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The capturecascade equations usually predict that at

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Unformatted text preview: velocity of an electron is 108 cm/s. Thus we might expect collisions to become important at densities exceeding ~108 cm−3. In fact, they become important at densities of ~105 cm−3 because the electric field from passing charged particles can induce a Stark effect. The 2s and 2p levels are (nearly) degenerate, so any perturbing electric field in (say) the z ­direction causes the energy eigenstates to be not 2s and 2pz but rather: 2s ± 2p z . 2 The electron then oscillates between the 2s and 2pz states. The net result of the collision is: € H(2s) + H + → H(2p) + H + . Note that the collisional transition is dominated by protons rather than electrons: the fact that the protons move more slowly enhances their ability to cause oscillations (see next € homework). 5 For the high ­n states, the same effect, driven by protons, can cause transitions between different l but the same n. At very high n, collisions with electrons (which produce a time ­dependent potential) can change the energy of the atom (hence the value of n). The qualitative effect can be understood as follows. In the capture ­cascade picture described in the previous sections, the number density of hydrogen atoms in the nl level is given by: n ( nl) = Tnl ∑ C ( n ' ' l' ', nl)α n '' l '' n e n p , n '' l '' where Tnl is the lifetime of the nl level. In comparison, at the very high n’s where collisions dominate, we expect the Saha equation to be valid: € ȹ h 2 ȹ 3 / 2 Ry / n 2 kT nSaha ( nl) = (2 l + 1)ȹ n e n p . ȹ e ȹ 2πme kT Ⱥ The ratio of the actual abundance to the Saha abundance is called the departure coefficient bnl. € The capture ­cascade equations usually predict that at large n, bnl<1. Thus at the critical n where coll...
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