PX282 - C2a - the boltzmann equation

• the boltzmann equation gives how many electrons are in each state:

derivation

• from statistical mechanics, the probability that an electron is in state 1 , requiring an energy, $E_1$, over state 2, with an energy, $E_2$, for $E_1>E_2$, is given by:

• if the temperature is low, $k_{B}T\to 0⟹\frac{P_1}{P_2}\to 0:$ the electrons go to the lowest state

• if the temperature is high, $k_{B}T\to \mathrm{\infty }⟹\frac{P_1}{P_2}\to 1:$ with infinite energy, all states are equally accessible

• these are quantum states, defined by:
  $n:$ energy
  $l=n-1:$ angular momentum
  $m_l\in [-l,+l],\mathbb{Z}:$ alignment
  $m_s=\pm \frac{1}{2}:$ spin

• consider quantum numbers for $H:$

  • at ground state:
    • $n=1$
    • $l=n-1=0$
    • $m_l=-l\to +l=0$
    • $m_s=\pm \frac{1}{2}$
    • there are two energy states
  • at $n=2:$
    • $l=0,1$
    • $m_0=0$, $m_1=-1,0,+1$
    • $m_s=\pm \frac{1}{2}$
    • there are 8 energy states

• states with the same $n$ have the same energy

• all possible states need to be included

• this gets done using 'statistical weights', $g$, such that $g_A$ is the number of states with the energy, $E_A$

• $\frac{P_1}{P_2}$ is for a given energy, not a given state, so accounting for states, the probability ratio becomes:

• a stellar atmosphere has a large number of atoms, $N$
• the probability ratio is equivalent to the number ratios of atoms in each state, therefore, the boltzmann equation is obtained:

• considering the atoms of a single element in a specified state of ionization with number of electrons, $N_x$, at an energy level, $E_x$