# Understanding the Spectrum What does a spectrum tell us? Ever since we've seen those dark gaps in the light of stars, planets, we've never stopped to wonder and consider the consequences of our discovers. Although researchers may not question the most fundamental of scientific tenets, many astrophysicists look over at these spectra for hours on end, wondering and dreaming about what may lie ahead. Spectra come in all shapes and sizes. Take a look at this image below for a quick example! ![[Pasted image 20230710101634.png]] This is the spectra of hydrogen, which dictates the photons it can absorb to bump itself up an orbital. Remember that the shorter the wavelength, the higher energy it is. To bump an electron up to higher and higher orbitals, photons with higher and higher energies must be used. These are given as **line numbers** - for example, bumping an electron up from N = 2 to N = 3 would be considered a H-2 line, while the base absorption from the ground state is known as a H-$\alpha$ line. Depending on the resolution/profoundness of the materials, we can work out the strength and concentration of the materials that make up the object emitting the light. Spectra can also give the ionisation states of different atoms - the number of electrons at one orbital compared to the number of electrons at another orbital. As higher energy photons are produced in stars with higher temperatures, the ratio of atoms in one state to another state depends on the temperature! We can create an equation to describe the relationship of the electrons occupying one state with the electrons occupying the other state as follows: $\frac{N_{1}}{N_{2}} = \frac{g_{1}}{g_{2}}e^{-(\chi_{1}-\chi_{2}) /(k_{b}T)}$ Where: - $g$ is the **degeneracy** of the atoms, the number of states which the electron can occupy at which they can't - $N$ is the number of atoms in one target orbital (one orbital level) - $T$ is the temperature - $\chi$ (chi) is the discrete energy of the photons at each orbital level. - $\chi_{1} - \chi_{2}$ in the equation gives the energy of the photon emitted/absorbed. >[!Example]- Example Question - Hydrogen Lyman Alpha Populations >A hydrogen gas is at 50000K. What is the ratio of the populations for the Lyman alpha transition? > >**Solution:** > >We know that the electrons are going to increase in orbital levels. This means that the second degeneracy coefficient $g_2$ will be given by 6. As the hydrogen atom is rising from the ground state, the coefficient $g_{1}$ will just be given by 2. This denotes the number of electrons in a similar configuration to the one in question. >Substitute into the population ratio equation NOW! >$\frac{N_{1}}{N_{2}} = \frac{g_{1}}{g_{2}}e^{\frac{-(\chi_{1} - \chi_{2})}{k_{B}T}}$ ## More on Degeneracy and Orbitals While this is more of a chemistry thing, it's best we stop for a bit and consolidate what we know about degeneracy, or $g$. above is a cross sectional, 3d visualisation of the geometry of an atom. there is one S shell (first orbital) and 3 different P orbitals. As an electron can have spin up or down, this gives it 2 possible configurations at the ground state S orbital and 6 possible configurations at the P orbital. Therefore, we can find the number of possible configurations (degeneracy) as a function of: $g = 2N$ Where: - $N$ is the number of orbitals - $g$ is the degeneracy of the electron >[!Example]- Case Study - The Neon Atom > >![[Pasted image 20230710102903.png]] >As shown in the diagram above, a neon atom has two levels of potential orbital - the S orbital, the ground state, and the P orbital, the orbital N = 2. This is because the P and S orbitals overlap. There are two ground states within the S orbital, while there is one extra shell above it. As a result, if electrons in the N=1 state absorb an electron, they will be bumped up to the P orbital. > >There are 3 P orbitals and 1 S orbital. Each electron can occupy 2 states (spin up or down, more on that later). This means that the degeneracy at N = 1 is 2, while the degeneracy at N = 2 is 6. So a hydrogen shell would only have the S orbital! meaning it's (somewhat) simple. # Absorption and Emission By this point, we're encroaching onto quantum physics - and fast! //describing a line ratio: //ABSRORPTION LINES: --> depending on the energy of the photon that has been absorbed, the higher and higher wavelength will allow you to bump up more and more electron levels (so the absorption line which enables a photon to jump from N=1 to N=2 will be much less energy than one that allows a jump from N=2 to N=5)