Photo-Ionization

where is the (negative) hydrogen ground-state energy, is assumed to be positive, which corresponds to an unbound state. In other words, the absorption of the photon causes the hydrogen atom to dissociate.

Let
and
**
**
be the wavevector and electric polarization vector of the photon, respectively. (Recall that
and
**
**
are both unit vectors.) It follows from Equations (8.136) and
(8.152) that the absorption cross-section of the hydrogen atom can be written

(8.222) |

where is the electron momentum, and . Here, we have written , rather than , because we are only considering final states in which the electron momentum, , is directed into the range of solid angles . As discussed in Section 8.6, we must operate on the previous expression with , where is the density of final states, to obtain the true absorption cross-section, which takes the form

Here, we have made use of the fact that is an Hermitian operator, and have also employed the result , where is a constant. (See Exercise 19.)

The electron is initially in the hydrogen ground state, whose wavefunction takes the form

where is the Bohr radius. (See Chapter 4.) On the other hand, the final electron state is assumed to be a plane-wave state whose wavefunction is

In writing this expression, we have conveniently assumed that the emitted electron is contained in a cubic box of dimension , and volume . Furthermore, the electron wavefunction is required to be periodic at the boundaries of the box. Of course, we can later take the limit to obtain the general case. (However, it turns out that this is not necessary, because the final result does not depend on , provided that .) The wavefunction of the final electron state is normalized such that the probability of finding the electron in the box is unity: that is,

(8.226) |

Note that the final state is an eigenstate of belonging to the eigenvalue

(This follows from the standard Schrödinger representation .) Furthermore, in writing the wavefunction (8.228), we have neglected any interaction between the emitted electron and the hydrogen nucleus. This neglect is only reasonable provided the final electron energy,

is much larger than the ionization energy (i.e., the binding energy) of the hydrogen atom,

(8.229) |

The condition yields

In other words, the de Broglie wavelength of the emitted electron must be much larger than the typical dimension of the hydrogen atom.

Let us calculate . Suppose that . The periodicity constraint on at the boundaries of the box imply that

(8.231) | ||

(8.232) | ||

(8.233) |

where , , and are integers. It follows that

(8.234) |

where , et cetera. Thus, the number of final electron states contained in a volume of momentum space is , where

(8.235) |

Note that is constant. Hence, we deduce that the number of final electron states for which lies between and , and is directed into the range of solid angles , is , where . In other words,

(8.236) |

Finally, the number of final electron states whose energies lie between and is , which yields

where use has been made of Equations (8.230) and (8.231).

Equations (8.226) and (8.240) imply that the differential cross-section for the photo-ionization of atomic hydrogen takes the form

Furthermore, making use of Equations (8.227) and (8.228), we can write

(8.239) |

where

(8.240) |

Note, by momentum conservation, that is the recoil momentum of the hydrogen nucleus after ionization. The usual Schrödinger representation reveals that

where we have made use of the standard electromagnetic result

Assuming that , we can write

(8.242) |

where . Here, , are spherical angles whose polar axis is parallel to . Thus, we obtain

However (see Exercise 20),

(8.244) |

which implies that

Thus, Equations (8.241), (8.244), and (8.248) yield

where use has been made of the standard electromagnetic dispersion relation [49].

It is convenient to orientate our coordinate system such that
**
**
and
.
Thus, we can specify the direction of the emitted electron in terms of the conventional
polar angles,
and
. In fact,

(8.247) |

and

where

(8.249) |

is the normalized ionization energy.

Now, energy conservation demands that

(8.250) |

[See Equation (8.224).] This expression can be rearranged to give

and

where

(8.253) |

Here, is the speed of the emitted electron. However, we have already seen, from Equation (8.233), that the approximations made in deriving Equation (8.251) are only accurate when . Hence, according to Equation (8.254), we also require that . Furthermore, because we are working in the non-relativistic limit, we need . From Equation (8.255), this necessitates

(8.254) |

Finally, the inequality can be combined with the previous inequality to give

(8.255) |

as the condition for the validity of Equation (8.251).

Now,

(8.256) |

which implies that

where use has been made of Equation (8.255), and we have neglected terms of order . Equations (8.251), (8.254), and (8.260) can be combined to give the following expression for the differential photo-ionization cross-section of atomic hydrogen:

Integrating over all solid angle, and neglecting terms of order , the total cross-section becomes

(8.259) |

(See Exercise 21.)

Note that the previous two expressions are only accurate in the limits and . Nevertheless, we have retained the factors in these formulae because they emphasize that there is a threshold photon energy for photo-ionization: namely, . For (i.e., ), the incident photons are not energetic enough to ionize the hydrogen atom, so there are no emitted photoelectrons. Consequently, the cross-section for photo-ionization tends to zero as approaches unity from below. We have retained the factor involving in Equation (8.261) because it makes clear that the photoelectrons are emitted preferentially in the directions , and . Thus, in the non-relativistic limit, , the electrons are emitted preferentially along the -axis (i.e., parallel or anti-parallel to the incident photon's electric polarization vector.) On the other hand, as relativistic effects become important, and consequently increases, the directions of preferentially emission are beamed forward (i.e., they acquire a component parallel to the wavevector of the incident photon.) An accurate expression for the photo-ionization cross-section close to the ionization threshold (i.e., ) can only be obtained using unbound positive energy hydrogen atom wavefunctions as the final electron states [106]. Likewise, an accurate expression for the cross-section in the finite- limit requires a relativistic treatment of the problem [96,97].