Lorentz–polarization correction

Definition

A multiplicative factor involved in converting diffracted radiation intensities to structure factors during the process of structure determination for X-ray diffraction experiments involving moving crystals.

Polarization correction

X-rays are electromagnetic radiation, and so are scattered with an amplitude proportional to the sine of the angle between the direction of the electric vector of the incident radiation and the direction of scattering. The radiation from an X-ray tube is unpolarized, but may be regarded as consisting of two components, one with the electric vector normal to the plane of scattering, and the other with the electric vector lying in this plane. The angle between the normal component and the scattering vector is [math]\pi/2[/math]. The angle for the other component is [math](\pi/2)-2\theta[/math]. The intensity is proportional to the square of the amplitude, so that the polarization factor is

[math]\{\sin^2(\pi/2)+\sin^2[(\pi/2)-2\theta)]\}/2 =(1+\cos^22\theta)/2.[/math]

If the radiation has been 'monochromatized' by reflection from a crystal, it will be partially polarized, and the two parts of the beam will be of unequal intensity. The intensity of reflection then depends on the angular relations between the original, the reflected, and the scattered beams, but in the commonest arrangements all three are coplanar. The polarization factor then becomes

[math](1+A\cos^2 2\theta)/(1+A)[/math],

where [math]A=\cos^2 2\theta_M[/math] and [math]\theta_M[/math] is the Bragg angle of the monochromator crystal.

Lorentz correction

Another geometric correction that accounts for the length of time that a moving crystal remains in the diffracting position for any particular scattered beam. Since different reciprocal lattice points correspond to different sets of diffracting planes with different geometries relative to the instrument, the Lorentz factor depends both on the Bragg angle and on the diffraction geometry.

Lorentz–polarization factor

Since both corrections are dependent on the experimental conditions but not on the structural model, they are normally computed together for a given experimental geometry and applied as a single correction factor, Lp. The form of the Lorentz factor depends on the diffraction geometry. For a common case, when the original X-ray beam, the monochromatized beam and the scattered beam are all coplanar, the Lorentz correction is simply [math]1/\sin (2\theta)[/math] and the Lp factor may be written

[math]{{(1+A\cos^2 2\theta)}\over{(1+A)\sin 2\theta}}[/math],

where, as before, [math]A=\cos^2 2\theta_M[/math] and [math]\theta_M[/math] is the Bragg angle of the monochromator crystal.

History

The Lorentz factor was first reported by P. Debye [(1913). Interferenz von Röntgenstrahlen und Wärmebewegung. Ann. Phys. 348, 49–95].

See also

• Trigonometric intensity factors. H. Lipson, J. I. Langford and H.-C. Hu. International Tables for Crystallography (2006). Vol. C, ch. 6.2, pp. 596-598