# On energy levels and emission of photons

This is a very basic question but I cannot seem to find the answer anywhere.

Say we have an atom in ground state. Its first energy level is 2 eV. An incoming photon of energy 2.5 eV hits an electron in the atom (with the lowest energy level) which is excited and moves up one enery level. However it does not have enough energy to reach the next energy level. When it loses energy it emits a photon of energy 2 eV. What has happened to the remaining 0.5 eV? Is it some other type of energy in the atom such as kinetic energy?

• Welcome to the wonderful world of molecular spectra! You are absolutely correct about purely electronic transitions, which can only occur at discrete energy levels. This, however, is no longer true when you have multiple atoms involved, because now there is a vast "forrest" of rotational and vibrational states that can also be excited in addition to the electronic states. Extend this from diatomic or triatomic molecules like $O_2$ or $CO_2$ to solids and now there are entire energy bands that can be excited, which (partly) explains the colors of solids. – CuriousOne Jun 1 '15 at 19:41

An incoming photon of energy 2.5 eV hits an electron in the atom (with the lowest energy level) which is excited and moves up one enery level.

In most cases, this won't happen. The incoming energy must match the transition that is being excited.

It's possible for a 2.5 eV photon to excite a 2 eV transition, but only if there is some other particle available to carry away the other 0.5 eV. For example there could be a phonon or there could be a new 0.5 eV photon generated. However these processes tend to be much less likely to occur, because they are three-particle interactions rather than two particles.

The probability of a photon of energy $E$ — or corresponding wavelength $\lambda = h c / E$ — being absorbed by an atom and bringing an electron from level $i$ to level $j$ is given by the cross section $\phi_{ij}(\lambda)$, which is a sharply peaked function of wavelength (a so-called Lorentzian function). For instance, to bring a hydrogen atom from its ground state to its first excited state, a photon of $\lambda$ = 1215.67 Å, or $E$ = 10.2 eV is needed. If the energy is just a little smaller, or a little larger, say 10.21 eV, corresponding the a wavelength half an Ångström shorter, the probability of being absorbed is many orders of magnitude smaller. The photon is said no longer to be in resonance with the energy level.

In reality, when a photon enters an ensemble of atoms, the temperature of the gas causes the atoms to have a (Gaussian) distribution of velocities. This means that although the incoming photon has a "wrong" energy of 10.21 eV, there are plenty of atoms that happen to be moving away from the photon at exactly the right speed, such that in the reference frame of the atom, the photon is redshifted into resonance. That is, the probability of being absorbed is large.

The effect is that the absorption cross section of the energy level is broadened, becoming a convolution of the "natural" (Lorentzian) and the thermal (Gaussian) profiles, called a Voigt profile.