====== Bragg Scattering ======

Although the proportional counter used to detect the X-rays in this experiment can distinguish different energies, its resolution is limited. Much better energy resolution can be obtained by diffracting x-rays with a crystal of known lattice spacing. Bragg reflection from a single crystal is analogous to the diffraction of visible light from an optical diffraction grating. As formulated by Bragg and von Laue, and as explained, for example, in Kittel's //Introduction to Solid State Physics//, the condition for constructive interference of diffracted rays is two-fold.

First, we must satisfy the equation

| $n\lambda = 2d \sin\theta$  | (4)  |

where //n// is an integer called the //order number//, $\lambda$ is the wavelength of the x-ray, $d$ is the distance between neighboring planes of atoms in the crystal, and $\theta$ is the angle between the incident x-rays and the surface of the crystal. (See Fig. 4).

If you need to propagate uncertainties in Eq. (4), make sure that $\Delta \theta$ is in radians.

| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:plane_spacing.png?400 |}} |
| **Figure 4**: The geometry of Bragg scattering. (Source: [[https://en.wikipedia.org/wiki/File:DiffractionPlanes.png|Wikipedia]]). |

Second, since the crystal planes form a three-dimensional “grating”, in order for phases to add constructively, the angle of incidence must equal the angle of diffraction. (This constraint is not present for an optical diffraction grating.)

If a parallel, polychromatic beam of x-rays is incident on a crystal, the only wavelengths that will be diffracted constructively will be the wavelengths satisfying the Bragg condition (both parts!) for that angle of incidence. Thus, diffraction can be used to separate different wavelengths into different angles for quantitative analysis.