Implied parentheses around the denominator "kT" are omitted for brevity. The normalization denominator "Q" (denoted by some authors by "Z") is the canonical partition function  It results from the constraint that the probabilities of all accessible states must add up to 1. The Boltzmann distribution is the distribution that maximizes the entropy  subject to the constraint that formula_9 equals a particular mean energy value (which can be proven using Lagrange multipliers). The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database. The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states "i" and "j" is given as  where "pi" is the probability of state "i", "pj" the probability of state "j", and "εi" and "εj" are the energies of states "i" and "j", respectively. The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over energy states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state "i" is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state "i". This probability is equal to the number of particles in state "i" divided by the total number of particles in the system, that is the fraction of particles that occupy state "i".  where "Ni" is the number of particles in state "i" and "N" is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state "i" as a function of the energy of that state is   This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line of atoms or molecules that we are interested in going from one state to another. In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state.