Subroutine: Chen
Comparison data generated by program: Chen-prog.exe
This model describes the scattering from polydisperse fractal clusters of monodisperse noninteracting subunits (specifically, aggregates formed by a variable number of identical rigid spheres –
primary ‘particles’ - that fill space according to the fractal power law RDf where 1 < Df < 3).
Note that small values of the polydispersity index  correspond to a broad size distribution. A
reaction-limited cluster aggregation (RLCA) process should lead to a  value of 1.5.
The scattered intensity I(Q) is calculated as:
I (Q) 
(  cluster   medium ) 2 .  . V primary . N agg
(2   )
2 2
 [ F (3   , Q ) (1  Q  )
 D f ( 3 ) / 2
 G (2   , Q )  Q 
 h 
where:
V primary  (4 / 3) .  . R03
; volume of a primary ‘particle’
R12  (3 / 5) . R02
; radius-of-gyration of a primary ‘particle’
  h . R1 . N agg
; correlation length of the fractal cluster
(1 / D f )
h
D f ( D f  1)
6
F ( a, x )   ( a )   ( a, u )
 h 2 . (1  Q 2 2 ) 
u

Q 2 2


Df / 2
  x Df
 a,  
(
D

1
)
.

 f
   h 
G (a, x)  sin 
.
2
( D f  1)






(a,u) is the incomplete Euler gamma function and (a) is the normal gamma function.
REFERENCES
S-H Chen, J Rouch & P Tartaglia, Croat. Chem. Acta, 65(2), (1992), 353-366
YC Liu, EY Sheu, S-H Chen & DA Storm, Fuel, 74(9), (1995), 1352-1356
Df
] B
E Fratini, M Bonini, A Oasmaa, Y Solantausta, J Teixeira & P Baglioni, Langmuir, 22, (2006), 306-312
TEST DATASET
This example dataset was produced using 300 data points, qmin=0.001 Å-1, qmax=0.3 Å-1 and the
default values:
Parameter name
scale
primary radius
aggregation number
cluster polydispersity
fractal dimension
background
Symbol
Units
cm-4
Å
None
None
None
cm-1
(  cluster   medium ) 2 . 
R0
Nagg

Df
B
Default value
3.0X1019
3.5
70.0
1.6
1.45
0.01
NB: The model could, of course, be parameterised by separating out the contrast term and volume
fraction in the scale parameter.
The returned intensity is scaled to units of [cm-1].
ChenFit
1.E+01
1.E+00
1.E-01
1.E-02
1.E-03
1.E-02
1.E-01
1.E+00