# Changeset 12f77e9 in sasmodels

Ignore:
Timestamp:
Oct 30, 2018 10:56:38 AM (10 months ago)
Branches:
master
Children:
4e96703
Parents:
1657e21 (diff), c6084f1 (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.
Message:

Merge branch 'master' into ticket-608-user-defined-weights

Files:
20 edited

Unmodified
Removed
• ## doc/guide/magnetism/magnetism.rst

 rbefe905 ===========   ================================================================ M0:sld       $D_M M_0$ mtheta:sld   $\theta_M$ mphi:sld     $\phi_M$ up:angle     $\theta_\mathrm{up}$ up:frac_i    $u_i$ = (spin up)/(spin up + spin down) *before* the sample up:frac_f    $u_f$ = (spin up)/(spin up + spin down) *after* the sample sld_M0       $D_M M_0$ sld_mtheta   $\theta_M$ sld_mphi     $\phi_M$ up_frac_i    $u_i$ = (spin up)/(spin up + spin down) *before* the sample up_frac_f    $u_f$ = (spin up)/(spin up + spin down) *after* the sample up_angle     $\theta_\mathrm{up}$ ===========   ================================================================ .. note:: The values of the 'up:frac_i' and 'up:frac_f' must be in the range 0 to 1. The values of the 'up_frac_i' and 'up_frac_f' must be in the range 0 to 1. *Document History*
• ## doc/guide/plugin.rst

 r2015f02 def random(): ... This function provides a model-specific random parameter set which shows model features in the USANS to SANS range.  For example, core-shell sphere sets the outer radius of the sphere logarithmically in [20, 20,000], which sets the Q value for the transition from flat to falling.  It then uses a beta distribution to set the percentage of the shape which is shell, giving a preference for very thin or very thick shells (but never 0% or 100%).  Using -sets=10 in sascomp should show a reasonable variety of curves over the default sascomp q range. The parameter set is returned as a dictionary of {parameter: value, ...}. Any model parameters not included in the dictionary will default according to This function provides a model-specific random parameter set which shows model features in the USANS to SANS range.  For example, core-shell sphere sets the outer radius of the sphere logarithmically in [20, 20,000], which sets the Q value for the transition from flat to falling.  It then uses a beta distribution to set the percentage of the shape which is shell, giving a preference for very thin or very thick shells (but never 0% or 100%).  Using -sets=10 in sascomp should show a reasonable variety of curves over the default sascomp q range. The parameter set is returned as a dictionary of {parameter: value, ...}. Any model parameters not included in the dictionary will default according to the code in the _randomize_one() function from sasmodels/compare.py. erf, erfc, tgamma, lgamma:  **do not use** Special functions that should be part of the standard, but are missing or inaccurate on some platforms. Use sas_erf, sas_erfc and sas_gamma instead (see below). Note: lgamma(x) has not yet been tested. or inaccurate on some platforms. Use sas_erf, sas_erfc, sas_gamma and sas_lgamma instead (see below). Some non-standard constants and functions are also provided: Gamma function sas_gamma\ $(x) = \Gamma(x)$. The standard math function, tgamma(x) is unstable for $x < 1$ The standard math function, tgamma(x), is unstable for $x < 1$ on some platforms. :code:source = ["lib/sas_gamma.c", ...] (sas_gamma.c _) sas_gammaln(x): log gamma function sas_gammaln\ $(x) = \log \Gamma(|x|)$. The standard math function, lgamma(x), is incorrect for single precision on some platforms. :code:source = ["lib/sas_gammainc.c", ...] (sas_gammainc.c _) sas_gammainc(a, x), sas_gammaincc(a, x): Incomplete gamma function sas_gammainc\ $(a, x) = \int_0^x t^{a-1}e^{-t}\,dt / \Gamma(a)$ and complementary incomplete gamma function sas_gammaincc\ $(a, x) = \int_x^\infty t^{a-1}e^{-t}\,dt / \Gamma(a)$ :code:source = ["lib/sas_gammainc.c", ...] (sas_gammainc.c _) sas_erf(x), sas_erfc(x): If $n$ = 0 or 1, it uses sas_J0($x$) or sas_J1($x$), respectively. Warning: JN(n,x) can be very inaccurate (0.1%) for x not in [0.1, 100]. The standard math function jn(n, x) is not available on all platforms. Sine integral Si\ $(x) = \int_0^x \tfrac{\sin t}{t}\,dt$. Warning: Si(x) can be very inaccurate (0.1%) for x in [0.1, 100]. This function uses Taylor series for small and large arguments: For large arguments, For large arguments use the following Taylor series, .. math:: - \frac{\sin(x)}{x}\left(\frac{1}{x} - \frac{3!}{x^3} + \frac{5!}{x^5} - \frac{7!}{x^7}\right) For small arguments, For small arguments , .. math::

• ## sasmodels/__init__.py

 re65c3ba defining new models. """ __version__ = "0.97" __version__ = "0.98" def data_files():
• ## sasmodels/compare.py

 r01dba26 # Limit magnetic SLDs to a smaller range, from zero to iron=5/A^2 if par.name.startswith('M0:'): if par.name.endswith('_M0'): return np.random.uniform(0, 5) magnetic_pars = [] for p in parameters.user_parameters(pars, is2d): if any(p.id.startswith(x) for x in ('M0:', 'mtheta:', 'mphi:')): if any(p.id.endswith(x) for x in ('_M0', '_mtheta', '_mphi')): continue if p.id.startswith('up:'): pdtype=pars.get(p.id+"_pd_type", 'gaussian'), relative_pd=p.relative_pd, M0=pars.get('M0:'+p.id, 0.), mphi=pars.get('mphi:'+p.id, 0.), mtheta=pars.get('mtheta:'+p.id, 0.), M0=pars.get(p.id+'_M0', 0.), mphi=pars.get(p.id+'_mphi', 0.), mtheta=pars.get(p.id+'_mtheta', 0.), ) lines.append(_format_par(p.name, **fields)) if suppress: for p in pars: if p.startswith("M0:"): if p.endswith("_M0"): pars[p] = 0 else: first_mag = None for p in pars: if p.startswith("M0:"): if p.endswith("_M0"): any_mag |= (pars[p] != 0) if first_mag is None:
• ## sasmodels/convert.py

 ra69d8cd if version == (3, 1, 2): oldpars = _hand_convert_3_1_2_to_4_1(name, oldpars) if version < (4, 2, 0): oldpars = _rename_magnetic_pars(oldpars) return oldpars def _rename_magnetic_pars(pars): """ Change from M0:par to par_M0, etc. """ keys = list(pars.items()) for k in keys: if k.startswith('M0:'): pars[k[3:]+'_M0'] = pars.pop(k) elif k.startswith('mtheta:'): pars[k[7:]+'_mtheta'] = pars.pop(k) elif k.startswith('mphi:'): pars[k[5:]+'_mphi'] = pars.pop(k) elif k.startswith('up:'): pars['up_'+k[3:]] = pars.pop(k) return pars def _hand_convert_3_1_2_to_4_1(name, oldpars):

• ## sasmodels/kernelpy.py

 r108e70e self.info = model_info self.dtype = np.dtype('d') logger.info("load python model " + self.info.name) logger.info("make python model " + self.info.name) def make_kernel(self, q_vectors):

• ## sasmodels/modelinfo.py

 r7b9e4dd self.is_asymmetric = any(p.name == 'psi' for p in self.kernel_parameters) self.magnetism_index = [k for k, p in enumerate(self.call_parameters) if p.id.startswith('M0:')] if p.id.endswith('_M0')] self.pd_1d = set(p.name for p in self.call_parameters if self.nmagnetic > 0: full_list.extend([ Parameter('up:frac_i', '', 0., [0., 1.], Parameter('up_frac_i', '', 0., [0., 1.], 'magnetic', 'fraction of spin up incident'), Parameter('up:frac_f', '', 0., [0., 1.], Parameter('up_frac_f', '', 0., [0., 1.], 'magnetic', 'fraction of spin up final'), Parameter('up:angle', 'degrees', 0., [0., 360.], Parameter('up_angle', 'degrees', 0., [0., 360.], 'magnetic', 'spin up angle'), ]) for p in slds: full_list.extend([ Parameter('M0:'+p.id, '1e-6/Ang^2', 0., [-np.inf, np.inf], Parameter(p.id+'_M0', '1e-6/Ang^2', 0., [-np.inf, np.inf], 'magnetic', 'magnetic amplitude for '+p.description), Parameter('mtheta:'+p.id, 'degrees', 0., [-90., 90.], Parameter(p.id+'_mtheta', 'degrees', 0., [-90., 90.], 'magnetic', 'magnetic latitude for '+p.description), Parameter('mphi:'+p.id, 'degrees', 0., [-180., 180.], Parameter(p.id+'_mphi', 'degrees', 0., [-180., 180.], 'magnetic', 'magnetic longitude for '+p.description), ]) Parameters marked as sld will automatically have a set of associated magnetic parameters (m0:p, mtheta:p, mphi:p), as well as polarization information (up:theta, up:frac_i, up:frac_f). magnetic parameters (p_M0, p_mtheta, p_mphi), as well as polarization information (up_theta, up_frac_i, up_frac_f). """ # control parameters go first result.append(expanded_pars[name]) if is2d: for tag in 'M0:', 'mtheta:', 'mphi:': if tag+name in expanded_pars: result.append(expanded_pars[tag+name]) for tag in '_M0', '_mtheta', '_mphi': if name+tag in expanded_pars: result.append(expanded_pars[name+tag]) # Gather the user parameters in order append_group(p.id) if is2d and 'up:angle' in expanded_pars: if is2d and 'up_angle' in expanded_pars: result.extend([ expanded_pars['up:frac_i'], expanded_pars['up:frac_f'], expanded_pars['up:angle'], expanded_pars['up_frac_i'], expanded_pars['up_frac_f'], expanded_pars['up_angle'], ]) info.structure_factor = getattr(kernel_module, 'structure_factor', False) info.profile_axes = getattr(kernel_module, 'profile_axes', ['x', 'y']) # Note: custom.load_custom_kernel_module assumes the C sources are defined # by this attribute. info.source = getattr(kernel_module, 'source', []) info.c_code = getattr(kernel_module, 'c_code', None) for k in range(control+1, p.length+1) if p.length > 1) for p in self.parameters.kernel_parameters: if p.length > 1 and p.type == "sld": for k in range(control+1, p.length+1): base = p.id+str(k) hidden.update((base+"_M0", base+"_mtheta", base+"_mphi")) return hidden
• ## sasmodels/models/bcc_paracrystal.py

 r2d81cfe r""" .. warning:: This model and this model description are under review following concerns raised by SasView users. If you need to use this model, please email help@sasview.org for the latest situation. *The SasView Developers. September 2018.* Definition ---------- I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q) where *scale* is the volume fraction of spheres, $V_p$ is the volume of the Authorship and Verification ---------------------------- --------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010
• ## sasmodels/models/be_polyelectrolyte.py

 ref07e95 r""" .. note:: Please read the Validation section below. Definition ---------- I(q) = K\frac{q^2+k^2}{4\pi L_b\alpha ^2} \frac{1}{1+r_{0}^2(q^2+k^2)(q^2-12hC_a/b^2)} + background \frac{1}{1+r_{0}^4(q^2+k^2)(q^2-12hC_a/b^2)} + background k^2 = 4\pi L_b(2C_s + \alpha C_a) r_{0}^2 = \frac{1}{\alpha \sqrt{C_a} \left( b/\sqrt{48\pi L_b}\right)} r_{0}^2 = \frac{b}{\alpha \sqrt{C_a 48\pi L_b}} where $K$ is the contrast factor for the polymer which is defined differently than in other models and is given in barns where $1 barn = 10^{-24} cm^2$.  $K$ is other models and is given in barns where 1 $barn = 10^{-24}$ $cm^2$.  $K$ is defined as: a = b_p - (v_p/v_s) b_s where $b_p$ and $b_s$ are sum of the scattering lengths of the atoms constituting the monomer of the polymer and the sum of the scattering lengths of the atoms constituting the solvent molecules respectively, and $v_p$ and $v_s$ are the partial molar volume of the polymer and the solvent respectively $L_b$ is the Bjerrum length(|Ang|) - **Note:** This parameter needs to be kept constant for a given solvent and temperature! $h$ is the virial parameter (|Ang^3|/mol) - **Note:** See [#Borue]_ for the correct interpretation of this parameter.  It incorporates second and third virial coefficients and can be Negative. $b$ is the monomer length(|Ang|), $C_s$ is the concentration of monovalent salt(mol/L), $\alpha$ is the ionization degree (ionization degree : ratio of charged monomers  to total number of monomers), $C_a$ is the polymer molar concentration(mol/L), and $background$ is the incoherent background. where: - $b_p$ and $b_s$ are **sum of the scattering lengths of the atoms** constituting the polymer monomer and the solvent molecules, respectively. - $v_p$ and $v_s$ are the partial molar volume of the polymer and the solvent, respectively. - $L_b$ is the Bjerrum length (|Ang|) - **Note:** This parameter needs to be kept constant for a given solvent and temperature! - $h$ is the virial parameter (|Ang^3|) - **Note:** See [#Borue]_ for the correct interpretation of this parameter.  It incorporates second and third virial coefficients and can be *negative*. - $b$ is the monomer length (|Ang|). - $C_s$ is the concentration of monovalent salt(1/|Ang^3| - internally converted from mol/L). - $\alpha$ is the degree of ionization (the ratio of charged monomers to the total number of monomers) - $C_a$ is the polymer molar concentration (1/|Ang^3| - internally converted from mol/L) - $background$ is the incoherent background. For 2D data the scattering intensity is calculated in the same way as 1D, q = \sqrt{q_x^2 + q_y^2} Validation ---------- As of the last revision, this code is believed to be correct.  However it needs further validation and should be used with caution at this time.  The history of this code goes back to a 1998 implementation. It was recently noted that in that implementation, while both the polymer concentration and salt concentration were converted from experimental units of mol/L to more dimensionally useful units of 1/|Ang^3|, only the converted version of the polymer concentration was actually being used in the calculation while the unconverted salt concentration (still in apparent units of mol/L) was being used.  This was carried through to Sasmodels as used for SasView 4.1 (though the line of code converting the salt concentration to the new units was removed somewhere along the line). Simple dimensional analysis of the calculation shows that the converted salt concentration should be used, which the original code suggests was the intention, so this has now been corrected (for SasView 4.2). Once better validation has been performed this note will be removed. References * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Paul Kienzle **Date:** July 24, 2016 * **Last Reviewed by:** Paul Butler and Richard Heenan **Date:** October 07, 2016 * **Last Modified by:** Paul Butler **Date:** September 25, 2018 * **Last Reviewed by:** Paul Butler **Date:** September 25, 2018 """ ["contrast_factor",       "barns",   10.0,  [-inf, inf], "", "Contrast factor of the polymer"], ["bjerrum_length",        "Ang",      7.1,  [0, inf],    "", "Bjerrum length"], ["virial_param",          "Ang^3/mol", 12.0,  [-inf, inf], "", "Virial parameter"], ["virial_param",          "Ang^3", 12.0,  [-inf, inf], "", "Virial parameter"], ["monomer_length",        "Ang",     10.0,  [0, inf],    "", "Monomer length"], ["salt_concentration",    "mol/L",    0.0,  [-inf, inf], "", "Concentration of monovalent salt"], def Iq(q, contrast_factor=10.0, bjerrum_length=7.1, virial_param=12.0, monomer_length=10.0, salt_concentration=0.0, ionization_degree=0.05, polymer_concentration=0.7): contrast_factor, bjerrum_length, virial_param, monomer_length, salt_concentration, ionization_degree, polymer_concentration): """ :param q:                     Input q-value :param contrast_factor:       Contrast factor of the polymer :param bjerrum_length:        Bjerrum length :param virial_param:          Virial parameter :param monomer_length:        Monomer length :param salt_concentration:    Concentration of monovalent salt :param ionization_degree:     Degree of ionization :param polymer_concentration: Polymer molar concentration :return:                      1-D intensity :params: see parameter table :return: 1-D form factor for polyelectrolytes in low salt parameter names, units, default values, and behavior (volume, sld etc) are defined in the parameter table.  The concentrations are converted from experimental mol/L to dimensionaly useful 1/A3 in first two lines """ concentration = polymer_concentration * 6.022136e-4 k_square = 4.0 * pi * bjerrum_length * (2*salt_concentration + ionization_degree * concentration) r0_square = 1.0/ionization_degree/sqrt(concentration) * \ concentration_pol = polymer_concentration * 6.022136e-4 concentration_salt = salt_concentration * 6.022136e-4 k_square = 4.0 * pi * bjerrum_length * (2*concentration_salt + ionization_degree * concentration_pol) r0_square = 1.0/ionization_degree/sqrt(concentration_pol) * \ (monomer_length/sqrt((48.0*pi*bjerrum_length))) term2 = 1.0 + r0_square**2 * (q**2 + k_square) * \ (q**2 - (12.0 * virial_param * concentration/(monomer_length**2))) (q**2 - (12.0 * virial_param * concentration_pol/(monomer_length**2))) return term1/term2 # Accuracy tests based on content in test/utest_other_models.py # Note that these should some day be validated beyond this self validation # (circular reasoning). -- i.e. the "good value," at least for those with # non zero salt concentrations, were obtained by running the current # model in SasView and copying the appropriate result here. #    PDB -- Sep 26, 2018 [{'contrast_factor':       10.0, 'bjerrum_length':         7.1, }, 0.001, 0.0948379], # Additional tests with larger range of parameters [{'contrast_factor':       10.0, 'bjerrum_length':       100.0, 'virial_param':           3.0, 'monomer_length':         1.0, 'salt_concentration':    10.0, 'ionization_degree':      2.0, 'polymer_concentration': 10.0, 'monomer_length':         5.0, 'salt_concentration':     1.0, 'ionization_degree':      0.1, 'polymer_concentration':  1.0, 'background':             0.0, }, 0.1, -3.75693800588], }, 0.1, 0.253469484], [{'contrast_factor':       10.0, 'bjerrum_length':       100.0, 'virial_param':           3.0, 'monomer_length':         1.0, 'salt_concentration':    10.0, 'ionization_degree':      2.0, 'polymer_concentration': 10.0, 'background':           100.0 }, 5.0, 100.029142149], 'monomer_length':         5.0, 'salt_concentration':     1.0, 'ionization_degree':      0.1, 'polymer_concentration':  1.0, 'background':             1.0, }, 0.05, 1.738358122], [{'contrast_factor':     100.0, 'bjerrum_length':       10.0, 'virial_param':        180.0, 'monomer_length':        1.0, 'virial_param':         12.0, 'monomer_length':       10.0, 'salt_concentration':    0.1, 'ionization_degree':     0.5, 'polymer_concentration': 0.1, 'background':             0.0, }, 200., 1.80664667511e-06], 'background':           0.01, }, 0.5, 0.012881893], ]
• ## sasmodels/models/fcc_paracrystal.py

 r2d81cfe #note - calculation requires double precision r""" .. warning:: This model and this model description are under review following concerns raised by SasView users. If you need to use this model, please email help@sasview.org for the latest situation. *The SasView Developers. September 2018.* Definition ---------- Calculates the scattering from a **face-centered cubic lattice** with paracrystalline distortion. Thermal vibrations are considered to be Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution. Definition ---------- The scattering intensity $I(q)$ is calculated as is the paracrystalline structure factor for a face-centered cubic structure. Equation (1) of the 1990 reference is used to calculate $Z(q)$, using equations (23)-(25) from the 1987 paper for $Z1$, $Z2$, and $Z3$. Equation (1) of the 1990 reference\ [#CIT1990]_ is used to calculate $Z(q)$, using equations (23)-(25) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and $Z3$. The lattice correction (the occupied volume of the lattice) for a ---------- Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 (Original Paper) .. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 (Original Paper) .. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 (Corrections to FCC and BCC lattice structure calculation) Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 (Corrections to FCC and BCC lattice structure calculation) Authorship and Verification --------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Paul Butler **Date:** September 29, 2016 * **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016 """
• ## sasmodels/models/sc_paracrystal.py

 r2d81cfe r""" .. warning:: This model and this model description are under review following concerns raised by SasView users. If you need to use this model, please email help@sasview.org for the latest situation. *The SasView Developers. September 2018.* Definition ---------- Calculates the scattering from a **simple cubic lattice** with paracrystalline distortion. Thermal vibrations are considered to be Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution. Definition ---------- The scattering intensity $I(q)$ is calculated as $Z(q)$ is the paracrystalline structure factor for a simple cubic structure. Equation (16) of the 1987 reference is used to calculate $Z(q)$, using equations (13)-(15) from the 1987 paper for Z1, Z2, and Z3. Equation (16) of the 1987 reference\ [#CIT1987]_ is used to calculate $Z(q)$, using equations (13)-(15) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and $Z3$. The lattice correction (the occupied volume of the lattice) for a simple cubic Reference --------- Hideki Matsuoka et. al. *Physical Review B,* 36 (1987) 1754-1765 (Original Paper) Hideki Matsuoka et. al. *Physical Review B,* 41 (1990) 3854 -3856 (Corrections to FCC and BCC lattice structure calculation) .. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 (Original Paper) .. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 (Corrections to FCC and BCC lattice structure calculation) Authorship and Verification --------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Paul Butler **Date:** September 29, 2016 * **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016 """
• ## sasmodels/models/spinodal.py

 r475ff58 where $x=q/q_0$, $q_0$ is the peak position, $I_{max}$ is the intensity at $q_0$ (parameterised as the $scale$ parameter), and $B$ is a flat background. The spinodal wavelength is given by $2\pi/q_0$. background. The spinodal wavelength, $\Lambda$, is given by $2\pi/q_0$. The definition of $I_{max}$ in the literature varies. Hashimoto *et al* (1991) define it as .. math:: I_{max} = \Lambda^3\Delta\rho^2 whereas Meier & Strobl (1987) give .. math:: I_{max} = V_z\Delta\rho^2 where $V_z$ is the volume per monomer unit. The exponent $\gamma$ is equal to $d+1$ for off-critical concentration H. Furukawa. Dynamics-scaling theory for phase-separating unmixing mixtures: Growth rates of droplets and scaling properties of autocorrelation functions. Physica A 123,497 (1984). Growth rates of droplets and scaling properties of autocorrelation functions. Physica A 123, 497 (1984). H. Meier & G. Strobl. Small-Angle X-ray Scattering Study of Spinodal Decomposition in Polystyrene/Poly(styrene-co-bromostyrene) Blends. Macromolecules 20, 649-654 (1987). T. Hashimoto, M. Takenaka & H. Jinnai. Scattering Studies of Self-Assembling Processes of Polymer Blends in Spinodal Decomposition. J. Appl. Cryst. 24, 457-466 (1991). Revision History * **Author:**  Dirk Honecker **Date:** Oct 7, 2016 * **Revised:** Steve King    **Date:** Sep 7, 2018 * **Revised:** Steve King    **Date:** Oct 25, 2018 """
• ## sasmodels/sasview_model.py

 rdf69efa The standard math function, tgamma(x) is unstable for $x < 1$ on some platforms. sas_gammaln(x): log gamma function sas_gammaln\ $(x) = \log \Gamma(|x|)$. The standard math function, lgamma(x), is incorrect for single precision on some platforms. sas_gammainc(a, x), sas_gammaincc(a, x): Incomplete gamma function sas_gammainc\ $(a, x) = \int_0^x t^{a-1}e^{-t}\,dt / \Gamma(a)$ and complementary incomplete gamma function sas_gammaincc\ $(a, x) = \int_x^\infty t^{a-1}e^{-t}\,dt / \Gamma(a)$ sas_erf(x), sas_erfc(x): from numpy import pi, nan, inf from scipy.special import gamma as sas_gamma from scipy.special import gammaln as sas_gammaln from scipy.special import gammainc as sas_gammainc from scipy.special import gammaincc as sas_gammaincc from scipy.special import erf as sas_erf from scipy.special import erfc as sas_erfc
 rd089a00 -------------------------------------------- For some models we can calculate the average intensity for a population of particles that possess size and/or orientational (ie, angular) distributions. In SasView we call the former *polydispersity* but use the parameter *PD* to parameterise both. In other words, the meaning of *PD* in a model depends on For some models we can calculate the average intensity for a population of particles that possess size and/or orientational (ie, angular) distributions. In SasView we call the former *polydispersity* but use the parameter *PD* to parameterise both. In other words, the meaning of *PD* in a model depends on the actual parameter it is being applied too. The resultant intensity is then normalized by the average particle volume such The resultant intensity is then normalized by the average particle volume such that P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an average over the distribution $f(x; \bar x, \sigma)$, giving .. math:: P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} f(x; \bar x, \sigma) F^2(q, x)\, dx + \text{background} Each distribution is characterized by a center value $\bar x$ or $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* the standard deviation, so read the description of the distribution carefully), the number of sigmas $N_\sigma$ to include from the tails of the distribution, and the number of points used to compute the average. The center of the distribution is set by the value of the model parameter. The distribution width applied to *volume* (ie, shape-describing) parameters is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$. However, the distribution width applied to *orientation* parameters is just $\sigma = \mathrm{PD}$. the standard deviation, so read the description carefully), the number of sigmas $N_\sigma$ to include from the tails of the distribution, and the number of points used to compute the average. The center of the distribution is set by the value of the model parameter. The meaning of a polydispersity parameter *PD* (not to be confused with a molecular weight distributions in polymer science) in a model depends on the type of parameter it is being applied too. The distribution width applied to *volume* (ie, shape-describing) parameters is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$. However, the distribution width applied to *orientation* (ie, angle-describing) parameters is just $\sigma = \mathrm{PD}$. $N_\sigma$ determines how far into the tails to evaluate the distribution, Users should note that the averaging computation is very intensive. Applying polydispersion and/or orientational distributions to multiple parameters at the same time, or increasing the number of points in the distribution, will require patience! However, the calculations are generally more robust with polydispersion and/or orientational distributions to multiple parameters at the same time, or increasing the number of points in the distribution, will require patience! However, the calculations are generally more robust with more data points or more angles. *  *Schulz Distribution* *  *Array Distribution* *  *User-defined Distributions* These are all implemented as *number-average* distributions. Additional distributions are under consideration. **Beware: when the Polydispersity & Orientational Distribution panel in SasView is** **This may not be suitable. See Suggested Applications below.** .. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace the term 'polydispersity' (see Pure Appl. Chem., (2009), 81(2), 351-353 _ in order to make the terminology describing distributions of chemical properties unambiguous. However, these terms are unrelated to the proportional size distributions and orientational distributions used in .. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace the term 'polydispersity' (see Pure Appl. Chem., (2009), 81(2), 351-353 _ in order to make the terminology describing distributions of chemical properties unambiguous. However, these terms are unrelated to the proportional size distributions and orientational distributions used in SasView models. or angular orientations, consider using the Gaussian or Boltzmann distributions. If applying polydispersion to parameters describing angles, use the Uniform distribution. Beware of using distributions that are always positive (eg, the If applying polydispersion to parameters describing angles, use the Uniform distribution. Beware of using distributions that are always positive (eg, the Lognormal) because angles can be negative! The array distribution allows a user-defined distribution to be applied. The array distribution provides a very simple means of implementing a user- defined distribution, but without any fittable parameters. Greater flexibility is conferred by the user-defined distribution. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ User-defined Distributions ^^^^^^^^^^^^^^^^^^^^^^^^^^ You can also define your own distribution by creating a python file defining a *Distribution* object with a *_weights* method.  The *_weights* method takes *center*, *sigma*, *lb* and *ub* as arguments, and can access *self.npts* and *self.nsigmas* from the distribution.  They are interpreted as follows: * *center* the value of the shape parameter (for size dispersity) or zero if it is an angular dispersity.  This parameter may be fitted. * *sigma* the width of the distribution, which is the polydispersity parameter times the center for size dispersity, or the polydispersity parameter alone for angular dispersity.  This parameter may be fitted. * *lb*, *ub* are the parameter limits (lower & upper bounds) given in the model definition file.  For example, a radius parameter has *lb* equal to zero.  A volume fraction parameter would have *lb* equal to zero and *ub* equal to one. * *self.nsigmas* the distance to go into the tails when evaluating the distribution.  For a two parameter distribution, this value could be co-opted to use for the second parameter, though it will not be available for fitting. * *self.npts* the number of points to use when evaluating the distribution. The user will adjust this to trade calculation time for accuracy, but the distribution code is free to return more or fewer, or use it for the third parameter in a three parameter distribution. As an example, the code following wraps the Laplace distribution from scipy stats:: import numpy as np from scipy.stats import laplace from sasmodels import weights class Dispersion(weights.Dispersion): r""" Laplace distribution .. math:: w(x) = e^{-\sigma |x - \mu|} """ type = "laplace" default = dict(npts=35, width=0, nsigmas=3)  # default values def _weights(self, center, sigma, lb, ub): x = self._linspace(center, sigma, lb, ub) wx = laplace.pdf(x, center, sigma) return x, wx You can plot the weights for a given value and width using the following:: from numpy import inf from matplotlib import pyplot as plt from sasmodels import weights # reload the user-defined weights weights.load_weights() x, wx = weights.get_weights('laplace', n=35, width=0.1, nsigmas=3, value=50, limits=[0, inf], relative=True) # plot the weights plt.interactive(True) plt.plot(x, wx, 'x') The *self.nsigmas* and *self.npts* parameters are normally used to control the accuracy of the distribution integral. The *self._linspace* function uses them to define the *x* values (along with the *center*, *sigma*, *lb*, and *ub* which are passed as parameters).  If you repurpose npts or nsigmas you will need to generate your own *x*.  Be sure to honour the limits *lb* and *ub*, for example to disallow a negative radius or constrain the volume fraction to lie between zero and one. To activate a user-defined distribution, put it in a file such as *distname.py* in the *SAS_WEIGHTS_PATH* folder.  This is defined with an environment variable, defaulting to:: SAS_WEIGHTS_PATH=~/.sasview/weights The weights path is loaded on startup.  To update the distribution definition in a running application you will need to enter the following python commands:: import sasmodels.weights sasmodels.weights.load_weights('path/to/distname.py') .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Note about DLS polydispersity ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and it should not be assumed that any of the following can be simply equated with Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and it should not be assumed that any of the following can be simply equated with the polydispersity *PD* parameter used in SasView. The dimensionless **Polydispersity Index (PI)** is a measure of the width of the distribution of autocorrelation function decay rates (*not* the distribution of particle sizes itself, though the two are inversely related) and is defined by The dimensionless **Polydispersity Index (PI)** is a measure of the width of the distribution of autocorrelation function decay rates (*not* the distribution of particle sizes itself, though the two are inversely related) and is defined by ISO 22412:2017 as PI = \mu_{2} / \bar \Gamma^2 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the intensity-weighted average value, of the distribution of decay rates. PI = \sigma^2 / 2\bar \Gamma^2 where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** to be defined as RP = \sigma / \bar \Gamma = \sqrt{2 \cdot PI} PI values smaller than 0.05 indicate a highly monodisperse system. Values PI values smaller than 0.05 indicate a highly monodisperse system. Values greater than 0.7 indicate significant polydispersity. The **size polydispersity P-parameter** is defined as the relative standard deviation coefficient of variation The **size polydispersity P-parameter** is defined as the relative standard deviation coefficient of variation .. math:: where $\nu$ is the variance of the distribution and $\bar R$ is the mean value of $R$. Here, the product $P \bar R$ is *equal* to the standard value of $R$. Here, the product $P \bar R$ is *equal* to the standard deviation of the Lognormal distribution.