Classification of Spectra, Spectra from the Hydrogen Atom The Bohr model of the hydrogen atom gave us the following results:Excitation, De-excitation, Ionization and Recombination When an electron jumps from a lower energy level to a higher energy level (smaller n to larger n), the atom is said to be excited, and the process is called excitation. Energy must be added to the atom in order for such a jump to occur. There are two ways for an atom to be excited:Heisenberg Uncertainty Principle We noted last time the relationships:DxDp ~ h/2pbut did not have time to show some effects that come from them. The n = 2 level of hydrogen lives for a very short time, only about 10-8 s. That is, an electron in the first excited state will, after about 10-8 s, spontaneously emit a photon of energy 10.2 eV and go back to the ground state (emitting a Lyman a photon). The second relation, above, means that the photons emitted have an energy uncertainty of This uncertainty in the wavelength shows up as a broadening of the spectral line (natural broadening).Spectral Classification
Adapted from data in the electronic version of "A Library of Stellar Spectra," by Jacoby G.H., Hunter D.A., Christian C.A. Astrophys. J. Suppl. Ser., 56, 257 (1984). The spectra were then reordered according to temperature using the continuum spectrum (blackbody curve) as a guide (Annie Jump Cannon), to arrive at a new ordering:
(early) (late) ...B7, B8, B9, A0, A1, A2, A3, ..., A7, A8, A9, F0, F1... etc.
He I = neutral helium He II = singly-ionized helium He III = doubly-ionized helium These questions can be answered by appealing to statistical mechanics. Imagine firing a beam of particles, all with the same energy, into a trap and allowing them to come into equilibrium by colliding elastically. Due to standard probability and statistics (akin to coin-tossing), they will wind up with a gaussian distribution of speeds (a bell curve). Any distribution of particles in equilibrium will have such a curve, which can be parametrized by a single number, the temperature. Due to the 3D nature of velocity, the gaussian when expressed in terms of speed becomes:Boltzmann Equationnv dv = n (m/2pkT)3/2exp(-1/2 mv2/kT )(4pv2) dvwhere the important dependence is the argument of the exponential, which is just the ratio of the kinetic energy (1/2 mv2) to thermal energy (kT ). This expression is the number of particles per unit volume having speeds between v and v + dv. As atoms collide, their electrons can be knocked up to the next higher energy level if the colliding atoms have enough energy (collisional excitation). The electron can even be knocked entirely away from the atom (ionization). Again, looking at the problem from a statistical standpoint, the probability of the atom's being in one energy state, sa, isSaha EquationP(sa) ~ exp(-Ea/kT )and the probability for state sb isP(sb) ~ exp(-Eb/kT )where Ea and Eb are the energies of the two states (e.g. = 13.6 eV for the ground state of the hydrogen atom). The ratio of these probabilities is then To get a hydrogen atom's electron into level 2, it required a collision with at least 10.2 eV of energy. However, once it is in level 2, it requires only 3.4 eV more energy to knock it completely away from the atom, so the first excited state is a very precarious place for an electron! The actual number of atoms in level 2 at any one time is a balance between collisions to get it there in the first place, and collisions to ionize the atom.Partition Function
Combining the Boltzmann and Saha Equations We can find the variation of the strength of the H-alpha line with temperature by combining the Boltzmann and Saha equations as follows: Ratio of hydrogen atoms in the n = 2 level to the total number of hydrogen atoms as a function of frequency. Figure from Carroll and Ostlie, Modern Astrophysics, Addison-Wesley Publishing Co., 1996. |