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					Physics PH15720 Laboratory Practical

Microwave optics: The Bragg Law.

Crystals of varying complexity comprise a periodic array of atoms in
three dimensions. The spacing of these atoms is such that they will act as
a diffraction grating for electromagnetic waves that have a wavelength
comparable to the atomic spacing. Indeed the first proof of the wave
nature of X-rays was the demonstration of diffraction o X-rays from a
thin crystal. A series of bright spots, corresponding to the scattering of X-
rays from the electrons surrounding atoms could be seen on photographic
film surrounding a central bright spot. This is Laué Diffraction and the
spots reflect the different geometrical arrangement of atoms in crystals. A
further application of the wave theory of light to the study of crystals was
the study of the atomic structure of crystals by X-rays started by Sir
William Bragg in 1914.

Because atoms have a regular arrangement in crystals Bragg found that
monochromatic beams of X-rays could be reflected from these planes.
When the atoms interact with X-rays, the X-rays are scattered as spherical
waves, using the Huygens construction, planar wavefronts can be drawn,
connecting the point scattering atoms. If the waves are in phase with each
other a strong reflected wave is produced. This condition can be
formulated into the so-called Bragg Law:

2d sin   n
Where  is the wavelength of the electromagnetic radiation, and n is an
integral number. The incident wave makes a glancing angle  with atom
planes that are spaced a distance d apart.

In this experiment we will explore the Bragg condition using
microwaves, since microwave radiation has a wavelength that is much
longer than the distance between atoms we will use a macroscopic cubic
crystal with metal spheres embedded in a foam cube. The arrangement of
these spheres is analogous to the spheres in a simple cubic crystal lattice.

Using the microwave equipment, place the cube on the rotating table and
to determine the intensity of the scattered radiation as a function of angle
().You will need:

    A microwave transmitter
    A microwave receiver
    Rotating table

Martin Wilding                         Page 1                  26 March 2010
Physics PH15720 Laboratory Practical

    Goniometer
    Foam cube with metal spheres (cubic lattice)

Using the rotating table, rotate the cube so that the incident radiation
makes a grazing angle, , with a plane of spheres. The Bragg condition is
satisfied when the angle of incidence is equal to the angle of Bragg
reflection. Using this geometry and from the cube, document the variation
of intensity as a function of angle for the cubic “crystal”. Assume that the
wavelength of the microwaves used is 2.85cm There are several families
of planes even in a simple cubic lattice, these have designations (100),
(110) and (210) in two-dimensions, the Miller Indices. Document the
intensity as a function of angle and plot these data, allocate a miller index
to each peak in your diffraction pattern. Comment on these data. What
other miller indices might you expect in a cubic lattice? If you had more
than one atom type, how would you distinguish between them in a
diffraction pattern?

Martin Wilding                         Page 2                  26 March 2010