Bragg's Law
When x-rays are scattered from a crystal lattice, peaks of scattered intensity are
observed which correspond to the following conditions:
1. The angle of incidence = angle of scattering.
2. The pathlength difference is equal to an integer number of wavelengths.
The condition for maximum intensity contained in Bragg's law above allow us to
calculate details about the crystal structure, or if the crystal structure is known, to
determine the wavelength of the x-rays incident upon the crystal.
Click on active symbol above to perform calculation
For a wavelength
=
a lattice spacing of d =
nm and order n =
,
nm
would give a diffraction maximum at
=
degrees.
This calculation is designed to calculate wavelength, crystal plane separation or
diffraction angle. After entering data, click on the symbol of the quantity you wish to
calculate in the active graphic above. Default data will be entered for any unspecified
quantity, but all values can be changed.
Bragg Spectrometer
Much of our knowledge about crystal structure and the structure of molecules as
complex as DNA in crystalline form comes from the use of x-rays in x-ray diffraction
studies. A basic instrument for such study is the Bragg spectrometer.
To obtain nearly monochromatic x-rays, an x-ray tube is used to produce characteristic
x-rays. In order to eliminate as much of the brehmsstrahlung continuum radiation as
possible, matched filters are used in the x-ray beam to optimize the fraction of the
energy which is in the K-alpha line. Such filters use elements just above and just below
the metal in the x-ray target, making use of the strong "absorption edges" just above
and below the K-alpha energy of the target metal.
The x-rays are collimated with apertures in a strong x-ray absorber (usually lead) and
the narrow resulting x-ray beam is allowed to strike the crystal to be studied. The
spectrometer arrangement couples the rotation of the crystal with the rotation of the
detector so that the angle of rotation of the detector is twice that of the crystal. This
satisfies the conditions of Bragg's law for diffraction of the x-rays from the crystal
lattice planes.
Bragg's Law refers to the simple equation:
(eq 1)
n = 2d sin
(eq 2)
n = AB +BC .
Recognizing d as the hypotenuse of the right triangle Abz, we can use
trigonometry to relate d and to the distance (AB + BC). The distance AB
is opposite so,
(eq 3)
AB = d sin .
Because AB = BC eq. (2) becomes,
(eq 4)
n = 2AB
Substituting eq. (3) in eq. (4) we have,
(eq 1)
n = 2 d sin
and Bragg's Law has been derived. The location of the surface does not
change the derivation of Bragg's Law.