Biomedical Engineering Reference
In-Depth Information
where
l
is the orbital angular momentum quantum number.
Combiningthese two equationsfor the angular momentum gives
l
ma
.
v
θ
=
(12.17)
We find the magnitude of the effective magnetic charge,
e
m
,by
writing the magneticcharge density along
z
axis as follows:
j
m
v
z
=
2
e
a
ω
0
r
v
θ
v
z
δ
(
r
−
r
e
(
t
)).
e
m
δ
(
r
−
r
e
(
t
))
=
(12.18)
Subsequently, weobtainthe effective magnetic charge:
2
el
ω
0
r
v
z
a
2
.
e
m
=
(12.19)
12.4.3
Vortex-EELS Scattering Loss Probability
The analysis to find the vortex-EELS scattering probability follows
the same approach as with conventional EELS with the duality
replac
emen
ts
E
→
Z H
,
Z H
→−
E
,
j
→
j
m
,and
e
→
e
m
,where
=
√
μ/
. With these transforms, the scattering probability can be
found from Eq. 12.9 above:
Z
exp
i
t
)
G
H
zz
}
e
m
v
z
0
π
dt dt
(12.20)
The
z
component of the magnetic dyadic Green's function is the
particular solution to the equation:
r
e
(
t
);
ω
=
{
ω
−
ω
(
)
(
t
r
e
(
t
),
1
r
∇×
G
z
(
r
,
r
)
−
k
2
G
z
(
r
,
r
)
=
δ
(
r
−
r
)
z
.
∇×
(12.21)
Equation 12.20 is the key result of this chapter, providing a
relation that may be used to calculate the electron energy loss
scattering expected from a vortex electron beam by means of
coupling to the local magnetic response. As with Eq. 12.9, the
integrals may take over the spatial co-ordinates
z
,
z
, by noting the
electron velocity. A more detailed derivation and discussion of this
equation is given elsewhere (Mohammadi et al., 2012).
