1 files changed, 53 insertions, 5 deletions
diff --git a/tutorials/module_4/Spectroscopy problem.md b/tutorials/module_4/Spectroscopy problem.md
index 57770bb..0e9af66 100644
--- a/tutorials/module_4/Spectroscopy problem.md
+++ b/tutorials/module_4/Spectroscopy problem.md
@@ -29,23 +29,23 @@ Requirements:
 # Example problem on Data Processing - Optical Emission Spectroscopy
 
 - Import xls data into Python
-- Plot the Intensity [a.u.]  vs pixels
+- 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 "I**ntensity_Calibration_Oxygen_Discharge_Solution.xlsx**")
+- 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. Since this is unit-less we will need to calibrate the experiment with a know light source to be able to obtain useful results.
+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 1: Plot the intensity of the mercury lamp as a function of pixel count. 
 
-Problem 2: Calibrate the wavelength readings (x-axis) by converting the pixel count to true wavelength ($nm$) by using linear regression. The following equation can be used:
+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:
 $$
 \lambda_p=I+C_1p+C_2p^2+C_3p^3
 $$
@@ -85,3 +85,51 @@ $$
 n_{1,2}=\frac{\int_{\lambda_0-\Delta\lambda}^{\lambda_0+\Delta\lambda} I(\lambda)d\lambda}{\frac{1}{4\pi}hv_{1,2}A_{1,2}l}\tag{1}
 $$
 
+
+
+
+---
+
+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 $[cm^{-3}]$
+$$
+n_{1,2}=\frac{\int_{\lambda_0-\Delta\lambda}^{\lambda_0+\Delta\lambda} I(\lambda)d\lambda}{\frac{1}{4\pi}hv_{1,2}A_{1,2}l}\tag{1}
+$$
+
+
+
+