Index

Modulated structures dictionary (msCIF) version 3.2.1

Category ATOM_SITE_DISPLACE_XHARM

Parent category:
MS_GROUP

Definition:

   
 
      The set of harmonic functions used in the Fourier series describing the 
      Modulation functions is orthogonal and complete in the interval [0,1). 
      However within of the x4 interval defined by a Crenel function orthogonality
      is no longer preserved and therefore the Fourier coefficients are correlated
      and the refinement becomes fragile. There are several ways to avoid this
      technical problem (see Petricek et al., 2016). One of them is to use
      orthogonal or orthogonalized sets of functions defined within the Crenel
      interval. This procedure is more robust than the orthogonalization of harmonics 
      described in *_ORTHO. categories. Moreover these sets of functions are 
      complete. Two different sets of orthogonal or orthogonalized functions have 
      been implemented in JANA2006: Legendre polynomials and the so-called 
      x-harmonics. x-harmonic functions are defined from the set (Petricek, Eigner, 
      Dusek & Cejchan, 2016):

      {1, x, sin(\p x), cos(\p x), ... , sin(n\p x), cos(n\p x)}

      and a subsequent orthogonalization (see the above reference and the 
      supplementary material) owing to the presence of x which is not orthogonal to 
      sin(n\p x) (for any n). Notice that x-harmonics are restricted to one-
      dimensional cases and include as a particular case the sawtooth modulation.

      Data items in the ATOM_SITE_DISPLACE_XHARM category record details about the 
      x-harmonic functions used to describe the displacive modulations when the 
      atomic domain of a given atom is restricted by a crenel function. In the case 
      of rigid groups, items in this category would only include the translational 
      part of the modulation. The rotational part would appear in a separate list of 
      items belonging to the ATOM_SITE_ROT_XHARM category. 

      References: Petricek, V., Van Der Lee & Evain, M. (1995).
              Acta Cryst. A51, 529-535. DOI 10.1107/S0108767395000365
              On the Use of Crenel Functions for Occupationally Modulated  
              Structures 

              Petricek, V., Eigner, V., Dusek, M. & Cejchan, A. (2016). Z. 
              Kristallogr. 231(5), 301-312. DOI 10.1515/zkri-2015-1913 
              Discontinuous modulation functions and their application for 
              analysis of modulated structures with the computing system JANA2006 

Category is of type Loop