Angular dependance of NEXAFS spectroscopy - derivation

[Fegito]

Hi all, this is my first time posting so I hope it's in the right place, if not I apologise. I'm trying to understand the angular dependence in NEXAFS spectroscopy for linearly polarised light.

So from what I understand, the quantum mechanical description of the excitation process for a single electron in the dipole approximation, is given as |<ψ_i|e.p|ψ_f>| where e is the unit electric field vector and p is the dipole transition operator. In all the books/sites I have read it then states that for linearly polarised light, the matrix elements of interest take a simplified form, and can be written as:

|<ψ_i|e.p|ψ_f>| = e.|<ψ_i|p|ψ_f>|

It then goes on with the derivation, which is fairly straightforward What I don't understand is this seemingly simple step, more specifically why the electric dipole vector can be taken out front (I have no prior knowledge of quantum mechanics so there's a good chance it's something really simple which I'm just not understanding!). Would someone please be able to shed some light on this? Thanks in advance.

 

[eljose79]

Hello and welcome to the forum! I'm happy to help you understand the angular dependence in NEXAFS spectroscopy for linearly polarised light.

First of all, let's start with the dipole approximation. This is a simplification that is often used in quantum mechanics to describe the interaction between an atom or molecule and an external electric field. In this approximation, the size of the atom or molecule is considered to be much smaller than the wavelength of the light, so the atom or molecule is treated as a point-like object.

Now, in the excitation process, the external electric field causes the electron to transition from an initial state |ψi> to a final state |ψf>. The dipole transition operator, denoted by p, describes the strength and direction of this transition. It is a vector operator, meaning that it has both magnitude and direction.

In the case of linearly polarised light, the electric field vector e is also a vector, with a specific direction and magnitude. This electric field vector is perpendicular to the direction of propagation of the light, and it oscillates along a single plane. This is why we call it linearly polarised light.

Now, when we take the inner product |<ψi|p|ψf>|, we are essentially calculating the overlap between the initial and final states, which gives us the probability of the electron transitioning from one state to the other. However, since both the electric field vector and the dipole transition operator are vector quantities, we can rewrite the inner product as a dot product, which is given by e.|<ψi|p|ψf>|.

This dot product essentially tells us the component of the dipole transition operator that is aligned with the direction of the electric field, which is what we are interested in for linearly polarised light. This is why we can take the electric field vector out front, as it represents the direction and magnitude of the electric field that is responsible for the transition.

I hope this explanation helps you understand why the electric dipole vector can be taken out front in the case of linearly polarised light. If you have any further questions, please don't hesitate to ask. Happy researching!