Novel study on the electron density distribution projection maps calculated via the discrete cosine transform

It is already known that a curve estimated via the discrete cosine transform (DCT) always passes through all the measured points without using imaginary numbers in the DCT coefficients, unlike the discrete Fourier transform. Moreover, owing to its character, the DCT can be used instead of the nonlinear least-squares method to express various theoretical curves. Because the DCT is a kind of Fourier transform, there is a possibility that the DCT could be employed to draw electron density distribution maps of crystals. If so, the probability that the DCT could be used to investigate the internal structure of materials by analysing the theoretical curves would increase. This article reports an attempt to draw the electron density distribution maps of the Mg₃BN₃ low-pressure phase [Mg₃BN₃(L)] by using the DCT in order to confirm the utility of the DCT for analysing the internal structure of materials. It is found that the DCT can provide mirror-symmetric electron density distribution projection maps and a modified DCT can be used to calculate whole standard electron density distribution projection maps for the surface plane of the unit cell. Moreover, a real crystal structure that has a centre of symmetry can be determined by the DCT by transforming a 1/4 part of the mirror-symmetric electron density distribution projection map.