K

Before

Before

Collisional excitation. In each case, the left frame shows the atoms before the collision and the right frame shows them after. In each frame, the occupied level is indicated by a heavier line. (a) To a lower state. After the collision, atom 1 is in a lower state and atom 2 is moving faster. (b) To a higher state. After the collision atom, 1 is in a higher state and atom 2 is moving slower.

The collisional excitation rates will depend on the kinetic temperature of the gas. The higher the temperature the faster the atoms are moving. For atoms of kinetic temperature Tk the average kinetic energy per atom is (3/2)kTk. As the temperature increases more energy is available for collisions. This makes higher energy states easier to reach. Also, since the particles are moving faster, they spend less time between collisions. There are more collisions per second.

When a gas is in thermodynamic equilibrium (which we discussed in the previous chapter), with a kinetic temperature Tk, the ratios of the level populations are given by a Boltzmann distribution. If we let n, and nj be the populations of levels i and j, respectively, their ratio is given by n, g,

ni gi

In this equation gf and gj are called statistical weights. They are needed because certain energy levels are actually groupings of sublevels that have the same energy. The statistical weight of a level is just a count of the number of sublevels in that level. Typically, g are small integers.

To help us understand the Boltzmann distribution, Fig. 3.7 shows how the ratio of populations for an atom with just two levels depends on temperature. When the temperature is zero, all

Temperature (K)

H Level populations as a function of temperature for a two-level system. In this case we have put in energies and statistical weights (3,5) for the n = 2 and n = 3 states of hydrogen (first Balmer transition).

the atoms are in the ground state, so the ratio is zero. As the temperature increases, the quantity in square brackets gets smaller, so the exponent becomes less negative, and the ratio increases. If we let Tk go to infinity the ratio of populations approaches the ratio of statistical weights. For a given temperature, increasing the energy separation between the two levels makes the exponent more negative, lowering the ratio. This makes sense, since the greater the energy separation, the harder it is to excite the atom to the higher level.

The Boltzmann distribution provides us with a convenient reference point, even for a system that is not in thermodynamic equilibrium. For any given population ratio n,/^, we can always find some value of T to plug into equation (3.9) to make the equation correct. We call such a temperature the excitation temperature. When they are not in equilibrium, each pair of levels can have a different excitation temperature. In thermodynamic equilibrium all excitation temperatures are equal to each other and to the kinetic temperature.

3.4.2 Ionization

If we know the temperature in the atmosphere of a star, we can use the Boltzmann equation to predict how many atoms will be in each state, i, and predict the strengths of various spectral lines. However, there is still an additional effect that we have not taken into account - ionization. If the temperature is very high, some of the colliding particles will have kinetic energies greater than the ionization energy of the atom, so the electron will be torn away in the collision. Once a hydrogen atom is ionized, it can no longer participate in line emission or absorption.

When the gas is ionized, electrons and positive ions will sometimes collide and recombine. When the total rate of ionizations is equal to the total rate of recombinations, we say that the gas is in ionization equilibrium. If the gas is in thermal equilibrium and ionization equilibrium, then the Saha equation tells us the relative abundances of various ions. We let n(Xr) and n(Xr+1) be the densities of the r and r + 1 ionization states, respectively, of element X. (For example, if r = 0, then we are comparing the neutral species and the first ionized state.) The ionization energy to go from r to r + 1 is Eion. The electron density is ne, and the kinetic temperature is Tk. Finally, gr and gr+1 are the statistical weights of the ground electronic states of Xr and Xr+1 (assuming that most of each species is in the ground electronic state). The Saha equation tells us that nen(Xr+i ) n(Xr )

2-nmr kTk h2

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