c2x and Radial Plots

Version 2.32 of c2x extends its 1D plotting capabilities to perform radial plots in which the quantity is averaged over all z and θ in a cylindrical co-ordinate system. This is intended to be of use in analysing nanotubes.

The cell must have the c axis as the axis of the cylinder, with a and b orthogonal to it, though they need not be the same length as each other, nor need they be orthogonal to each other.

The syntax is similar to that when plotting an unaveraged line through a system,

  -P=P1:P2:n

with P1 defining the origin and being of the form "(x,y,z)" (fractional co-ordinates), or "atom[number]", as before, and n being the number of points along the line, as before. However, the string P2 is now an "r" (for a radial plot) followed by a length. If no units are given the length will be in Angstroms, and, if the number is suffixed with a B, in Bohr.

The plot is created by first averaging over the c axis to create a 2D plane. Then, for each radius, a number of equispaced points along the circumference of a circle of that radius are chosen, and the average on sampling at these points taken. Bilinear interpolation is used for the sampling, and the spacing of the points on the circle is approximately the same as the spacing of the points along the radial direction.

As an example, from the Abinit nanotube calculation:

  $ c2x -P='(0.5,0.5,0.5):r6:100' --gnu tube_VCLMB pot.gnu
  $ gnuplot pot.gnu

results in

The slight difference in potential between the inside (left) and outside (right) is visible, along with a large dip where the nanotube wall is.

One can also average densities, such as the charge density. In this case one could use:

  $ c2x -vc -P='(0.5,0.5,0.5):r6:100' --gnu tube_DEN pot.gnu
  $ gnuplot pot.gnu

As usual, with -v c2x reports some basic information about the data read, including "int=112" stating that the integral of the density was 112 (28 atoms each with four valence electrons). For a radial plot, it also integrates the final radial data, with the expected 2πr weighting, and, in this case, reports

  Radial integral: 111.994804

Of course this value should be very similar to the integral over the whole cell, provided that the data integrated are confined to the overlap between the unit cell and the cylinder defined by the plot. and provided that a reasonable number of samples is taken.

From version 2.40d, c2x can weight these plots by 2πr (suffix a "w" to the line specification), and both weight and accumulate (suffix an "a"). From 2.41 these factors are 2πrc, where c is the length of the c-axis, so that the per unit volume part of a density is wholly removed, as it is in the spherical case. More details are found in the description of spherical plots.