Biomedical Engineering Reference
In-Depth Information
Fig. 7.1
Vector notation used
in this section. The centre of
mass of the cluster is
indicated by CM, the circles
labelled n and n
0
represent
two generic sub-units whose
centre of mass positions are
identified by vectors
e
−
→
r
n
→
r
n
r
→
R
n
and
n
R
n
0
respectively. The vectors
→
R
n,n
r,
r
n
0
indicate the
position of the scattering
electron with respect to the
different centres of mass
E
E
r
n
and
E
n
→
R
n
→
R
n
CM
7.2.1
Theory
The asymptotic form of the total wavefunction describing the electron scattered
from the whole cluster may be written as:
r/
j
l
0
.kr/ı
L;L
0
2
h
l
0
.kr/T
L
0
;L
X
1
f
L
;
.
E
r/
D
Y
L
0
.
O
C
(7.1)
L;L
0
where the first ter
m
in the square brackets represents the incident plane wave of
momentum k
D
p
E, with E the kinetic energy of the scattering electron. The
second term represents the scattered wave having the same energy (since the
scattering process is elastic). The index L indicates collectively .l;m/, Y
L
.
r/ are
spherical harmonics and j
l
and h
l
are, respectively, the spherical Bessel and Hankel
functions of the first kind. Here, the coordinates are referred to the centre of mass of
the cluster. The matrix
T
G
contains the information of the scattering process with
the whole target (the cluster); collisional data, for example cross sections, can be
easily obtained from it.
The incident plane wave from the above equation:
O
X
r/j
l
0
.kr/f
L
0
;
in;PW
.
E
r/
D
Y
L
0
.
O
(7.2)
L
0
needs to be re-expanded around the centre of mass of each molecular sub-unit.
Following the vector notation indicated in Fig.
7.1
, we can write:
