# Note: model title and parameter table are inserted automatically r""" Calculates the interparticle structure factor for a hard sphere fluid with a narrow, attractive, square well potential. **The Mean Spherical Approximation (MSA) closure relationship is used, but it is not the most appropriate closure for an attractive interparticle potential.** However, the solution has been compared to Monte Carlo simulations for a square well fluid and these show the MSA calculation to be limited to well depths $\epsilon < 1.5$ kT and volume fractions $\phi < 0.08$. Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential "shoulder", which may or may not be physically reasonable. The :ref:`stickyhardsphere` model may be a better choice in some circumstances. Computed values may behave badly at extremely small $qR$. .. note:: Earlier versions of SasView did not incorporate the so-called $\beta(q)$ ("beta") correction [2] for polydispersity and non-sphericity. This is only available in SasView versions 4.2.2 and higher. The well width $(\lambda)$ is defined as multiples of the particle diameter $(2 R)$. The interaction potential is: .. math:: U(r) = \begin{cases} \infty & r < 2R \\ -\epsilon & 2R \leq r < 2R\lambda \\ 0 & r \geq 2R\lambda \end{cases} where $r$ is the distance from the center of a sphere of a radius $R$. In SasView the effective radius may be calculated from the parameters used in the form factor $P(q)$ that this $S(q)$ is combined with. For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the $q$ vector is defined as .. math:: q = \sqrt{q_x^2 + q_y^2} References ---------- .. [#] R V Sharma, K C Sharma, *Physica*, 89A (1977) 213 .. [#] M Kotlarchyk and S-H Chen, *J. Chem. Phys.*, 79 (1983) 2461-2469 Source ------ `squarewell.py `_ Authorship and Verification ---------------------------- * **Author:** * **Last Modified by:** * **Last Reviewed by:** Steve King **Date:** March 27, 2019 * **Source added by :** Steve King **Date:** March 25, 2019 """ import numpy as np from numpy import inf name = "squarewell" title = "Square well structure factor with Mean Spherical Approximation closure" description = """\ [Square well structure factor, with MSA closure] Interparticle structure factor S(Q) for a hard sphere fluid with a narrow attractive well. Fits are prone to deliver non- physical parameters; use with care and read the references in the model documentation.The "beta(q)" correction is available in versions 4.2.2 and higher. """ category = "structure-factor" structure_factor = True single = False #single = False # ["name", "units", default, [lower, upper], "type","description"], parameters = [ # [ "name", "units", default, [lower, upper], "type", # "description" ], ["radius_effective", "Ang", 50.0, [0, inf], "volume", "effective radius of hard sphere"], ["volfraction", "", 0.04, [0, 0.08], "", "volume fraction of spheres"], ["welldepth", "kT", 1.5, [0.0, 1.5], "", "depth of well, epsilon"], ["wellwidth", "diameters", 1.2, [1.0, inf], "", "width of well in diameters (=2R) units, must be > 1"], ] # No volume normalization despite having a volume parameter # This should perhaps be volume normalized? form_volume = """ return 1.0; """ Iq = """ // single precision is very poor at extreme small Q, would need a Taylor series double req,phis,edibkb,lambda,struc; double sigma,eta,eta2,eta3,eta4,etam1,etam14,alpha,beta,gamm; double x,sk,sk2,sk3,sk4,t1,t2,t3,t4,ck; double S,C,SL,CL; x= q; req = radius_effective; phis = volfraction; edibkb = welldepth; lambda = wellwidth; sigma = req*2.; eta = phis; eta2 = eta*eta; eta3 = eta*eta2; eta4 = eta*eta3; etam1 = 1. - eta; etam14 = etam1*etam1*etam1*etam1; // temp borrow sk for an intermediate calc sk = 1.0 +2.0*eta; sk *= sk; alpha = ( sk + eta3*( eta-4.0 ) )/etam14; beta = -(eta/3.0) * ( 18. + 20.*eta - 12.*eta2 + eta4 )/etam14; gamm = 0.5*eta*( sk + eta3*(eta-4.) )/etam14; // calculate the structure factor sk = x*sigma; sk2 = sk*sk; sk3 = sk*sk2; sk4 = sk3*sk; SINCOS(sk,S,C); SINCOS(lambda*sk,SL,CL); t1 = alpha * sk3 * ( S - sk * C ); t2 = beta * sk2 * 2.0*( sk*S - (0.5*sk2 - 1.)*C - 1.0 ); t3 = gamm*( ( 4.0*sk3 - 24.*sk ) * S - ( sk4 - 12.0*sk2 + 24.0 )*C + 24.0 ); t4 = -edibkb*sk3*(SL +sk*(C - lambda*CL) - S ); ck = -24.0*eta*( t1 + t2 + t3 + t4 )/sk3/sk3; struc = 1./(1.-ck); return(struc); """ def random(): """Return a random parameter set for the model.""" pars = dict( scale=1, background=0, radius_effective=10**np.random.uniform(1, 4.7), volfraction=np.random.uniform(0.00001, 0.08), welldepth=np.random.uniform(0, 1.5), wellwidth=np.random.uniform(1, 1.2), ) return pars demo = dict(radius_effective=50, volfraction=0.04, welldepth=1.5, wellwidth=1.2, radius_effective_pd=0, radius_effective_pd_n=0) # tests = [ [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0, 'volfraction': 0.04, 'welldepth': 1.5, 'wellwidth': 1.2, 'radius_effective_pd': 0}, [0.001], [0.97665742]], ] # ADDED by: converting from sasview RKH ON: 16Mar2016