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model = sqw_sine3d(p, h,k,l,w, {signal}) : sine dispersion(HKL) with DHO(energy)


function signal=sqw_sine3d(varargin)


 model = sqw_sine3d(p, h,k,l,w, {signal}) : sine dispersion(HKL) with DHO(energy)

   iFunc/sqw_sine3d: a 4D S(q,w) with a 3D HKL sine dispersion, and a DHO line
      shape. The sine dispersion can be tuned with energy gaps at zone centre
      and boundary, as well as a periodicity of the sine wave.

 WARNING: Single intensity and line width parameters are used here.

 Along a principal Q axis, the dispersion has the form:
       w = E0 + (E1-E0)*sin(Q_freq*pi*(Q-Q0));
 This is a sine wave which goes from w=E0 at Q=Q0, up to w=E1 at 1/2Q_freq.
 The inter-atomic distance between the scattering units (atoms) is thus:
       d=a*Q_freq/2 [in Angs, with a=lattice parameter]

 A spin wave could for instance mostly use Q0=0, Q_freq=1,  E0=0, E1>0
   For a simple ferromagnetic magnon the gap width is E1-E0=4JS 
     with J=exchange energy and S=magnetic moment of spins.
 An acoustic branch could use           Q0=0, Q_freq=.5, E0=0, E1>0
   The sound velocity in acoustic branches is:
       c=E1*Q_freq*pi*1.519e2*a/2 [in m/s, with a=lattice parameter]
 An optical branch could use            Q0=0, Q_freq=.2, E0>E1 E1>0

 To shift the minimum/maximum Q of the dispersion, move QH0,QK0,QL0
 To change the extent of the dispersion in Q, vary QH_freq,QK_freq,QL_freq
 To change to minimum and maximum energy, move E0 and E1_qh,E1_qk,E1_ql
 To model an incomensurate dispersion, move both QH0,QK0,QL0 and the
   QH_freq,QL_freq,QK_freq to incomensurate (non rational) values.

 To quickly create predefined models, use:
   model=sqw_sine3d(Emax)      creates an acoustic dispersion up to Emax
   model=sqw_sine3d([ E0 E1 ]) creates an optical dispersion from E0 to E1
   model=sqw_sine3d([ E0 E1 Q_freq ])    creates a dispersion from E0 to E1
                                      with given Q frequency, e.g. .5, 1 or 2
 Of course, as for any iFunc model, parameter values can be changed afterwards.

 To model more than one branch, just add these models together, e.g.:
     disp3 = sqw_sine3d(5) + sqw_sine3d([ 10 8 ]) + sqw_sine3d([ 2 4 1 ])
 which is an acoustic plus optical branch, and a spin wave with 2 meV gap.

   s=sqw_sine3d(5); qh=linspace(0,1,50);qk=qh; ql=qh'; w=linspace(0.01,10,50);
   f=iData(s,s.p,qh,qk,ql,w); plot(log(f(:,:,1,:)));

 Reference: https://en.wikipedia.org/wiki/Phonon

 input:  p: sqw_sine3d model parameters (double)
             p(1)= E1_qh   energy at QH half period [meV]
             p(2)= E1_qk   energy at QK half period [meV]
             p(3)= E1_ql   energy at QL half period [meV]
             p(4)= E0      zone-centre energy gap [meV]
             p(5)= QH0     QH zone-centre [rlu]
             p(6)= QK0     QK zone-centre [rlu]
             p(7)= QL0     QL zone-centre [rlu]
             p(8)= QH_freq QH frequency [multiples of pi]
             p(9)= QK_freq QK frequency [multiples of pi]
             p(10)=QL_freq QL frequency [multiples of pi]
             p(11)=Gamma   dispersion DHO half-width in energy [meV]
             p(12)=Temperature of the material [K]
             p(14)=Background (constant)
          or p='guess'
         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: Nov. 26, 2018
 See also iData, iFunc/fits, iFunc/plot, gauss, sqw_phonons, sqw_cubic_monoatomic, sqw_vaks
   <a href="matlab:doc(iFunc,'Models')">iFunc:Models</a>
 (c) E.Farhi, ILL. License: EUPL.


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