Chemistry Reference
In-Depth Information
analysis procedure involves comparing
N
points {(
x
i
,
y
i
)} from an experimental
profile to
N
homologous points {(
u
i
,
v
i
)} from a theoretical profile. The theoretical
points must be rotated by an angle
y
, translated by a vector (
a
,
b
), and scaled by a
factor
t
in order to effect this comparison. The transformed theoretical coordinates
are given by:
¼
þ t
u
i
v
i
u
i
v
i
a
b
cos
y
sin
y
(4)
sin
y
cos
y
and are compared to {(
x
i
,
y
i
)} for each value of the shape parameter
B
. The value of
the shape parameter, which yields the minimum overall error, provides the optimal
fit. The interfacial tension is, then, obtained from the associated optimal scaling
factor
t
, recognizing that:
1
=
2
p
¼
g
t ¼
1
=
(5)
g
Dr
In least squares regression methods, the values of all the shape parameters
(i.e.,
t
,
a
,
b
,
y
) must be chosen simultaneously in order to minimize the sum of
the squared residuals:
h
i
X
N
2
2
u
i
v
i
sum
¼
x
i
þ
y
i
(6)
i
¼
1
In contrast, with the robust shape comparison method, each of the optimal
parameter values can be evaluated independently. In the case of rotation and
magnification variables, this is accomplished using the concept of repeated medians
as represented by the relationships:
t
ij
t
¼
med
i
med
i
(7a)
where
h
i
1
=
2
2
2
x
j
x
i
þ
y
j
y
i
t
ij
¼
(7b)
h
i
1
=
2
2
2
u
j
u
i
þ
v
j
v
i
and
y
ij
y
¼
med
i
med
i
(8)
