Interactive Tutorial about Diffraction
Interactive example: solutions

Reciprocal space
Square lattice
Rectangular lattice
Oblique lattice
Translation of lattice
Rotation lattice

Interactive examples
2D crystal builder
Planes and HKL's
Finite size effect


Example 1:
  • The axes in real and reciprocal space are perpendicular to each other. The a axis stands normal to the b* axis and the b axis stands normal to the a* axis.
  • As you increase the length of the b axis, the corresponding length of the b* axis decreases.
  • In our two-dimensional example, the lengths of the axes are:
    a* = 1/(a sin(γ) ), b* = 1/(b sin(γ) ), γ* = 180° - γ
  • As you rotate the complete structure, the diffraction pattern is rotated as well.
  • Shifting the complete pattern has no effect on the scattering intensities.
  • The number of atoms influences the width on the Bragg peaks in the corresponding direction. As in the 1D example, increasing number of atoms results in a decreasing Bragg peak width. Try building a crystal where the number of atoms in each direction is quite different and observe the diffraction pattern. The examples in this section cover all those topics in more detail.

Example 2:

  • A plane 'hk' intersects the unit cell edges at the points 1/h and 1/k, respectively. If h or k are equal to zero, 1/h (1/k) becomes infinity and the plane is parallel to the corresponding axis. Note the difference between 'h' and '-h'!
  • As |h| and |k| increase, the distance d between the planes decreases. In a square lattice d is simply d=a/√(h*h+k*k), while for a general triclinic lattice you have to take care of the angles as well.
  • Since the planes intersect the unit cell at values 1/h, 1/k measured in multiples of the base vectors a and b, the planes move along as you increase or decrease the b/a ratio.
  • Since the planes are fixed to the unit cell, they strictly follow a rotation and a shift of the unit cell. Note, that as you shift the unit cell, the distance between the planes does not change at all.
  • The corresponding Bragg reflection is a vector in reciprocal space with indices (h,k). Its direction is normal to the planes, its length is inversly proportional to the distance d between the planes.
  • The indices (h,k) of the Bragg reflection are inversely proportional to the axis intersections in direct space. Correspondingly, the effect of changing b/a is inverse to the corresponding effect in direct space. As you shorten the b-axis, the length of the reciprocal vector along the reciprocal b-axis, i.e. the k-axis becomes longer.
  • As you increase the angle between the axes in direct space, the corresponding angle between the reciprocal axes becomes smaller. The Bragg reflection follows this distortion of the reciprocal lattice, yet remains normal to the planes.
  • The Bragg reflection follows the rotation of the direct lattice, while the shift does not affect the position of the Bragg reflections.
  • The radiation type does not affect the position of the planes or the Bragg reflection.
© Th. Proffen and R.B. Neder, 2003