Biomedical Engineering Reference
In-Depth Information
with R
n;n
0
the distance between the centres of mass of sub-units n and n
0
. Equation
(
7.8
) expresses the wave scattered off sub-unit n and incoming onto sub-unit n
0
.
We now need to find a relation linking g
L
(the total incoming wave amplitude) to
b
g
L
1
(the plane wave amplitude), so we will re-write the total incoming wavefunction
from equation (
7.6
) explicitly as:
X
in;n
.
r
n
/
E
D
in;PW
.
r
n
/
E
C
sc;n
0
.
E
r
n
/:
(7.10)
n
0
¤
n
Substituting equations (
7.3
)for
in;PW
.
E
r
n
/ and (
7.8
)for
sc;n
0
.
E
r
n
/:
2
3
X
X
X
X
n;n
0
L
1
;L
0
T
n
0
L
0
;L
g
n
L
4
b
g
L
1
C
5
;
in;n
.
E
r
n
/
D
Y
L
1
.
r
n
/j
l
1
.kr
n
/
O
(7.11)
n
0
¤
L;L
0
L
1
n
and comparing this expression with equation (
7.6
) we obtain the following relation:
X
X
X
n;n
0
L
1
;L
0
T
n
0
L
0
;L
g
n
L
:
g
L
1
D b
g
L
1
C
(7.12)
n
0
¤
n
L;L
0
This equation can be rewritten in matrix form:
G D G C XTG
;
(7.13)
if one defines the following vectors and matrices:
2
4
3
5
I G
2
4
3
5
I T
2
4
3
5
I
g
1
b
g
1
g
2
:
:
:
g
N
T
1
b
0
0
g
2
:
:
:
b
0T
2
0
G
(7.14)
:
:
:
:
:
:
00
g
N
T
N
2
4
3
5
0X
1;2
X
1;N
X
2;1
X
2;N
0
X
;
(7.15)
:
:
:
:
:
:
X
N;1
X
N;2
0
where N is the number of sub-units in the cluster. Notice that
T
n
and
X
n;n
0
in these
expressions are matrices with dimensions
1
.l
max
C
1/
2
, while
1/
2
;.l
max
C
b
g
n
and
g
n
are vectors with dimension .l
max
C
1/
2
, since the index L
D
.l;m/ runs from
1
0
has the same dimensions.
