** Next:** The Atomic Scattering Factor. The Oscillator
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In the classical theory of dispersion the atom is assumed to scatter radiation
as if it was formed by dipole oscillators whose natural frequencies are those of
the absorption edges of the electronic shells. These oscillators may be thought
of as originated by simple harmonic vibrations of the electronic charges, as for
instance, the movement of an electron of mass *m* around the positive nucleus
assumed to be at rest. An electromagnetic wave falling on the atom and having
an instantaneous electric field at
the position of the dipole, sets the electron in oscillation, the displacement
of the atom satisfies then the differential equation:

(1) |

The forced solution of (1) is:

(2) |

(3) |

This oscillating dipole radiates with the same frequency of oscillation; the amplitude of the wave at a unit distance in the equatorial plane being

(4) |

The scattering factor of the dipole is defined, as usual, as the ratio of the
scattered amplitude *A* to that scattered by a free electron *A*_{e} under the
same conditions. In this case, *A*_{e}, the Thomson amplitude, is obtained by
taking = 0, *k* = 0:

(5) |

The dipole scattering factor is then given by

(6) |

Expression (6) is very important since the atomic scattering factor results from
a superposition of similar terms by considering the atom as made up of a
distribution of dipole oscillators. Let us denote by () and
() respectively the real and the imaginary parts of
*f*:

(7) |

If a medium is composed of *N* similar dipoles per unit volume, it can be shown
that the refractive index *n* is also complex and given by

(8) |

which we rewrite as

with

(9) |

The fact that *n* is complex and particularly so when approximates
, indicates that the medium is absorbent. In fact by taking the
origin of phases at an arbitrary origin *O*, the phase, after the wave has
travelled a distance *r*, is

(10) |

After substitution of and division by *N* one obtains the
linear absorption coefficient per dipole in the medium for the circular
frequency:

(11) |

Reciprocally, the imaginary component of the dipole scattering factor of an absorbent medium is given by:

(12) |

Obviously and have their respective maximum values
for a frequency close to ; the value of the damping coefficient *k*
being the breadth of the absorption peak at half height. The smaller the value
of *k* the sharper the absorption peak which then becomes a line. If a beam of
white radiation is passing through the medium only frequencies near to
will be significantly absorbed.

**Copyright © 1981, 1998 International Union of
Crystallography**