Changes in sasmodels/models/polymer_excl_volume.py [2d81cfe:aa90015] in sasmodels

File:

• sasmodels/models/polymer_excl_volume.py (modified) (7 diffs)

sasmodels/models/polymer_excl_volume.py

@@ -17 +17 @@
 $a$ is the statistical segment length of the polymer chain,
 and $n$ is the degree of polymerization.
-This integral was later put into an almost analytical form as follows
+
+This integral was put into an almost analytical form as follows
 (Hammouda, 1993)
+
+.. math::
+
+    P(Q)=\frac{1}{\nu U^{1/2\nu}}
+    \left\{
+        \gamma\left(\frac{1}{2\nu},U\right) -
+        \frac{1}{U^{1/2\nu}}\gamma\left(\frac{1}{\nu},U\right)
+    \right\}
+
+and later recast as (for example, Hore, 2013; Hammouda & Kim, 2017)
 
 .. math::
@@ -29 +40 @@
 .. math::
 
-    \gamma(x,U)=\int_0^{U}dt\ exp(-t)t^{x-1}
+    \gamma(x,U)=\int_0^{U}dt\ \exp(-t)t^{x-1}
 
 and the variable $U$ is given in terms of the scattering vector $Q$ as
@@ -37 +48 @@
     U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6}
 
+The two analytic forms are equivalent. In the 1993 paper
+
+.. math::
+
+    \frac{1}{\nu U^{1/2\nu}}
+
+has been factored out.
+
+**SasView implements the 1993 expression**.
+
 The square of the radius-of-gyration is defined as
 
@@ -43 +64 @@
     R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)}
 
-Note that this model applies only in the mass fractal range (ie, $5/3<=m<=3$ )
-and **does not apply** to surface fractals ( $3<m<=4$ ).
-It also does not reproduce the rigid rod limit (m=1) because it assumes chain
-flexibility from the outset. It may cover a portion of the semi-flexible chain
-range ( $1<m<5/3$ ).
+.. note::
+    This model applies only in the mass fractal range (ie, $5/3<=m<=3$ )
+    and **does not apply** to surface fractals ( $3<m<=4$ ).
+    It also does not reproduce the rigid rod limit (m=1) because it assumes chain
+    flexibility from the outset. It may cover a portion of the semi-flexible chain
+    range ( $1<m<5/3$ ).
 
 A low-Q expansion yields the Guinier form and a high-Q expansion yields the
@@ -73 +95 @@
 .. math::
 
-    P(Q) = \frac{2}{Q^4R_{g}^4} \left[exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right]
+    P(Q) = \frac{2}{Q^4R_{g}^4} \left[\exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right]
 
 For 2D data: The 2D scattering intensity is calculated in the same way as 1D,
@@ -89 +111 @@
 
 B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview,
-Advances in Polym. Sci.* 106(1993) 87-133
+Advances in Polym. Sci.* 106 (1993) 87-133
+
+M Hore et al, *Co-Nonsolvency of Poly(n-isopropylacrylamide) in Deuterated
+Water/Ethanol Mixtures* 46 (2013) 7894-7901
+
+B Hammouda & M-H Kim, *The empirical core-chain model* 247 (2017) 434-440
 """
 
@@ -124 +151 @@
     with errstate(divide='ignore', invalid='ignore'):
         upow = power(usub, -0.5*porod_exp)
+        # Note: scipy gammainc is "regularized", being gamma(s,x)/Gamma(s),
+        # so need to scale by Gamma(s) to recover gamma(s, x).
         result = (porod_exp*upow *
                   (gamma(0.5*porod_exp)*gammainc(0.5*porod_exp, usub) -