know the collection volume (note that the column length enters the density calculation formula). For ease, take l=250 μm
3.7. Extra-credit: I would like you to think about how you would determine the collection volume of the light from the plasma knowing: the diameter of the lenses used (2 inches), the focal lengths (f1=125mm, f2=200mm) and the diameter of the optical fiber (d=200 μm).
[To solve this problem use the data from the Excel document titled: “Calibrare Intensitate Oxigen.xlsx”]
**Problem 4:** Identify the types of errors/uncertainties that occur in an OES experiment.
---
# Example problem on Data Processing - Optical Emission Spectroscopy
- Import xls data into Python
- Plot the Intensity [a.u.] vs pixels
- Interpolate and convert x-axis from pixels to nm (true wavelength) using Hg lamp data (using data in file: **Lampa_Calibrare_Mercur.xlsx**)
- Find response function of the spectrometer using the tungsten lamp data from file: "**Calibrare Intensitate Oxigen.xlsx**)": $R=\frac{I_{measured}}{I_{true}}$ (where True is computed by Planck's law of radiation (see notes in the pptx above)
- Convert y-axis from Intensity [a.u.] into Intensity in [W/(cm^2*sr*nm)] by dividing the measured Oxygen spectrum with the response function: $I_{oxygen, true}=\frac{I_{oxygen, measured}}{R}$
- Once the spectra is in real units: compute the density of one of the oxygen lines by integrating underneath one of the peaks (see equation from Slide 39 - bottom). We will give all of the constants that are in this equation (see the "**Intensity_Calibration_Oxygen_Discharge_Solution.xlsx**")
---
# Spectroscopy
A spectroscopy experiment is set up to collect the spectra discharges of Argon and Oxygen (figure 1). The data obtained from the spectrometer represent the light intensity measured across different pixel positions on the detector. Where both the light intensity and the pixel position don't have any units. To solve this problem we will need to calibrate the set-up by comparing light sources of know wavelengths and intensity. This will allow us to obtain useful results for other unknown light sources.
<img src="image_1762363220163.png" width="530">
*Fig 1: Spectroscopy experiment set-up for characterization for a plasma source in Argon and Oxygen.*
### Calibration of spectrometer
Problem 1: Plot the intensity of the mercury lamp as a function of pixel count.
Problem 2: The goal want to convert our distance dimension units from pixel to wavelength. To calibrate the wavelength readings (x-axis) by matching the peaks of the spectra to literature data. To calibrate the wavelength readings (x-axis), use linear regression to convert the pixel count to true wavelength ($nm$). The following equation can be used:
$$
\lambda_p=I+C_1p+C_2p^2+C_3p^3
$$
where:
$\lambda =$ wavelength of pixel $p$,
$I =$ wavelength of pixel 0,
$C_1 =$ first coefficient ($\frac{nm}{pixel}$),
$C_2 =$ second coefficient $(\frac{nm}{pixel^2})$,
$C_3 =$ third coefficient $(\frac{nm}{pixel^3})$
Problem 3: The accuracy of the spectrometer is limited by the sensitivity and resolution of the instrument. We can model these imperfection by comparing the measured values with "true" literature data. Find response function of the spectrometer using a tungsten (W) lamp data. The data has been recorded and saved in "calibrare_intensity_oxygen".
$$
R(\lambda)=\frac{I_{measured}}{I_{true}}
$$
For $I_{measured, W}$ use $I_{W} - I_{background}$.
$I_{true}$ is obtained by using plancks law at T=1800 K.
$$
I_{\lambda,\Omega}(T)= \epsilon (\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/kT}-1})
$$
$I_{measure}^W = R(\lambda) * I_{true}^W(\lambda)$
$I_{measure}^{plasma} = R(\lambda) * I_{true}^{plasma}(\lambda)$
$I_{meas}^{plasma}(\lambda) = \frac{I_{meas}^{W}(\lambda)}{I_{true}^{W}(\lambda)} * I_{true}^{plasma}(\lambda)$
Problem 4: Calibrate the light intensity from the arbitrary units to real units of $[{\frac{W}{cm^2*sr*nm}}]$
$I_{oxygen, true}=\frac{I_{oxygen, measured}}{R}$
### Calculating the density of oxygen
Problem 5: Once the spectra is in real units, compute the density of one of the oxygen lines by integrating underneath one of the peaks.
$$
n_{1,2}=\frac{\int_{\lambda_0-\Delta\lambda}^{\lambda_0+\Delta\lambda} I(\lambda)d\lambda}{\frac{1}{4\pi}hv_{1,2}A_{1,2}l}\tag{1}
$$
---
Calibrate wavelength by converting the length dimension from pixels to nm.
- Use polymonomial interpolation to find the coefficients to the function that converts pixels $[N]$ -> $\lambda$ $[nm]$
$$
\lambda_p=I+C_1p+C_2p^2+C_3p^3
$$
- Solve for C1 C2 and C3, use $I=195.54$
Calibrate the y-axis
- Find the response function of the spectrometer
$$
R(\lambda)=\frac{I_{measured}}{I_{true}}
$$
- $I_{measured}$ => Intensity array from Mercury (Hg)
- $I_{true}$ => Data from tungsten lamp using planks law of radiation
$$
I_{\lambda,\Omega}(T)= \epsilon (\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/kT}-1})
$$
- T=1800K
- Use $R(\lambda)$ to correct spectra for plasma
$$
I_{measure}^W = R(\lambda) * I_{true}^W(\lambda)
$$
$$
I_{meas}^{plasma}(\lambda) = R(\lambda) * I_{true}^{plasma}(\lambda)
$$
$$
I_{meas}^{plasma}(\lambda) = \frac{I_{meas}^{W}(\lambda)}{I_{true}^{W}(\lambda)} * I_{true}^{plasma}(\lambda)
$$
Measure the densities of an excited state of oxygen using $I(\lambda)$
$$
I(\lambda)=\frac{1}{4\pi}hvAnl\phi(\lambda-\lambda_0)
$$
- We can re-arrange to solve for $n$ in units of $[cm^{-3}]$
$$
n_{1,2}=\frac{\int_{\lambda_0-\Delta\lambda}^{\lambda_0+\Delta\lambda} I(\lambda)d\lambda}{\frac{1}{4\pi}hv_{1,2}A_{1,2}l}\tag{1}
$$
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