# Introduction to Fiber Diffraction

#### Diffraction from a helix

Calculating the x-ray diffraction pattern from a helix was of central significance in the development of molecular biology. It was first described by Francis Crick in his doctoral thesis. He wished to understand the diffraction to be expected from an [alpha]-helix. However, the theory was very quickly applied to determining the structure of DNA. Fourier used Bessel functions to calculate the flow of heat in cylindrical objects. Bessel functions characteristically begin with a strong peak and then oscillate like a damped sine wave as x increases. The position of the first strong peak depends on the order n of the Bessel function. A Bessel function of order zero begins in the middle of the pattern, a Bessel function of order 5 has its first peak at about x = 7, a Bessel function of order 10 does everything roughly twice as far out. © Kenneth C. Holmes

Crick showed that the diffraction from a helix occurs along a series of equidistant lines rather than the Bragg spots one obtains from a three dimensional crystal. These lines (known as layer-lines) are at right angles to the axis of the fiber and the scattering along each layer-line is made up from Bessel functions. In helical diffraction Bessel functions take the place of sines and cosines one uses for crystals: Bessel functions (written Jn(x), where n is called the order and x the argument) are the form that waves take in situations of cylindrical symmetry (e.g. the waves you get if you throw a pebble into the middle of a pond). Bessel was a German astronomer who calculated accurately the orbits of the planets. Fourier used Bessel functions to calculate the flow of heat in cylindrical objects. Bessel functions characteristically begin with a strong peak and then oscillate like a damped sine wave as x increases. The position of the first strong peak depends on the order n of the Bessel function. A Bessel function of order zero begins in the middle of the pattern, a Bessel function of order 5 has its first peak at about x = 7, a Bessel function of order 10 does everything roughly twice as far out.