Question about measuring the FWMH for non-zero values of a and c

[tim.hbn]

Hi Everyone

As some of you know, when callibrating a detector for gamma spectroscopy, you need to find the coefficients of the following equation:

Energy = ax^2 + bx + c (where x is the channel number)

It seems that a lot of the time, people will assume that both a & c will be zero.

However, when the true values of a and/or c are not zero, measuring the FWHM may not be straightforward.

The question is, should the FWHM be measured from the perspective of the energies or the channel numbers? My guess is the energies.

If my guess is correct then if you attempt to measure the FWHM visually by looking at how wide a peak looks on the spectrum then you may get the wrong answer.

I would be very grateful for any comments on everything I have said in this post.

Thank you very much.

[Sesselmann]

The question is, should the FWHM be measured from the perspective of the energies or the channel numbers? My guess is the energies.

Tim,

It is not 100% clear to me why you are asking about FWHM resolution in conjunction with linearity, although the two can be related, for most reasonably linear detectors it is a non issue.

First of all, most detectors should be close to linear, but for various reasons some detectors fall short when capturing the higher energy peaks and as a result these peaks appear a bit lower than they should. A common reason for this when using high impedance voltage dividers is a lack of current flow in the upper stages of the voltage divider. This can be improved by adding capacitors across the last 3-4 stages of the voltage divider.

In any case there are detector setups where some non linearity is inevitable, and due to the way detectors are designed it just so happens that the non linearity can be closely imitated by a second order polynomial function as the one you refer to above. Although some experts in the trade such as Marek who is the Author of PRA sit's in the other camp and prefers using multiple calibration points with interpolation.

The objective here is nothing more than being able to accurately determine the energy of a peak which is not aligned with your calibration peak.

Resolution on the other hand is a measure of how good your detector can resolve energy from a gamma source, this is a whole science in itself. Why do the gamma rays, which all have identical energy, result is a gaussian distribution of energies in our spectrum? The answer is complicated and involves thermal movement of atoms within the crystal structure plus a lot of other factors which you can look up.

The short answer is we detect the gamma energy with an uncertainty and this follows a normal distribution resulting in a peak the shape of a bell curve. FWHM (full width at half maximum) is just a way to describe the spread of the bell curve. Another common way to describe resolution is in percentage, where we take the energy spread at half maximum and divide it by the peak energy, the result is then expressed as a percentage.

Here is a simple example, say we have recorded a spectrum of a Cs137 source and our peak is 1000 counts on the Y axis and 662 keV on the x axis, then if we draw a line at 500 counts and take the maximum energy value less the minimum energy value in our wide peak it shows a difference of 66.2 keV then dividing 66.2 by 662 gives 10% resolution at 662 keV.

It should be noted that resolution is not linear, so if you measure a peak higher up on the energy scale the resolution will be lower and likewise at the lower energy part of the scale it will be higher. This intuitively makes sense from a signal to noise ratio point of view.

Even if this did not directly answer your question I hope someone learns something from it.

Steven

[tim.hbn]

Hi Steven

Thank you very much indeed for your very informative and helpful reply.

[NuclearPhoenix]

The question is, should the FWHM be measured from the perspective of the energies or the channel numbers? My guess is the energies.

Hi Tim

just to add on what Steven said. As you guessed, if you wanted to get your energy resolution, you should always only do that with the energies.
If you just used the channel numbers, these might not necessarily be linear (already mentioned) and also you might often have an offset in your bins (c coefficient in your example), which moves the spectrum around as a whole.
That would mean your individual peaks are basically at arbitrary bin numbers since moving the spectrum as a whole doesn't change the spectrum itself of course. But it would completely change the relative energy resolution you get (%), because you need the peak energy for that.

[tim.hbn]

Hi Matthias

Thank you very much indeed for your answer.