Photodetachment Cross Section Formulas: Explained

[Dahaka14]

I know the formulas for the length and velocity forms of the photodetachment cross section for an electron with energy $$\hbar\omega$$ are, respectively,

$$\sigma_{L}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}z_{j}|\Psi_{i}\rangle|^{2}$$

and

$$\sigma_{V}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}\frac{\nabla_{j}^{z}}{i\omega}|\Psi_{i}\rangle|^{2}.$$

Could someone please explain what these actually mean and how they are derived, or even better, where I can find a good reference on them? I asked my research professor about them and he was no help (these are where I got them from), and I can't find anything about them on the internet.

 

[olgranpappy]

I know the formulas for the length and velocity forms of the photodetachment cross section for an electron with energy $$\hbar\omega$$ are, respectively,

$$\sigma_{L}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}z_{j}|\Psi_{i}\rangle|^{2}$$

and

$$\sigma_{V}(\omega)=\frac{4\pi^{2}\alpha a_{0}^{2}\omega}{3}\sum_{f}|\langle\Psi_{f}|\sum_{j=1}^{n}\frac{\nabla_{j}^{z}}{i\omega}|\Psi_{i}\rangle|^{2}.$$

Could someone please explain what these actually mean and how they are derived, or even better, where I can find a good reference on them? I asked my research professor about them and he was no help (these are where I got them from), and I can't find anything about them on the internet.

try Sakurai's quantum mechanics book. Look up Fermi's Golden Rule.

 

[charng]

I am happy to provide an explanation of these photodetachment cross section formulas.

First, let's define some terms. The photodetachment cross section is a measure of the probability of an electron being detached from an atom or molecule when it interacts with a photon of a particular energy, represented by \hbar\omega. This is an important quantity in understanding the behavior of electrons in various systems.

The two formulas you have provided are the length and velocity forms of the photodetachment cross section. The length form, \sigma_{L}(\omega), is used when the electron is considered to be in a bound state and is calculated using the transition matrix elements between the initial state of the electron, |\Psi_{i}\rangle, and the final state after detachment, |\Psi_{f}\rangle. The sum over f indicates that the formula takes into account all possible final states of the electron. The term \alpha is the fine structure constant, a_{0} is the Bohr radius, and z_{j} represents the position of the jth electron in the system.

The velocity form, \sigma_{V}(\omega), is used when the electron is considered to be in a continuum state, meaning it is not bound to the atom or molecule. This formula takes into account the velocity of the electron and is calculated using the transition matrix elements between the initial state, |\Psi_{i}\rangle, and the final state, |\Psi_{f}\rangle, after detachment. The term \nabla_{j}^{z} represents the derivative of the position of the jth electron in the system with respect to z, and the term i\omega in the denominator accounts for the frequency of the photon.

Both of these formulas are derived from the basic principles of quantum mechanics and involve complex mathematical calculations. They are commonly used in theoretical studies and can also be experimentally measured to validate theoretical predictions.

If you are interested in learning more about these formulas, I suggest consulting a textbook on quantum mechanics or atomic and molecular physics. Additionally, you can find more information in research papers and articles related to photodetachment processes. I hope this explanation has been helpful and I wish you success in your research.