Notes on the second derivative method for spectrum analysis

• 1. Basics

Second derivative is often used to roughly estimate peak positions in spectra.

A general idea behind it is shown below.

In the right figure, the red curve is a Lorentzian function which represents typical peak shape in photoelectron spectra. The first derivative of the Lorentzian is shown in blue. The second derivative of the Lorentzian is shown in black.

Now a dip appears at the Lorentzian peak position (250).

It looks like just a conversion from peak to dip. However, the shape becomes sharper, and the value of d^2f/dx^2 around peak region is negative. These make peak position determination much easier.

Another good point is a subtraction of linear and parabolic backgrounds. Such background components in the spectra disappear by the second derivative. It is easily shown below.

Suppose a spectrum is expressed as

f(x) = Lorentzian(x, x0, w) + a x^2 +bx +c.

The second derivative of this function is

f''(x) = ((d^2/dx^2) Lorentzian(x, x0, w)) + 2a

There is no background components depending on x anymore.

This makes the detection of peak buried in linear and parabolic background much easier.

2. some issues

• • quantitative discussion

The peak position detection by second derivative method is not always quantitatively correct. An example is shown below.

Suppose we have two lorentzian peaks, Lorentzian0 (blue) and Lorentzian1(light blue). The sum of these two Lorentzian peaks is the red curve below.

It is no more a symmetric single Lorentzian peak, but an asymmetric peak. One can see the right hand side of the peak is somewhat broader then the left hand side.

Peak fitting can nicely detect the two components as shown in the right.

However, applying the second derivative gives us "not correct" result.

Here the red curve is the sum of the blue ad light blue Lorentzian peaks. The beak position is at 200 and 250. However, the dip of the second derivative curve (black) appears at 200 and at around "280".