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PSCF v1.1
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Commands - pscf_pc (Up/Prev) Commands - pscf_pg (Next)
THE WRITE_WAVES command accepted by pscf_pc and pscf_pg programs writes a detailed description of the symmetry-adapted basis used by these programs. Closely related information about the basis is also provided by the WRITE_STARS command.
Conceptual background : As background, before reading this description of the output of the WRTIE_WAVES command, we strongly recommend that users read the discussion of the relationships between "stars" and basis functions given in the documentation of the basis file format. Users who seek a more complete discussion can also consult appendices that discuss crystallographic space group symmetry operations and symmetry-adapted bases.
The type of symmetry-adapated Fourier basis used by the pscf_pc and pscf_pg programs is constructed by grouping all of the reciprocal lattice wavevectors used in a discrete Fourier transform of periodic functions into subsets of wavevectors that are related to one another by space group symmetry operations. We refer to each such subset of symmetry-related wavevectors as a star of wavevectors. A basis function that is invariant under all operations of the space group, known as a star function, is created as a linear superposition of waves with wavevectors that belong to each uncancelled star. The report created by the WRITE_WAVES command writes a list of all of wavevectors that belong to uncancelled stars, in which the row associated with each wavevector gives integer indices for the wavevector, the integer index of the star to which that wavevector belongs, and the coefficients of the corresponding wave within the associated star function
For users who examine the source code : The format of this report is defined by the outputWaves function of class template Basis<int D>, which is defined in the file src/pscf/crystal/Basis.tpp.
The file created by the MAKE_WAVES command contains a header section similar to that used for field files followed by a long data section. The last line of the data section contains the value for N_wave, which is the total number of wavevectors listed in the data section.
The data section contains one row for each reciprocal lattice wavevector that belongs to an uncancelled star. Wavevectors that belong to cancelled stars are omitted. Waves that belong to the same star are are assigned a continuous block of values for the index waveId, and are listed as a consecutive block within this file. Because all of the wavevectors in a star are related by symmetry operations such as rotations and reflections, they are all reciprocal lattice vectors of equal magnitude. Stars appear in increasing (or non-decreasing) order of the magnitude of the constituent wavevectors, as computed using the unit cell parameters that were available when the basis was constructed.
The format of each row of the data section is as follows
in which:
Please see the resources recommended above for definitions of what we mean by a star, a cancelled star, and a star function.
An example of the file created by a WRITE_WAVES command is shown below for a system with space group \( I a\overline{3} d \), or PSCF space group identifier I_a_-3_d. This is the space group of the double gyroid phase of a diblock copolymer. A long section has been removed from the data section of this file, as indicated by vertical dots.
The first row of the data section in this example (waveId = 0) represents the zero wavevector, with integer components (0,0,0). This is the only wave in the first star, which has a starId = 0. Every space group has a star consisting of just the zero vector, which is always the first line in the output of the WRITE_WAVES command.
The next 24 rows (with i = 1, ..., 24 and waveId = 57, ..., 80) represent wavevectors that belong to the {2,1,1} family of wavevectors, which all belong to a star with starId = 6. This is the first star of nonzero wavevectors that is allowed by the rules for systematic cancellations in crystals with this space group.
The next 12 rows (with i = 25, ..., 36 and waveId = 81, ..., 92) represent wavevectors that belong to the {2,2,0} family of wavevectors, which all belong to a star with starId = 6. This is the first star of nonzero wavevectors that is allowed by the rules for systematic cancellations in crystals with this space group.
Stars of wavevectors for a cubic crystal are always sets of wavevectors with indices that are related to a list (h, k, l) by permutation and changes in sign of the indices. The number of wavevectors in such a star is equal to 48 if h, k, l and all distinct and nonzero, and can be less than 48 if some of the indices are equal and/or zero, as is true for the {211} and {220} stars.
Waves belonging to stars with starId values 1,...,5, which have waveId values 1, ..., 56, are not listed in this file because these stars are all cancelled stars (corresponding to families of wavevectors that exhibit systematic cancellation in scattering experiments), and thus cannot be used to construct symmetry-adapted basis functions.
The space group used in this example has an inversion center at the origin. The existence of an inversion center at the origin can be shown to always leads to star functions for which the coefficient of every star is a real number. This is why the last column, which gives the imaginary part of this coefficient for each wave, is zero in every row of the data section.
Commands - pscf_pc (Up/Prev) Commands - pscf_pg (Next)
1.9.5