Category ATOM_SITE_DISPLACE_LEGENDRE

Name:
'ATOM_SITE_DISPLACE_LEGENDRE'

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 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. Legendre polynomials are orthogonal in the Crenel interval and can 
      be easily calculated by the recurrence relation:

      P~0~(x) = 1
      P~1~(x) (x) = x
      (n+1)P~n+1~(x) = (2n+1)x P~n~(x) - nP~n-1~(x)

      Notice that Legendre polynomials are restricted to one-dimensional cases and 
      include as a particular case the sawtooth modulation.

      Data items in the ATOM_SITE_DISPLACE_LEGENDRE category record
      details about the Legendre polynomials 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_LEGENDRE 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

Category key:
_atom_site_displace_Legendre.id