We will now work with Raman spectroscopy data from the luminescence
of a ruby particle under high pressure: Data/rubis_xx.txt.
This measurement is performed in a diamond anvil cell with a laser of
wavelength \(\lambda_L\)=568.189 nm,
and the pressure is known by calibrating the frequency of a diamond
line.
The ruby luminescence contains two lines, and we want to use the position of the R1 line, which is the most intense and has the highest energy, to determine the pressure. We therefore need to determine the parameters \(A\) and \(B\) of the equation of state linking wavelength (in nm) and pressure (in GPa):
\[ P~[GPa] = \frac{A}{B}\left[\left(\frac{\lambda}{\lambda_0}\right)^B-1\right], \] where \(\lambda_0\) = 694.24 nm is the wavelength of the R1 line at zero pressure, and \(\lambda\) is the wavelength of the R1 line at \(P\) pressure. For a given line, the relationship between its measured Raman shift, \(\omega\), and its wavelength is given by \(\lambda = \frac{10^7}{\omega_L - \omega}\), with \(\omega_L= \frac{10^7}{\lambda_L}\) the frequency of the probe laser.
Open one of the files, for example Data/rubis_01.txt to see its structure -
they’re all the same. They are made up of two columns, which are the
Raman shifts in cm-1 and the intensities. The nomenclature of
the files "Data/rubis_xx.txt" is such that xx
is the pressure in GPa.
First, consider the steps involved in determining the parameters \(A\) and \(B\) of the equation of state.
- What data do you need?
- What steps do you need to take to obtain them?
Prubis(w,A,B),
which returns the pressure in GPa as a function of the Raman shift in
cm-1, and the parameters \(A\) and \(B\).Find in the folder “Data” the list of files with the pattern “rubis_”
in their name (check out the list.files() function) and
store it in flist
What is the length of flist?
Initialize an empty tibble called spec that will store
all spectra:
Using a for loop, load each file in flist
and store them into spec by making it tidy.
Another version of this that does not use a for loop would be like so (try to understand what’s happening in this code by running each line one after the other and looking at the help on the functions used):
Modify spec to get the pressure in GPa from the file
name, and so that the intensity column is normalized to [0,1].
Plot the first spectrum in spec to see the x range
Using ggplot2, plot all spectra normalized to [0,1]
(with points) and stacked on top of each other with a vertical shift
corresponding to the pressure in GPa
What we want to do now is to fit this data by the sum of two
Lorentzians to find the precise position of the R1 line as a function of
pressure. To do this, we’ll use the function nls(),
which allows us to make a non-linear fit of the data. We also need to
define the function lor(x,x0,gamma) which returns a Lorentzian
centered at x0 and of width at half-height
gamma. Start by defining the function
lor(x,x0,gamma).
Now create an empty tibble called results, in which
we’ll store all the fitting parameters in a tidy tibble. Then fit each
spectrum by the sum of two Lorentzians, and save all fit parameters in
results, along with the file name and pressure. For the fit
to work, you’ll need to give reasonable initial values for the fit
parameters using the start=list(...) argument of
nls(). For example, you can use the
which.max() function to find the index of the maximum of a
vector.
Another way of doing that is to use the map() function
from the purrr package, which allows us to apply a function
to each element of a list. Here, we want to apply the nls()
function to each spectrum in spec, and store the result in
a new column of spec. We can do this like so:
Plot the result of the fit on the graph above.
Now all you have to do is plot the position of the peak as a function of pressure, and model this data by the equation of state defined at the beginning. What values of A and B do you obtain? Compare with the literature values: A = 1876 GPa and B = 10.71.