Assignment – Phyc 6250 Module 2 – UPS and IPES
Due February 14, 2022
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Question 1:
Convolution is a mathematical operation that is extremely useful in error analysis, statistics,
understanding experimental data, image analysis, signal process … and the list goes on.
The convolution of two functions f(x) and g(x) is also a function of x, and is defined as:
(𝑓 ∗ 𝑔)(𝑥) = ∫ 𝑓(𝑥′)𝑔(𝑥 − 𝑥
Note that the integration is over the variable x’, and following the integration we are left with a function
In the context of UPS and IPES (and many, many other experimental techniques), the function f(x) is
what we are trying to measure experimentally, for instance the intensity of photoelectrons as a function
of kinetic energy. The function g(x) is the instrument pass, or transfer, function.
Ideally, our instrument would only measure electrons with a specific kinetic energy, and would reject all
others. In this case, g(x) would be a Dirac delta function, (x-KE), where KE represents the kinetic
energy we are trying to detect, and g(x) is only non-zero when x=KE. In this ideal case, the measured
spectrum, (𝑓 ∗ 𝑔), would equal f(x).
In reality, our pass function tends to be a peak with finite width, and is often modelled as a gaussian. In
the question below, we will use the “full-width at half-max (FWHM)”, w, to characterize the width of our
pass functions. Note that in the case of a gaussian, w=2.35, where is the standard deviation.
Consider a hypothetical photoelectron spectrum, f(x):
This is a step function, and represents the idealized density of occupied states in a metal at absolute
zero (where the Fermi-Dirac distribution becomes a step function). X can be thought of as the energy
scale (usually in eV), where 0 represents the energy of a photoelectron excited from a state at the Fermi
Now, consider three possible instrument pass functions, g1-3 : a square window of width w, a symmetric
triangular window of FWHM=w, and a gaussian peak with FWHM=w:
(𝑤 + 𝑥),−𝑤 ≤ 𝑥 < 0
(𝑤 − 𝑥), 0 ≤ 𝑥 < 𝑤
Note that g3 is just a standard gaussian, where we are using the FWHM, w, instead of the standard
deviation. Also note that all three functions are normalized, such that their integrated intensities are 1.
For this question, please show your work, and submit your solutions along with the graphs, preferably in
electronic form. Photos of handwritten solutions are fine.
a) Calculate, algebraically, the three convolutions 𝑓 ∗ 𝑔1, 𝑓 ∗ 𝑔2, 𝑓 ∗ 𝑔3 (note that for the case of
the gaussian, you will need to express it in terms of the gaussian error function, erf(x))
b) Make three graphs, one for each of the convolutions above. In each graph use the range -5