(spinw-fourier)=

# spinw.fourier method

## Syntax
  
`F = fourier(obj,Q,Name,Value)`
  
## Description
  
`F = fourier(obj,hkl,Name,Value)` calculates the following Fourier sum:
 
$$J(\mathbf{k}) = \sum_{i,j} J_{i,j} * \exp(i \mathbf{k}\cdot \mathbf{d}_{i,j})$$
 
The code is optimised for calculating the sum for large number of wave
vectors and alternatively for a large number of $d_{i,j}$ vectors (large
system size). The single ion anisotropy is not included in the sum.
  
## Input Arguments
  
`obj`
: [spinw](#spinw) object.
  
`Q`
: Defines the $Q$ points where the spectra is calculated, in reciprocal
  lattice units, size is $[3\times n_{Q}]$. $Q$ can be also defined by
  several linear scan in reciprocal space. In this case `Q` is cell type,
  where each element of the cell defines a point in $Q$ space. Linear scans
  are assumed between consecutive points. Also the number of $Q$ points can
  be specified as a last element, it is 100 by defaults. 
    
  For example to define a scan along $(h,0,0)$ from $h=0$ to $h=1$ using
  200 $Q$ points the following input should be used:
  ```matlab
  Q = {[0 0 0] [1 0 0]  50}
  ```
 
  For symbolic calculation at a general reciprocal space point use `sym`
  type input. 
 
  For example to calculate the spectrum along $(h,0,0)$ use:
  ```matlab
  Q = [sym('h') 0 0]
  ```
  To calculate spectrum at a specific $Q$ point symbolically, e.g. at
  $(0,1,0)$ use:
  ```matlab
  Q = sym([0 1 0])
  ```
  
## Name-Value Pair Arguments
  
`'extend'`
: If `true`, the Fourier transform will be calculated on the
  magnetic supercell, if `false` the crystallographic cell will
  be considered. Default is `true.`
  
`'isomode'`
: Defines how Heisenberg/non-Heisenberg Hamiltonians are
  treated. Can have the following values:
  * `'off'`   Always output the $[3\times 3]$ form of the
              Hamiltonian, (default).
  * `'auto'`  If the Hamiltonian is Heisenberg, only output
              one of the diagonal values from the $[3\times 3]$
              matrices to reduce memory consumption.
  
`'fid'`
: Defines whether to provide text output. The default value is determined
  by the `fid` preference stored in [swpref](#swpref). The possible values are:
  * `0`   No text output is generated.
  * `1`   Text output in the MATLAB Command Window.
  * `fid` File ID provided by the `fopen` command, the output is written
          into the opened file stream.
  
## Output Arguments
  
`res` struct type with the following fields:
* `ft`        contains the Fourier transform in a matrix with dimensions
              $[3\times 3\times n_{magExt}\times n_{magExt}\times
              n_{hkl}]$ or $[1\times 1\times n_{magExt}\times n_{magExt}\times n_{hkl}]$
              for Heisenberg and non-Heisenberg Hamiltonians respectively
              (if isomode is `'auto'`). Here $n_{magExt}$ is the number of
              magnetic atoms in the magnetic cell and $n_{hkl}$ is the number
              of reciprocal space points.
* `hkl`       Matrix with the given reciprocal space points stored in a
              matrix with dimensions $[3\times n_{hkl}]$.
* `isiso`     True is the output is in Heisenberg mode, when the `ft`
              matrix has dimensions of $[1\times 1\times n_{magExt}\times n_{magExt}\times n_{hkl}]$,
              otherwise it is `false`.
  
## See Also
  
[spinw.optmagk](#spinw-optmagk)