diff --git a/docs/source/pythonapi/index.rst b/docs/source/pythonapi/index.rst index fc01b99e75..857826e097 100644 --- a/docs/source/pythonapi/index.rst +++ b/docs/source/pythonapi/index.rst @@ -375,6 +375,7 @@ Core Classes openmc.data.DataLibrary openmc.data.Decay openmc.data.FissionProductYields + openmc.data.WindowedMultipole Core Functions -------------- diff --git a/openmc/data/__init__.py b/openmc/data/__init__.py index c361a204d3..9e7a44f0d7 100644 --- a/openmc/data/__init__.py +++ b/openmc/data/__init__.py @@ -3,6 +3,9 @@ HDF5_VERSION_MAJOR = 1 HDF5_VERSION_MINOR = 0 HDF5_VERSION = (HDF5_VERSION_MAJOR, HDF5_VERSION_MINOR) +# Version of WMP nuclear data format +WMP_VERSION = 'v0.2' + from .data import * from .neutron import * @@ -24,3 +27,4 @@ from .urr import * from .library import * from .fission_energy import * from .resonance import * +from .multipole import * diff --git a/openmc/data/multipole.py b/openmc/data/multipole.py new file mode 100644 index 0000000000..e4049c33c2 --- /dev/null +++ b/openmc/data/multipole.py @@ -0,0 +1,655 @@ +from numbers import Integral, Real + +import h5py +import numpy as np +from six import string_types + +from . import WMP_VERSION +from .data import K_BOLTZMANN +import openmc.checkvalue as cv +from openmc.mixin import EqualityMixin + + +# Formalisms +_FORM_MLBW = 2 +_FORM_RM = 3 + +# Constants that determine which value to access +_MP_EA = 0 # Pole + +# Reich-Moore indices +_RM_RT = 1 # Residue total +_RM_RA = 2 # Residue absorption +_RM_RF = 3 # Residue fission + +# Multi-level Breit Wigner indices +_MLBW_RT = 1 # Residue total +_MLBW_RX = 2 # Residue compettitive +_MLBW_RA = 3 # Residue absorption +_MLBW_RF = 4 # Residue fission + +# Polynomial fit indices +_FIT_T = 0 # Total +_FIT_A = 1 # Absorption +_FIT_F = 2 # Fission + + +def _faddeeva(z): + r"""Evaluate the complex Faddeeva function. + + Technically, the value we want is given by the equation: + + .. math:: + w(z) = \frac{i}{\pi} \int_{-\infty}^{\infty} \frac{1}{z - t} + \exp(-t^2) \text{d}t + + as shown in Equation 63 from Hwang, R. N. "A rigorous pole + representation of multilevel cross sections and its practical + applications." Nuclear Science and Engineering 96.3 (1987): 192-209. + + The :func:`scipy.special.wofz` function evaluates + :math:`w(z) = \exp(-z^2) \text{erfc}(-iz)`. These two forms of the Faddeeva + function are related by a transformation. + + If we call the integral form :math:`w_\text{int}`, and the function form + :math:`w_\text{fun}`: + + .. math:: + w_\text{int}(z) = + \begin{cases} + w_\text{fun}(z) & \text{for } \text{Im}(z) > 0\\ + -w_\text{fun}(z^*)^* & \text{for } \text{Im}(z) < 0 + \end{cases} + + Parameters + ---------- + z : complex + Argument to the Faddeeva function. + + Returns + ------- + complex + :math:`\frac{i}{\pi} \int_{-\infty}^{\infty} \frac{1}{z - t} \exp(-t^2) + \text{d}t` + + """ + from scipy.special import wofz + if np.angle(z) > 0: + return wofz(z) + else: + return -np.conj(wofz(z)) + + +def _broaden_wmp_polynomials(E, dopp, n): + r"""Evaluate Doppler-broadened windowed multipole curvefit. + + The curvefit is a polynomial of the form :math:`\frac{a}{E} + + \frac{b}{\sqrt{E}} + c + d \sqrt{E} + \ldots` + + Parameters + ---------- + E : Real + Energy to evaluate at. + dopp : Real + sqrt(atomic weight ratio / kT) in units of eV. + n : Integral + Number of components to the polynomial. + + Returns + ------- + numpy.ndarray + The value of each Doppler-broadened curvefit polynomial term. + + """ + sqrtE = np.sqrt(E) + beta = sqrtE * dopp + half_inv_dopp2 = 0.5 / dopp**2 + quarter_inv_dopp4 = half_inv_dopp2**2 + + if beta > 6.0: + # Save time, ERF(6) is 1 to machine precision. + # beta/sqrtpi*exp(-beta**2) is also approximately 1 machine epsilon. + erf_beta = 1.0 + exp_m_beta2 = 0.0 + else: + erf_beta = np.erf(beta) + exp_m_beta2 = np.exp(-beta**2) + + # Assume that, for sure, we'll use a second order (1/E, 1/V, const) + # fit, and no less. + + factors = np.zeros(n) + + factors[0] = erf_beta / E + factors[1] = 1.0 / sqrtE + factors[2] = (factors[0] * (half_inv_dopp2 + E) + + exp_m_beta2 / (beta * np.sqrt(np.pi))) + + # Perform recursive broadening of high order components. range(1, n-4) + # replaces a do i = 1, n=3. All indices are reduced by one due to the + # 1-based vs. 0-based indexing. + for i in range(1, n-4): + if i != 1: + factors[i+2] = (-factors[i-2] * (i - 1.0) * i * quarter_inv_dopp4 + + factors[i] * (E + (1.0 + 2.0 * i) * half_inv_dopp2)) + else: + # Although it's mathematically identical, factors[0] will contain + # nothing, and we don't want to have to worry about memory. + factors[i+2] = factors[i]*(E + (1.0 + 2.0 * i) * half_inv_dopp2) + + return factors + + +class WindowedMultipole(EqualityMixin): + """Resonant cross sections represented in the windowed multipole format. + + Attributes + ---------- + num_l : Integral + Number of possible l quantum states for this nuclide. + fit_order : Integral + Order of the windowed curvefit. + fissionable : bool + Whether or not the target nuclide has fission data. + formalism : {'MLBW', 'RM'} + The R-matrix formalism used to reconstruct resonances. Either 'MLBW' + for multi-level Breit Wigner or 'RM' for Reich-Moore. + spacing : Real + The width of each window in sqrt(E)-space. For example, the frst window + will end at (sqrt(start_E) + spacing)**2 and the second window at + (sqrt(start_E) + 2*spacing)**2. + sqrtAWR : Real + Square root of the atomic weight ratio of the target nuclide. + start_E : Real + Lowest energy in eV the library is valid for. + end_E : Real + Highest energy in eV the library is valid for. + data : np.ndarray + A 2D array of complex poles and residues. data[i, 0] gives the energy + at which pole i is located. data[i, 1:] gives the residues associated + with the i-th pole. There are 3 residues for Reich-Moore data, one each + for the total, absorption, and fission channels. Multi-level + Breit Wigner data has an additional residue for the competitive channel. + pseudo_k0RS : np.ndarray + A 1D array of Real values. There is one value for each valid l + quantum number. The values are equal to + sqrt(2 m / hbar) * AWR / (AWR + 1) * r + where m is the neutron mass, AWR is the atomic weight ratio, and r + is the l-dependent scattering radius. + l_value : np.ndarray + A 1D array of Integral values equal to the l quantum number for each + pole + 1. + w_start : np.ndarray + A 1D array of Integral values. w_start[i] - 1 is the index of the first + pole in window i. + w_end : np.ndarray + A 1D array of Integral values. w_end[i] - 1 is the index of the last + pole in window i. + broaden_poly : np.ndarray + A 1D array of boolean values indicating whether or not the polynomial + curvefit in that window should be Doppler broadened. + curvefit : np.ndarray + A 3D array of Real curvefit polynomial coefficients. curvefit[i, 0, :] + gives coefficients for the total cross section in window i. + curvefit[i, 1, :] gives absorption coefficients and curvefit[i, 2, :] + gives fission coefficients. The polynomial terms are increasing powers + of sqrt(E) starting with 1/E e.g: + a/E + b/sqrt(E) + c + d sqrt(E) + ... + + """ + def __init__(self): + self.num_l = None + self.fit_order = None + self.fissionable = None + self.formalism = None + self.spacing = None + self.sqrtAWR = None + self.start_E = None + self.end_E = None + self.data = None + self.pseudo_k0RS = None + self.l_value = None + self.w_start = None + self.w_end = None + self.broaden_poly = None + self.curvefit = None + + @property + def num_l(self): + return self._num_l + + @property + def fit_order(self): + return self._fit_order + + @property + def fissionable(self): + return self._fissionable + + @property + def formalism(self): + return self._formalism + + @property + def spacing(self): + return self._spacing + + @property + def sqrtAWR(self): + return self._sqrtAWR + + @property + def start_E(self): + return self._start_E + + @property + def end_E(self): + return self._end_E + + @property + def data(self): + return self._data + + @property + def pseudo_k0RS(self): + return self._pseudo_k0RS + + @property + def l_value(self): + return self._l_value + + @property + def w_start(self): + return self._w_start + + @property + def w_end(self): + return self._w_end + + @property + def broaden_poly(self): + return self._broaden_poly + + @property + def curvefit(self): + return self._curvefit + + @num_l.setter + def num_l(self, num_l): + if num_l is not None: + cv.check_type('num_l', num_l, Integral) + cv.check_greater_than('num_l', num_l, 1, equality=True) + cv.check_less_than('num_l', num_l, 4, equality=True) + # There is an if block in _evaluate that assumes num_l <= 4. + self._num_l = num_l + + @fit_order.setter + def fit_order(self, fit_order): + if fit_order is not None: + cv.check_type('fit_order', fit_order, Integral) + cv.check_greater_than('fit_order', fit_order, 2, equality=True) + # _broaden_wmp_polynomials assumes the curve fit has at least 3 + # terms. + self._fit_order = fit_order + + @fissionable.setter + def fissionable(self, fissionable): + if fissionable is not None: + cv.check_type('fissionable', fissionable, bool) + self._fissionable = fissionable + + @formalism.setter + def formalism(self, formalism): + if formalism is not None: + cv.check_type('formalism', formalism, string_types) + cv.check_value('formalism', formalism, ('MLBW', 'RM')) + self._formalism = formalism + + @spacing.setter + def spacing(self, spacing): + if spacing is not None: + cv.check_type('spacing', spacing, Real) + cv.check_greater_than('spacing', spacing, 0.0, equality=False) + self._spacing = spacing + + @sqrtAWR.setter + def sqrtAWR(self, sqrtAWR): + if sqrtAWR is not None: + cv.check_type('sqrtAWR', sqrtAWR, Real) + cv.check_greater_than('sqrtAWR', sqrtAWR, 0.0, equality=False) + self._sqrtAWR = sqrtAWR + + @start_E.setter + def start_E(self, start_E): + if start_E is not None: + cv.check_type('start_E', start_E, Real) + cv.check_greater_than('start_E', start_E, 0.0, equality=True) + self._start_E = start_E + + @end_E.setter + def end_E(self, end_E): + if end_E is not None: + cv.check_type('end_E', end_E, Real) + cv.check_greater_than('end_E', end_E, 0.0, equality=False) + self._end_E = end_E + + @data.setter + def data(self, data): + if data is not None: + cv.check_type('data', data, np.ndarray) + if len(data.shape) != 2: + raise ValueError('Multipole data arrays must be 2D') + if data.shape[1] not in (4, 5): # 4 for RM, 5 for MLBW + raise ValueError('The second dimension of multipole data arrays' + ' must have a length of either 4 or 5') + if not np.issubdtype(data.dtype, complex): + raise TypeError('Multipole data arrays must be complex dtype') + self._data = data + + @pseudo_k0RS.setter + def pseudo_k0RS(self, pseudo_k0RS): + if pseudo_k0RS is not None: + cv.check_type('pseudo_k0RS', pseudo_k0RS, np.ndarray) + if len(pseudo_k0RS.shape) != 1: + raise ValueError('Multipole pseudo_k0RS arrays must be 1D') + if not np.issubdtype(pseudo_k0RS.dtype, float): + raise TypeError('Multipole data arrays must be float dtype') + self._pseudo_k0RS = pseudo_k0RS + + @l_value.setter + def l_value(self, l_value): + if l_value is not None: + cv.check_type('l_value', l_value, np.ndarray) + if len(l_value.shape) != 1: + raise ValueError('Multipole l_value arrays must be 1D') + if not np.issubdtype(l_value.dtype, int): + raise TypeError('Multipole l_value arrays must be integer' + ' dtype') + self._l_value = l_value + + @w_start.setter + def w_start(self, w_start): + if w_start is not None: + cv.check_type('w_start', w_start, np.ndarray) + if len(w_start.shape) != 1: + raise ValueError('Multipole w_start arrays must be 1D') + if not np.issubdtype(w_start.dtype, int): + raise TypeError('Multipole w_start arrays must be integer' + ' dtype') + self._w_start = w_start + + @w_end.setter + def w_end(self, w_end): + if w_end is not None: + cv.check_type('w_end', w_end, np.ndarray) + if len(w_end.shape) != 1: + raise ValueError('Multipole w_end arrays must be 1D') + if not np.issubdtype(w_end.dtype, int): + raise TypeError('Multipole w_end arrays must be integer dtype') + self._w_end = w_end + + @broaden_poly.setter + def broaden_poly(self, broaden_poly): + if broaden_poly is not None: + cv.check_type('broaden_poly', broaden_poly, np.ndarray) + if len(broaden_poly.shape) != 1: + raise ValueError('Multipole broaden_poly arrays must be 1D') + if not np.issubdtype(broaden_poly.dtype, bool): + raise TypeError('Multipole broaden_poly arrays must be boolean' + ' dtype') + self._broaden_poly = broaden_poly + + @curvefit.setter + def curvefit(self, curvefit): + if curvefit is not None: + cv.check_type('curvefit', curvefit, np.ndarray) + if len(curvefit.shape) != 3: + raise ValueError('Multipole curvefit arrays must be 3D') + if curvefit.shape[2] != 3: # One each for sigT, sigA, sigF + raise ValueError('The third dimension of multipole curvefit' + ' arrays must have a length of 3') + if not np.issubdtype(curvefit.dtype, float): + raise TypeError('Multipole curvefit arrays must be float dtype') + self._curvefit = curvefit + + @classmethod + def from_hdf5(cls, group_or_filename): + """Construct a WindowedMultipole object from an HDF5 group or file. + + Parameters + ---------- + group_or_filename : h5py.Group or str + HDF5 group containing multipole data. If given as a string, it is + assumed to be the filename for the HDF5 file, and the first group is + used to read from. + + Returns + ------- + openmc.data.WindowedMultipole + Resonant cross sections represented in the windowed multipole + format. + + """ + if isinstance(group_or_filename, h5py.Group): + group = group_or_filename + else: + h5file = h5py.File(group_or_filename, 'r') + version = h5file['version'].value[0].decode() + if version != WMP_VERSION: + raise ValueError('The given WMP data uses version ' + + version + ' whereas your installation of the OpenMC ' + 'Python API expects version ' + WMP_VERSION) + group = h5file['nuclide'] + + out = cls() + + # Read scalar values. Note that group['max_w'] is ignored. + + length = group['length'].value + windows = group['windows'].value + out.num_l = group['num_l'].value + out.fit_order = group['fit_order'].value + out.fissionable = bool(group['fissionable'].value) + + if group['formalism'].value == _FORM_MLBW: + out.formalism = 'MLBW' + elif group['formalism'].value == _FORM_RM: + out.formalism = 'RM' + else: + raise ValueError('Unrecognized/Unsupported R-matrix formalism') + + out.spacing = group['spacing'].value + out.sqrtAWR = group['sqrtAWR'].value + out.start_E = group['start_E'].value + out.end_E = group['end_E'].value + + # Read arrays. + + err = "WMP '{}' array shape is not consistent with the '{}' value" + + out.data = group['data'].value + if out.data.shape[0] != length: + raise ValueError(err.format('data', 'length')) + + out.pseudo_k0RS = group['pseudo_K0RS'].value + if out.pseudo_k0RS.shape[0] != out.num_l: + raise ValueError(err.format('pseudo_k0RS', 'num_l')) + + out.l_value = group['l_value'].value + if out.l_value.shape[0] != length: + raise ValueError(err.format('l_value', 'length')) + + out.w_start = group['w_start'].value + if out.w_start.shape[0] != windows: + raise ValueError(err.format('w_start', 'windows')) + + out.w_end = group['w_end'].value + if out.w_end.shape[0] != windows: + raise ValueError(err.format('w_end', 'windows')) + + out.broaden_poly = group['broaden_poly'].value.astype(np.bool) + if out.broaden_poly.shape[0] != windows: + raise ValueError(err.format('broaden_poly', 'windows')) + + out.curvefit = group['curvefit'].value + if out.curvefit.shape[0] != windows: + raise ValueError(err.format('curvefit', 'windows')) + if out.curvefit.shape[1] != out.fit_order + 1: + raise ValueError(err.format('curvefit', 'fit_order')) + + # Note that all the file 3 data (group['reactions/MT...']) are ignored. + + return out + + def _evaluate(self, E, T): + """Compute total, absorption, and fission cross sections. + + Parameters + ---------- + E : Real + Energy of the incident neutron in eV. + T : Real + Temperature of the target in K. + + Returns + ------- + 3-tuple of Real + Total, absorption, and fission microscopic cross sections at the + given energy and temperature. + + """ + + if E < self.start_E: return (0, 0, 0) + if E >= self.end_E: return (0, 0, 0) + + # ====================================================================== + # Bookkeeping + + # Define some frequently used variables. + sqrtkT = np.sqrt(K_BOLTZMANN * T) + sqrtE = np.sqrt(E) + invE = 1.0 / E + dopp = self.sqrtAWR / sqrtkT + + # Locate us. The i_window calc omits a + 1 present in F90 because of + # the 1-based vs. 0-based indexing. Similarly startw needs to be + # decreased by 1. endw does not need to be decreased because + # range(startw, endw) does not include endw. + i_window = int(np.floor((sqrtE - np.sqrt(self.start_E)) / self.spacing)) + startw = self.w_start[i_window] - 1 + endw = self.w_end[i_window] + + # Fill in factors. Because of the unique interference dips in scatering + # resonances, the total cross section has a special "factor" that does + # not appear in the absorption and fission equations. + if startw <= endw: + twophi = np.zeros(self.num_l, dtype=np.float) + sigT_factor = np.zeros(self.num_l, dtype=np.cfloat) + + for iL in range(self.num_l): + twophi[iL] = self.pseudo_k0RS[iL] * sqrtE + if iL == 1: + twophi[iL] = twophi[iL] - np.arctan(twophi[iL]) + elif iL == 2: + arg = 3.0 * twophi[iL] / (3.0 - twophi[iL]**2) + twophi[iL] = twophi[iL] - np.arctan(arg) + elif iL == 3: + arg = (twophi[iL] * (15.0 - twophi[iL]**2) + / (15.0 - 6.0 * twophi[iL]**2)) + twophi[iL] = twophi[iL] - np.arctan(arg) + + twophi = 2.0 * twophi + sigT_factor = np.cos(twophi) - 1j*np.sin(twophi) + + # Initialize the ouptut cross sections. + sigT = 0.0 + sigA = 0.0 + sigF = 0.0 + + # ====================================================================== + # Add the contribution from the curvefit polynomial. + + if sqrtkT != 0 and self.broaden_poly[i_window]: + # Broaden the curvefit. + broadened_polynomials = _broaden_wmp_polynomials(E, dopp, + self.fit_order + 1) + for i_poly in range(self.fit_order+1): + sigT += (self.curvefit[i_window, i_poly, _FIT_T] + * broadened_polynomials[i_poly]) + sigA += (self.curvefit[i_window, i_poly, _FIT_A] + * broadened_polynomials[i_poly]) + sigF += (self.curvefit[i_window, i_poly, _FIT_F] + * broadened_polynomials[i_poly]) + else: + temp = invE + for i_poly in range(self.fit_order+1): + sigT += self.curvefit[i_window, i_poly, _FIT_T] * temp + sigA += self.curvefit[i_window, i_poly, _FIT_A] * temp + sigF += self.curvefit[i_window, i_poly, _FIT_F] * temp + temp *= sqrtE + + # ====================================================================== + # Add the contribution from the poles in this window. + + if sqrtkT == 0.0: + # If at 0K, use asymptotic form. + for i_pole in range(startw, endw): + psi_chi = -1j / (self.data[i_pole, _MP_EA] - sqrtE) + c_temp = psi_chi / E + if self.formalism == 'MLBW': + sigT += ((self.data[i_pole, _MLBW_RT] * c_temp * + sigT_factor[self.l_value[i_pole]-1]).real + + (self.data[i_pole, _MLBW_RX] * c_temp).real) + sigA += (self.data[i_pole, _MLBW_RA] * c_temp).real + sigF += (self.data[i_pole, _MLBW_RF] * c_temp).real + elif self.formalism == 'RM': + sigT += (self.data[i_pole, _RM_RT] * c_temp * + sigT_factor[self.l_value[i_pole]-1]).real + sigA += (self.data[i_pole, _RM_RA] * c_temp).real + sigF += (self.data[i_pole, _RM_RF] * c_temp).real + else: + raise ValueError('Unrecognized/Unsupported R-matrix' + ' formalism') + + else: + # At temperature, use Faddeeva function-based form. + for i_pole in range(startw, endw): + Z = (sqrtE - self.data[i_pole, _MP_EA]) * dopp + w_val = _faddeeva(Z) * dopp * invE * np.sqrt(np.pi) + if self.formalism == 'MLBW': + sigT += ((self.data[i_pole, _MLBW_RT] * + sigT_factor[self.l_value[i_pole]-1] + + self.data[i_pole, _MLBW_RX]) * w_val).real + sigA += (self.data[i_pole, _MLBW_RA] * w_val).real + sigF += (self.data[i_pole, _MLBW_RF] * w_val).real + elif self.formalism == 'RM': + sigT += (self.data[i_pole, _RM_RT] * w_val * + sigT_factor[self.l_value[i_pole]-1]).real + sigA += (self.data[i_pole, _RM_RA] * w_val).real + sigF += (self.data[i_pole, _RM_RF] * w_val).real + else: + raise ValueError('Unrecognized/Unsupported R-matrix' + ' formalism') + + return sigT, sigA, sigF + + def __call__(self, E, T): + """Compute total, absorption, and fission cross sections. + + Parameters + ---------- + E : Real or Iterable of Real + Energy of the incident neutron in eV. + T : Real + Temperature of the target in K. + + Returns + ------- + 3-tuple of Real or 3-tuple of numpy.ndarray + Total, absorption, and fission microscopic cross sections at the + given energy and temperature. + + """ + + fun = np.vectorize(lambda x: self._evaluate(x, T)) + return fun(E) diff --git a/src/math.F90 b/src/math.F90 index eac4ea94cc..2fa0a6ce1e 100644 --- a/src/math.F90 +++ b/src/math.F90 @@ -772,11 +772,11 @@ contains !=============================================================================== ! BROADEN_WMP_POLYNOMIALS Doppler broadens the windowed multipole curvefit. The ! curvefit is a polynomial of the form -! a/En + b/sqrt(En) + c + d sqrt(En) ... +! a/E + b/sqrt(E) + c + d sqrt(E) ... !=============================================================================== - subroutine broaden_wmp_polynomials(En, dopp, n, factors) - real(8), intent(in) :: En ! Energy to evaluate at + subroutine broaden_wmp_polynomials(E, dopp, n, factors) + real(8), intent(in) :: E ! Energy to evaluate at real(8), intent(in) :: dopp ! sqrt(atomic weight ratio / kT), ! kT given in eV. integer, intent(in) :: n ! number of components to polynomial @@ -788,10 +788,10 @@ contains real(8) :: beta ! sqrt(atomic weight ratio * E / kT) real(8) :: half_inv_dopp2 ! 0.5 / dopp**2 real(8) :: quarter_inv_dopp4 ! 0.25 / dopp**4 - real(8) :: erfbeta ! error function of beta + real(8) :: erf_beta ! error function of beta real(8) :: exp_m_beta2 ! exp(-beta**2) - sqrtE = sqrt(En) + sqrtE = sqrt(E) beta = sqrtE * dopp half_inv_dopp2 = HALF / dopp**2 quarter_inv_dopp4 = half_inv_dopp2**2 @@ -799,30 +799,30 @@ contains if (beta > 6.0_8) then ! Save time, ERF(6) is 1 to machine precision. ! beta/sqrtpi*exp(-beta**2) is also approximately 1 machine epsilon. - erfBeta = ONE + erf_beta = ONE exp_m_beta2 = ZERO else - erfBeta = erf(beta) + erf_beta = erf(beta) exp_m_beta2 = exp(-beta**2) end if ! Assume that, for sure, we'll use a second order (1/E, 1/V, const) ! fit, and no less. - factors(1) = erfbeta / En + factors(1) = erf_beta / E factors(2) = ONE / sqrtE - factors(3) = factors(1) * (half_inv_dopp2 + En) & + factors(3) = factors(1) * (half_inv_dopp2 + E) & + exp_m_beta2 / (beta * SQRT_PI) ! Perform recursive broadening of high order components do i = 1, n-3 if (i /= 1) then factors(i+3) = -factors(i-1) * (i - ONE) * i * quarter_inv_dopp4 & - + factors(i+1) * (En + (ONE + TWO * i) * half_inv_dopp2) + + factors(i+1) * (E + (ONE + TWO * i) * half_inv_dopp2) else ! Although it's mathematically identical, factors(0) will contain ! nothing, and we don't want to have to worry about memory. - factors(i+3) = factors(i+1)*(En + (ONE + TWO * i) * half_inv_dopp2) + factors(i+3) = factors(i+1)*(E + (ONE + TWO * i) * half_inv_dopp2) end if end do end subroutine broaden_wmp_polynomials