# The effect of instrumental broadening on FWHM in Raman peak

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1. May 17, 2018

### AVRA

I would like to know how to extract the instrumental broadening effect in Raman spectrometer using solid angle of the objectives, slit width or spectrometer length. I am calculation the FWHM in Raman peaks and I would like to know the effect of instrumental broadening on Raman line width and how to get accurate measurement.

2. May 17, 2018

For a diffraction grating spectrometer, $m \lambda=d (\sin(\theta_i)+\sin(\theta_r))$ is the condition for constructive interference and getting a primary maximum. At first order ($m=1$), this means that $\Delta \lambda \approx d \, \Delta \theta$. A slit width of $b$ will introduce a $\Delta \theta=\frac{b}{f}$. This results in a $\Delta \lambda \approx \frac{d \, b}{f}$. $\\$ There is a lower limit that can be achieved though by narrowing the slit widths: $\frac{\Delta \lambda}{ \lambda}>\frac{1}{Nm }$ where $N$ is the number of lines on the grating that are used in creating the spectral line and $m$ is the order.

3. May 17, 2018

### AVRA

Thank you Charles for the response. I have some questions regarding the formula you mentioned.
I use 1800gr/mm for the Raman measurement. Does that mean N will be equals 1800 here ? How will I determine the 'm' ? and how will I get the slit width 'd' ?

Thanks.

4. May 17, 2018

### AVRA

What is the value 'f' here ?

5. May 17, 2018

$d=\frac{1}{1800}$mm. Meanwhile $b$ is the slit width and should be adjustable=entrance and exit slits, usually set to be nearly equal width. For $N$, you need to know what the illuminated width $w$ of the diffraction grating is. $N=1800 \, w$ when $w$ is in mm. $\\$ $f$ is the focal length of the collimating optics that makes parallel rays incident onto the grating. The same $f$ is usually also used to make the far field diffraction pattern image in the plane of the exit slit. $f$ is often the "length of the spectrometer" when such a number is provided by the manufacturer.

6. May 17, 2018

### AVRA

I would like to know, what is the formula of the instrument broadening here ? I mean is it
Δλ/λ ? and the value should be as small as possible for accurate linewidth measurement in Raman peaks ? and lastly, are 'w' and slit width 'b' instrument specific ?

7. May 17, 2018

Please see additions to the post 5 above about $f$. $\\$ The instrument broadening will normally be $\Delta \lambda \approx \frac{d \, b}{f}$. Generally, if you narrow the slits to achieve minimum resolution $\Delta \lambda=\frac{\lambda}{Nm}$, (maximum resolving power), you get very little throughput=very low signal. $\\$ Spectrometers often have adjustable slit widths $b$. $\\$ Meanwhile, the $w$ can be improved by optimally focusing the beam onto the entrance slit. The rays emerging from the entrance slit should nearly fill the collimating optic/diffraction grating rather than only illuminating a small portion across the grating. (We're referring here to filling it horizontally=vertical filling isn't necessary).

8. May 17, 2018

### AVRA

Thank you again for the explanation. Actually I am new in this subject, so trying to connect the dots. I already measured the Raman peaks and calculated the FWHM from those measurements. Now, I am trying to figure out if the instrumental broadening is small enough not to influence the variation in FWHMs.
So, according to you Δλ≈db/f , is the formula of the instrumental broadening. In the equation, d is equal to 1/1800nm. What does 'f' represent here ? and how will I know the value of the slit width ? Can you give me any idea about the general values of d and f ?

Thanks.

9. May 17, 2018

$d=\frac{1}{1800}$ mm. The slit widths $b$ should be adjustable and a micrometer (hand turning to adjust) type control is often used. And see post 5 above for $f$. If you can open up the instrument, $f$ is the distance of the concave mirror from the slit. Sometimes the same concave mirror is used both for entrance and exit slit=in other cases it will be two separate mirrors. The slits are always in the focal plane of these mirrors, so the focal length is readily measured by measuring the distance from slit to mirror. And a typical slit width (for low resolution work) might be around $b=1$ mm.

10. May 17, 2018

### AVRA

Thank you for the response. The concept is getting clearer for me. So the instrument broadening unit will be 1/nm according to the formula ? and if the value of instrumental broadening is very small, can I say that the FWHM measurement is nearly accurate ?

11. May 17, 2018

$\Delta \lambda$ is in nm. $\\$ If $\Delta \lambda =\frac{ d \, b}{f}$ << FWHM , e.g. a factor of 10 less, then the instrument broadening would be considered to be of little significance.

12. May 18, 2018

### AVRA

Ok, I got it. There was a mistake in my calculation. So, if the value of instrument broadening is small, can I say that the linewidth measurement is correct ? And can you give me any reference of these formula, it will be helpful for me to dig deep into these. Thank you so much for helping me solving this problem.

Thanks.

13. May 18, 2018

You will always have some error bars on your measured line widths, but those error bars can basically be $\Delta \lambda=\frac{d \, b}{f}$. I really don't have a good reference for this=you can try googling "Ebert Spectrometer" or "Czerny-Turner spectrometer". I can also look for a couple of "links" to PF homework questions, where I may have derived a couple of the equations: Let me see if I can find one or two...

14. May 18, 2018

15. May 18, 2018

16. May 18, 2018

### AVRA

Thank you very much for the reply.

One additional query I would like to know from you and that is
Thank you very much for the link and response. I would like to know mention here one thing that is that the Rama
Thank you very much for the link and response. I would like to know mention here one thing that is that the Raman intensity is usually plotted against Raman shift whose unit is cm-1 and accordingly the FWHM of Raman peak is also expressed in cm-1. Now, does the formula of the instrument broadening applies same here as the unit there is nm and in case of Raman shift it is cm-1 ? I am asking these, because, if I want to plot the error bars for measuring raman shifts, then the units should be in cm-1, but using the formula you mentioned, the value will be in nm. and 1nm=10000000cm-1. So, the number will be huge compared with nm.

17. May 18, 2018

In spectroscopy $\nu=\frac{1}{\lambda}$. This means, using a little calculus, that $\Delta \nu=\frac{\Delta \lambda}{\lambda^2}$. If $\lambda=500 \, nm=.5 \, um$, and $\Delta \lambda=1 \, nm$, that gives $\Delta \nu=1 \cdot 10^{-7} cm/(.5 \cdot 10^{-4})^2 cm^2=40 \, cm^{-1}$.

18. May 18, 2018

### AVRA

Thank you so much for the response. It really helped me clearing my concept regarding this. i will figure out the slit width and the length of the spectrometer to calculate the instrument broadening.

I would like to seek your help in resolving another query. Do you know how to determine the laser spot size in Raman spectrometer if I use a 10X objective with NA equal to 0.25 ?

19. May 18, 2018