Biomedical Engineering Reference
In-Depth Information
where each Drude-type ELF,
Im
W
2
i
1
!
i
D
.!
i
/
2
;
(15.8)
D
I
.W
2
i
"
D
.k
0; !
W
i
;
i
/
!
2
/
2
C
is characterized by the position W
i
and width
i
of its peak [
37
], whose intensity is
quantified by the constant A
i
.
Besides a satisfactory agreement with the available optical data, the consistency
of the fitting procedure must be ensured by the fulfilment of physical constrains,
such as several sum rules [
38
,
39
]. The f -sum rule gives the effective number of
electrons per molecule that can be excited and guarantees a good behaviour of the
ELF at high energy transfers:
Z
d!!
Im
Im
m
e
2
2
e
2
N
1
1
valence
C
D
Z
2
;
".k
D
0; !/
".k
D
0; !/
K
-
shell
0
(15.9)
where Z
2
is the number of electrons of the water molecule and m
e
is the
electronmass.
The Kramers-Kronig or perfect screening sum rule is an important test for the
accuracy of the ELF at low energy transfer:
Z
!
Im
d!
2
1
1
n
2
.!
C
D
0/
D
1;
(15.10)
".k
D
0; !/
0
where n.ยจ
0/ represents the refractive index at the static limit.
All the procedures to describe the ELF of a material, to be discussed in what
follows, satisfy better than 99% both sum rules, Eqs.
15.9
and
15.10
. Besides, these
methods guarantee the fulfilment of the sum rules for every momentum transfer,
provided it is satisfied at k
D
0.
In Fig.
15.2
a we show the fitting curves resulting from applying Eq.
15.7
to
the experimental OELF derived from both sets of data, REF. [
15
]andIXSS[
16
],
for the valence-electron excitations of liquid water. The right panel of Fig.
15.2
corresponds to higher energy transfer, where the OELF values have been obtained
from the FFAST database of NIST for the water molecule [
40
], and from the x-ray
scattering factors of H and O [
41
]. The contribution of the inner-shell electrons to
the OELF of liquid water, Eq.
15.6
, has been added to the valence contribution,
Eq.
15.7
, for energy transfers greater than the K-shell binding energy of oxygen
.E
K
D
D
540 eV/.
The fitting curves in Fig.
15.2
exhibit a satisfactory agreement with the exper-
imental OELF, whose main trends are well reproduced. This is one of the main
advantages of the approach based on using optical data specific to the material
under consideration, which automatically accounts for electronic-structure effects
in a realistic manner not always possible within electron gas models.
