Translation of lattice
2D crystal builder
Planes and HKL's
Finite size effect
- 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
- 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.
- 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
- 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.