Replace the half-sum with numerical integration when calculating the bremsstrahlung spectrum

CHANGELOG.md

@@ -9,6 +9,8 @@ New:
 * Add custom line shape support to BeamCXLine model. (#394)
 * Add PeriodicTransformXD and VectorPeriodicTransformXD functions to support the data simulated with periodic boundary conditions. (#387)
 * Add CylindricalTransform and VectorCylindricalTransform to transform functions from cylindrical to Cartesian coordinates. (#387)
+* Add numerical integration of Bremsstrahlung spectrum over a spectral bin. (#395)
+* Replace the coarse numerical constant in the Bremsstrahlung model with an exact expression. (#409)
 * Add the kind attribute to RayTransferPipelineXD that determines whether the ray transfer matrix is multiplied by sensitivity ('power') or not ('radiance'). (#412)
 
 

cherab/core/model/plasma/bremsstrahlung.pxd

@@ -19,20 +19,27 @@
 # cython: language_level=3
 
 from numpy cimport ndarray
+from cherab.core.math cimport Function1D
+from cherab.core.math.integrators cimport Integrator1D
 from cherab.core.atomic cimport FreeFreeGauntFactor
 from cherab.core.plasma cimport PlasmaModel
 
 
-cdef class Bremsstrahlung(PlasmaModel):
+cdef class BremsFunction(Function1D):
 
     cdef:
-        FreeFreeGauntFactor _gaunt_factor
-        bint _user_provided_gaunt_factor
-        ndarray _species_charge, _species_density
-        double[::1] _species_density_mv, _species_charge_mv
+        double ne, te
+        FreeFreeGauntFactor gaunt_factor
+        ndarray species_density, species_charge
+        double[::1] species_density_mv
+        double[::1] species_charge_mv
 
-    cdef double _bremsstrahlung(self, double wvl, double te, double ne)
 
-    cdef int _populate_cache(self) except -1
+cdef class Bremsstrahlung(PlasmaModel):
 
+    cdef:
+        BremsFunction _brems_func
+        bint _user_provided_gaunt_factor
+        Integrator1D _integrator
 
+    cdef int _populate_cache(self) except -1

cherab/core/model/plasma/bremsstrahlung.pyx

@@ -21,58 +21,144 @@
 import numpy as np
 from raysect.optical cimport Spectrum, Point3D, Vector3D
 from cherab.core cimport Plasma, AtomicData
-from cherab.core.atomic cimport FreeFreeGauntFactor
+from cherab.core.math.integrators cimport GaussianQuadrature
 from cherab.core.species cimport Species
-from cherab.core.utility.constants cimport RECIP_4_PI, ELEMENTARY_CHARGE, SPEED_OF_LIGHT, PLANCK_CONSTANT
-from libc.math cimport sqrt, log, exp
+from cherab.core.utility.constants cimport RECIP_4_PI, ELEMENTARY_CHARGE, SPEED_OF_LIGHT, PLANCK_CONSTANT, ELECTRON_REST_MASS, VACUUM_PERMITTIVITY
+from libc.math cimport sqrt, log, exp, M_PI
 cimport cython
 
 
-cdef double PH_TO_J_FACTOR = PLANCK_CONSTANT * SPEED_OF_LIGHT * 1e9
+cdef double EXP_FACTOR = PLANCK_CONSTANT * SPEED_OF_LIGHT * 1e9 / ELEMENTARY_CHARGE
 
-cdef double EXP_FACTOR = PH_TO_J_FACTOR / ELEMENTARY_CHARGE
+cdef double BREMS_CONST = (ELEMENTARY_CHARGE**2 * RECIP_4_PI / VACUUM_PERMITTIVITY)**3
+BREMS_CONST *= 32 * M_PI**2 / (3 * sqrt(3) * ELECTRON_REST_MASS**2 * SPEED_OF_LIGHT**3)
+BREMS_CONST *= sqrt(2 * ELECTRON_REST_MASS / (M_PI * ELEMENTARY_CHARGE))
+BREMS_CONST *= SPEED_OF_LIGHT * 1e9 * RECIP_4_PI
+
+
+cdef class BremsFunction(Function1D):
+    """
+    Calculates bremsstrahlung spectrum.
+
+    :param FreeFreeGauntFactor gaunt_factor: Free-free Gaunt factor as a function of Z, Te and
+                                             wavelength.
+    :param object species_density: Array-like object wiyh ions' density in m-3.
+    :param object species_charge: Array-like object wiyh ions' charge.
+    :param double ne: Electron density in m-3.
+    :param double te: Electron temperature in eV.
+    """
+
+    def __init__(self, FreeFreeGauntFactor gaunt_factor, object species_density, object species_charge, double ne, double te):
+
+        if ne <= 0:
+            raise ValueError("Argument ne must be positive.")
+        self.ne = ne
+
+        if te <= 0:
+            raise ValueError("Argument te must be positive.")
+        self.te = te
+
+        self.gaunt_factor = gaunt_factor
+        self.species_density = np.asarray(species_density, dtype=np.float64)  # copied if type does not match
+        self.species_density_mv = self.species_density
+        self.species_charge = np.asarray(species_charge, dtype=np.float64)  # copied if type does not match
+        self.species_charge_mv = self.species_charge
+
+    @cython.cdivision(True)
+    @cython.boundscheck(False)
+    @cython.wraparound(False)
+    cdef double evaluate(self, double wvl) except? -1e999:
+        """
+        :param double wvl: Wavelength in nm.
+        :return:
+        """
+
+        cdef double ni_gff_z2, radiance, pre_factor, ni, z
+        cdef int i
+
+        ni_gff_z2 = 0
+        for i in range(self.species_charge_mv.shape[0]):
+            z = self.species_charge_mv[i]
+            ni = self.species_density_mv[i]
+            if ni > 0:
+                ni_gff_z2 += ni * self.gaunt_factor.evaluate(z, self.te, wvl) * z * z
+
+        # bremsstrahlung equation W/m^3/str/nm
+        pre_factor = BREMS_CONST / (sqrt(self.te) * wvl * wvl) * self.ne * ni_gff_z2
+        radiance = pre_factor * exp(- EXP_FACTOR / (self.te * wvl))
+
+        return radiance
 
 
 # todo: doppler shift?
 cdef class Bremsstrahlung(PlasmaModel):
     """
     Emitter that calculates bremsstrahlung emission from a plasma object.
 
-    The bremmstrahlung formula implemented is equation 2 from M. Beurskens,
-    et. al., 'ITER LIDAR performance analysis', Rev. Sci. Instrum. 79, 10E727 (2008),
+    The bremmstrahlung formula implemented is equation 5.3.40
+    from I. H. Hutchinson, 'Principles of Plasma Diagnostics', second edition,
+    Cambridge University Press, 2002, ISBN: 9780511613630,
+    https://doi.org/10.1017/CBO9780511613630
+
+    Note that in eq. 5.3.40, the emissivity :math:`j(\\nu)` is given in (W/m^3/sr/Hz) with respect
+    to frequency, :math:`\\nu`. Here, the emissivity :math:`\\epsilon_{\\mathrm{ff}}(\\lambda)`
+    is given in (W/m^3/nm/sr) with respect to wavelength, :math:`\\lambda = \\frac{10^{9} c}{\\nu}`,
+    and taking into account that :math:`d\\nu=-\\frac{10^{9} c}{\\lambda^2}d\\lambda`.
 
     .. math::
-        \\epsilon (\\lambda) = \\frac{0.95 \\times 10^{-19}}{\\lambda 4 \\pi} \\sum_{i} \\left(g_{ff}(Z_i, T_e, \\lambda) n_i Z_i^2\\right) n_e T_e^{-1/2} \\times \\exp{\\frac{-hc}{\\lambda T_e}},
+        \\epsilon_{\\mathrm{ff}}(\\lambda) = \\left( \\frac{e^2}{4 \\pi \\varepsilon_0} \\right)^3
+        \\frac{32 \\pi^2}{3 \\sqrt{3} m_\\mathrm{e}^2 c^3}
+        \\sqrt{\\frac{2 m_\\mathrm{e}^3}{\\pi e T_\\mathrm{e}}}
+        \\frac{10^{9} c}{4 \\pi \\lambda^2}
+        n_\\mathrm{e} \\sum_i \\left( n_\\mathrm{i} g_\\mathrm{ff} (Z_\\mathrm{i}, T_\\mathrm{e}, \\lambda) Z_\\mathrm{i}^2 \\right)
+        \\mathrm{e}^{-\\frac{10^9 hc}{e T_\\mathrm{e} \\lambda}}\\,,
 
-    where the emission :math:`\\epsilon (\\lambda)` is in units of radiance (ph/s/sr/m^3/nm).
+    where :math:`T_\\mathrm{e}` is in eV and :math:`\\lambda` is in nm.
+
+    :math:`g_\\mathrm{ff} (Z_\\mathrm{i}, T_\\mathrm{e}, \\lambda)` is the free-free Gaunt factor.
 
     :ivar Plasma plasma: The plasma to which this emission model is attached. Default is None.
     :ivar AtomicData atomic_data: The atomic data provider for this model. Default is None.
     :ivar FreeFreeGauntFactor gaunt_factor: Free-free Gaunt factor as a function of Z, Te and
                                             wavelength. If not provided,
                                             the `atomic_data` is used.
+    :ivar Integrator1D integrator: Integrator1D instance to integrate Bremsstrahlung radiation
+                                   over the spectral bin. Default is `GaussianQuadrature`.
     """
 
-    def __init__(self, Plasma plasma=None, AtomicData atomic_data=None, FreeFreeGauntFactor gaunt_factor=None):
+    def __init__(self, Plasma plasma=None, AtomicData atomic_data=None, FreeFreeGauntFactor gaunt_factor=None, Integrator1D integrator=None):
 
         super().__init__(plasma, atomic_data)
 
+        self._brems_func = BremsFunction.__new__(BremsFunction)
         self.gaunt_factor = gaunt_factor
+        self.integrator = integrator or GaussianQuadrature()
 
         # ensure that cache is initialised
         self._change()
 
     @property
     def gaunt_factor(self):
 
-        return self._gaunt_factor
+        return self._brems_func.gaunt_factor
 
     @gaunt_factor.setter
     def gaunt_factor(self, value):
 
-        self._gaunt_factor = value
+        self._brems_func.gaunt_factor = value
         self._user_provided_gaunt_factor = True if value else False
 
+    @property
+    def integrator(self):
+
+        return self._integrator
+
+    @integrator.setter
+    def integrator(self, Integrator1D value not None):
+
+        self._integrator = value
+        self._integrator.function = self._brems_func
+
     def __repr__(self):
         return '<PlasmaModel - Bremsstrahlung>'
 
@@ -84,13 +170,12 @@ cdef class Bremsstrahlung(PlasmaModel):
 
         cdef:
             double ne, te
-            double lower_wavelength, upper_wavelength
-            double lower_sample, upper_sample
+            double lower_wavelength, upper_wavelength, bin_integral
             Species species
             int i
 
         # cache data on first run
-        if self._species_charge is None:
+        if self._brems_func.species_charge is None:
             self._populate_cache()
 
         ne = self._plasma.get_electron_distribution().density(point.x, point.y, point.z)
@@ -100,57 +185,28 @@ cdef class Bremsstrahlung(PlasmaModel):
         if te <= 0:
             return spectrum
 
+        self._brems_func.ne = ne
+        self._brems_func.te = te
+
         # collect densities of charged species
         i = 0
         for species in self._plasma.get_composition():
             if species.charge > 0:
-                self._species_density_mv[i] = species.distribution.density(point.x, point.y, point.z)
+                self._brems_func.species_density_mv[i] = species.distribution.density(point.x, point.y, point.z)
                 i += 1
 
-        # numerically integrate using trapezium rule
-        # todo: add sub-sampling to increase numerical accuracy
+        # add bremsstrahlung to spectrum
         lower_wavelength = spectrum.min_wavelength
-        lower_sample = self._bremsstrahlung(lower_wavelength, te, ne)
         for i in range(spectrum.bins):
-
             upper_wavelength = spectrum.min_wavelength + spectrum.delta_wavelength * (i + 1)
-            upper_sample = self._bremsstrahlung(upper_wavelength, te, ne)
 
-            spectrum.samples_mv[i] += 0.5 * (lower_sample + upper_sample)
+            bin_integral = self._integrator.evaluate(lower_wavelength, upper_wavelength)
+            spectrum.samples_mv[i] += bin_integral / spectrum.delta_wavelength
 
             lower_wavelength = upper_wavelength
-            lower_sample = upper_sample
 
         return spectrum
 
-    @cython.cdivision(True)
-    @cython.boundscheck(False)
-    @cython.wraparound(False)
-    cdef double _bremsstrahlung(self, double wvl, double te, double ne):
-        """
-        :param double wvl: Wavelength in nm.
-        :param double te: Electron temperature in eV
-        :param double ne: Electron dencity in m^-3
-        :return:
-        """
-
-        cdef double ni_gff_z2, radiance, pre_factor, ni, z
-        cdef int i
-
-        ni_gff_z2 = 0
-        for i in range(self._species_charge_mv.shape[0]):
-            z = self._species_charge_mv[i]
-            ni = self._species_density_mv[i]
-            if ni > 0:
-                ni_gff_z2 += ni * self._gaunt_factor.evaluate(z, te, wvl) * z * z
-
-        # bremsstrahlung equation W/m^3/str/nm
-        pre_factor = 0.95e-19 * RECIP_4_PI * ni_gff_z2 * ne / (sqrt(te) * wvl)
-        radiance = pre_factor * exp(- EXP_FACTOR / (te * wvl)) * PH_TO_J_FACTOR
-
-        # convert to W/m^3/str/nm
-        return radiance / wvl
-
     cdef int _populate_cache(self) except -1:
 
         cdef list species_charge
@@ -159,31 +215,31 @@ cdef class Bremsstrahlung(PlasmaModel):
         if self._plasma is None:
             raise RuntimeError("The emission model is not connected to a plasma object.")
 
-        if self._gaunt_factor is None:
+        if self._brems_func.gaunt_factor is None:
             if self._atomic_data is None:
                 raise RuntimeError("The emission model is not connected to an atomic data source.")
 
             # initialise Gaunt factor on first run using the atomic data
-            self._gaunt_factor = self._atomic_data.free_free_gaunt_factor()
+            self._brems_func.gaunt_factor = self._atomic_data.free_free_gaunt_factor()
 
         species_charge = []
         for species in self._plasma.get_composition():
             if species.charge > 0:
                 species_charge.append(species.charge)
 
         # Gaunt factor takes Z as double to support Zeff, so caching Z as float64
-        self._species_charge = np.array(species_charge, dtype=np.float64)
-        self._species_charge_mv = self._species_charge
+        self._brems_func.species_charge = np.array(species_charge, dtype=np.float64)
+        self._brems_func.species_charge_mv = self._brems_func.species_charge
 
-        self._species_density = np.zeros_like(self._species_charge)
-        self._species_density_mv = self._species_density
+        self._brems_func.species_density = np.zeros_like(self._brems_func.species_charge)
+        self._brems_func.species_density_mv = self._brems_func.species_density
 
     def _change(self):
 
         # clear cache to force regeneration on first use
         if not self._user_provided_gaunt_factor:
-            self._gaunt_factor = None
-        self._species_charge = None
-        self._species_charge_mv = None
-        self._species_density = None
-        self._species_density_mv = None
+            self._brems_func.gaunt_factor = None
+        self._brems_func.species_charge = None
+        self._brems_func.species_charge_mv = None
+        self._brems_func.species_density = None
+        self._brems_func.species_density_mv = None