Basic HTML Version
235
sin(
qR
i
,
j
)
qR
i
,
j
≈
1
−
(
qR
i
,
j
)
2
6
+
(
qR
i
,
j
)
4
120
+
.......
Keeping only the first terms (limit of low angles) and substituting for P(
θ
) we
obtain:
P
(
θ
)
=
1
−
q
2
6
N
2
R
i
,
j
2
j
=
1
N
∑
i
=
1
N
∑
The expression looks complicated, but in fact it contains information about the
radius of gyration (R
G
), since the latter can be defined as:
R
G
2
=
1
2
N
2
i
=
1
N
∑
R
i
,
j
2
j
=
1
N
∑
[
Check the definition of R
G
used earlier in the course and convince yourself
they are mathematically equivalent
]
. Taking all this into the original equation
we obtain:
P
(
θ
)
=
1
−
q
2
R
G
2
3
+
...
=
1
−
16
π
2
R
G
2
3
λ
2
sin
2
θ
2
⎛
⎝⎜
⎞
⎠⎟ +
...
Since 1/1-x
≈
1+x for small x, we rearrange and obtain the following
expression for P(
θ
)
-1
:
P
(
θ
)
−
1
=
1
+
16
π
2
R
G
2
3
λ
2
sin
2
θ
2
⎛
⎝⎜
⎞
⎠⎟ +
...
The beauty of the equation is that with light scattering
we can determine R
G
without any assumptions of the size or shape of the molecule
. On the
contrary, knowledge of R
G
provides valuable information of the size and
shape of the molecule. For instance, denaturation of a protein changes R
G
,
but M is the same. Alternatively, two proteins with the same molecular weight
but widely differing R
G
must have very different shapes.
It remains to state the complete light scattering equation: