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model = sqw_vaks(p, h,k,l,w, {signal}) : dispersion(HKL) in perovskites ABX3 with DHO(energy)


function signal=sqw_vaks(varargin)


 model = sqw_vaks(p, h,k,l,w, {signal}) : dispersion(HKL) in perovskites ABX3 with DHO(energy)

   iFunc/sqw_vaks: a 4D S(q,w) with a HKL dispersion, and a DHO line
      shape, based on a parameterisation by Vaks, suited for ABX3 perovskite 
      compounds. This parameterisation is based on a quadratic expansion of the
      Hamiltonian, valid for e.g. |k|<.3 rlu in a cubic 
      crystal and provides energies of TA1,TA2,LA,TO1 and TO2 modes.

      Single intensity and line width parameters are used here.
      The HKL position is relative to the closest Bragg peak. 

   The acoustic parameters can be related to the elastic constants as:
     At= (C12+2C44)/rho [meV2/rlu2]
     Al=  C44/rho
     Aa= (C11-C12-2C44)/rho
   with rho=density of the crystal [g/cm3]

 s=sqw_vaks('KTaO3'); qh=linspace(0,.5,50);qk=qh; ql=qh; w=linspace(0.01,10,51);
 f=iData(s,[],qh,qk,ql,w); scatter3(log(f(:,:,1,:)),'filled');

 References: E. Farhi et al, EPJB, 15 (2000) pp 615-623. DOI: 10.1007/s100510051164 
             A.K. Tagantsev et al, Nature Comm, 4 (2013) 2229. DOI: 10.1038/ncomms3229
             V.G. Vaks, Introduction to the Microscopic Theory of Ferroelectrics (Nauka, Moscow, 1973)

 input:  p: sqw_vaks model parameters (double)
           p(1)=At transverse acoustic [meV2/rlu2]
           p(2)=Al longitudinal acoustic [meV2/rlu2]
           p(3)=Aa anisotropic acoustic [meV2/rlu2]
           p(4)=St soft mode  transverse soft optical [meV2/rlu2]
           p(5)=Sa soft mode  anisotropic soft optical [meV2/rlu2]
           p(6)=Vt transverse acoustic-optical coupling [meV2/rlu2]
           p(7)=Va anisotropic acoustic-optical coupling [meV2/rlu2]
           p(8)=w0 soft mode frequency at q=0, depends on temperature [meV] 
           p(9)=Gamma   dispersion DHO half-width in energy [meV]
           p(10)=Temperature of the material [K]
           p(12)=Background (constant)
          or 'KTaO3','BaTiO3','SrTiO3' for predefined settings
         qh: axis along QH in rlu (row,double)
         qk: axis along QK in rlu (column,double)
         ql: axis along QL in rlu (page,double)
         w:  axis along energy in meV (double)
    signal: when values are given, a guess of the parameters is performed (double)
 output: signal: model value

 Version: Aug. 22, 2017
 See also iData, iFunc/fits, iFunc/plot, gauss, sqw_phonons, sqw_sine3d
   sqw_cubic_monoatomic, <a href="matlab:doc(iFunc,'Models')">iFunc:Models</a>
 (c) E.Farhi, ILL. License: EUPL.


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