Chemistry Reference
In-Depth Information
Fig. 3.5
Scattering geometry
in HREELS experiments
and wave-vector parallel to the surface.
E
loss
=
E
p
−
E
S
(3.1)
h
(
k
i
sin
θ
i
−
k
S
sin
θ
S
)
h
q
||
=
¯
¯
Where
q
||
is the parallel momentum transfer,
k
i
is the wave-vector of incident elec-
trons,
k
s
is the wave-vector of scattered electrons,
θ
i
and
θ
s
are the angles formed with
the normal to the surface by incident and scattered electrons, respectively (Fig.
3.5
).
Thus an expression linking
q
||
with E
p
,E
loss
, and
θ
i
,
θ
s
could be obtained:
sin
θ
i
−
1
2
mE
p
¯
E
loss
E
p
q
||
=
−
sin
θ
S
(3.2)
h
Likewise, it is possible to obtain the indeterminacy on
q
||
, that is the window in the
reciprocal space which also depends on the angular acceptance of the apparatus
α ·
(Rocca
1995
), usually ranging between 0.5 and 1.0 degrees:
cos
θ
i
−
1
2
mE
p
¯
E
loss
E
p
q
||
=
−
cos
θ
S
·
α
(3.3)
h
Thus
q
||
is minimized for low impinging energies and for grazing scattering
conditions.
Three scattering mechanisms for impinging electrons are possible:
dipole
,
impact
and
resonant scattering
(Ibach and Mills
1982
). The last mechanism is prevalent for
molecules in gaseous phase (see Ref. (Ibach and Mills
1982
) for more details) and
thus it is not important for plasmonic excitations.
Concerning
dipole scattering
, it is worth remembering that the Coulombian field
produced by incoming electrons interacts at long range (about 100 Å) with the surface.
Loss events may occurs both before and after diffusion from the surface (Rocca
1995
),
as shown in Fig.
3.6
.
According to Mills (Mills
1975
), the differential cross section d
2
S
/
d
ω
d
,isgiven
by:
v
⊥
q
||
(
R
s
+
v
||
q
||
)
2
d
2
S
d
ω
d
=
(
mev
⊥
)
2
2
π
2
R
i
)
+
i
(
R
s
−
R
i
)(
ω
−
k
s
k
i
P
(
q
||
,
ω
)
q
||
v
||
q
||
)
2
]
2
h
5
cos
θ
i
2
[(
v
⊥
q
||
)
2
¯
+
(
ω
−
(3.4)
