text: '''Collection of classes and functions for loading, interpolation and processing of atomic data. 
  Refer also to the adas_files.py script. 
  '''
  import numpy as np
  import matplotlib.pyplot as plt
  from scipy.interpolate import RectBivariateSpline, interp1d
  from matplotlib import cm
  import os, sys, copy
  import scipy.ndimage
  from scipy.linalg import svd
  from scipy import constants
  
  from . import adas_files
  
  
  def get_adas_file_types():
      '''Obtain a description of each ADAS file type and its meaning in the context of Aurora.
  
      Returns
      ------------
      dict
          Dictionary with keys given by the ADAS file types and values giving a description for them.
  
      Notes
      ---------
      For background on ADAS generalized collisional-radiative modeling and data formats, refer to 
      [1]_.
  
      References
      -----------------
      
      .. [1] Summers et al., "Ionization state, excited populations and emission of impurities 
         in dynamic finite density plasmas: I. The generalized collisional-radiative model for 
         light elements", Plasma Physics and Controlled Fusion, 48:2, 2006
  
      '''
      
      return {'acd':'effective recombination',
              'scd':'effective ionization',
              'prb':'continuum radiation',
              'plt':'line radiation',
              'ccd':'thermal charge exchange',
              'prc':'thermal charge exchange continuum radiation',
              'pls':'line radiation in the SXR range',
              'prs':'continuum radiation in the SXR range',
              'brs':'continuum spectral bremstrahlung',
              'fis':'sensitivity in the SXR range',
              'pbs':'impurity bremsstrahlung in SXR range, also included in prs files'
      }
  
  
  class adas_file:
      '''Read ADAS file in ADF11 format over the given density and temperature grids. 
      Note that such grids vary between files, and the species they refer to may too.
  
      Refer to ADAS documentation for details on each file.
  
      Parameters
      ----------
      filepath : str
          Path to location where ADAS file is located.
      '''
      def __init__(self, filepath):
          
          self.filepath = filepath
          self.filename=filepath.split('/')[-1]
          self.file_type = self.filename[:3]
  
          if self.file_type not in ['brs','sxr']:
              self.imp = self.filename.split('_')[1].split('.')[0]
  
          # get data
          self.load()
  
          # settings for plotting
          self.n_ion  = self.data.shape[0]
          self.ncol = np.ceil(np.sqrt(self.n_ion))
          self.nrow = np.ceil(self.n_ion/self.ncol)
  
  
      def load(self):
  
          with open(self.filepath) as f:
              header = f.readline()
              n_ions, n_ne, n_T = header.split()[:3]
              details = ' '.join(header.split()[3:])
  
              f.readline()
  
              n_ions,n_ne,n_T = int(n_ions),int(n_ne),int(n_T)
              logT = []
              logNe = []
              while len(logNe)< n_ne:
                  line = f.readline()
                  logNe = logNe+[float(n) for n in line.split()]
              while len(logT)< n_T:
                  line = f.readline()
                  logT = logT+[float(t) for t in line.split()]
  
              logT = np.array(logT)
              logNe = np.array(logNe)
  
              data = []
              for i_ion in range(n_ions):
                  f.readline()
                  plsx = []
                  while len(plsx)< n_ne*n_T:
                      line = f.readline()
                      plsx = plsx+[float(L) for L in line.split()]
                  plsx = np.array(plsx).reshape(n_T, n_ne)
                  data.append(np.array(plsx))
  
          self.logNe = logNe; self.logT = logT; self.data = np.array(data)
  
  
      def plot(self, fig=None, axes=None):
          '''Plot data from input ADAS file. If provided, the arguments allow users to overplot
          and compare data from multiple files. 
  
          Parameters
          ----------
          fig : matplotlib Figure object
              If provided, add specification as to which ADAS file is being plotted.
          axes : matplotlib Axes object (or equivalent)
              If provided, plot on these axes. Note that this typically needs to be a set of axes
              for each plotted charge state. Users may want to call this function once first to get
              some axes, and then pass those same axes to a second call for another file to compare with.
          '''
          if fig is None or axes is None:
              fig,axes = plt.subplots(int(self.ncol),int(self.nrow), sharex=True, sharey=True)
  
          axes = np.atleast_2d(axes)
          colormap = cm.rainbow
          if fig is not None:
              fig.suptitle(self.filename+'  '+ get_adas_file_types()[self.file_type])
  
          for i,ax in enumerate(axes.flatten()):
              if i >= self.n_ion: break
              if all(self.data[i,:,:].std(axis=1)==0): #independent of density
                  ax.plot(self.logT, self.data[i,:,0])
              else:
                  ax.set_prop_cycle('color', colormap( np.linspace(0,1,self.data.shape[2])))
                  ax.plot(self.logT, self.data[i])
              ax.grid(True)
              if self.file_type != 'brs':
                  charge = i+1 if self.file_type in ['scd','prs','ccd','prb'] else i+2
                  ax.set_title(self.imp+'$^{%d\!+}$'% (charge-1))   # check?
  
          for ax in axes[-1]:
              ax.set_xlabel('$\log T_e\ \mathrm{[eV]}$')
          for ax in axes[:,0]:
              if self.file_type in ['scd','acd','ccd']:
                  ax.set_ylabel('$\log('+self.file_type+')\ \mathrm{[cm^3/s]}$')
              elif self.file_type in ['prb','plt','prc','pls','brs','prs']:
                  ax.set_ylabel('$\log('+self.file_type+')\ \mathrm{[W\cdot cm^3]}$')
  
  
  def read_filter_response(filepath, adas_format=True, plot=False):
      '''Read a filter response function over energy. 
  
      This function attempts to read the data checking for the following formats (in this order):
      
      #. The ADAS format. Typically, this data is from obtained from http://xray.uu.se and produced
         via ADAS routines. 
  
      #. The format returned by the `Center for X-Ray Optics website  <https://henke.lbl.gov/optical_constants/filter2.html>`__ .
  
      Note that filter response functions are typically a combination of a filter transmissivity 
      and a detector absorption. 
  
      Parameters
      ----------
      filepath : str
          Path to filter file of interest.
      plot : bool
          If True, the filter response function is plotted. 
      
      '''
      E_eV=[]
      response=[]
      try:
          # Attempt to read ADAS format
          with open(filepath) as f:
  
              header = f.readline()
              num = int(header.split()[0])
  
              # *****
              f.readline()
  
              while len(E_eV)< num:
                  line = f.readline()
                  E_eV += [float(n) for n in line.split()]
              while len(response)< num:
                  line = f.readline()
                  response += [float(t) for t in line.split()]
  
          # energy and response function are written in natural logs
          E_eV = np.concatenate(([0.,], np.array(np.exp(E_eV))))
          response = np.concatenate(([0.,], np.array(np.exp(response))))
      except ValueError:
          try:
              # Attempt to read CXRO format
              with open(filepath) as f:
                  contents = f.readlines()
  
              for line in contents[2:]:
                  tmp = line.strip().split()
                  E_eV.append(float(tmp[0]))
                  response.append(float(tmp[1]))
              E_eV=np.concatenate(([0.,], np.array(E_eV)))
              response=np.concatenate(([0.,],np.array(response)))
          except ValueError:
              raise ValueError('Unrecognized filter function format...')
      
      if plot:
          fig,ax = plt.subplots()
          ax.semilogx(E_eV, response)
          ax.set_xlabel('Photon energy [eV]')
          ax.set_ylabel('Detector response efficiency')
          plt.tight_layout()
  
      return E_eV, response
  
      
  def get_atom_data(imp, files=['acd','scd']):
      ''' Collect atomic data for a given impurity from all types of ADAS files available or
      for only those requested. 
  
      Parameters
      ----------
      imp : str
          Atomic symbol of impurity ion.
      files : list or dict
          ADAS file types to be fetched. Default is ["acd","scd"] for effective ionization 
          and recombination rates (excluding CX) using default files, listed in :py:func:`~aurora.adas_files_adas_files_dict`.
          If users prefer to use specific files, they may pass a dictionary instead, of the form
  
          .. code-block:: python
  
              {'acd': 'acd89_ar.dat', 'scd': 'scd89_ar.dat'}
  
          or
  
          .. code-block:: python
  
              {'acd': 'acd89_ar.dat', 'scd': None}
  
          if only some of the files need specifications and others (given as None) should be 
          taken from the default files.
      
      Returns
      -------
      atom_data : dict
          Dictionary containing data for each of the requested files. 
          Each entry of the dictionary gives log-10 of ne, log-10 of Te and log-10 of the data
          as attributes atom_data[key].logNe, atom_data[key].logT, atom_data[key].data
      '''
  
      try:
          # default files dictionary
          all_files = adas_files.adas_files_dict()[imp] # all default files for a given species
      except KeyError:
          raise KeyError(f'ADAS data not available for ion {imp}!')
      
      if isinstance(files, dict):
          # user provided files choices as a dictionary
          for filename in files:
              if files[filename] is not None:
                  all_files[filename] = files[filename]
  
      # exclude files not of interest
      keys_to_delete = [key for key in all_files if key not in files]
      for key in keys_to_delete:
          del all_files[key]
  
      for filecheck in files:
          if filecheck not in all_files:
              raise ValueError(f'Could not fetch {imp} {filecheck.upper()} file!')
      
      atom_data = {}
      for filetype in all_files:
  
          # find location of required ADF11 file
          fileloc = adas_files.get_adas_file_loc(all_files[filetype],filetype='adf11')
      
          # load specific file and add it to output dictionary
          res = adas_file(fileloc)
          atom_data[filetype] = res.logNe, res.logT, res.data
  
      return atom_data
  
  
  def null_space(A):
      '''Find null space of matrix `A`.
      '''
      u, s, vh = svd(A, full_matrices=True)
      Q = vh[-1,:].T.conj()    # -1 index is after infinite time/equilibration
      # return the smallest singular eigenvalue
      return Q, s[-2]  # matrix is singular, so last index is ~0
  
  
  
  def superstage_rates(R, S, superstages,save_time=None):
      '''Compute rate for a set of ion superstages. 
      Input and output rates are log-values in arbitrary base.
  
      Parameters
      ----------
      R : array (time,nZ,space)
          Array containing the effective recombination rates for all ion stages, 
          These are typically combinations of radiative and dielectronic recombination, 
          possibly also of charge exchange recombination.
      S : array (time,nZ,space)
          Array containing the effective ionization rates for all ion stages.
      superstages : list or 1D array
          Indices of charge states of chosen ion that should be included. 
      save_time : list or 1D array of bools
          Indices of the timeslcies which are actually returned by AURORA
      
      Returns
      -------
      superstages : array
          Set of superstages including 0,1 and final stages if these were missing in the input.
      R_rates_super : array (time,nZ-super,space)
          effective recombination rates for superstages
      S_rates_super : array (time,nZ-super,space)
          effective ionization rates for superstages
      fz_upstage : array (space, nZ, time_
          fractional abundances of stages within superstages
  
      '''
      Z_imp = S.shape[1]
      
      # check input superstages
      if 1 not in superstages:
          print('Warning: 1th superstage was included')
          superstages = np.r_[1,superstages]
      if 0 not in superstages:
          print('Warning: 0th superstage was included')
          superstages = np.r_[0,superstages]
      if np.any(np.diff(superstages)<=0):
          print('Warning: sorted superstages in increasing order')
          superstages = np.sort(superstages)
      if superstages[-1] > Z_imp:
          raise Exception('The highest superstage must be less than Z_imp = %d'%Z_imp)
      superstages = np.array(superstages)
  
      # indexing that accounts for no recomb of neutral stage or ionization of full-stripped stage
      R_rates_super = R[:,superstages[1:]-1]
      S_rates_super = S[:,superstages[1:]-1]
      
      if len(S) == 1 or save_time is None: # time averaged kinetic profiles
          t_slice = slice(None,None)
          nt = 1
      else: # time resolved kinetic profiles
          t_slice = save_time
          nt = save_time.sum()
      
      # fractional abundance of supestages used for upstaging. 
      fz_upstage = np.ones((R.shape[-1], Z_imp+1, nt))
  
      # add fully-stripped charge state
      _superstages = np.r_[superstages, Z_imp+1]
  
      for i in range(len(_superstages)-1):
          if _superstages[i]+1!= _superstages[i+1]:
              sind = slice(_superstages[i]-1, _superstages[i+1]-1)
  
              # calculate fractional abundances within the superstage
              rate_ratio =  S[:,sind]/R[:,sind]
              fz = np.cumprod(rate_ratio, axis=1) 
              fz /= np.maximum(1e-60,fz.sum(1))[:,None] # prevents zero division
  
              # superstage rates
              R_rates_super[:,i-1] *= np.maximum(fz[:,0],1e-60)
  
              if i<len(_superstages)-2: # last superstage cannot ionize further
                  S_rates_super[:,i] *= np.maximum(fz[:, -1],1e-60)
  
              # fractional abundances inside of each superstage
              fz_upstage[:,_superstages[i]:_superstages[i+1]] = fz.T[:,:,t_slice]
  
      return superstages, R_rates_super, S_rates_super, fz_upstage
  
  
  def get_frac_abundances(atom_data, ne_cm3, Te_eV=None, Ti_eV=None, n0_by_ne=0.0, superstages=[],
                          ne_tau=np.inf, plot=True, ax = None, rho = None,
                          rho_lbl=None):
      r'''Calculate fractional abundances from ionization and recombination equilibrium.
      If n0_by_ne is not 0, radiative recombination and thermal charge exchange are summed.
  
      This method can work with ne,Te and n0_by_ne arrays of arbitrary dimension, but plotting 
      is only supported in 1D (defaults to flattened arrays).
  
      Parameters
      ----------
      atom_data : dictionary of atomic ADAS files (only acd, scd are required; ccd is 
          necessary only if include_cx=True)
      ne_cm3 : float or array
          Electron density in units of :math:`cm^{-3}`
      Te_eV : float or array, optional
          Electron temperature in units of eV. If left to None, the Te grid given in the 
          atomic data is used.
      Ti_eV : float or array, optional
          Bulk ion temperature in units of eV. If left to None, Ti is set to be equal to Te
      n0_by_ne: float or array, optional
          Ratio of background neutral hydrogen to electron density. If not 0, CX is considered.
      superstages : list or 1D array
          Indices of charge states of chosen ion that should be included. If left empty, all ion stages
          are included. If only some indices are given, these are modeled as "superstages".
      ne_tau : float, opt
          Value of electron density in :math:`m^{-3}\cdot s` :math:`\times` particle residence time. 
          This is a scalar value that can be used to model the effect of transport on ionization equilibrium. 
          Setting ne_tau=np.inf (default) corresponds to no effect from transport. 
      plot : bool, optional
          Show fractional abundances as a function of ne,Te profiles parameterization.
      ax : matplotlib.pyplot Axes instance
          Axes on which to plot if plot=True. If False, it creates new axes
      rho : list or array, optional
          Vector of radial coordinates on which ne,Te (and possibly n0_by_ne) are given. 
          This is only used for plotting, if given. 
      rho_lbl: str, optional
          Label to be used for rho. If left to None, defaults to a general "x".
  
      Returns
      -------
      Te : array
          electron temperatures as a function of which the fractional abundances and
          rate coefficients are given.
      fz : array, (space,nZ)
          Fractional abundances across the same grid used by the input ne,Te values. 
  
      '''
      # if input arrays are multi-dimensional, flatten them here and restructure at the end
      _ne = np.ravel(ne_cm3)
      _Te = np.ravel(Te_eV) if Te_eV is not None else None
      _Ti = np.ravel(Ti_eV) if Ti_eV is not None else _Te
      _n0_by_ne = np.ravel(n0_by_ne)
      if superstages is None: superstages = []
  
      include_cx = False if not np.any(n0_by_ne) else True
  
      Te, S, R, cx = get_cs_balance_terms(atom_data, _ne, _Te, _Ti, include_cx=include_cx)
      Z_imp = S.shape[1]
      
      if include_cx:
          # Get an effective recombination rate by summing radiative & CX recombination rates
          R += n0_by_ne[:,None]*cx
  
      rate_ratio = np.hstack((np.ones_like(Te)[:, None], S/R))
      fz_full = np.cumprod(rate_ratio, axis=1)
      fz_full /= fz_full.sum(1)[:, None]
      
      # Enable use of superstages
      if len(superstages):
          
          # superstage_rates expects values in shape (time,nZ,space)
          superstages, R, S,_ = superstage_rates(R.T[None], S.T[None], superstages)
          R = R[0].T; S = S[0].T
  
          rate_ratio = np.hstack((np.ones_like(Te)[:, None], S/R))
          fz_super = np.cumprod(rate_ratio, axis=1)
          fz_super /= fz_super.sum(1)[:, None]
          
          # bundled stages can have very high values -- clip here
          R = np.clip(R, 1e-25, 1) 
          S = np.clip(S, 1e-25, 1)
  
          _superstages = np.r_[superstages, Z_imp+1]
  
      if plot:
          # plot fractional abundances (only 1D)
          if ax is None:
              fig,axx = plt.subplots()
          else:
              axx = ax
  
          if rho is None:
              x = Te
              axx.set_xlabel('T$_e$ [eV]')
              axx.set_xscale('log')
          else:
              if rho_lbl is None: rho_lbl='x'
              x = rho
              axx.set_xlabel(rho_lbl)
  
          axx.set_prop_cycle('color',cm.plasma(np.linspace(0,1,fz_full.shape[1])))
  
          css=0
          for cs in range(fz_full.shape[1]):
              l = axx.semilogy(x, fz_full[:,cs], ls='--')
              imax = np.argmax(fz_full[:,cs])
              axx.text(np.max([0.1,x[imax]]), fz_full[imax,cs], cs,
                       horizontalalignment='center', clip_on=True)
  
              if len(superstages) and cs in _superstages:
                  axx.semilogy(x, fz_super[:,css], c=l[0].get_color(), ls='-')
                  imax = np.argmax(fz_super[:,css])
                  
                  if _superstages[css]==Z_imp:
                      lbl = r'\{'+f'{_superstages[css]}'+r'\}'
                  else:
                      if _superstages[css]!=_superstages[css+1]-1:
                          lbl = r'\{'+f'{_superstages[css]},{_superstages[css+1]-1}'+r'\}'
                      else:
                          lbl = r'\{'+f'{_superstages[css]}'+r'\}'
                  axx.text(np.max([0.05,x[imax]]), fz_super[imax,css], lbl,
                           horizontalalignment='center', clip_on=True, backgroundcolor='w')
                  css += 1
  
          axx.grid('on')
          axx.set_ylim(1e-2,1.5)
          axx.set_xlim(x[0],x[-1])
          plt.tight_layout()
  
      if np.size(ne_cm3)>1:
          # re-structure to original array dimensions
          Te = Te.reshape(np.array(ne_cm3).shape)
          fz_full = fz_full.reshape(*np.array(ne_cm3).shape, fz_full.shape[1])
          if len(superstages):
              fz_super = fz_super.reshape(*np.array(ne_cm3).shape, fz_super.shape[1])
      
      return [Te,] + [fz_super if len(superstages) else fz_full,]
   
  
  
  
  
  
  def get_cs_balance_terms(atom_data, ne_cm3=5e13, Te_eV=None, Ti_eV=None, include_cx=True):
      '''Get S, R and cx rates on the same logTe grid. 
      
      Parameters
      ----------
      atom_data : dictionary of atomic ADAS files (only acd, scd are required; ccd is 
          necessary only if include_cx=True)
      ne_cm3 : float or array
          Electron density in units of :math:`cm^{-3}`
      Te_eV : float or array
          Electron temperature in units of eV. If left to None, the Te grid
          given in the atomic data is used.
      Ti_eV : float or array
          Bulk ion temperature in units of eV, only needed for CX. If left to None, Ti is set equal to Te.
      include_cx : bool
          If True, obtain charge exchange terms as well. 
      
      Returns
      -------
      Te : array (n_Te)
          Te grid on which atomic rates are given
      S, R (,cx): arrays (n_ne,n_Te)
          atomic rates for effective ionization, radiative+dielectronic
          recombination (+ charge exchange, if requested). All terms will be in units of :math:`s^{-1}`. 
      '''
      
  
      if Te_eV is None:
          # find smallest Te grid from all files
          _, logTe1,_ = atom_data['scd']
          _, logTe2,_ = atom_data['acd']
  
          minTe = max(logTe1[0],logTe2[0])
          maxTe = min(logTe1[-1],logTe2[-1])  # avoid extrapolation
  
          if include_cx:
              _, logTe3,_ = atom_data['ccd']  # thermal cx recombination
              minTe = max(minTe,logTe3[0])
              maxTe = min(maxTe,logTe3[-1])  # avoid extrapolation
  
          Te_eV = np.logspace(minTe,maxTe,200)
   
  
      logne = np.log10(ne_cm3)
      logTe = np.log10(Te_eV)
  
      S = interp_atom_prof(atom_data['scd'], logne, logTe, x_multiply=True)
      R = interp_atom_prof(atom_data['acd'], logne, logTe, x_multiply=True)
      if include_cx:
          logTi = np.log10(Ti_eV) if Ti_eV is not None else logTe
          x,y,tab = atom_data['ccd']         
          # select appropriate number of charge states
          # this allows use of CCD files from higher-Z ions because of simple CX scaling
          cx = interp_atom_prof((x,y,tab[:S.shape[1]]), logne, logTi, x_multiply=True)
      else:
          cx = None
      
      return Te_eV, S, R, cx
  
  
  
  def get_atomic_relax_time(atom_data, ne_cm3, Te_eV=None, Ti_eV=None, n0_by_ne=0.0, superstages=[],
                          ne_tau=np.inf, plot=True, ax = None, ls='-'):
      r'''Obtain the relaxation time of the ionization equilibrium for a given atomic species.
  
      If n0_by_ne is not 0, thermal charge exchange is added to radiative and dielectronic recombination.
      This function can work with ne,Te and n0_by_ne arrays of arbitrary dimension.
      It uses a matrix SVD approach in order to find the relaxation rates, as opposed to the simpler
      approach of :py:func:`~aurora.get_frac_abundances`, but fractional abundances produced by the two
      methods should be the same.
  
      Parameters
      ----------
      atom_data : dictionary of atomic ADAS files (only acd, scd are required; ccd is 
          necessary only if n0_by_ne is not 0).
      ne_cm3 : float or array
          Electron density in units of :math:`cm^{-3}`.
      Te_eV : float or array, optional
          Electron temperature in units of eV. If left to None, the Te grid given in the atomic data is used.
      Ti_eV : float or array
          Bulk ion temperature in units of eV, only needed for CX. If left to None, Ti is set equal to Te.
      n0_by_ne: float or array, optional
          Ratio of background neutral hydrogen to electron density. If set to 0, CX is not considered.
      superstages : list or 1D array
          Indices of charge states of chosen ion that should be included. If left empty, all ion stages
          are included. If only some indices are given, these are modeled as "superstages".
      ne_tau : float, opt
          Value of electron density in :math:`m^{-3}\cdot s` :math:`\times` particle residence time. 
          This is a scalar value that can be used to model the effect of transport on ionization equilibrium. 
          Setting ne_tau=np.inf (default) corresponds to no effect from transport. 
      plot : bool, optional
          If True, the atomic relaxation time is plotted as a function of Te. Default is True.
      ax : matplotlib.pyplot Axes instance
          Axes on which to plot if plot=True. If False, new axes are created.
      ls : str, optional
          Line style for plots. Continuous lines are used by default. 
  
      Returns
      -------
      Te : array
          electron temperatures as a function of which the fractional abundances and
          rate coefficients are given.
      fz : array, (space,nZ)
          Fractional abundances across the same grid used by the input ne,Te values. 
      rate_coeffs : array, (space, nZ)
          Rate coefficients in units of [:math:`s^{-1}`]. 
      '''
      # if input arrays are multi-dimensional, flatten them here and restructure at the end
      _ne = np.ravel(ne_cm3)
      _Te = np.ravel(Te_eV) if Te_eV is not None else None
      _Ti = np.ravel(Ti_eV) if Ti_eV is not None else _Te
      _n0_by_ne = np.ravel(n0_by_ne)
  
      include_cx = False if not np.any(n0_by_ne) else True
  
      Te, S, R, cx = get_cs_balance_terms(atom_data, _ne, _Te, include_cx=include_cx)
      
      if include_cx:
          # Get an effective recombination rate by summing radiative & CX recombination rates
          R += cx*_n0_by_ne
          
      # Enable use of superstages
      if len(superstages):
          _, R, S,_ = superstage_rates(R, S, superstages)
  
      # numerical method that calculates also rate_coeffs
      nion = R.shape[1]
      fz  = np.zeros((Te.size,nion+1))
      rate_coeffs = np.zeros(Te.size)
  
      for it,t in enumerate(Te):
          A = - np.diag(np.r_[S[it], 0] + np.r_[0, R[it]] + 1e6 / ne_tau) + np.diag(S[it], -1) + np.diag(R[it], 1)
  
          N,rate_coeffs[it] = null_space(A)
          fz[it] = N/np.sum(N)
  
   
      if np.size(ne_cm3)>1:
          # re-structure to original array dimensions
          Te = Te.reshape(np.shape(ne_cm3))
          fz = fz.reshape(*np.shape(ne_cm3), fz.shape[1])
          rate_coeffs = rate_coeffs.reshape(np.shape(ne_cm3))
  
      if plot:
          # Now plot relaxation times
          if ax is None:
              fig,ax = plt.subplots()
              
          ax.loglog(Te,1e3/rate_coeffs,'b' )
          ax.set_xlim(Te[0],Te[-1])
          ax.grid('on')
          ax.set_xlabel('T$_e$ [eV]')
          ax.set_ylabel(r'$\tau_\mathrm{relax}$ [ms]')
          
      return Te, fz, rate_coeffs
  
  
  
  class CartesianGrid:
      """Fast linear interpolation for 1D and 2D vecctor data on equally spaced grids.
      This offers optimal speed in Python for interpolation of atomic data tables such
      as the ADAS ones.
  
      Parameters
      ----------
      grids: list of arrays, N=len(grids), N=1 or N=2
          List of 1D arrays with equally spaced grid values for each dimension
      values: N+1 dimensional array of values used for interpolation
          Values to interpolate. The first dimension typically refers to different ion stages, for which 
          data is provided on the input grids. Other dimensions refer to values on the density and temperature grids.
      """
      def __init__(self, grids, values):
  
          values = np.ascontiguousarray(np.moveaxis(values, 0, -1))
          self.N = values.shape[:-1]
  
          if len(self.N) > 2:
              raise OMFITexception('Only 1 and 2 dimensional interpolation is supported')
  
          for g, s in zip(grids, self.N):
              if len(g) != s:
                  raise OMFITexception('wrong size of values array')
  
          self.offsets = [g[0] for g in grids]
          self.scales = [(g[-1] - g[0]) / (n - 1) for g, n in zip(grids, self.N)]
  
          A = []
          if len(self.N) == 1:
              A.append(values[:-1])
              A.append(values[1:] - A[0])
  
          if len(self.N) == 2:
              A.append(values[:-1, :-1])
              A.append(values[1:, :-1] - A[0])
              A.append(values[:-1, 1:] - A[0])
              A.append(values[1:, 1:] - A[2] - A[1] - A[0])
  
          self.A = A
  
      def __call__(self, *coords):
          """Evaluate the interpolation at the input `coords` values.
          
          This offers optimally-fast linear interpolation for 1D and  2D dimensional vector data 
          on a equally spaced grids
          
          Parameters
          ----------
          coords:  list of arrays
              List of 1D arrays for the N coordines (N=1 or N=2). These arrays must be of the same shape.
   
          """
          coords = np.array(coords).T
          coords -= self.offsets
          coords /= self.scales
          coords = coords.T
  
          # clip dimension - it will extrapolation by a nearest value
          for coord, n in zip(coords, self.N):
              np.clip(coord, 0, n - 1.00001, coord)
  
          #  en.wikipedia.org/wiki/Bilinear_interpolation#Unit_square
  
          # get indicies x and weights dx
          x = np.int16(coords)  # fast floor(x) function
          coords -= x  # frac(x)
          dx = coords[..., None]
  
          # prepare coefficients
          if len(self.N) == 1:  # linear interpolation
              # inplace evaluate linear  interpolation
              inter_out = self.A[1][x[0]]
              inter_out *= dx[0]
              inter_out += self.A[0][x[0]]
  
          else:  # bilinear interpolation
              # inplace evaluate linear  interpolation
              inter_out = self.A[1][x[0], x[1]]
              inter_out *= dx[0]
              inter_out += self.A[0][x[0], x[1]]
  
              _tmp = self.A[3][x[0], x[1]]
              _tmp *= dx[0]
              _tmp += self.A[2][x[0], x[1]]
              _tmp *= dx[1]
              inter_out += _tmp
  
          return np.moveaxis(inter_out, -1, 0)
  
  
  class CartesianGrid_Ndim:
      """
      Linear multivariate Cartesian grid interpolation in arbitrary dimensions
      This is a regular grid with equal spacing.
  
      Use :py:class:`~aurora.atomic.CartesianGrid` for a version that is optimally fast
      in 2 dimensions only.
  
      Parameters
      ----------
      grids: list of arrays, N=len(grids), arbitrary N
          List of 1D arrays with equally spaced grid values for each dimension
      values: N+1 dimensional array of values used for interpolation
          Values to interpolate. The first dimension typically refers to different ion stages, for which 
          data is provided on the input grids. Other dimensions refer to values on the N-dimensional grids.
  
      """
      def __init__(self, grids, values):
  
          self.values = np.ascontiguousarray(values)
          for g,s in  zip(grids,values.shape[1:]):
              if len(g) != s: raise Exception('wrong size of values array')
  
          self.offsets = [g[0] for g in grids]
          self.scales  = [(g[-1]-g[0])/(n-1) for g,n in zip(grids, values.shape[1:])]
          self.N = values.shape[1:]
  
      def __call__(self, *coords):
          '''Transform coords into pixel values and provide interpolated result.
  
          Parameters
          ----------
          coords:  list of arrays
              List of 1D arrays for the N coordines (N=1 or N=2). These arrays must be of the same shape.
          '''
          out_shape = coords[0].shape
  
          coords = np.array(coords).T
          coords -= self.offsets
          coords /= self.scales
          coords = coords.T
  
          # clip dimension - gives extrapolation by nearest value
          for coord, n in zip(coords, self.N):
              np.clip(coord,0,n-1,coord)
  
          #prepare output array
          inter_out = np.empty((self.values.shape[0],)+out_shape, dtype=self.values.dtype)
  
          # fast interpolation on a N-dimensional regular grid
          for out,val in zip(inter_out,self.values):
              scipy.ndimage.map_coordinates(val, coords, order=1, output=out)
  
          return inter_out
  
  
  def interp_atom_prof(atom_table,xprof, yprof,log_val=False, x_multiply=True):
      r''' Fast interpolate atomic data in atom_table onto the xprof and yprof profiles.
      This function assume that xprof, yprof, x,y, table are all base-10 logarithms,
      and xprof, yprof are equally spaced.
  
      Parameters
      ----------
      atom_table : list
          List with x,y, table = atom_table, containing atomic data from one of the ADAS files. 
      xprof : array (nt,nr)
          Spatio-temporal profiles of the first coordinate of the ADAS file table (usually 
          electron density in :math:`cm^{-3}`)
      yprof : array (nt,nr)
          Spatio-temporal profiles of the second coordinate of the ADAS file table (usually 
          electron temperature in :math:`eV`)
      log_val : bool
          If True, return natural logarithm of the data
      x_multiply : bool
          If True, multiply output by :math:`10^{xprof}`. 
  
      Returns
      -------
      interp_vals : array (nt,nion,nr)
          Interpolated atomic data on time,charge state and spatial grid that correspond to the 
          ion of interest and the spatiotemporal grids of xprof and yprof. 
      '''
      x,y, table = atom_table
  
      if (abs(table-table[...,[0]]).all()  < 0.05) or xprof is None:
          # 1D interpolation if independent of the last dimension - like SXR radiation data
  
          reg_interp = CartesianGrid((y, ),table[:,:,0]*np.log(10))
          interp_vals = reg_interp(yprof) 
  
          # multipling of logarithms is just adding
          if x_multiply and xprof is not None:
              interp_vals += xprof*np.log(10)
  
      else: # 2D interpolation
          if x_multiply: #multipling of logarithms is just adding
              table += x
          # broadcast both variables in the sae shape
          xprof, yprof = np.broadcast_arrays(xprof, yprof)
          #perform fast linear interpolation
          reg_interp = CartesianGrid((x, y),table.swapaxes(1,2)*np.log(10))
          interp_vals = reg_interp(xprof,yprof) 
      
      # reshape to shape(nt,nion,nr)
      interp_vals = interp_vals.swapaxes(0,1)
  
      if not log_val:
          # return actual value, not logarithm
          np.exp(interp_vals, out=interp_vals)
  
      return interp_vals
  
  
  
  
  
  def gff_mean(Z,Te):
      '''
      Total free-free gaunt factor yielding the total radiated bremsstrahlung power
      when multiplying with the result for gff=1.
      Data originally from Karzas & Latter, extracted from STRAHL's atomic_data.f.
      '''
      from scipy.constants import e,h,c,Rydberg
      thirteenpointsix = h*c*Rydberg/e
  
      log_gamma2_grid =[-3.000,-2.833,-2.667,-2.500,-2.333,-2.167,-2.000,-1.833,-1.667,-1.500,
                          -1.333,-1.167,-1.000,-0.833,-0.667,-0.500,-0.333,-0.167, 0.000, 0.167,
                          0.333, 0.500, 0.667, 0.833, 1.000]
  
      gff_mean_grid = [1.140, 1.149, 1.159, 1.170, 1.181, 1.193, 1.210, 1.233, 1.261, 1.290,
                       1.318, 1.344, 1.370, 1.394, 1.416, 1.434, 1.445, 1.448, 1.440, 1.418,
                       1.389, 1.360, 1.336, 1.317, 1.300]
  
      # set min Te here to 10 eV, because the grid above does not extend to lower temperatures
      Te = np.maximum(Te, 10.0)
  
      log_gamma2 = np.log10(Z**2*thirteenpointsix/Te)
  
      # dangerous/inaccurate extrapolation...
      return np.interp(log_gamma2, log_gamma2_grid,gff_mean_grid)
  
  
  
  def impurity_brems(nz, ne, Te):
      '''Approximate bremsstrahlung in units of :math:`mW/nm/sr/m^3 \cdot cm^3`, or
      equivalently :math:`W/m^3` for full spherical emission. 
  
      Note that this may not be very useful, since this contribution is already included 
      in the continuum radiation component in ADAS files. The bremsstrahlung estimate in 
      ADAS continuum radiation files is more accurate than the one give by this function, 
      which uses a simpler interpolation of the Gaunt factor with weak ne-dependence. 
      Use with care!
  
      Parameters
      ----------
      nz : array (time,nZ,space)
          Densities for each charge state [:math:`cm^{-3}`]
      ne : array (time,space)
          Electron density [:math:`cm^{-3}`]
      Te : array (time,space)
          Electron temperature [:math:`cm^{-3}`]
  
      Returns
      -------
      brems : array (time,nZ,space)
          Bremsstrahlung for each charge state 
      '''
      # neutral stage doesn't produce brems
      Z_imp = nz.shape[1]-1
      Z = np.arange(Z_imp)[None,:,None]+1
  
      gff = gff_mean(Z,Te[:,None])
      brems = Z**2 * nz[:,1:] * gff * (1.69e-32 *np.sqrt(Te) * ne) [:,None]
  
      return brems
         
          
  
  
  def plot_norm_ion_freq(S_z, q_prof, R_prof, imp_A, Ti_prof,
                         nz_profs=None, rhop=None, plot=True, eps_prof=None):
      r'''Compare effective ionization rate for each charge state with the characteristic
      transit time that passing and trapped impurity ions take to travel a parallel distance 
      :math:`L = q R`, defining
  
      .. math::
  
          \nu_{ion}^* \equiv \nu_{ion} \tau_t = \nu_{ion} \frac{q R}{v_{th}} = \frac{\sum_z n_z \nu_z^{ion}}{\sum_z n_z} q R \sqrt{\frac{m_{imp}}{2 k_B T_i}}
  
      following Ref.[1]_. If the normalized ionization rate (:math:`\nu_{ion}^*`) is less than 1, 
      then flux surface averaging of background asymmetries (e.g. from edge or beam neutrals) may 
      be taken as a good approximation of reality; in this case, 1.5D simulations of impurity transport
      are expected to be valid. If, on the other hand, :math:`\nu_{ion}^*>1` then local effects 
      may be too important to ignore. 
  
      Parameters
      ----------
      S_z : array (r,cs) [:math:`s^{-1}`]
           Effective ionization rates for each charge state as a function of radius. 
           Note that, for convenience within aurora, cs includes the neutral stage.
      q_prof : array (r,)
           Radial profile of safety factor
      R_prof : array (r,) or float [m]
           Radial profile of major radius, either given as an average of HFS and LFS, or also
           simply as a scalar (major radius on axis)
      imp_A : float [amu]
           Atomic mass number, i.e. number of protons + neutrons (e.g. 2 for D)
      Ti_prof : array (r,)
           Radial profile of ion temperature [:math:`eV`]
      nz_profs : array (r,cs), optional
           Radial profile for each charge state. If provided, calculate average normalized 
           ionization rate over all charge states.
      rhop : array (r,), optional
           Sqrt of poloidal flux radial grid. This is used only for (optional) plotting. 
      plot : bool, optional
           If True, plot results.
      eps_prof : array (r,), optional
           Radial profile of inverse aspect ratio, i.e. r/R, only used if plotting is requested.  
  
      Returns
      -------
      nu_ioniz_star : array (r,cs) or (r,)
           Normalized ionization rate. If nz_profs is given as an input, this is an average over
           all charge state; otherwise, it is given for each charge state.
  
      References
      ----------
      .. [1] R.Dux et al. Nucl. Fusion 60 (2020) 126039
  
      '''
      nu = np.zeros((S_z.shape[0],S_z.shape[1]-1))  # exclude neutral states, which have no parallel transport
      for cs in np.arange(nu.shape[1]):
          nu[:,cs] = S_z[:,cs+1] * q_prof * R_prof * np.sqrt((imp_A * constants.m_p)/(2*Ti_prof*constants.e))
  
      if nz_profs is not None:
          # calculate average nu_ioniz_star 
          nu_ioniz_star = np.sum(nz_profs[:,1:]*nu,axis=1)/np.sum(nz_profs[:,1:],axis=1)
      else:
          # return normalized ionization rate for each charge state
          nu_ioniz_star = nu
  
      if plot:
          if rhop is None:
              rhop = np.arange(nu.shape[0])
              
          fig,ax = plt.subplots()
          if nu_ioniz_star.ndim==1:
              ax.semilogy(rhop,nu_ioniz_star, label=r'$\nu_{ion}^*$')
          else:
              for cs in np.arange(nu.shape[1]):
                  ax.semilogy(rhop, nu_ioniz_star[:,cs], label=f'q={cs+1}')
              ax.set_ylabel(r'$\nu_{ion}^*$')
  
          ax.set_xlabel(r'$\rho_p$')
  
          if eps_prof is not None:
              ax.semilogy(rhop, np.sqrt(eps_prof), label=r'$\sqrt{\epsilon}$')
  
          ax.legend().set_draggable(True)
          ax.set_xlim([0,1])
  
  
  
  def read_adf12(filename, block, Ebeam, ne_cm3, Ti_eV, zeff):
      '''Read charge exchange effective emission coefficients from ADAS ADF12 files.
  
      See https://open.adas.ac.uk/adf12 for details and to download files.
  
      Parameters
      ----------
      filename : str
          adf12 file name/path
      block : int
          Source block selected
      Ebeam : float
          Energy of the neutral beam population of interest, in units of :math:`eV/amu`.
      ne_cm3 : float or 1D array
          Electron densities at which to evaluate coefficients, in units of :math:`cm^{-3}`.
      Ti_eV : float or 1D array
          Bulk ion temperature at which to evaluate coefficients, in units of :math:`eV`.
      Zeff : float or 1D array
          Effective background charge.
  
      Returns
      -------
      float or 1D array :
          Interpolated coefficients, units of :math:`cm^3/s`.
  
      '''
      with open(filename, 'r') as f:
          nlines = int(f.readline())
  
          for iline in range(block):
              cer_line = {}
              params = []
              first_line = '0'
              while not first_line[0].isalpha():
                  first_line = f.readline()
  
              cer_line['header'] = first_line
              cer_line['qefref'] = np.float(f.readline()[:63].replace('D', 'e'))
              cer_line['parmref'] = np.float_(f.readline()[:63].replace('D', 'e').split())
              cer_line['nparmsc'] = np.int_(f.readline()[:63].split())
  
              for ipar, npar in enumerate(cer_line['nparmsc']):
                  for q in range(2):
                      data = []
                      while npar > len(data):
                          line = f.readline()
                          if len(line) > 63:
                              name = line[63:].strip().lower()
                              cer_line[name] = []
                              if q == 0:
                                  params.append(name)
  
                          values = np.float_(line[:63].replace('D', 'E').split())
                          values = values[values > 0]
                          if not len(values):
                              continue
                          data += values.tolist()
                      cer_line[name] = data
  
      # interpolate in logspace
      lqefref = np.log(cer_line['qefref'])
      
      lnq = np.zeros(np.broadcast(Ebeam, ne_cm3, Ti_eV, zeff).shape)
      lnq += lqefref * (1 - 4)
      lnq += np.interp(np.log(Ti_eV), np.log(cer_line['tiev']), np.log(cer_line['qtiev']))
      lnq += np.interp(np.log(ne_cm3), np.log(cer_line['densi']), np.log(cer_line['qdensi']))
      lnq += np.interp(np.log(Ebeam), np.log(cer_line['ener']), np.log(cer_line['qener']))
      lnq += np.interp(np.log(zeff), np.log(cer_line['zeff']), np.log(cer_line['qzeff']))
      
      return np.exp(lnq)
  
  
          
  def read_adf21(filename, Ebeam, ne_cm3, Te_eV):
      '''Read ADAS ADF21 or ADF22 files.
  
      ADF21 files contain effective beam stopping/excitation coefficients.
      ADF22 contain effective beam emission/population coefficients.
  
      See https://open.adas.ac.uk/adf21 and https://open.adas.ac.uk/adf22 for details
      and to download files.
  
      Parameters
      ----------
      filename : str
          adf21 or adf22 file name/path
      Ebeam : float
          Energy of the neutral beam, in units of :math:`eV/amu`.
      ne_cm3 : float or 1D array
          Electron densities at which to evaluate coefficients.
      Te_eV : float or 1D array
          Electron temperature at which to evaluate coefficients.
  
      Returns
      -------
      float or 1D array :
          Interpolated coefficients. For ADF21 files, these have units of :math:`cm^3/s`for ADF21 files.
          For ADF22, they correspond to n=2 fractional abundances.
      '''
      
      with open(filename, 'r') as f:
          line = f.readline()
          ref = float(line.split()[1].split('=')[1])
          f.readline()
          line = f.readline()
          nE, nne, Teref = line.split()
          nE, nne = int(nE), int(nne)
          Teref = float(Teref.split('=')[1])
          f.readline()
  
          E = []
          while len(E) < nE:
              line = f.readline()
              E.extend([float(f) for f in line.split()])
          E = np.array(E)
  
          ne = []
          while len(ne) < nne:
              line = f.readline()
              ne.extend([float(f) for f in line.split()])
          ne = np.array(ne)
          f.readline()
  
          Q2 = []
          while len(Q2) < nne * nE:
              line = f.readline()
              Q2.extend([float(f) for f in line.split()])
          Q2 = np.reshape(Q2, (nne, nE))
  
          f.readline()
          line = f.readline()
          nTe, Eref, Neref = line.split()
          nTe, Eref, Neref = int(nTe), float(Eref.split('=')[1]), float(Neref.split('=')[1])
  
          f.readline()
  
          Te = []
          while len(Te) < nTe:
              line = f.readline()
              Te.extend([float(f) for f in line.split()])
          Te = np.array(Te)
  
          f.readline()
  
          Q1 = []
          while len(Q1) < nTe:
              line = f.readline()
              Q1.extend([float(f) for f in line.split()])
          Q1 = np.array(Q1)
  
      # clip data in available range to avoid extrapolation
      Ebeam = np.clip(Ebeam, *E[[0, -1]])
      ne_cm3 = np.clip(ne_cm3, *ne[[0, -1]])
      Te_eV = np.clip(Te_eV, *Te[[0, -1]])
  
      lref = np.log(ref)
  
      # interpolate on the requested values
      RectInt1 = interp.interp1d(np.log(Te), np.log(Q1) - lref, assume_sorted=True, kind='quadratic')
      RectInt2 = interp.RectBivariateSpline(np.log(ne), np.log(E), np.log(Q2) - lref, kx=2, ky=2)
  
      adf = RectInt1(np.log(Te_eV)) + RectInt2.ev(np.log(ne_cm3), np.log(Ebeam))
      return np.exp(adf + lref)
  
  
  
  def get_natural_partition(ion, plot=True):
      '''Identify natural partition of charge states by plotting the variation of ionization energy
      as a function of charge for a given ion, using the ADAS metric :math:`2 (I_{z+1}-I_z)/(I_{z+1}+I_z)`.
      
      Parameters
      ----------
      ion : str
          Atomic symbol of species of interest.
      plot : bool
          If True, plot the variation of ionization energy.
  
      Returns
      -------
      q : 1D array
          Metric to identify natural partition.
      
      Notes
      -----
      A ColRadPy installation must be available for this function to work.
      '''
  
      try:
          # temporarily import this here, until ColRadPy dependency can be set up properly
          import colradpy
      except ImportError:
          raise ValueError('Could not import colradpy. Install this from the Github repo!')
  
      # find location of files containing NIST data
      colradpy_dist = os.sep.join(colradpy.__file__.split(os.sep)[:-2])
      loc = colradpy_dist+os.sep+'atomic'+os.sep+'nist_energies'
  
      # find energies for each charge state of interest
      _E_eV = []
      cs = []
      for filename in os.listdir(loc):
          if not filename.startswith('#') and not filename.endswith('~'):
              _ion = filename.split('_')[0]
  
              if _ion!=ion.lower():
                  continue
              charge = int(filename.split('_')[1])
  
              # read energy from file
              with open(loc+os.sep+filename,'r') as f:
                  cont = f.readlines()
  
              # last value is energy in cm^-1
              E_cm_val = float(cont[-1].split(',')[-1].strip('/n'))
              E_eV_val = constants.h*constants.c/constants.e*(E_cm_val*100.)
  
              cs.append(charge)
              _E_eV.append(E_eV_val)
  
      idx = np.argsort(cs)
      E_eV = np.array(_E_eV)[idx]
  
      # compute ADAS natural partitioning metric
      q = []
      for i in np.arange(1,len(E_eV)-1):
          q.append( 2*(E_eV[i+1] - E_eV[i])/(E_eV[i+1]+E_eV[i]))
  
      if plot:
          fig,ax = plt.subplots()
          
          ax.plot(np.arange(len(q)), q)
          ax.set_xlabel('Z')
          ax.set_title(r'$2\times (I_{z+1}-I_z)/(I_{z+1}+I_z)$')
  
          # take running mean over 7 adjacent charge states as indicated by Foster's thesis
          q_mean = np.convolve(q, np.ones(7)/7, mode='same')
          plt.plot(q_mean)
  
      return q
  
  